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ProQuest Information and Learning 300 North Zeeb Road, Ann Arbor, Ml 48106-1346 USA 800-521-0600 IM
AN EIGHTEENTH CENTURY PIETIST THEORY OF MUSIC PSYCHOLOGY; im . BREVISSIMA THEORIAEMUSICAEANALYSIS OF JOHANN FRICKER AND FRIEDRICH OETINGER
DISSERTATION
Presented in Partial Fulfillment of the Requirements for
the Degree Doctor of Philosophy in the
Graduate School of The Ohio State University
By
Peter John Knapp, B.M., M A., M.M.
*****
The Ohio State University 2001
Doctoral Examination Committee:
Professor Burdette Green, Adviser Approved by
Professor Kirk Freudenburg Adviser Professor Graeme Boone Music Graduate Program UMI Number; 3011096
UMI*
UMI Microform 3011096 Copyright 2001 by Bell & Howell Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code.
Bell & Howell Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Ml 48106-1346
/ ABSTRACT
Johann Flicker and Friedrich Oetinger were eighteenth century Pietist pastors from Wuertemberg, Germany. Oetinger was one of the leading minds in the Pietist movement, and is well known among religious scholars for his contributions to that movement. Among music scholars, he is virtually unknown, though he did make a music-theoretic contribution. He appended to his treatise Inquisitio in Sensum
Communem et Rationem (1753) a short music treatise by Fricker for which he provided an extensive commentary. This appendix served to illustrate the application of the sensus communis (literally, “common sense”)—the God-given organ that allows humans to make various types of perceptions.
In the music theory portion, Fricker draws on the ideas of both Leonhard Euler and Johann Mattheson, among others. The mathematical complexity of ratios in eighteenth-century music led Fricker to develop a psychological theory wherein the mind simplifies the ratios through the use of exponents and thereby makes possible the rapid perception of the various aspects of a composition: intervals, melody, harmony, rhythm, and text. Oetinger’s commentary deals extensively with the metaphysical issues surrounding the perception of music through the application of the sensus communis.
11 Dedicated to my wife, Lisa, whose love, patience, and generous spirit made this work possible.
lU ACKNOWLEDGMENTS
I wish to thank my adviser. Burdette Green, for his generous support, remarkable expertise, and most especially, his personal guidance. His broad knowledge, extensive experience, and excellent writing skills enabled me to create a truly scholarly document.
1 thank Kirk Fruedenburg for his patient, thorough, and expert work on the Latin translation. His commitment to this project was above and beyond the call of duty, and his work was greatly appreciated.
1 am grateful to Graeme Boone for his thoughtful comments and excellent writing. His personal enthusiasm for the work was a great encouragement.
1 also wish to thank Mr. Thomas Nahrmann of Frommer Verlag Publishing for his permission to use extensive quotes of the Latin manuscript.
IV VITA
Peter J. Knapp
1969 Bom: Chicago, IL. 1992 Bachelor of Music, Education, Elmhurst College. 1998 Master of Music, Composition, The Ohio State University. 1996 Master of Arts, Music Theory, The Ohio State University. 1995 Graduate Research Associate, Music Psychology, The Ohio State University. 1995-2000 Graduate Teaching Associate, Music Theory, The Ohio State University.
FIELDS OF STUDY
Major Field: Music
Studies in Music Theory: Burdette Green, Gregory Proctor, Lora Gingerich Dobos Studies in Music Psychology: David Butler, Caroline Palmer, David Huron Studies in Music Composition: Jan Radzynski, Richard Smoot, Thomas Wells TABLE OF CONTENTS
DISSERTATION...... i ABSTRACT...... ii DEDICATION...... üi ACKNOWLEDGMENTS...... iv VITA...... V FIELDS OF STUDY...... v UST OF TABLES...... vii LISTOFHGURES...... ix PREFACE...... xi INTRODUCTION...... 1 Biographies ...... 1 A Brief History of Pietism ...... 6 Opposition to Pietism and the Enlightenment ...... 10 Conclusion ...... 12 Sources ...... 13 Translation Issues ...... 16 PART I TRANSLATION...... 18 IFRICKER! SECTION II...... 18 rOETINGER; SECTION 111...... 80 IFRICKER; SECTION IIIl...... 97 lOETlNGER: SECTION IV1...... 114 PART n COMMENTARY ON THE VIEWS OF FRICKER AND OETINGER 151 CHAPTER 1...... 151 MATHEMATICS APPLIED TO MUSIC THEORY...... 151 CHAPTER 2...... 168 THEORIES OF MUSIC PERCEPTION...... 168 Psychological Thought of the eighteenth-century ...... 168 The Psychological Theory of Music: Fricker and Oetinger ...... 174 CHAPTERS...... 188 EPISTEMOLOGY AND AESTHETIC POSITION...... 188 Epistemology ...... 188 The Mind-Body Issue ...... 191 Aesthetic Position ...... 195 CHAPTER 4...... 199 THE LEGACY OF THEIR THOUGHT...... 199 Oetinger’s Influence on Romanticism: Schenker and Goethe ...... 199 Gadamer’s Debt to Oetinger. Christensen’s Debt to Gadamer...... 206
yi BIBLIOGRAPHY...... 210 Primary Sources ...... 210 INDEX...... 215
VII LIST OF TABLES
Table 1: Explanatory table of Figures 17 and 18...... 52 Table 2: Ratios for two tuning systems and Pricker’s modified compilation...... 152
Vlll LIST OF FIGURES
Figure 1: String ratios from Pricker’s experiment in Section three...... 23 Agedum igitur: properemus ad fontem harmoniae: hie videmus quidem ...... 27 Figure 2: Derivation of ratios through multiplication...... 27 Figure 3: Summary of kev ratios from Pricker’s experiment, section three...... 34 Figure 4: Pricker’s modifed text of Graun’s Artaxerxes along with Oman's original and the translation of both...... 37 Figure 5: Pricker’s division of the syllables from the "pensa" motive...... 38 Figure 6: Musical notation of Figure 5...... 39 Figure 7: Repetition of the “pensa” motive with half-note in the bass...... 39 Figure 8: Examples of sixteenth-note triplet division of the eighth-note...... 40 Figure 9: Example of thirtv-second-note division of the eighth-note...... 41 Figure 10: Musical notation of previous text...... 42 Figure 11: Pricker’s representation of the note values from the preceding paragraph. ...43 Figure 12: Musical notation of immediate repetition of the “pensa” motive...... 44 Figure 13: Musical notation of the sequence of the “pensa” motive...... 44 Figure 14: Musical notation of the sequence and use of secondary function...... 45 Figure 15: Musical notation of the previous analysis...... 46 Figure 16: Musical notation of the “march” motive...... 48 Figure 17: Flicker's representation of the song. "The spiritual Bride.” ...... 50 Figure 18: Musical notation of Figure 17...... 51 Figure 19: Pricker’s application of the doctrine of the affections to the prime numbers with translation...... 58 Figure 20: Another application of the affections with translation...... 59 Figure 21: Graphic representation of how the mind compares quantities...... 61 Figure 22: Graphic representation of ratios on a string...... 62 Figure 23: Flicker’s representation of the explanation of the dissonant fourth...... 63 Figure 24: Musical notation of Figure 23...... 64 Figure 25: An arithmetic series representation of the ratios of the fifth (ratios 1-4) and the fourth (ratio 51...... 65 Figure 26: Representation of the radii from the center...... 67 Figure 27: Pricker’s table of consonances and dissonances...... 72 Figure 28: An example of a series of consonances interspersed with dissonances...... 73 Figure 29: A comparison of two series...... 73 Figure 30: A comparison of three series...... 74 Figure 31: Several possible series generated from the three prime powers...... 75 Figure 32: Graphic representation of the ratios on a string...... 90
ix Figure 33: Inversion of bases and exponents in the ratio 8:9. expressed as 2^: 3^...... 103 Figure 34: Graphie representation of simple arithmetic progression conceived as points on a line...... 106 Figure 35: Francis Bacon’s representation of the muscial ratios...... 131 Figure 36: Graphic representation of the 9:8 ratio as a division of a string: then the individual eight and nine parts of the string, respectively...... 132 Figure 37: Graphic representation of the individual notes in the chord C - E - G as parts of a string...... 133 Figure 38: Table representing the physiological position of the various ratios...... 138 Figure 39: Inversion of bases and exponents in the ratio 8:9. expressed as 2 : 3^...... 166 Figure 40: Example of visual Gestalt demonstrating the principle of similarity...... 184 PREFACE
The Brevissima Theoria Musicae of Johann Fricker (1729-1766), published in
1753 and here translated into English, provides the only known music treatise associated with a significant Pietist figure.' While Fricker was the primary author of this work, it was actually published as an addendum to a larger work by Fricker’s teacher, Friedrich
Christoph Oetinger (1702-1782): Inquisitio Sensum Communem et Rationem (1753).^
Neither Oetinger nor Fricker is well known within studies of music theory, in spite of a number of works dealing with Pietism and music, and although Oetinger was one of the major Pietist leaders of the eighteenth century, his ideas on music are likewise unknown to theologians, historians, and even to musicologists. This dissertation seeks to rectify that situation. Pietism (which will be discussed in detail later) was a branch of
Lutheranism. Its effect on various musicians and theorists has received limited investigation by modem musicologists. Analyses of the music of Georg Phillip
Telemann, Dietrich Buxtehude, Georg Boehm, and J.S. Bach,^ the Pietist leanings of
* * Johann Fricker, “Brevissima Theoriae Musicae Analysis.” Inquisitio in Sensum Communem et Rationem. Ed. Friedrich Oetinger. Stuttgart: Friedrich Frommann Verlag, 1964. ^ Friedrich Oetinger, Inquisitio in Sensum Communem et Rationem. Stuttgart: Friedrich Frommann Verlag, 1964. ^ See Hans Homer, G. P. Telemanns nassionmusik. ein beitrae zur geschichte der nassionmusik in Hamburg. Boma-Leipzig Universitütsverlag von R. Noske, 1933. Martin Geek, Die Vokalmusik Dietrich Buxtehudes und der fruhe Pietismus. Basel: Barenreiter Verlag Kassel, 1965.
XI Johann Georg Sulzer and Johann Mattheson/ as well as the writings, poetry, and compositions of Erdmann Neumeister are among the topics covered/
Shortly after Pricker’s death, Oetinger also wrote a brief article outlining Pricker’s musical views and comparing them directly with Leonhard Euler’s music theory, entitled
“Die Eulerische und Prickerische Philosophie über die Musik’’ (1767)/ Getinger’s article from 1767 enjoyed several references in music-theoretical works, such as in Georg
Sulzer’s Allgemeine Theorie der schdnen KUnste (1793), under the entry for “Klang.”
Even Oetinger himself (along with this article reference) earned an entry in Petis’
Biographie universelle des musiciens et bibliographie générale de la musique (1875). In the Brevissima Pricker mentions Mattheson in three separate passages (with reference to his theories of melody). It is of some interest that Mozart possessed a copy of Oetinger’s
Die Metanhvsik in Connexion mit der Chemie (1770), which contained a restatement of
Pricker’s musical ideas.^ Notwithstanding these connections, no studies have
Carol Ann Crumrine, ‘The keyboard and vocal settings of Georg Boehm: an analysis of style as dictated by text.” Diss. Syracuse University, 1972. Jaraslov Pelican, Bach Among the Theologians. Philadelphia: Fortress Press, 1986. '* See Nancy Kovaleff Baker and Thomas Christensen, Eds. Aesthetics and the Art of Musical Composition in the German Enlightenment: Selected Writings of Johann Georg Sulzer and Heinrich Christoph Kock. New York: Cambridge University Press, 1995. Thomas Christensen, “Sensus, Ratio, and Pthongos.” Musical Transformation and Musical Intuition: Eleven Essavs in Honor of David Le win. Eds. Raphael Atlas and Michael Cherlin. Roxbury, Mass.: Ovenbird Press, 1994.1-22. Ian David Pearson, “Johann Mattheson’s Das forschende Orchestre: The Influence of Early Modem Philosophy on an Eighteenth-Century Theorist.” Diss. University of Kentucy, 1993. ^ Neumeister represents well the internal conflict which many composers felt regarding Pietism. Neumeister was orthodox in his theology, as seen in his anti-Pietist writings, yet his poetry and music had decidedly Pietist leanings. ‘ Reproduced in Rheinhard Breymayer, “Zu Friedrich Christoph Getingers emblematischer Musiktheorie: Getingers wiedergefundene Sctuift Die Eulerische und Frickerische Philosophie über die Musik,’ Mit einem Ausblick auf Friedrich Hdlderlin.” Blatter Für Württembergische Kirchengeschichte 76 (1976): 130- 175.
XII investigated the ideas or influences of Fricker or Oetinger, and there have been no music- theoretical treatises have been considered that are specifically Pietist in perspective.
The goal of this document is twofold: to provide a translation of the overlooked music treatise (the Brevissima from 1753) and to furnish an assessment of its contents.
Although Fricker and Oetinger present nothing fundamentally new within the context of eighteenth-century music theory, they do offer novel analyses and applications of their
musical ideas. They also provide detailed and original thinking in their perceptual
theories, which are of interest in the context of eighteenth-century psychology and the
history of music psychology as a whole.
It is also important to realize that many German literary scholars rightly view
Oetinger as a major source of Romantic thought, especially for Goethe’s views on
“organicism.” The well-known modem philosopher Hans-Georg Gadamer looked to
Oetinger (among others) as a source of phenomenologist thought in his work Wahrheit
und Methode (1960), which was, in turn, influential on the ideas of the current
musicologist Thomas Christensen. This dissertation will argue that Oetinger should be
credited for some of the “organicism” found in much of nineteenth and twentieth-century
music aesthetics, particularly in the analytical theories of Heinrich Schenker.
The commentary on the translation will consider four major areas of the
Brevissima: Fricker’s and Oetinger’s approach to music mathematics and theory, music
perception, their epistemological and aesthetic position, and the legacy of their thought.
^ See Herbert Henck, “Vom Monochord zur vierten Dimension, Johann Ludwig Prickers irdische und hinunlishce Musik.” Neue Zeitschrift fiir Musik 162 (2001), p. 48.
xiii Because the treatise has been ignored, this commentary will focus on an evaluation, background and contemporary context.
XIV INTRODUCTION
Biographies
Johann Ludwig Fricker was a Pietist pastor who lived from 1729 to 1766. Aside from Karl Eberhard Ehmann’s nineteenth-century biography of Fricker (1864), and
Fricker’s popular devotional book, Weisheit im Staube (“Wisdom in Dust, ” published posthumously in 1820),' he is a somewhat obscure figure. This obscurity is in keeping with his deeply religious, serious, and rather reticent nature. He was not interested in self-promotion, and, according to the historian Julius Roessle, the only religiously significant publication issued during his lifetime was, “UnvollstSndige, jedoch brauchbare Überbleibsel” (“Incomplete, yet useful remnants”).^
Fricker was a talented keyboard player, and he developed his own musical theory while studying theology at the university. Ehmann remarks in his biography that what is remarkable about this theory is not the fact that Fricker can account for the musical intervals through mathematical calculations, but the fact that Fricker, “...departing from the way of music and mathematics, attempted to penetrate into the realm of philosophy
' A collection of Pricker’s unpublished writings, edited by Ludwig Christoph Huzlein and published in 1820,1855, and 1963. See J. Roessle, Introduction. Weisheit im Staube. By Johann Fricker. Metzingen: Franz-Verlag, 1963. ^ Roessle, p. 19. and into the mystery of theosophy.”^ This mysticism, and his association with Pietism, made it difficult for Fricker’s theory to gain much recognition or acceptance. During his studies with Oetinger (beginning in 1749), Fricker primarily discussed his theory of music. Oetinger had a high opinion of Fricker; “He [Fricker] surpassed me by far, his teacher, especially in mathematics and physics. He invented a new theory of music which is, however, so deep, that few understand it."^ In this same quote, Oetinger also acknowledges the fact that Fricker’s musical ideas were published in his book on the sensus communis.
Admittedly, Fricker is not well known for his musical ideas, and he is even somewhat unknown among religious scholars. This obscurity is due, somewhat, to his being overshadowed by Oetinger, even though, as J. Roessle writes he was Oetinger’s most significant student. He made a sincere effort to understand the views of other theologians, and thus enabled him to act as a “middleman" between Swabian and
Rheinland Pietism. During his life, he was well-loved for his powerful and sincere sermons and for the personal attention he paid to his congregation.
Another reason for the obscure status of Fricker’s musical ideas is that he published so little. Most of his writings exist in manuscript form, and are only available in libraries in Germany. A recent article by Herbert Henck reveals that Fricker’s musical thoughts were “scattered about ’’ in various manuscripts of his own and in publications by
Oetinger between 1752 to 1770.
^ Roessle, p. 7. * Roessle, p. 9. Friedrich Christoph Oetinger was well-known during his lifetime, and has also been the subject of numerous religious, sociological, and historical studies. His interests ranged from theology to mysticism to theosophy and his associations were with well known religious and political figures such as Emmanuel Swedenborg.
According to the historian Ernest Stoeffler, Oetinger was, in his writings, primarily interested in developing a complete philosophical system to support his theological views. He recognized that Protestant thought could not afford to ignore differing viewpoints on reality, especially those of Enlightenment thinkers, and strove to embrace their knowledge as much as he could within the constraints of his biblical beliefs. Oetinger, although not a literalist, felt that Scripture should be relied on as the ultimate standard of truth. As sociologist George Becker points out:
In all matters pertaining to science and secular knowledge, the critical referent is always compatibility with biblical truth. As Oetinger pointedly argues, no philosophy or system of knowledge merits acceptance “that does not fit like the key to the lock of the Holy Scriptures.”^
Oetinger’s motivation was to show that the new discoveries of science could be put to the service of displaying God’s handiwork. However, he distrusted many of the methods of science, because they could not be proved with the unaided senses. As
Becker discuses, Oetinger advised against using microscopes and other such technology,
^ Sec George Becker, The Merton Thesis: Oetinger and German Pietism, A Significant Negative Case.” Sociological Forum. 7 (1992), p. 645. as well as quantification. The reason was that these activities would promote the “sin of pride” and lead to “deadly abstraction,” and, thereby, reduce one’s awe for God’s mysteries.**
Getinger s philosophy promotes a more subjective investigation, called the phenomenological approach, which favors an intuitive understanding of truth aided by
Christian understanding.^ In keeping with this method is Oetinger’s esteem for the ancients, whom he believed had superior understanding. This position convinced him to investigate the knowledge of the Greeks (specifically Aristotle), the church fathers.
Renaissance mystics and alchemists, and (given his penchant for the scriptures) the
Proverbs of Solomon.®
All of these factors led Oetinger to adopt a unique methodology for his scientific experiments. Becker reports that Oetinger performed one experiment in which he boiled dried, cut-up leaves of balm mint, and then claimed to see in the oils the restored shape of the leaf. Additionally, he held to an emblematic theory of classification, which grouped plants according to their resemblance to the human body. Accordingly, since the walnut resembles the human head, it should, therefore, contain properties useful in the treating of head injuries.^
Another integral part of Oetinger’s philosophy of science is the quest for practicality. The motivation behind this is spiritual, because Oetinger sees the need to
* See George Becker, “Pietism’s Confrontation with Enlightenment Rationalism: An Examination of the Relation Between Ascetic Protestantism and Science.” Journal for the Scientific Study of Religion 30 (1991), p. 152. ^ Becker, “Pietism’s Confrontation with Enlightenment Rationalism,” p. 153. * Becker, “Pietism’s Confrontation with Enlightenment Rationalism, ” p. 153. ’ Becker, “The Merton Thesis ”, p. 652. make all discoveries a reinforcement of the spiritual life {Geistleiblichkeii)}^
Consequently, Oetinger’s phenomenological approach always strives for “the simplest, the most useful, and the most necessary."" This viewpoint led to two tenets: philosophical ideas must be kept simple and less abstract, and truth can be made accessible to even the most simple-minded. Oetinger and Fricker state in the music theory treatise that any theory which claims to explain music should be able to explain how uneducated people can appreciate a phenomenon as complex as music.
Although Oetinger held to some strange notions, and was even censured at one point,'' he should not be viewed as a complete outsider. Several scholars have noted that
Oetinger expressed sentiments shared by a number of contemporary philosophers and
religious leaders, namely, a concern about the overly rationalistic methodology of the
Enlightenment. Consider the comments of sociologist George Becker and Germanicist
Priscilla Hayden-Roy:
His was not a lone and aberrant voice calling out in the wilderness against the modem scientific heresy, but a natural outgrowth of a chorus of previously articulated dissatisfaction.”'^
Oetinger represents a vital undercurrent of thought within the IS"* century that responded critically to tendencies within the Enlightenment...'^
...It is most helpful...to place Oetinger within a tradition of hermetic and theosophical thought to which many intellectuals in the second half
See Priscilla Hayden-Roy, “Sensate Language and the Hermetic Tradition in Friedrich Christoph Oetinger’s Biblishes und Embletnatisches Worterbuch.” Subversive Sublimities: Undercurrents of the German Enlightenment Ed. Eitel Friedrich Timm. Columbia, SC: Camden House, 1992, p. 59 “ Becker, “The Merton Thesis,” p. 650. Hayden-Roy, p. 61. " Becker, “The Merton Thesis,” p. 654. Hayden-Roy, p. 67. of the 18* century turned to find a corrective to the rationalistic Wolffian philosophy of the German Enlightenment...
Although Fricker is known primarily through Ehmann’s biography and for his work Weisheit im Staube. Oetinger, on the other hand, was a prolific and well-known
writer in his time. His works span a period of over fifty years and treat numerous
theological and metaphysical topics.
A Brief Historv of Pietism
Recent scholarship has illuminated the religious movement called Pietism.
Historian Arthur McCardle provides a succinct summary of its development:
Pietism in Germany came into being as a reaction against the orthodoxy of the Lutheran Church which, it was felt, had betrayed the aims and achievements of the Reformation by transforming Luther’s doctrine of spiritual renewal into a rigid and scholastic dogmatism. Correctness of belief and the mere hearing of the Word were considered to be suffîcient for justification by faith, while the inner experience of this justification by the individual was all but ignored. Pastors were trained only in systematic theology, and the importance of pastoral responsibilities to the individual members of the congregation was neglected.*®
More specifically, historian William Petig outlines four general tenets of Pietism.
Primarily, they were concerned with a more personal and emotional experience and
expression of the Christian life. There were two particular terms related to this aspect,
namely, Wiedergeburt (Rebirth) and the Durchhruch (Breakthrough) to spiritual insight.
Hayden-Roy, p. 68. See Arthur W. McCardle, Friedrich Schiller and Swabian Pietism. New York: Peter Lang Publishing, Practical Christianity, that is, an emphasis on good works was also important, in contrast with Orthodoxy’s more aloof position towards the congregation. Another strongly agreed upon tenet was the importance of the scriptures. However, many Pietists leaders did not agree upon the proper method of achieving scriptural understanding, whether it should be through exegesis or personal revelation. Finally, all Pietists shared an opposition to the religious establishment, or more correctly, any establishment that limits individual expression.'^
Peter Erb furnishes a more detailed account of Pietism’s development. While it is beyond the scope and expertise of this work to consider Pietism’s history in great detail,
Erb provides a useful overview, and even traces some of the more important branches of
Pietism.
After Luther’s death, there were a number of doctrinal controversies which needed to be resolved: the role of good works in one’s salvation, the nature of the conversion experience, the role of the sacraments, sin and man’s essence, and justification. These matters were settled in 1580 with the Formula of Concord: as expected, this concord did not ultimately satisfy all members of the Lutheran church.
The legalistic nature of the Concord and its lack of “practical piety’’ were direct factors leading to the development of Pietism. Johann Arndt (1555-1621), with the publication of Wahres Christentum (“True Christianity ”) in 1606, led the opposition to the orthodox
position. Erb points out that, along with Arndt, Calvinism and other Puritan groups also
played a role in the developing rift within mainstream Protestantism.
1986, p. 7. See William E. Petig, Literary Antipietism in Germany During the First Half of the Eighteenth-Century. Philipp Jakob Spener (1635-1705) is most directly associated with the beginning of Pietism. Spener's treatise, Pia Desideria (“Pious Desires”), published in 1676,‘* outlined his ideas for a much needed reform within the church, ideas he had already started to put into practice in 1670. He was concerned with practical piety and moral reform; he was particularly troubled by the societal decline and “disillusionment” following the Thirty Years War (1618-1648). Spener proposed renewed attention to
Bible study, the convening of conventicles (small prayer and study groups), and the establishing of a “priesthood of all believers.” Dedication to these practices should lead
to spiritual enlightenment and an put end to theological rancor. Due to the separatist and
subjective, even emotional, nature of these tenets, the movement was labeled,
pejoratively, by its opponents as “Pietism.”
August Hermann Francke (1663-1727) was one of Spener’s leading defenders.
Using his position at the University of Halle, Francke spread the influence of Pietism
through his teachings. His teachings and support were also instrumental in establishing
numerous social (e.g., the building of an orphanage) and missionary programs. Halle was
also the source of a large and popular printing program that produced inexpensive Bibles,
songbooks, prayer books, and devotional books. Franke’s theology was more ecumenical
than Spener’s, and this fact lead to greater acceptance of theosophical (i.e., reliance on
direct revelations) and mystical ideas among certain Pietists.
Adherents to such theosophical and mystical ideas are categorized as “Radical
Pietists.” Although the Haile Pietists were more tolerant and open, ironically, the
New York: Peter Lang Publishing, 1984, pp. 11,12. It was originally published as an introduction to Arndt’s work in 1675.
8 mystical viewpoint lead others, such as Gottfried Arnold (1666-1714), to separatism.
True believers, those who had experienced a Wiedergeburt, would attain a mystical union with Christ and be able to have a true understanding of the scriptures; those trapped within orthodoxy were considered a part of spiritual Babel. Arnold’s views, published in several works, influenced the leaders of numerous separatist groups, both in Europe and
America.
Johann Albrecht Bengel (1687-1752) and Friedrich Christoph Oetinger (1702-
1782) were the two most prominent leaders of what is referred to as “Württemberg
Pietism." Oetinger was a student and life-long friend of Bengel. Bengel leaned more towards Spener’s concerns with scriptural enlightenment through scriptural interpretation, and consequently produced a large study on the New Testament (Gnomon on the New Testament. 1742). Petig describes Württemberg Pietism in this way:
A brand of Pietism slightly different from the Spener-Halle variety, but nevertheless influenced by it, emerged in Württemberg around the University of Tübingen...Württemberg Pietism was less stem than the Halle variety...put less emphasis on the conversion experience, and was more a movement of the people than of the clergy and nobility.
Pietism continued well into the nineteenth century, and became more socially oriented. Johann Blumhardt (1805-1880) and his son Christoph Blumhardt (1842-1919) are recognized as the most prominent leaders of Pietist thought during this time. Their goal (highly influenced by Württemberg Pietism) was to bring the kingdom of God to oppressed people, and therefore they became more involved in political causes affecting the social conditions of the poor. Opposition to Pietism and the Enliehienment
Petig notes that, ‘The spread of Pietism was accompanied by frequent controversy and often strong and bitter opposition from Lutheran orthodoxy.”^" He cites nine different edicts against the Pietists, in ten different cities, ranging across a period of fîfty- five years (1692-1747). Certain edicts were quite severe; one particular edict of 1735, in
Switzerland, threatened imprisonment for holding or promulgating such heretical views.^'
Musicologist Ian Pearson suggests that the opposition to Pietist thought caused Johann
Mattheson to be critical of certain Pietist leaders, even though he actually supported much of their thought.^^
The Pietists were not always on the receiving end of persecution, because the severity of persecution depended on the local influence of the Orthodox leadership. In
Hamburg, the Pietists exercised a great deal of control over the opera, to the point of limiting its production. The resulting “opera wars” affected Mattheson.
Pietism, by emphasizing Scriptural understanding as the path to true knowledge, soon found itself (in addition to orthodoxy) in opposition to the Enlightenment. The interaction between Pietism, orthodox Lutheranism, and the Enlightenment was not always clear-cut and direct. At times, those who were enemies became allies, and vice versa. Certain individuals, most commonly identified with one movement, often shared various sentiments with another seemingly antagonistic movement. William Petig
” Petig, p. 21. “ Petig, p. 25. Petig. p. 25.
10 describes how two of the Enlightenment’s leading thinkers, Christian Thomasius (1655-
1728) and Christian Wolff (1679-1754), “sympathized with Pietism, but both also came
into conflict with it.”“
In order to understand this incongruity, it is important to note the unique character
of the German Enlightenment. As science historian Ronald Calinger states, “The German
Classical Weltanschauung (world view) was unique in Europe. Newtonian thought did
not dominate here as it did in Western Europe. Instead, scientific eclecticism, or
selective synthesis of different systems, held sway.”^'* Rather than a Newtonian view
Calinger states that a Leibnizian view was dominant. This coincided with Pietism's
rejection of a “mechanistic" view of the universe, devoid of God’s active interaction and
control; however, not all of Leibniz’s views were satisfactory to the Pietists, as will be
seen in the ‘Wolff controversy.’
Pietism and the Enlightenment had a “love-hate ” relationship in Germany.
Despite the apparent surface contradiction, this notion is reasonable, given the flux of
human thought, and the strong flood of new, profound, and even conflicting ideas
swirling throughout Europe at this time. Petig asserts that, “Both movements shared a
common impetus, namely, the revolt against an arbitrary, institutional authority. ”^ Petig
develops this thought more, drawing on the work of another scholar, Horst Stephan.
“ Pearson, p. 357. “ Petig, p. 31. See Ronald Calinger, “The German Classical Weltanschauung in the Physical Sciences.” The influence of early enlightenment thought upon German Classical science and letters. Ed. Ronald Calinger. New York: Science and History Publications, 1972, p. 1. “ Petig, p. 30.
11 Stephan argues that Enlightenment philosophy surpassed Pietist influence in Germany not by the rejection of its ideals, but by the adaptation of them.
The absence in Pietism of an organized program or systematic theology undoubtedly aided this confluence of ideas. Both Pietism and the German Enlightenment opposed arbitrary institutional authority, whether in the guise of religious dogmatism or political absolutism, and both advocated freedom of conscience.^®
This sympathy of ideals was, eventually, not enough to hold the two together—
not that they were ever comfortable bed-fellows. The ideas of the Enlightenment
continued to become more “heretical” until confrontation became inevitable. This
conflict came to head in the ‘Wolf controversy’ at the University of Halle. Christian
Wolff, who was influenced by Leibniz’s philosophy, was teaching mathematics at the
University of Halle. Due to his dependence on Leibniz’s ideas, which could be
interpreted as rejecting the need for repentance and grace (by an over reliance on reason).
Wolf found himself squaring off against the Halle leadership.
Ironically Franke (a Pietist) and Lange (an orthodox Lutheran) came together to
request of Frederick William I that Wolf be forced out of Halle. Charged with teaching
“fatalism,” Wolff was forced to leave Halle, by the decree of November 8,1723, on pain
of death."^ He was later reinstated by Frederick the Great.
Conclusion
While this summary of Pietism is not exhaustive, it sufficiently delineates some of
Pietism’s complexities of thought and interaction within German society. As with any
“ Peüg.p.28.
12 movement of influential ideas, Pietism should not be overly simplified in its characterization. Much of its character depended on the strength of individual leaders and their singular ideas. These thoughts should be borne in mind while considering the musical treatise of Oetinger and Fricker.
Sources
The 1964 Frômmer Verlag edition of Inouisitio in Sensum Communem et
Rationem is a facsimile of the 1753 original print. The main portion of the book, by
Oetinger alone (270 pages), is on the sensus communis. In contrast, the shorter music* theory portion comprises a total of eighty-seven pages, and is divided into two main sections, wherein Flicker’s writings are presented with Oetinger’s commentary following each section.
The entire treatise is in Latin, with the exception of numerous sentences, or portions of sentences, in Greek. The facsimile introduction to the 1964 edition is written by the well known twentieth-century German hermeneutical and phenomenological philosopher Hans Gadamer. Since the scope of the present study prohibits consideration the full content of Oetinger’s main philosophical treatise, Gadamer introduction is relied upon for a summary and estimation of that work. This will be considered later in Part n.
Chapter 3.
The music-theoretic treatise itself is actually the work of Fricker, and Oetinger provides substantial commentary on Fricker’s ideas. As Oetinger’s student, many of
Petig, p. 33.
13 Flicker’s views conform to the theological and philosophical ideas of the teacher.
Interestingly, Oetinger admits that Fricker is the better mathematician, and so relies on him for the details of the mathematical portion of the theory. There is a brief presentation of the same musical ideas within the body of Oetinger’s main treatise, but there are no new ideas presented therein. In fact, his focus there is more metaphysical than musical.
Despite the fact that the music portion of the treatise is mentioned in the title page of the work, there is no attribution of Fricker as the author. This has caused confusion about who is writing the analyses. Oetinger does not present the work as a quote, rather
he mentions Fricker (as the originator of these ideas) at the end of the first portion when
he offers his commentary. Following this, the second part of Fricker’s work is presented;
Oetinger then mentions Fricker again at the start of his second commentary.
Richard Breymayer examines this treatise as a footnote to his study of Oetinger’s
1767 monograph comparing the theories of Fricker and Euler,*® and judges Fricker to be
the author of a larger amount of the 1753 music treatise than is correct. Breymayer
considers the authorship to be as follows:
Part I (Theory): Fricker (pp. 1-36) Commentary: Oetinger (pp. 37-48) Part 2 (Psychology): Fricker (pp. 49-87)
Actually, the authorship should be attributed as follows:
Part 1 (Theory): Fricker (pp. 1-36) Commentary: Oetinger (pp. 37-48) Part 2 (Psychology): Fricker (pp. 49-61) Commentary: Oetinger (pp. 61-87)
^ See Rheinhard Breymayer, “Zu Friedrich Christoph Oetingers emblematischer Musiktheorie; Oetingers wiedergefiindene Schrift ‘Die Eulerische und Frickerische Philosophie über die Musik,' Mit einem Ausblick auf Friedrich Hôlderlin.” Blatter Ftir Wflrttembereische Kirchengeschichte 76 (1976): 130-175.
14 The primary reason for this correction is based on the internal evidence. First of all, Oetinger mentions Fricker in the third person in his commentary immediately following the theoretical part of the treatise. This identifies Oetinger, more logically, as the current writer. Oetinger then states that the preceding ideas were given to him by
Fricker, yet there is no indication that Oetinger has been writing the preceding as a paraphrase of Fricker’s ideas. In addition, Fricker begins the psychological part of his treatise with a reference to his theoretical part as well as to the “author” (Oetinger) of the main treatise, the Inquisitio.^^
In addition, after the psychological portion, Oetinger introduces his commentary on the preceding part by referring to Fricker. Although it can be misleading at first glance, the Latin refers to the psychological part “a Dn. Frickero”; the preposition a should not be understood as agent, but as reference, that is, a second part of commentary on Flicker’s work.^® Oetinger states that Fricker wrote this second portion after having read his (Oetinger’s) thoughts on the sensus communis. Fricker makes the same statement at the beginning of his psychological treatment. This view makes the flow of the sections more logical: treatise, commentary, treatise, commentary. Furthermore,
Oetinger, in his commentary, directly refers to Fricker again in two separate passages.
^ Breymayer’s original reads as follows: “Nach S.270 des Haupneils foigt mit eigener Paginierung (S. 1- 48): [Johann Ludwig Fricker (Hauptverfasser) / Friedrich Christoph Oetinger (Mitverfasser und Redaktor),] ‘BREVISSIMA’...Fricker sind folgende Abschnitte zuzuschreiben: S. (1), Z. 4 bis S.36, Z. 23; S. 49-87: ‘PARS SECUNDA’...Von Fricker stanunt hier der Abschnitt S. 49, Z 5 bis S. 61, Z. 10. S.88-96 foIgt ein Oetinger zuzuschreibendes anonymes ‘SUPPLEMENTUM...’ Fricker hat auch die von Oetinger stammenden Abschnitte beeinflufit.” Notice that on the one hand, Breymayer attributes pp. 49-87 to Fricker, then ammends it to pp. 49-61 without explanation. See Breymayer, pp. 158,9. ^ “Accipe, Lector benevole, partem alteram Analyseos Musicae a Dn. Frickero post lectum de sensu communi librum exaratam. Dabo & hie notas quasdam.” Page 61 in the original; see p. 114 in the current document.
15 Finally, each section in the psychological commentary is labeled “ad 1,” “ad 2," and so on; that is, it is a commentary on (ad) the corresponding section from the previous portion.
Gadamer mentions that a popular German edition was available from the same
year (1753). The title of this work is Die Wahrheit des Sensus Communis (“The Verity of the sensus communis”). However, this work is a more general version of Oetinger’s
thoughts on the sensus communis, as well as additional material on biblical interpretation
of the book of Solomon, and contains no portion of Pricker’s music theory treatise.
Richard Breymayer presents a critical edition of the 1767 treatise by Oetinger,
“Die Eulerische und Frickerische Philosophie über die Musik, ” which is a ten point
comparison of Euler’s and Pricker’s theory. This work provides some helpful insights
about both writers’ ideas, but again, it presents no new thoughts about Pricker’s essential
musical thesis.
Translation Issues
While it can be rather cumbersome to present both the original and translation of a
treatise in parallel columns, in this case it seemed a judicious approach. First, this work
is not readily available to many, so, for scholarly benefit, it is herein presented. Second,
the use of Latin in the eighteenth century, particularly this somewhat Germanic Latin,
offers some problematic translation issues. Finally, Oetinger and Fricker present a
number of obscure ideas and analyses that do not yield an easy translation or explanation.
Therefore, if there is any doubt or question as to this author’s method of translation, the
original is ready for examination.
16 Translation is often more art than technique. Initially, a literal translation was completed to assure accuracy. Afterwards, it was necessary to provide something that could actually be called English. To accomplish this, a certain degree of liberty was
taken. While trying to be accurate to the grammar, it was often necessary to change the
tense, voice, word order, and sentence length for clarity. Of course, it is always more
important to translate ad sensum, rather than hold strictly to the grammar. Taking this
liberty should not be problematic to the reader, since the original is presented as well.
17 PART I TRANSLATION
[BREVISSIMA THEORIAE MUS!CAE\
[FRICKER: SECTION I]
Numbers in brackets [ ] refer to page numbers in the Inquisitio. 1964 edition.
The music treatise, in the original, begins a new pagination alter p. 270 of the main treatise, therefore this section begins with page one. The text in braces | | is marginal commentary from the original. Occasionally, when footnotes are appended to a number, they are placed within superscript brackets [ ] to avoid confusing them with exponents.
Brevissima Theoriae Musicae Analysis A Very Brief Analysis of the Theory of Music
§1 §1
Complexus 12. Tonorum ah experientia The compass of 12 tones is supported suppositus. by proof.
Musici 12. Tonos agnoscunt intra spatium Musicians discern 12 pitches within the Octavae comprehensos: nam ultra illud interval of the octave: for beyond the iidem Toni; quoad qualitatem Musicam octave, the same tones occur again with seu adfectum. occurrunt; suntque illi. respect to their musical quality or effect. longitudinibus chordarum. quae in By the lengths of the strings (which on the monochordo ejusdem densitatis & eodem monochord have the same density and are pondéré tensae sunt, ad soni principalis stretched uniformly by weight), these
18 seu finalis, quern C adpellare lubet, pitches are related to the string of the chordam relatis, sequentes: principal sound or fînal (which we will call C) as follows:
C, 1. Sonus finalis (ex quo cani dicunt C 1 Final note (from which the Germans Germani) say one sings)' Cis, 'V|6. Semitonus seu Secunda minor. C# 'V|6 semitone or minor second^ D, %. Tonus seu Secunda. D % tone or [major] second Dis, %. Sesquitonus seu Tertia minor. D# % sesquitonus or minor third E, %. Ditonus seu Tertia major. E Vs ditonus or major third F, %. Diatessaron seu Quarta. F V4 diatessaron or fourth Fis, Quinta falsa seu minor. F#‘‘^64 false fifth or diminished fifth 45 ’/64 Quarta major. ^^4s false fourth or augmented fourth^ G, 7 3 . Diapente seu Quinta. G V3 diapente or fifth Gis, %. Semitonus cum Diapente seu G# Vg semitone-plus-diapente or minor Sexta minor. sixth A, V5. Tonus cum Diapente seu Sexta A Vs tone plus the dipente or major sixth major. Bb V16 sesquitonus plus the diapente or B, /i6. Sesquitonus cum Diapente, seu minor seventh** Septima minor. B Vis ditonus plus diapente or major H, Vis. Ditonus cum Diapente seu seventh Septima major. c '/z diapason or octave c, '/}. Diapason seu Octava.
[2] Not. 1. Plures quidem Tonos modemi Note 1. Indeed, modem musicians seek a Musici quaerunt, immo adhibent, sed greater number of tones—no, actually adfectus plane extraordinarios iis they add more, but with these additional expressuri, ita ut & ipsi naturam hie tones they try to express emotions that are limitati vel terminari facile concesserint. clearly extraordinary. Thus, as a result, they would concede that, in regard to the ratios, nature is restricted or limited.
' Flicker lists all ratios here as vibrations (eg. 15/16), not string lengths (16/15). See Chapter 1, page 152 for a table providing the calculations of these intervals and their comparison to the Pythagroean and Just systems. ^ Actually, the augmented second, C-C#, is 25/24, but Fricker is employing an older German notation convention, which also explains D# instead of Eb. F.T. Arnold notes that it was a common German convention to write e flat as dis (d sharp) "well on in the eighteenth century.” See F.T. Arnold, The Art of Accompaniment from a Thorough-bass, as Practised in the XVÜth & XVnith Centuries. London: Holland Press, 1961, p. 205. ^ These ratios are incorrectly reversed in the original. '* I believe Fricker chooses to notate this as Bb instead of A# in accordance with the German tradition of using the modal hexachord.
19 2. Tono minore ^/lo & Septima illi 2. In theory they use, even now, the respondente % etiam utuntur in theoria, smaller whole tone, 9/10, and the seventh sc. ita, ut isti numeri loco horum % & ^/i6 corresponding to the ratio 5/9, so that in in explicanda aliqua hannonia debeant explaining a given harmony those adsumi; sed in praxi coincidunt isti Toni numbers ought to be adopted in place of cum his % & ^/i6; quemadmodum & hi 9/8 and 9/16, respectively. In practice, Toni & ‘‘*/64 ab auribus nostris however, those tones coincide with 9/8 confunduntur. and 9/16^ just as these tones—32/45 and 45/64— are blended by our ears.
3. Temperatura, qua rationes non nihil a 3. Temperament (by which the ratios, at praecisione datorum numerorum least in practice, are similarly bent a bit deflectuntur, similiter in praxi saltem, from the given numbers) is allowed in inque iis instrumentis, ubi transpositio those instruments where transposition has locum habet, ut in clavichordio, & propter a place, as on the clavichord.^ This is hanc ipsam suscipitur, quia ilia puritas seu because a clean and accurate expression adcurata expressio rationum inC = 1 of the ratios on C = 1 would produce the maximam inferret impuritatem, si ex D greatest impurity, if one were to sing from caneretur, nam tum E deberet esse D, because then E would be the second of Secunda Secundae pro finali assum[p]tae a second (i.e. 8/9 x 8/9 = 64/81) according % % =.64 cum tamen revera esset % = to the adopted final [C]. In reality, ^/go, qui defectus in puro Tono % aurem however, E is 4/5 = 64/80. That nimis feriret. difference in the pure tone 4/5 strikes the ear too much.
§2 §2
Ex definitione Toni, utpote rationis From the definition of a tone as the duoroum sonorum, eruitur methodus ratio of two sounds, a method for investigandi harmoniam plurium investigating the harmony of many sonorum per reductionem fractionum sounds is elicited through the reduction sub eundem denominatorem. of the fractions to the same denominator.
Toni isti singuli non sunt nisi relationes Tones do not exist individually, but as the duorum sonorum (inde & a Musicis relations of two sounds. This is the reason vocantur intervalla), atque adeo rationes they are called intervals by musicians. geometricis proportionibus fundamentum Moreover, the ratios furnish the basis for praebiturae, nam ex aequalitate rationum geometric proportions. Indeed, proportion oritur proportio; quapropter & per arises from the similarity of ratios. On this fractiones exprimuntur. account, they are expressed through fractions.
* 9/8 with 10/9 and 9/16 with 5/9. ‘ Tha is, fixed-pitch instruments.
20 [3] Ex.gr. in § 1. D dicitur esse 8/9, h e. si For example, in section 1, D is said to be chorda, quae C sonat, habet 9 partes in 8/9. That is, if the string that sounds C longitudine sua, tum, si hujus chordae has nine parts in its length, and if, then, magada mobilis ad immobilem tantum this string’s moveable bridge is moved accedat, quantum opus est ad sonum 8 nearer the fixed bridge by the amount partium, nona ilia parte neglecta, chordae needed to obtain a sound of eight parts invariato pondéré tensae obtinendum, erit minus that ninth part, (and the string’s ille sonus D: itaque sonus C ad sonum D, tension is unvaried), that sound will be D. ut longitudo 9 partium ad longitudinem 8 Therefore, the sound C is to the sound D partium ejusdem, quoad ceteros respectus, as the length of nine parts of the string is chordae, h.e. C D = 9:8 seu sonus D est to the length of the eight identical parts of pars % soni C. the same string. Thus, C D = 9:8, or the sound D is 8/9ths of the sound C.
Ergo si plures auri sonos vel successive Therefore, if you will present several vel simul obtuleris, anima pariter exquiret sounds to the ear either successively or horum, quotquot sint, rationem, qui labor simultaneously, the mind^ will seek their est Arithimeticus; plurium enim ratio, however many there are. This task fractionum ratio indicator a singulis is arithmetical. Indeed, the ratio of the ipsarum numeratoribus, quam primum ad many fractions is indicated precisely by eandem reductae fuerint denominationem: their individual numerators as soon as quo labore suscepto & peracto judicium they have been reduced to the same de Consonantia illorum Tonorum eo modo denominator. Once that task has been feret anima, quo ferre solet in allegatis undertaken and completed, the mind will jam Tonis, C, D, E, F, G, A, H, c, ex proceed to judge the consonance of those quibus intervalla v. gr. Tertiae minoris C tones, in the same way it is accustomed to & Dis, majoris C & E, & Sextae minoris judge collected tones, C, D, E, F, G, A, C & Gis, majoris C & A vocantur valde B, c, the intervals from which, for jucunda, Quarta autem seu Quinta falsa C example, the minor third (C - D#), the & Fis, item Septima major C & H, major third (C - E), and the minor sixth (C Secunda minor C & Cis, valde dissonantes - G#), the major sixth (C - A) are deemed sunt. very agreeable, whereas the false fourth and fifth (C - F#), likewise the major
^ Throughout the entire treatise, the word for mind or soul appears 186 times. O f those, 166 times is the feminine word anima, and only twenty times is the masculine version, animus, used. Anima carries more the thought of “mind,” “soul,” “seat of affections.” Animus can be soul also, but is closer to the idea of the “rational mind,” the “intellect.” All occurrences are simply translated as mind, and those cases which use the masculine are marked in the translation with an asterisk [*]. N.B. Prof. Freudenburg; T h e anima and animus division you describe works for the classical usage. And I believe it works well enough for the way the terms are usol in your treatise. I think you can keep it as is. If anything, Oetinger and Fricker are a bit too free in their use of anima in the sense of mind. Anima is more the life-force/ soul, animus the 'mind' in our sense as rational principle. Often Fricker seems to use the former in the latter sense. Not [critical] though, as the Rotnans themselves used these terms very loosely.” That is, the Major scale
21 seventh (C and B), the minor second (C • C#), are very dissonant.
§3 §3
Generalis idea inferlus arithmetice Below is a general arithmetic demonstranda de operatione animae in demonstrating the mind’s operation in reducenda & resolvenda consonantia & reducing and resolving the consonance dissonantia Tonorum. and dissonance of tones.
Dabo Exemplum; In Clavichordio tange For example: on the clavichord, play uno ictu claves C & c infimae Octavae simultaneously the keys C and c of the sinistra, & dextra tertiae Octavae claves c, bottom octave with the left hand and the e, g, b (liceat enim his litteris minutioribus keys c, e, g, of the third octave with the [4] uti loco illarum, quae pro Discantu r i^ t hand. (Indeed, in place of these alias soient lineolis supemis notari); hunc smaller letters, one may use those little tactum excipe tactu altero clavium F & f superscripts that are customarily marked in Basso, & in Discantu clavium c.f. a: in the treble.) After this stroke play F and audies ita consonant!am ex dissonantia f in the bass and c, / a in the treble. You resultantem; Atqui fac hie, quod in § will hear a consonance [F major] antecedente dictum, & cognosces resolving a dissonance [C7]. But now, operationem animae, quam ita consider here what was written in the repraesentamus: preceding section, and you will recognize the operation of the mind, which we represent thus:
22 [tactus] Stroke I. C c c e S b A [vibrationes] vibrations: V4 V2 1 % \ % B [rationes] ratio: 9 18 36 45 54 64
[tactus] Stroke 2. F f c / a C [vibrationes] vibrations: V3 % I % % D [rationes] ratio: 12 24 36 48 60 12- E [rationes] ratio: 1 2 3 4 5[’]
Figure 1; String ratios from Pricker’s experiment in Section three.
Nimirum anima sonos tactus prioris Evidently, in order to understand this ratio percipiens in ratione datarum in serie A more clearly, the mind perceives the fractionum, quoad vibrationes, eamque sounds of the first stroke as vibrations'®— clarius intellectura redigit fractiones sub given in the ratio of the fractions in series communem denominatorem 36; quo facto A . It reduces the fractions under the rationem sonorum non ex minoribus common denominator 36. That being intelligere potest numeris, quam ex datis done, the mind understands the ratio of in ferie B; at cum subsequens tactus, cujus the sounds not from the smaller numbers, vibrationes pro singulis sonis anima in as much as from the numbers given in datarum iterum in serie C fractionum series B. And while the tone c does not ratione percipit, reducendarum pariter ad change in the subsequent stroke, of course numéros integros, tonum c non mutet, sed the others do. The mind perceives these reliquos saltim; vibrations as individual sounds in the ratio of the fractions given a second time in series C, which are reduced to integer numbers.
propter repetitionem ejusdem soni pro hoc Because of the repetition of c, the mind eundem in prima reductione fractionum will adopt the same number, 36 (which tactus prioris ex serie B numerum 36 occurrs in the first reduction of the
' An explanation of the chart:
C c c e S h* C c c / a 1/4 1/2 1 5/4 3/2 16/9 1/3 2/3 1 4/3 5/3 multiplied by 9 18 36 9 18 4 m ult by 12 12 36 12 12 equals 9/36 18/36 36/36 45/36 54/36 64/36 equals 12/36 24/36 36/36 48/36 60/36 leaves 9 18 36 45 54 64 leaves 12 24 36 48 60 div. by 12 = 1 2 3 4 5 Fricker is calculating vibrations, not string lengths; therefore, C = 1/4 is vibrating 1/4 as fast as c = 1; g = S/4 is vibrating 5/4’s faster. Fricker provides his explanation in § 4, n. 2.
23 adsciscet anima in seriem D numerorum fractions of the first stroke), in series D of tactus posterions, qui numeri, posito pro c the numbers of the second stroke. With seu 1 hoc 36, erunt, ut feries D indicat: the number 36 replacing c or 1, the other sed jam intelliget anima, rationem eandem numbers will be indicated as in series D. seriei D minoribus multo numeris in serie But now the mind understands that the E posse exprimi; id quod sequente statim same ratio of series D is able to be [5] §. Arithmeticae non adeo peritis expressed in the smaller numbers in series proHxius explicatur. Nonne vero labor E. This is explained more explicitly in the Arithmeticus in primo tactu magnus fuit, next § for those not so skilled in cum 6 . fractiones deberent ad communem arithmetic. However, was it not a great reduci denominationem numeri 36? arithmetical feat in the first touch, when six fractions were reduced to a common denominator of 36? nonne ex eo nati in altero tactu numeri As a result, are the simple numbers that simplices per abbreviationem arise in the second stroke not pleasing by suavissimam delectant? Facile me the most gratifying abbreviation? intelligent Musici, facilius Arithmetici Musicians will easily understand me; practici, qui spontaneis numerorum practical mathematicians—who regularly maximorum ad minimos resolutionibus delight in the spontaneous resolutions of delectari soient. Providit autem heic larger numbers into smaller numbers— Deus, ne analyseos multitudine anima more easily. Moreover, in this matter, opprimeretur, ut infra patebit. God provided that the mind is not overwhelmed by a multitude of analysis, as it will be made obvious below.
§4 §4
Explicatio Caiculationis in § antec. An explanation of the calculations institutae. established in the preceding §.
Calculaturus autem duos illos tactus Moreover, he who will calculate those two supponit strokes supposes:
1) in utroque c= 1, ut ratio ceterorum 1) that c = 1 for both, so that the ratio of sonorum facilius determinetur ex datis in the rest of the sounds is more easily §. 1. fractionibus: determined from the fractions given in the first §:
2) erit ergo e in tactu primo = A; nam 2) Therefore, e will = % in the first stroke. ffactio % §. 1. invertenda est, quia The fraction % from § 1 must be inverted vibrationes inverse se habent, ut because the vibrations are the inverse of longitudines chordarum ita, ut dimidia the string lengths, such that the divided monochordi chorda bis vibret, dum tota string of the monochord vibrates twice.
24 semel, atque adeo pars ejus % quinquies, while the whole string vibrates once. And dum tota quater: so, precisely one part of % vibrates five times, while the whole string vibrates four times.
3) ex eadem ratione erunt & in primo & in 3) from this same reasoning, the ratios secundo tactu (quia in utroque c= 1),/= will be inverted both in the first and in the g = ^ 2, a = 6 ,6 = *%: second stroke (because in both c=l), thus f =% , g = Vi, a = ^ 3, b = ‘Vg.
4) at ex eadem quoque ratione c erit V2 , 4) and also from this reasoning, c will be nam ut vibrationes c (quae pro duplici 72 , for as the vibrations c (or c” ) is to c, lineola notata valeat) ad c, sic & c ad c, sc. thus also is c to C. Of course, it is the intervallum Octavae est in utroque casu: interval of an octave in both cases: atqui ex dicta ratione est c ad c, ut ad I, however, from the previously stated seu, quod idem, ut 1 ad ‘6 , ergo c = V 2. reason it is c to c, as 7i is to 1, similarly, Et sic quoque erit f = V3, F = 73, C = 74 . as 1 is to 72 , therefore c = 72 . And thus also f = ^ 3, F = 73 , C = 74 .
5) Redige fractiones tactus primi inventas 5) Reduce the fractions discovered in the ad communem deno-[ 6 ]minatorem 36, qui first stroke to the common denominator producitur ex multiplicatione 36. This is produced by multiplication of denominatoris 9, qui est in 6 = in the denominators 9, which is included in b denominatorem 4, qui est in C = 74 , e = = 'Vg, in the denominator 4, which is %, & sub quo jam tertius 2, qui est in c = included in C = 74 , e = ^U, and under 7%, & g = ^ 2, continetur; which moreover the third denominator 2, which in c = 72 , and g = ^ 2.
sic habebis C = 74 (h.e. multiplicande Thus you will have C = 74 (that is, by utrumque fractionis numerum per 9) = multiplying each number of the fraction %6, c = 72 (mult, per 18) = c = 1 by 9) = % 6, c = 72 (multiplied by 18) = (h.e. 7i seu mult, per 36) = e = % * / 36, c = 1 ( that is, 7i or multiplied by 36) (mjp. 9) = ‘‘736 ,8 = y 2 (ra.p. 18) = ^ 65 , b = ^736, e = ^4 (multiplied by 9) = g = = * h (m.p. 4) = ®^/36; h.e. Cx:c:e:g:b = ^2 (multiplied by 18) = b = '7 , 9:18:36:45:54:64, qui numeri nullo (multiplied by 4) = ^ / 36: that is—C : c : c : quocunque simul invicem possunt dividi e:g:b= 9 :1 8 : 36:45 : 54:64. These aut deprimi, quia 45 & 64 sunt inter se numbers are not able to be similarly primi. divided or reduced reciprocally, because 45 and 64 are, to each other, primes."
6 ) Pari ratione inverties pro altero tactu & 6) By the same rationale, you will communi denominatore 36, in c iterum discover the second touch and its common
" They are not reducible in relation to each other, i.e., do not share a smaller Least Common Multiple (LCM)than36.
25 adsumendo, F = V3 (m.p. 12) = f = \ denominator 36. In adopting c again, F = (m.p. 12) = % , c = 1 (V , m.p. 36) = V3 (multiplied by 12) = /3e, f= 6 / = h (m.p. 12) = ^/36, a - % (m.p. 12) = (multiplied by 12) = c = 1 (V; % , h.e. ? ± c ta = 12:24:36:48:60 = multiplied by 36) = ^1^, f = % 1:2:3:4:5, nam tota ilia series per 12 est (multiplied by 12) = “*^ 36, a = V3 divisibilis. (multiplied by 12) = “ /36, that is—F : f : c : f : a = 12:24:36:48:60 = 1:2:3:4:5, for that whole series is divisible by 12.
§5 §5
Specialior de musicis animae A more special deduction concerning operationibus deductio. the musical activities of the mind.
Quodnam igitur est animae in audienda & Therefore, what is the activity of the mind uno actu leflexivo dijudicanda Musica in hearing and judmng music in one negotium? Multiplicatio & Divisio in reflexive motion? [It is this:] the continua simultaneorum & successivorum multiplication and division of Tonorum collatione, seu reductione simultaneous and successive tones in a fractionum suarum ad maiores & minores continuous comparison, or the reduction termines, prout id distincta singularum of its fractions to larger and smaller perceptio ex communi denominatore limits—in so far as a distinct perception of haurienda requirit. Quodnam inde illi individual [ratios] to be drawn out from excitatur molestiae? the common denominator requires it. For what [reason] then is it excited to that trouble?
difficilis fit labor in Tonis minus congruis The task becomes diffrcult in tones that seu in fractionibus denominatorum are less congruent, or in fractions with diversae originis. Quid delectationis? [7] denominators of diverse origin. Where sponte Tonorum rationes, quae ad does pleasure arise from? On their own, maximos numéros excreverant, se the tones’ ratios that had increased to componunt iterum, statimque ad large numbers, arrange themselves a simplicissimos redeunt, uno vel altero second time, and immediately reduce Tono mutato. Quaenam denique radix themselves to the simplest numbers by harmoniae? origo ipsarum fractionum, ex alteration of one or the other tone. qua quippe vel consentientes vel Finally, what then is the root of harmony? dissentientes fluunt. Et quae ad illam via? The origin of the fractions themselves, from which, of course, flows either consonances or dissonances. And what is the way to that origin?
That is, the mind performs the demonstration of section 3 in one step.
26 idem nimiram Arithmeticae genus, Doubtless, it is the very same type of Multiplicatio & Divisio; prout enim arithmetic—multiplication and division. denominatores ex iisdem aut aliis numeris For in so far as the denominators are primis sunt producti, vel concinunt, vel produced riom the same numbers or from dissonant. Sic in exemplo dato Tonus b = other prime numbers they either produce totius laboris illius culpam sustinet, consonant or dissonant sounds. Thus, in quia nominatori 9 non convenit cum the example given, the tone b = ceteris 4 & 2. sustains the responsibility of that entire task, because with respect to the 9, it does not agree with the other [denominators], 4 and
§6
Inventio basées Musicae posita in Devising the basis of music rests on the resolutione fractionum per divisionem reduction of fractions by division to the ad numéros primes 2 , 3, & S. prime numbers 2,3 , and 5.
Agedum igitur: properemus ad fontem Come then, let's hasten to the source of harmoniae; hie videmus quidem harmony. Indeed, this is where we see it:
1 = I, V2 = V2, "6 = V3, ^/s = Vs. but likewise [sed etiam] % = V i. 2. V5 = ^ V5, Ve =
Vz . 3, Vg = V2 . 2 2, Vg = V3 .3, Vg = ^ ^ V3.3, V|0 = ^ V2.5. Vi6=^ V2.2.22, V|S :
32 2 - 2 -2 -2 -2•/; _ , 45 43// _ 3 - J3 '5 3/ /]. S, /l6 = /2 .2 2 2, Us- '3 3 5, ' 6 4 = '2 2 . 2 2 . 2 . :
Figure 2: Derivation of ratios through multiplication.
Ergo tota Musica pendet a Aequenti Therefore, all music depends on the Multiplicatione & Divisione numerorum repeated multiplication and division of the trium primorum 2,3, & 5, in ordine suo three prime numbers 2,3, and S, flowing ex unitate fluentium. En observationem in their order from unity. Granted, this is facilem, sed tamen totius Theoriae an easy observation, yet nevertheless, it Musicae fundamentum merito rightfully should be called the basis of all adpellandam! music theory!
In other words, without 16/9, the LCM for the first series would have been four, instead of 36.
27 §7 § 7
[8] Limites illius baseos, per quos The limits of that basis according to numéros istos non uitra potentias 2*, 3 \ which those numbers may not property 5 \ elevari par est be raised beyond the powers 2^ 3 , 5*.
Sed ita vel trium numerorum in infinitum But thus indeed, the multiplication of posset extendi Multiplicatio, nisi suis these three numbers could be extended fmibus contineretur: scilicet numerus indefinitely were it not contained within primus maximus S nunquam in se ductus its own confines. Of course the largest conspicitur, ut 25 in f[r]actionibus §.1. prime number, 5, is never seen to be occurreret; sed 9 ex 3 ortum superat ilium reckoned into itself, as 25 might have 5; immo in *^1^ 64 ex solo 2 ortum occurred in the factors of § iT'"*' But 9, excedit ipsum productum 45 = 5 9. having arisen from 3 surpasses 5; indeed, in '*^ 64.64, having arisen from 2 alone, exceeds the product 45 = 5 9.
Régula limitum videlicet haec est: Minor The rule of limits, evidently, is this: every quisque numerus eousque [sic, eo usque] smaller number, should be raised to the elevandus est, donee factarum ex degree that it surpasses, by size, the majoribus reliquis omnibus elevationum product of the elevated factors from all the productum superaverit magnitudine; remaining larger numbers.
{Basis Musica quid?} Hinc hoc schema (What is the musical basis?} Henceforth, r , 2^, 3 \ 5% 7®, vocaverim basin I will call this scheme— 1", 2^ 3 \ 5‘, f — Musicae, cujus occulta radix est divinus the basis of music, whose hidden root is ille septenarius, sed sub unitate latens, & that divine septenario. However, hiding cum eadem coincidens, nam 1“ = 7° = 1. underneath the septenario, and at the same time agreeing with it, is unity, for r=7°=i.
§8
Quomodo basis Musica totam Musicam Clarification of how the musical basis exhauriat, ostenditur. fulfills all music.
Facile nunc quivis Arithmeticae peritus Easily now, anyone skilled in 26 . 32. 5i. intelliget, productum hoc 1 mathematics will understand that this = 45 64 = 2880, si in omnes suos product r 2*^ 3^ 5‘ 7° = 45 64 = possibiles divisores versos resolvatur. 2880.^*“**^ This is so if it is reduced into
Five to the 2nd power occurs in the Just scale in the ratio 25:24, which Flicker does not use (Section 1). This procedure is based on Euler’s Tentamen treatise (1739), which demonstrates the Lowest Common Multiple (LCM) for a series of numbers. Fricker is presenting this as the LCM for all the ratios between I and 1/2. See Section 10, also the discussion of Euler in Chapter 1. See Charles Smith, “Leonhard Euler’s
28 atque ex omnibus istis ac singulis ail of its true, possible divisors together componantur fractiones quotquot possint, with the fractions—however many are magnitudine sua intra spatium Octavae 1 possible—composed from all of those & V2 cadentes, non daturum esse plures divisors [together] or individually. The aut pauciores, quam illos 15. in §. 1 & 6 . fractions, falling according to their size Tonos; within the interval of an octave — 1 and V2—will not give more or less tones, than those fifteen in sections one and six.'^
id quod rudioribus ita ostendo; Pone ex I make it clear, thus, to the uninitiated: numeris ordine naturali 1, 2,3, &c. place fractions derived from numbers sequentibus fractiones ex.gr. '/% (at V3, V4 following in the natural order 1, 2,3, etc., &c. jam sunt minores, quam '/%, itaque in for example Vi (Whereas V3, V4, etc. are Octava eadem, cu-[9]jus Toni quaeruntur, smaller than V2, and therefore, they are non comprehenduntur), porro l^ (at % = not included in the octave, of which we V2 jam adfuit, V5 & sequentes sunt justo seek the tones); hereafter, V3 (whereas V4 minores); is already = V 2, V5 and its following [fractions] are, with propriety, smaller [than 1/2).
pergo igitur ad %, V 5 (at iterum desino, I proceed therefore to V4, V5 (whereas I quia % = V2 &c.); sequitur itaque % (% = stop again, because Vo = V 2, etc.); and thus 6 adfuit, transcenderet basin, quae Vs follows (Vo already is = V3, V? goes septenarium verum non admittit, !% = V2 beyond the basis, which the true iterum terminât usum numeri 4); %, Vg, septenario does not allow, Vg = V2 again, V9 (Vio = V2, % habet septenarium terminating the use of the number 4); Vo, vetitum, Vg = V4, V9 = "6, V10 = V5, Vn Vg, V9 (Vio = V2, V? has the forbidden 7th, habet 11 vetitum, V12 = V2. Vg % Vio &c. Vg = V4, V9 = V3, Vio = Vs, V ii has the excluduntur a basi); V9 (Vio = V5, Vu forbidden 11th, V12 = V2, Vg, V9, Vio and excluditur, V12 = V3, V13 & V14 = V? so on are excluded by the basis); V9 (Vio = transcendant basin); V15, Vio, Vie; sicque Vs, Vii is excluded. V u = V3, V13 and V u = rite pergendo non invenies plures V? go beyond the basis); Vis, Vio, Vio; and fractiones licitas, quam ‘Vie, & “*Ve 4. thus, duly proceeding, you will not discover more permissible bigger fractions, than Vio, and /04.
Tentamen Novae Theoriae Musicae: A Translation and Commentary.” Diss. Indiana University, 1960. There is a total of fifteen tones given in section six, and the bottom o f Section eight (not including the unision. I). To find IS tones in section one, one must add the alternate ratios (10/9, etc.) which Fricker does not include in the scale.
29 §9 §9
Cel. Euleri régula explicatur a nostra, We explain the basic principle of the sc. quatenus rationes simplices ex celebrated Euler, namely, that, insofar compositis oriuntur, eatenus delectant. as the ratios that arise from composites are simple, they are pleasant.
Adparet hinc, regulam celeberimi Euleri It is apparent that the rule of the in Tentamine Theoriae Musicae, qua ex celebrated Euler in his Tentamen Novae simplicitate rationum suavitatem Tonorum Theoriae Musicae [1739]—whereby he atque harmoniam Musicam aestimat, a judges the sweetness of the tones, as well priori traditam, ita a nostra, ab experientia as the musical harmony, from the desum[p]ta, explicari & determinari: simplicity of the ratios, handed down a Musicam esse omnium animi adfectuum priori—is thus explained and defined by non saltim laetitiae pictorem, & hanc us, by select experiment. Music is, of all quidem ea lege ex simplicitate nasci the a^ections of the mind*, not in the solere, quatenus ilia ex opposite suo least the painter of pleasure, and pleasure résultat, ac magis elucescit. is accustomed to originate from simplicity by that principle, in so far as simplicity results ftom its opposite [i.e., complexity], and also shines out all the more from it.
Sic numeri 2,3,4, &c. ex suis compositis Thus the numbers 2 ,3 ,4 and so on, arise oriundi, sic maximus primus S ex according to their composites; thus the ceterorum primorum productis seu largest prime 5 [arising] according to the elevationibus (5* ex 2 \ 2®, 3 \ 2 3,2^ 3, products or powers of the other primes (5‘ &c.), Musicam fere omnem according to 2*, 2®, 3^, 2 3,2“^,2 . 3, . and so exhaurientibus, magnisque illis & ad 9, on), they charm nearly all music 64, immo propter Octavas Musicas ad according to their drawing out, and those 512,1024 & ultra excurrentibus numeris large numbers, even to 9,64, more resultans delectant. correctly, on account of musical octaves, to 512,1024,^'"'^ reverberating beyond their extended numbers.
S12 = 2”, 1024 = 2'°; Fricker is simply showing that adding octaves does not change the fundamental reliance on the prime numbers.
30 §10 10
[10] Cur rationes ex 7,11, &c. numeris Why are the ratios produced from the factae extra Musicum concentum non numbers 7,11, etc. displeasing? (That delectant? Rx. animae organum In is, those beyond the musical concord.) praesenti aevo nondum eo est elevatum, Answer: At the present time, the organ ut septenarium aetemitati reservatum, of the mind has not been si^ciently multo minus reliquos, plene elevated in order to pursue the number adsequatur. seven, which is reserved for the future—let alone the remaining ones.
At cur numeri 7, II, etiam simplices, si But why are the numbers 7, II—actually vel extra compositionem Musicam ex.gr. simple numbers if they are considered in singulis Tonis %, ®/n, %, ’/n &c. apart from musical composition, for considerentur, non magis delectant, quam example, as individual tones '*/?, %i, ^/?, qui ad Musicam pertinent majores etiam ’/ii etc.—not more pleasing, than larger sine Musico usu, sed saltem v. gr. in numbers which do pertain to music— singulis audiendis Tonis */is, ^/le, *^/i 6, even without a musical use, but at least, &c. adhibiti? Respondeo: for example, must be heard in the individual tones ®/is, ’/le, ’^/le etc.? I respond:
1) Delectatio non potest quaeri ex 1) Delight is not able to be sought from a sensationibus paucis; si animus ab few sensations. If the mind* is aroused to extemis objectis sit ad delectationem delight by external objects, then infinite incitandus, infmitae impressiones in impressions produced in the body are corpus factae vix sufficient ad effectum hardly sufficient to be responsible for this hunc praestandum, itaque ex singulis effect, and thus, we can judge that Tonis non satis delectationem aestimare individual tones are not adequately possumus. pleasing.*®
2) Anima basin Musicam in ipso suo 2) The mind carries the musical basis in fundo gerit, ex hac ergo dijucUcat Tonos, its core, therefore, according to this it perque eam ipsa quoque limitata est, vid. judges tones, and by it, the mind itself is §. 7. & 9. & haec est organon seu limited, as seen in § 7 and 9. And this instrumentum sensus illius communis basis is the organ or instrument of the musici, quo videmus, e.gr. sequelam common musical sense, by which we see, illorum duorum tactuum §. 3 decem for example, the sequence of those two rusticis placituram, cum vix unus sit, cui strokes in § 3, which would be pleasing to displiceat. ten peasants, since there is hardly one to whom it would be displeasing.
That is, any complex ratio is possible by itself, but the mind cannot understand it in context. This also is forwarded as an explanation of the unusable ratios 4/7,6/11, etc. Even though they are simpler than 8/15, 9/16, etc., which are used, they do not mean anything musically by themselves.
31 3) Ponamus, dari Musicam, quae 3) Let us propose that the musical basis is secundum eandem harmoniae legem etiam given to them that, following the same septenarium admittat, videlicet in potentia law of harmony, admits even the prima l \ ut nostra quinarium 5*; hoc vero septenarium—namely, the prime power ilia utetur in secunda, quia 5 < 7,5^ ( = 7*, as our fifth 5'. However, 5 will use 25) > 7; porro 3 elevabitur ad 3^, nam 3^ ( that [power] to the second, because 5 <7, = 81) < 5^ 7‘ ( = 175), sed 3* ( = 243) > 5^ (= 25) > 7; furthermore, 3 is raised to 175; denique sic 2 adscendet usque ad 2'^, 3^ for 3^ (=81) <5^ f (= 175), but 3^ nam 2'^ ( = 32768) <3^ 5^ 7 '( = 42525) (= 243) > 175; thus finally, 2 raises all the sed2“^( = 65536)[ll]>3^ 5^ 7'. way up to 2 '^ for 2‘* (= 32768) <3^ 5^ 7' (42525) but 2*“ (65536) > 3^ 5^ 7*.
(Basis Musicae coelestis.) ( Basis of heavenly music.}
Sed quis no videt, talem Musicam ex hac But who does not see, that the mind basi r , 2‘^, 3*, 5^, 7‘, 11°, conficiendam cannot grasp such music to be constructed capi nondum posse ab anima, quia ea vel from this basis 1“, 2'°, 3 \ 5 \ 7*, 11°, in una Octava 157. Tonos comprehendit, because it actually includes 157 tones in quorum aliqui sibi proximi vix /eoo parte one octave, of which some of the very chordae totius inter se differunt, ut adeo close ones differ by scarcely ^ 6oo of the ab aure nostra plane debeant confundi. whole strin g ,so that truly they are clearly confused by our ears.
Sine dubio igitur coelestis Musica hac Without a doubt, therefore, the music of basi fundatur. Hoc vero non obstante heaven is founded on this basis. equidem suspicor, etiam in hoc tempore However, this does not keep me from usum septenarii non plane esse suspecting that even in this time, the use impossibilem, sed ei, qui se in tali Musica of seven is clearly not impossible. But to frequenter exercitaverit, paullo him, who frequently trains himself in such comprehensibilem, quia infra §. 17. music, it is somewhat comprehensible, observabimus, Italos pariter eam Musicae because as we will observe below in § basin, quae ad Tactum pertinet, ipsamque 17,^° that the Italians in like manner adeo naturam transcendere. surpass the basis of music which pertains to the tactus,^* and indeed nature itself.
” The number 137 is a permutation based on this new series, similar to Pricker's permutation of 15 tones based on the original series (Henck comfirms this view). Euler’s “degree” method can obtain the 15 tones from Section 8. However, applying this method to the new series does not yield the 157 number. It is also not clear how he is calculating the number 5/600. See the discussion in Chapter 1. ^ This refers to the example of the augmented 6 '*', at the end of Section 17. See explanatory footnotes in Section 17. This refers to the example of the “military march” rhythm, given in Section 17. See explanatory footnotes in Section 17.
32 §11 §11
Generalis Idea de concentu melodiarum The general idea about the concord of plurimum In Musica. Definitio most melodies in music. The definition Melodiae & Harmoniae. of melody and harmony.
Hactenus de Tonis extra Compositionem To this point, we have considered tones Musicam consideratis; cum vero Musica outside of musical composition. sit ars & successives per Melodiam & However, since music is an art of simultanées per Harmoniam dextre composing tones—both successive tones componendi Tonos; inquirendus nunc est through melody and simultaneous tones uterque modus Compositionis; & through harmony, that is, composing them posterior quidem in Harmonia consistens skillfully—now, both modes of quasi fundamentalis §. 2. & 3. jam aliquo composition must be investigated. In fact, modo est consideratus; licet enim the latter, consisting in harmony as harmonia etiam aliqua ad successives fundamental to the basis in § 2 and 3 is Tonos pertineat, is tamen modus, quo ad now considered by another manner. For eos referenda est, ultimum fere in Musica although actually any harmony relates to erit considerandum: Contra Melodia, successive tones, nevertheless this quasi Musicae proprium, nunc in manner, which must be referred back to considerationem venit. those tones, will be considered just about the ultimate in music: we now consider counterpoint as special music.
Nimirum ante omnia hanc tradit regulam Evidently, before everything else, the cel. Mattheson in dem voll[l2]-kommenen celebrated Mattheson delivers this rule in Capellmeister. In Musica omnia debent the Vollkommenen Kapellmeister [1739]: concinere, h.e. quantum equidem in music, everything ought to harmonize, intelligo, Toni simultanei diversi diversas, that is, as much as I understand, sed inter se consentientes, in successione simultaneous diverse tones make diverse sua conficiunt melodias, quapropter & in melodies, but between themselves—in uno pleno cantu a Musicis genera their succession—consonances. melodiarum constituuntur quatuor ilia: Wherefore, in one complete song by Bassus, Altus, Tenoris & Discantus. musicians, those types of melodies being arranged are four: Bass, Alto, Tenor, and Soprano.
Ergo in Exemplo §. 3. dato non saltim Therefore in the example given in § 3, we consideranda erat magnitude numerorum not only considered the size of the 9,18,36,45,54,64, & simplicitas numbers 9,18,36,45,54,64, and the sequentium 1, 2,3 ,4 ,5 ; sed etiam simplicity of the following numbers 1, 2, aequalitas rationum in duobus tactibus: 3,4,5, but also the equalities of the ratios in the two strokes.
33 C :F = c : f =c\f ~e:a 9 :1 = 18:2 = 36:4 = 45:5
Figure 3: Summary of key ratios from Flicker’s experiment, section three.
ubi etiam numéro 3 = c respondebat 54 = Where, furthermore, the number 3 = c g, duplum numeri 27 = g, nam 9:1 = 27:3, corresponded to 54 = g, twice the number & ubi denique numerus 64 = 6 illam 27 = g, for 9:1 = 27:3, and finally, where dissonantiam sustinebat, propter quam the number 64 = 6 flat supported that nullus ei in consonante posteriore tactu dissonance.^ Because of that dissonance respondere poterat. no ratio was able to correspond to it with a consonance by the later stroke.
§12 §12
Investigatur basis Musicae melodica, An investigation of music’s melodic quae constat ex potentils 2", 3*. basis, which comes from the powers 2", and 3 \
Sed jam de Melodia ipsa, quae vere But now we are concerned with melody artificium Musicae omne constituit, & in itself, which truly constitutes the whole art successivorum Tonorum compositione of music, and results from the collocatur. Successiva inferunt composition of successive tones. durationem, & vere duratio uniuscujusque Successions infer duration, and indeed, Toni dimensa, quae per varias Tonorum the measured duration of each and every notas (quae Latinis ut maxima, longa, tone, which, through the notes, dots, and semibrevis, brevis &c. Germanis autem rests of the tones—properly observed in sola Tactus, cujus duratio per notam O in the composition of the songs—completes contextu & ab initio per litteram C the entire melody. (Notes are designated indigitatur, divisione per fractiones Vz, V 4, by the Latins as the maxima, longa, Vg, V16, V32, ‘/64, quae sunt in ratione semibreve, breve, etc., they are designated potentiarum 2®, 2 , 2*, 2^, 2^, 2% by the Germans however as one denominantur), puncta (quae duratione measure,^ whose duration is indicated by valent dimidiam [13] partem antecedentis the whole note in the context and from the notae, ut cum ea semper sint 3 partes per beginning through letter C, by dividing it secantem numerum 2 factae), pausas according to fractions Vz, V4, Vg, V ie, In, (quae sunt in ratione notarum), probe in Ve4, which are in the ratio of powers 2®,
^ All of the other numbers in the C series can reduce through three (3) into the lower F series (musical demonstration. Section three), but 64 cannot. ^ Additive versus divisive rhythmic notation systems.
34 scriptione cantionim observatur, confîcit 2®, 2^, 7?, 2^, 2*; dots, with respect to melodiam totam, cujus partes aequales duration, are worth a half part of the tempore sunt tactus, qui mensurantur preceding note, so that with dots, there iterum partibus inter se aequalibus (talibus are always three parts made by the sc. quae possint, si opus sit, uno signo seu dividing number 2; rests are in the ratio of nota dictarum Tactus partium exprimi), the notes.) The melody’s equal parts, with vel secundum numerum temarium vel respect to time, are beats, which are binarium numerandi. measured again into equal parts in relation to themselves (which are able, if it is necessary, by the equal parts to be expressed by one sign or note of the designated parts of the beat), numbered according to either a ternary number or a binary number.
Ergo Melodia est nova Musica, cujus Therefore melody is a new music, whose basis duobus tantum constat numeris basis consists of only the two prime primis cardinalibus 2 & 3, ita tamen, ut cardinal numbers 2 and 3,^'^ but the more simplicior 2 ad infînitas fere potentias simple number 2 is able to expand by multiplicande excrescere possit, 3 vero multiplication almost to infinite powers, non facile ultra primam, h.e. 2", 3*. however, 3 cannot easily expand beyond one, that is 2“, 3'.
§13 §13
Eadem basis adcuratlus nunc Likewise, the same basis is now demonstratur ab experientia & demonstrated with greater precision on Musicorum regulis. the basis of a test and by the rules of musicians.
Quod prolixius nunc ita demonstro: I now demonstrate this more extensively Omnis Musica seu cantilena est oratio as follows; all music or song is—as it quasi aliqua persuasoria, ut preclare docet were—somehow persuasive oration, as peritus Mattheson in dem Kem the skillful Mattheson clearly pointed out Melodischer Wissenschafft, quae oratio ex in the Kem Melodischer Wissenschafft. suis constat propositionibus s[cilicet]. This oration derives from its propositions, collectionibus Tactuum aut duorum 2, aut that is, collective clauses of either two 2 trium 3, aut quatuor 4, aut 6,8,12,16,24, beats, or of three 3 beats, or 4, or 6,8,12, 32. &c., h.e. 2 \ 3% 2 \ 2' 3% l \ 2^ 3 \ 2 \ 16,24,32, etc., that is, l \ 3% 2 \ 2* 3% 2^
Euler had a similar thought concerning these two numbers: “In the discrimination of highness and lowness the proportions use 2,3, or S as factors; in the discrimination of duration, musicians have not reached beyond this point but have drawn all agreeableness of this kind from the numbers 2 and 3 alone. In this durational category, the hearing caimot comprehend such composite ratios as it can in the category of intervals” (see Smith, p. 29.)
35 2^ 3‘, 2*, &c. ubi vides 3 non ultra 22.31, 24,23 3«, 2^, etc. You see that 3 primam, 2 vero ad indefînitum does not go beyond prime, but 2 can be potentiarum numerum eievari. elevated to an indefinite number of powers.
Tactus iterum singuli sunt, hoc est, Tactus Once again, the meters are distinct, that is, ipse cantionis est vel rectus (gerade the meter of a song is either simple mensur), ex.gr. hie C, Vi (h.e. una nota (straight measure), 01 compound.^ talis O, quae pro I valet), %, %, %, 'Vi6 Simple meter, for example, [includes] C, &c. item % (h.e. duae notae, quarum V i (that is, a whole note equals 1), *U, unaquaeque pro V 4 notae O valet), %, */i 6 \ etc.; similarly % (that is, two &c. vel [14] obliquus seu triplus notes, each worth V 4 of the whole note), (trippeltact), quales sunt U, %, %, %, %, */i 6, etc. Compound or triple /12. */l2&C. (triplebeat), includes such ones as V 2, %, %, %, % 2, etc.
Cum vero omnes secundum régulas Since, according to the rules of musicians, Musicorum allegati numeri contineantur all the linked numbers are based on 2°°and in hac basi 2°°, 3'; vera haec est basis 3', this is the true melodic basis of music. Musicae melodica. Ceterum Melodiam But, it is apparent to the senses that quasi novum esse Musicae genus, melody is a new type of music from that sensibus patet ab eo, quod Germani which the Germans are accustomed to call adpellare soient risum ligneum, qui non dry laughter,^® which is merely a kind of est nisi adcurate secundum dimensiones singing accurately following the singularum Musicae notarum expressa measurements of individual musical notes cantio qualiscunque in sonis surdis, ubi expressed as if in muted sounds where nulla certe Tonorum Harmonia adeoque certainly [there is] no harmony of tones, sola Melodia locum habet. and for that reason only melody takes place.
§14 §14
Consideratio & calculatio aliquarum Consideration and reckoning of some minlmarum melodiae italicae very small parts of an Italian melody. particulanim.
Ut vero non nihil Compositionis In order that we might subject a bit of melodicae calculo subjiciamus, seligamus composed music to reckoning, let’s select cantilenam illam, ut appelant, italicam in that well-known song, as they call it.
^ Translating rectus and obliquus as “simple” and “compound,” respectively, addresses the basic issue; however, there could be a more subtle distinction, given the “fluid” nature of rhythmic theory, particularly its mensural background. ^ A very interesting term, quite likely refers to the coloratura singing style he is o f which he is so fond.
36 opera de Aitaxerxe occurrentem, ita occurring in the Italian opera about tamen, ne animi laedantur, textu mutate: Artaxerxes.^ However, in order that minds* be not offended it is [presented] with the text changed:
[Text from Fricker:] [Translation of Pricker:]
Conservatifedele Keep faithful (to me) Pensa, ch’ilSignor viene, And think about how E la mercede tiene The Lord is coming. garbat’al vincitor Fine. And His mercy is appropriate
Ch 'il Maestro della fede For the victor Doppo le pene fiede, For the Master of faith. Etemo Redentor. Da Capo After suffering is over, will reveal himself as Eternal Redeemer.
[Original Text from Graun:] [Translation of Graun:]
Conservati fedele Keep faithful (to me) Pensa, ch’il resto epena, And think about how E qual che volta almeno I will be suffering here. ricordati di me. And think of me at least sometimes.
Ch 'il per virtu d ’amore Because I, thanks to love's power. Will be parlando col mio core conversing with you ragionero con te. When I consult my heart.
Figure 4: Pricker's modifed text of Graun's Artaxerxes along with Gruan’s original and the translation of both.
Heic propter artifîcium Melodiae in Here, because of the high degree of art in Discantu summum Bassus propriam non the melody of the soprano, the bass does habet melodiam, nisi tacente Discantu, sed not have its own melody, except when the tantum animum in dividenda & secundum soprano is silent. The bass greatly assists
^ The opera Arfajcrjc (1743) by Carl Heinrich Graun (1704-1759), libretto by Pietro Metastasio. See Carl Heinrich Graun, Artaserse. New York: Garland Publishing, 1978.
37 singulas partes numeranda duratione the mind* by dividing and counting the alicujus in Discantu occurrentis soni ad duration according to the individual notes magnum excrescente numerum sublevat, of any sound of those occurring in the v:c: soprano,^^ increasing to a large number— v[oice]:c[ontinuo]:^ [15] 1) in voce pen-fa, ch’il, vbi [ubi] 1) in the voice pen - sa, ch 'il, where the durationes harum trium syllabarum sunt in durations of the three syllables are in a ratione numerorum 12:3:1, nam ch’il ratio of the numbers 12:3:1, for instance durât 1/32 Tactus (quem in posterum ch'il lasts V32 note (which hereafter we indicabimus per T) vi notae suae, -sa 1/16 indicate by T)^“. By note value, -sa lasts T ex nota & 1/32 T ex puncto, h.e. 3/32 T, V16 note and Vszdot, that is, V 32 notes, pen -1/4 T ex nota & ex alia confluente pen- is V4 note and tied to another Vg note, 1/8 T, h.e. 3/81 =12/321: heic that is Vg notes = 'V32 notes: here, the bass intercurrunt in Basso voces h & a, ita ut tones B and run in between, so that syllaba pen - longior dividatur, ut the longer syllable pen- is divided thus, as conspicis: you see:^^
pen—— — -fij d fil
\ ^ H Sea. 1 a 3
Figure 5: Pricker’s division of the syllables from the “pensa” motive.
28 The bass helps the mind make sense of the florid soprano by moving in slower note divisions. 29 This is a suggestion for what “v:c” stands for: vox:continuo. ^ Actually, simply translated here as note. Actually, C and B are in the bass, but it is not critical to his argument. This chart shows how the bass parses the soprano line in order to make it perceptible.
38 J J - ^ — 8— — * * ------J
pen flI chU
------ÎV ■ ■ ■"^” 1 ""
Figure 6 : Musical notation of Figure 5.
ita per variationem Toni in Basso, cum a Thus, through variation of the bass tone, sequitur h, bisecatur quasi terminus pe - at the point when A follows B [sic^^], pe nsa, ch’il, & tactus ipse, uti postea id fit in nsa. ch’il— and the beat itself—is divided repetitione per notam 16 T in Basso; as though a boundary, with the result that afterwards it is repeated with a Vz note in the bass.^‘‘
pen ft chio
Figure 7: Repetition of the “pensa” motive with half-note in the bass.
in repraesentatione autem hac vides, Moreover, you see in this representation sectionem primam fieri ab inceptione that the section goes, firstly, from the syllabae pen -, 2. [secun]dam ab beginning of the syllable pen-\ secondly, inceptione soni Bassus h, 3. [ter]tiam soni fiom the beginning of the sound in the Bassus a, 4. [quar]tam ab inceptione bass B [sic, C]; thirdly, from the syllabae -sa, 5. [quin]tam syllabae Signor. beginning of the sound in the bass A [sic.
Actually, B follows C in the music. ^ After helping the mind understand the soprano the fîrst time, it can have a simpler rhythm.
39 B]; fourthly, from the beginning of the syllable -sa\ fifthly, from the syllable Signor.
2) Eadem sublevatio animae a Basso 2) Likewise, the mind is assisted by the praestatur in usu colUgationum trium bass, accomplished in using three sonorum, notuia 3 indicatarum connected sounds, by the little sign 3 of (Dreyerlein), & quasi peculiares Tactus the indicated notes {threefold) [triplets], triplos minores constituentium, ubi Bassus and as it were constituting three especially simplicissimo pede ingreditur, small beats, where the bass advances by the simplest step.
Figure 8: Examples of sixteenth-note triplet division of the eighth-note.
& uno illis tribus respondente sono And, with one [bass] note corresponding numerat veri Tactus partes, ne is amittatur to those three, the bass numbers the parts ab anima, très istos brevissimos sonos of the true beat—by connecting the colligente; facile enim audiendo continuas triplets—lest it be lost by the mind. For V24 Tactus partes posset seduci ad easily, by hearing continuous ‘/24th parts adsumendum Tactum ex numéro 3 of the beat [1/24 equals 1/16* triplet], aestimandum, quia 24 = 3 8 = 3* 2^. the mind could be seduced into judging according to the assumed beat—from the number three, because 24 = 3 8 = 3‘ 2^.
Idem fit in altematione colligationum 4 The same thing happens in the alternation brevissimarum [16] V32 T, quibus of the smallest four connected V 32 notes, instrumentum quasi excellere vult cantori, by which notes the instrument as it were cum una illis respondente V4 T. wishes to distinguish the voice, with the one V4 note corresponding to them.^®
Either the V* is incorrect or the 1/32 is incorrect, because four 1/32’s do not equal one V* note.
40 Figure 9: Example of thirty-second-note division of the eighth-note.
§15 §15
Similis totius aiicujus Tactus A similar consideration by numbers of consideratio per numéros. a compiete measure.
Ac facile ex hac cantilena potest ostendi, And moreover, it can easily be shown quam Melodia nova sit Musica, ex.gr. from this song, how melody is a new Durationes sonorum tertii tactus in music. For example, the durations of Exordio ab instrumentis dicendo se habent notes in the third measure of the invicem ut VgT ex nota + VieT ex puncto, introduction, stated by the instruments, V32T ex nota + V^T ex puncto, V 64T ex maintain themselves in turn as Vg note + nota, (V3 ex Vg h.e.) V24T ex nota, V24T ex V16 dot, V32 note + V ^ dot, V « note, V24 nota, V24T ex nota, V24T ex nota, V24T ex note (V3 from Vg that is), V24 note, V24 nota, V24T ex nota, Viel ex nota + V 32T ex note, V24 note V24 note, V24 note. Vie note puncto, V32T ex nota, VgT ex nota, VgT ex + V32 dot, V32 note, Vg note, Vg rest, Vg pausa, VgT ex nota: note:^^
The first version is the original, the second what Fricker actually presents. He has changed it to make the mathematics fît his theory better.
41 ¥ w B
Figure 10: Musical notation of previous text. The first measure (A) is the original from Graun; the second measure (B) is Pricker’s modified version.
atqui: rather:
Vg + Vi6 = Vi6 Vg + Vi6 = Vj6 V32 + Vfi4 = Vô4 V32 + */64 = Vô4 V16 + V32 = V32; V16 + */32 = V 32;
itaque ut and thus as
Vi6, Ve4, Ve4, V24, V24, V24, V24, V14, V24, V16, Vô4, Vw, V24, V24, V24, V24, V24, V24, V32, V32, Vg, Vgp, Vg V32, V32, Vg, Vgp, Vg
seu (dividendo communem fractionum or (by dividing the individual denominator harum denominatorem 192 per singularem of each one by the common denominator cuj usque denominatorem, & quotientem of these fractions— 192, and by in unaquaque per ejusdem fractionis multiplying the quotient in each and every numeratorem multiplicande) ut numeri 36, one by the numerator of the same fraction) 9 ,3 , 8, 8 ,8 ,8 ,8 ,8 ,1 8 ,6 ,2 4 ,24p, 24 inter the result is the numbers 36 ,9 ,3 ,8 ,8 ,8 , se. Heic velim conspicias 8,8,8,18,6,24, 24p^’, 24 [as a ratio] between themselves.^® I wish you to notice:
1) producta numerorum ex 2 & 3 1) that the products of the resulting oriundorum, 2) rationes quasi dissonantes numbers come from 2 and 3. 2) the ratios, 9:8,8:18 seu 4:9,3:8; immo multo magis, as it were, the dissonances 9:8,8:18 or si colligaveris addendos duos primos 4:9,3:8. Or rather, even more so, if you
He used the German here to abbreviate the word “pause” for an 1/8"’ note rest. This chart converts the ratios he gives into a common denominator o f 192, which then allows one to derive his final series of numbers. 3/16 3/64 1/64 1/24 1/24 1/24 1/24 1/24 1/24 3/32 1/32 1/8 I/8p 1/8
36/ 3/ 3/ 8/ 8/ 8/ 8/ 8/ 8/ 18/ 6/ 24/ 24p/ 24/ 192 192 192 192 192 192 192 192 192 192 192 192 192 192 36 9 3 8 8 8 a 8 8 18 6 24 24p 24
42 numéros 36 + 9 = 45, rationem 45:3 seu will add the two increased prime numbers 15:1, quam sequuntur numeri 8, itaque & 36 + 9 = 45, notice the ratio 45:3 or 15:1. rationes 8:15,15:16, immo 32:45: denique which the numbers 8 follow. Therefore, 3) colligationes s. additiones veras, sed notice the ratios 8:15.15:16, more diversas; nam proprie très primi numeri properly, 32:45.^^ Finally, 3) notice the sunt summandi. deinde bis très connections, that is. the true additions, but subséquentes, iterum semel très deinceps diverse additions: that is. specifically, the positi, & postremo reliqui [17] duo. nam tlrst three numbers must be added, then, 36 + 9 + 3 = 48. 2(8 + 8 + 8) = 48. 18 + two times the following three. Again, the 6 + 24 = 48. 24 + 24 = 48. Atqui hoc three successive numbers in order, and Bassus indicat: finally the remaining two. for 36+9+3 = 48; 2 ( 8+8+8) = 48; 18+6+24 = 48; 24+24 = 48. Indeed, this bass indicates:
3 6 :9 :3 •Q*n»Q»Q»0*O •□•O*O*0*O*0 :i8:6 1 2 4 1 : 0 :14 48 — 4 8 ------:24- :2 + 1 :24 :Z 4.
Figure 11 : Flickers representation of the note values from the preceding paragraph.
Nunc inspice animum tuum. quaere istos Now examine your mind*, seek for those numéros, et exclama: 0 PaGoç nXou. t o u numbers, and exclaim: "O the depth of the Ktti aoipia KOI yvoioso) Geou! richness of both the wisdom and knowledge of God!'"*”
§16 §16
Consideratio totius alicujus Cantionis A consideration of another complete per conclusionem ex uno tactu facta. song as the consequence of one measure accomplished.'"
Atque hoc est praecipue incredibile ac And this is especially incredible and also divinum. animum magnam aliquam divine, that the mind* is able to hold—by
In this second observation. Fricker is demonstrating that it is allowable to consider any two pairs of numbers as adjacent in perception, even if not literally so in the music. He states this explicitly in the first sentence of the next section, regarding the mind's ability to "hold” a series of numbers in memory. This is somewhat reminiscent of Krumhansl’s theory of tone profiles. See discussion in Chapter I and Carol Krumhansl. "Perceptual Structures for Tonal Music." Music Perception. 1:32-54. The Bible, NT. Romans 11:33. In Section 14, he considered a fiagment of melody, in Section 15 a complete measure of the same, and now in Section 16 a more complete analysis of the song.
43 seriem tantorum numerorum posse means of the memory—some large series memoria tenere, &, ubi similis aliqua of so many numbers. And where one priori occurrit, cum hac conferre; sic non similar to an earlier one occurs, the mind tantum repetuntur statim propositiones: compares it with the latter. Thus not only are the phrases repeated immediately:
111 L f T i t It 1“ L L / t 'z
pen s« ch'A
Figure 12: Musical notation of immediate repetition of the “pensa” motive.
pensa, ch’il Signor viene, garbat’al From Pensa, ch'il Signor viene, to vincitor; sic non tantum melodia garbat’al vincitor, and not only is the propositionis altera: e la mercede tiene; second melody of the phrase [repeated]: e etiam de propositione: garbat’al vincitor; la mercede tiene —actually applied to the canitur, &c. phrase—garbat’al vincitor—it is sung [again], etc. sed tota stropha prior simili sibi melodia But the entire first strophe is repeated by a repetitur, brevi per repetitam melody similar to itself, briefly through a propositionem: pensa, ch’il Signor viene; repeated phrase: pensa, ch ’il Signor viene [the pensa phrase is sequenced from G nat., then Rf].
I,* Y a
ptn - M chio m to • • pm - o pm • m chio tuto t • pm - o
Figure 13: Musical notation of the sequence of the “pensa” motive.
44 in Sextam majorem, sed mollem, This happens [not] by a major 6 th [from digressione facta, qua restituitur Tonus G# down to B], but a minor [ 6 th, from G finalis, ex quo in dominantem Quintam Nat. down to B], by a digression having antea discessum erat; been made [to vii“7/ii, partly through the G Nat.]. By this the final tone is restored [by the next phrase from F# to A], from which it had previously departed into the fifth scale degree [i.e., secondary dominant function: the vii“7/ii].
‘ftMV.-iVMtlplnH*, Ota IK.
«UaTAi
Figure 14: Musical notation of the sequence and use of secondary function.
immo cum haec ipsa repetitio iterum in Indeed, since this very repetition [a large Quinta desinat, tertia sit repetitio a Quinta phrase repetition] cadenced a second time molli incipiens & turn demum in verum on the 5th, a third repetition happens by Tonum evadens. beginning [in] a minor 5th [actually, pointing at A minor, but he is correct that dominant notes are in the bass and melody] and then finally escaping into the true key [A major].
45 #» M MM..."
I V
M: W 3 3 #
Figure 15: Musical notation of the previous analysis.
{Explicatio Cantus mollis.} (Explanation of the minor song.}
Et, quod satis mirari nequit, cantionis And, here is something which one cannot altera pars adeo similis est priori, sed in be amazed at enough: the second part of cantu molli, ut fere non nisi remissior, the song is so similar to the first, but in the prout id hie cantus requirit, & ab [18] minor melody, that it is only more artificio extemo, quod nunc in harmonia subdued, just as this song requires it— Tonorum interna, rationibus plane inversis both by the external art, which is now in ac difficilioribus gaudentium, requiescit the internal**" harmony of the tones, atque occultatur, remotior esse videatur. clearly by the inverse ratios—and of those [tones] rejoicing in the more difficult ratios, it rests or goes into hiding, it seems to be more remote.
nam C, Dis, G, c, sunt ut 1:%: ^/z: Vi = For C, D# [Eb], G, c, areas 1: % Vz:V, (inverse) Vg: Vg: V4: V3, unde tota natura = (inverse) Vg : V5 : V4 : Vs,**^ from which
A reference to the harmonic series. Fricker presents an interesting justification for the minor triad here: C, D# [Eb], G, c equals a minor triad, that is, 1 [C, PI]: Eb [6/S, m3rd]: G [3/2, PS] c [2/1, P8], based on the arithmetic series (equivalent to the ratios presented in Section 1). Fricker then states that this series is equal to the inverse of the following harmonic series: 1/6:1/S: 1/4:1/3. This is explained as follows: 1/6 = C to G; 1/S = C to E; 1/4 = C to C; 1/3 = C to G, resulting in a major triad. Inverting this does not mean literally inverting the remainder of the octave, for example, PS = PS, PS = P4, M3 = m6; rather, it means instead of going up a fifth from C to G for the ratio 1/6, go down a fifth to F. This is a technique used by eighteenth-century theorists to explain the minor chord. This results in this series becoming: F Ab C F, a minor triad. Therefore, the only way that these two series are equivalent is in quality. Why does Fricker bother? Why not simply assert the
46 Cantus mollis aestimanda, quod sc. in the whole nature of the minor song ought contrariam Cantui dure plagam abeat, sc. to be judged, because, of course, it departs in intensitatem, cum ille potius in into the opposite plagal area—with extensitatem excurrat. respect to the major song—namely in intensity, since the former would rather hasten into extension.^
§17 §17
Consideratio singularis gustus Itaiici, Consideration of the extraordinary naturam & in Harmonia & in Melodia Italian taste, which to some degree aliquantum transcendentis. transcends nature in hoth harmony and melody.
Quae vero circa Melodiam §. 14. & 15. However, the things that we subjected to numeris subjecimus, ea omnino numbers—concerning the melody which pertinebant ad singularem gustum we dealt with in §14 and 15—pertains to Italicum h.e. dexteritatem in calculatione the unique Italian style, that is, to a skill in difficiliori, ubi etiam Ÿ occurrit, uti a more difficult calculation, where 3^ also praecipue in certo ejusdem Operae Cantu occurs. So that, for instance, in a certain militari (Marche) haec fere formula VgT military song (march) of the same opera, ex nota + V,oT ex puncto, VieT ex nota, this formula— Vg note + Via dot. Via note, (V4T constans ex 3 (VgT, VgT, VgT) notis, (V4 corresponding to 3 (Vg, Vg, Vg) notes, h.e.) V12T ex nota, V^T ex nota, VuT ex that is) V12 note, V12 note, V12 note"**—as a nota, regnat. rule, reigns.
minor triad arising from the arithmetic ratios? Because, Rameau stated that deriving intervals from the arithmetic division is not as theoretically satisfactory as deriving them from the harmonic series. Therefore, Fricker has shown (quite indirectly) that the arithmetic minor triad is based in the harmonic series. That is, continuing in the major would be too bright, go too far, the temporary use of minor tempers the effect. In other words, three Vg-note triplets.
47 P'P p i
Figure 16: Musical notation of the “march” motive.
quae in numeris integris est 9:3:4:4:4, & This formula in the integral numbers is in alia ejusdem Operae Cantilena S tactus 9:3:4:4:4,'*® and in another song of the propositionem principalem ac maxime same opera, S beats [tactus] make the adfectuosam conficiunt: principal phrase and also the greatest affection. hinc gustus Italicus in melodia quasi Hence, the Italian style in melody, as it sequitur basin Musicae harmonicam were, follows the true harmonic basis of veram 2®, 3^ 5', sed tamen cum aliqua music 2®, 3V 5‘, but nevertheless with discretione; in Harmonia autem ipsam some distinction. In the case of harmony eandem saepius transcendit, ex.gr. si in however, it more often transcends that peregrinum aliquem Tonum mollem very basis itself. For example, if digressurus per Septimam ejus majorem someone—about to digress into some seu inferius Semitonium Vis in Discantu foreign minor key by the raised seventh of cadit, inque eundem una in Basso per it or the lower semitone Vis—falls in the Semitonium superius, quod vere non est soprano, then he descends in the same Secunda minor 'Vie, sed novus Tonus ^Vzs [manner] together in the bass by the (ubi 5^ occurrit), descendit. higher semitone, which truly is not the minor second 'Vie, but a new tone ^Vzs (where 5‘ occurs).'*^
46 A Lowest Common Denominator (LCD) o f 48. That is:
1/8 l/16-dot 1/16 1/12 1/12 1/12 x6 x3 x3 x4 x4 x4 equals 6/48 3/48 3/48 4/48 4/48 4/48 remove demon. 6+ 3 (dot) 3 4 4 4 equals 9: 3: 4: 4: 4
This is the “tactus” example he referred to at the end of Section 10. Having argued that the musical basis of rhythm (“melody”) makes use of 2“ and 3‘ in Section 12, this provides an example of the use of 3^ (which goes beyond the basis), as he just stated. Interestingly, Section IS provides another example of the use of 3 , or 9, but Fricker skirts this example by his conclusion of adding 9 into the surrounding numbers. As the example below illustrates, Fricker is describing an Augmented 6""" resolving out to an octave. His use of the term “raised seventh of it” refers to the raised fourth scale degree acting as a leading tone to the dominant He goes on to say “or the lower semitone 8/15,” which I take to be another way o f referring to the same pitch, by referencing the ratio given in section one: in other words, the M7 above the fundamental
48 §18 §18
[19] Harmonia totalis in Cantione Tbe compiete barmony in the part song Seelen-Briiutigamze vocibus plena, Seelen Brautigamze is subjected to calculis subjecta. calculations.
Nunc ad Harmcniam Melodiarum: ilia Now to the harmony of the melodies: for enim §. 2. & 3. tantum ad singulos that harmony in § 2 and 3—only pertinens Tones praemittenda fuit. Huis pertaining to tones taken one at a time— ergo gratia consideremus Cantionem had to be postponed. Therefore, on Seelen-Brautigamze cujus artificiosior account of this, let us consider this song: compositio erit: The Spiritual Bride, the composition of which will be more artistic:
note, in this case the dominant. He concludes by referring to the bass resolving down a half-step — something most often seen in an augmented 6* chord, especially in conjuntion with an upper leading tone function—but he specifies that it be the more complex 25:24 half-step, not the 16:15. 8/15 (a description of the F# pitch as a leading tone, not a description of the % step up to G) F#-»G T Aug. 6th i Ab—)G 24/25
49 SiniAra in Baflb DextrainDi&ntu: è t t 0 G g : h d 2 4 : r 6 3 2 : 4 0 48 P e : If 3 0 : £ e : h : e : g f 1 0 : I f : 2 0 : 2 4 C c : 4 8 : lefolv. I 2 4 : f : 6 ( B r d m a d D d a : d : 9 I 2 3 : 4 : f 9 1 8 2 7 : 3^ : 4 f C c i 6 8 ad h t u é H : d : g I f f 1 0 2 0 : 2 4 : 3 2 3 0 6 0 1 2 0 :i44 :i9 2
Figure 17: Pricker’s representation of the song, “The spiritual Bride.’
50 3E *$-4- m
m 3 - * - 3 Ï3 :
""" 3^4 #19 $« ^3 30 g2 ^ 93 16:13 23
r 'J i 9 Fumtom«nUlHot«
Figure 18: Musical notation of Figure 17.
51 Syllable: Fundamental: Multiplicand: Pitches: 1 A BC D E F GH
2 See- GG g h d 8
3 These X 8 2 4 5 6 8
4 = -> 16 32 40 48 64
5 (F#)9 GF# f# - - -
6 to 15 30 ---
7 -len C E e h e 8 8 (E) 8 5 10 15 20 24
9 (C )8 C c c i i 10 to These -r by 4 4 8 16 (20) (24)
11 - -- - -
12 Resolve = -» 1 2 4 5 6
13 Brau- D D d a d J» 14 (D )9 These x 9 1 2 3 4 5
15 = —¥ 9 18 27 36 45
16 (C) 16 C c -- -
17 to 8 16 ---
18 "ti' G B H h d 8 19 (B) 15 These x 6 5 10 20 24 32
20 = -» 30 60 120 144 192
21 (G )2 (A) G G c# e - 22 to These -r 4 24 48 135 160 -
23 -gam D D d d # a 24 (D )3 These f 36 36 72 144 180 216
25 Resolve = -» 1 2 4 5 6
Table I: Explanatory table of Figures 17 and 18. For clarification, see footnote on page 53.
52 [20] (ex qua sequitur, latere in omni (from which it follows, that in every symphonia melodiam aliquam symphony there lies hidden a certain very simplicissimam} simple melody.}
Continuet, quicunque velit: nobis Let it continue in whatever way it should suffecerit, demonstrasse, latere in omni wish: it will be sufficient for us to have symphonia seu concentu plurium vocum shown, that hidden in every symphony or seu melodiarum unam aliquam concord of many voices or melodies is simplicissimam melodiam, cujus semper one certain very simple melody.'** Of this audiuntur rationes quamplurimae simple melody, as many simple ratios as simplicissimae, in dissonantiis vero etiam possible are always heard—to be sure, nonnullae magis compositae, prout even more so in the dissonances of some compositio est artificiosa. composition, accordingly as a composition is artistic.
V . gr. in syllaba See- radix est Tonus Gl, For example in the syllable See- the root imus in longioribus solito clavichordiis; is the tone G l, the lowest pitch in the deinde mutato G in F* idem G numéro 8 common, longer clavichords. Then, by est multiplicandus, & ratio quidem 8: IS moving 0 to F#, that same G is multiplied satis gravem hinc infert dissonantiam; tum by the number 8, and certainly the ratio in syllaba -len illud G8 ascendit usque ad 8:15, from this, sufficiently introduces a C l, atque adeo, cum G:C = 3:8, quasi harsh dissonance.'*^ Then on the syllable numerus 8 8 3 = 192 ad unitatem se -len that G8 continues all the way to Cl,^° composuit. and indeed, since G:C = 3:8, as though the number 192 (the product of 8 8 3) compared itself to unity,
(Régula pro calculanda harmonia totali IA rule for calculating a complete seu plurium vocum.} harmony or multiple voices.}
nam ispe adscensus ex G in C erat 8:3, at for that very approach from G to C was etiam ex 8G in C erat 8 8 in 3, denique ex 8:3, and also from 8G in relation to C was 8G in 1 C erat 3 8 8 in 1, scilicet 8 8 in relation to 3, finally from 8G in
** (The following explanatory footnotes refer to Table I by referencing Rows 1-25 and Coliunns A-H.) Row 12, Columns D-H as well as Row 25, Columns D-H, 1:2:4:5:6. This is similar to the procedure in section 3. Rows 2-6, Column D, The lowest G is multiplied by 8 in order to raise it an octave, which allows one to more easily see that the motion from 0 to F#, a half step, creates the dissonant ratio 16:15. This is also seen from the music notation that shows the G major triad holding over the descending motion of the bass. In order to understand Flicker’s method, it is necessary to understand that he is positing a “frmdamental” tone beneath each of these calculations. Therefore, the first G is labelled G2, which implies that a lower G l exists below, although not literally present in the music. This becomes more relevant to the calculations in the next few sentences. ^ As previously noted, the G major chord holds until the statement of the C major chord.
53 attendendum hic est ad singula, quae ad relation to 1C was 3 8 8 in relation to 1. simplicatatem proferendam faciunt, Of course, this is applied to each eorumque numeri sunt in se multiplicandi. [harmony] individually, which [calculations] are made for the purpose of simplicity, and of them the numbers are multiplied into themselves.**
Sequens syllaba Brau- habet D pro The following syllable Brau- has D for unitate, est ergo in harmonia priori syllaba unity,*^ there exists, therefore, a more modo deferra elegantior, quantum 8 elegant syllable in the prior harmony just superat 9 simplicate; sed in hujus D locum left, in so far as 8 surpasses 9 by statim succetüt 8 C, quod quidem révéra simplicity.** But in the place of this D, ad simplicissimam melodiam non pertinet, 8C immediately follows, which certainly, quippe quae potius ita sonat: on the contrary, does not pertain to the most simple melody, which of course, rather sounds thus:
Seelen-Brautigam Seelen—Brautigam G O D C D G 0 D 0 D
sed in artificiosiori hac compositione ita But in this more skilled art it holds thus: habet:
Seelen - Brautigam Seelen—Brautigam G C D A D G C D A D55
The basic point he is making is to compare the motion of G to C in the bass (and their harmonies). This is literally stated as G:C = 3:8. In order to state this as a ratio in relation to the fundamental C—that is, “unity”, 1— he divides 3 by 3, to get 1, but must multiply 8 by itself and 3 in order to get 192, the smallest multiplicand that can be divided by 3. This results in: 8:3 8 (8 X 3) = 192:3 reduced 192/3 -» 192+3=64 3+3=1 results in 64:1 The various calculations are octave elaborations of this basic calculation, which allows him to reduce this progression to Row 12, Column D, the unity number 1. The remaining simple numbers (2,3,5,6), Row 12, Columns E-H, as shown on the table are derived by reducing the C major chord of Row 9 by four. The E minor chord intervening between the G major and C major chords is ignored for calculations and regarded as a passing harmony. ” In other words, D as the fundamental, 1, used for calculations. ” The bass moved from C to D, which in relation to eachother is the ratio 8:9, taking C as the fundamental (expressed as vibrations); therefore, literally the whole string moving in 8 parts (C) versus 9 parts (D), so 8 is more “simplicity” than 9. Notice, also, on the table that he multiplies the D chord by 9, in order to have a 9:8 ratio comparison. ^ That is, this C harmony is another passing harmony not considered in the final reduction to the most simple ratios of Row 25. The first version should actually read G C D G D, so that the only difference is the substitution of the A for the G. He is here discussing the progression of the fundamental notes for the entire excerpt, not the
54 quae po- [21] stea in syllabis reinem Which afterwards descends in the Triebe in Tonos H* descendit. syllables reinem Triebe in the tones B#.*®
§19 § 1 9
Cur Quintae & Octacae [sic, Octavae] Why might fifths and octaves in se insequentes a Musicis sint vetitae? sequence with themselves he prohibited Quia impuritas Tonorum aurem nimis hy musicians? Because the impurity of feriret. the tones would punish the ear too much.
Sed jam intelligi potest (confer §. 11. fin.), But now it is possible to understand quomodo in omni fere cantu pleno (compare § 11, end),^^ how in nearly singularum syllabarum Toni una every full song of the individual syllables, concinentes se habeant inter se, ut plurimi the tones harmonizing together hold ex numeris; 1,2,3,4,5,6,8,10,12,16, themselves between themselves, so that &c. & in dissonantiis ex.gr. pro 8 many [tones come] from the numbers: intercedere IV2 , pro 10 numerum 9, pro 16 1 ,2 ,3,4,5,6,8,10,12,16, and so on numerum 15 &c.; even into the dissonances, for example, 7 V2 intercedes before 8, the number 9 before 10, the number 15 before 16 and so on.^*
Quintarum autem & Octavarum sequelas a However, parallels of the fifths and of the Musicis ea propter vetari, quia totae octaves are forbidden by musicians for syllabae se haberent, ut 8 (1:2:3:4:5) ad 9 these reasons, because entire syllables (1:2:3:4:5) h.e. ut (8:16:24:32:40) ad would contain [dissonance] within (9:18:27:36:45), ubi se insequerentur themselves, as 8 (1: 2 :3 :4 :5 ) to 9 (1 :2 numeri dissonatissimi & amousikoi 16:27, : 3 :4 : 5), that is as (8 : 16 : 24: 32:40) item 27:32,27:40,32:45,45:64, &c. to (9:18 : 27 : 36 :45), where the most dissonant and amusical numbers would follow themselves, such as 16:27, likewise 27:32,27:40,32:45,45:64, etc.*’ literal bass notes (as noted in the table). The (corrected) first version shows a more simple fundamental progression (based on the actual pitches), but Fricker suggests a more “skilled” version that works better for two reasons. Substituting the “A” fits the upper voices better, especially the C#. Secondly, as an A chord, this is more logical as a secondary dominant resolving to the dominant, the last chord. Without the original music to consider, it is not clear what he means here. ” Where he discusses the reduction of the series of numbers from section three. These numbers are the harmonic series. Notice how he discusses the dissonant terms as “interceding” numbers, and the 7*'' harmonic as 7Vi, accounting for its out-of-tuneness. His argument is that the mind does not perceive simply a motion of 8 to 9, but also the concurrent harmonics above each, even if they are not parallel terms in the series.
8(12 3 4 5) = (8 1624 32 40) 9(12345) = (9 1827 3645)
55 Praecipua vero illius legis ratio sine dubio in eo sita est, quod voces duae inter However, the chief reason for that law, ceteras extremae, ima & summa, aurem without a doubt, is situated in the fact that prae reliquis verberantes, impuritatem two voices, extreme among the rest, low rationum Quintae 2:3 = 16:27 nimis and high, striking the ear before the proderent, & falsitatem propositionum others,*® produce too great an impurity of harum 8:9 = 9:10 & 2:3 = 27:40 nimis the ratios of the fifth 2:3 = 16:27, and they dare auri ostenderent, qua ilia sane exhibit to the ear too clearly the falseness laeditur. of these propositions 8:9 = 9:10** and 2:3 = 27:40, by which falsefalseness is the ear certainly offended.
Pone enim in ima melodia sequi se Tonos For instance, imagine that the tones C and C & D, in summa g & a, erunt numeri D follow themselves in the bass, in the priores 8:9, posteriores ergo hoc respectu soprano g and a follow themselves, the 48: **®/ 3; & utrique sane satis magni, & former numbers will be 8:9, therefore, the rationem 2:3 in 9: **®/3 fallentes. later numbers, with this respect, [will be] 48: **®/ 3. Either will be certainly large enough, and mistake the ratio 2:3 in g . 160^^ (fa 63)
He seems to mean that the Bass and Soprano, respectively, strike the ears more prominently than the others; however, this would mean that parallel fifths should only be a problem when occurring in these voices. He could be referring to voices represented by high and low numbers in the series. That is, 10 does not occur in the series based off of eight, so they cannot be compared. His choice of 27:40 and 16:27 (non-parallel terms in the series) to illustrate the dissonance seems random. 63 Thus:
g a 48 to 160/3 C D 8 to 9
That is, the perfect fifth C-g moving up a major second to the perfect fifth D a. As C D is a major second (8:9), g-a is also a major second, however, this second is the 10:9, based on a Just scale. Therefore, the note C up to A is not D + 3:2, but is G + 10:9, or otherwise, A is equal to C + 5:3 (M6th), which he expresses A = 160/3. How? He has just multiplied this ratio times the series 1-5 to show the dissonances among the upper harmonics, and he now takes it one step further, multiplying it by 6, which explains 48 as 8 x 6 (six equals a fifth just as the first partial to the 6"’ partial is a fîfth). Next, adding a fifth to 9 (again, the ninth partial above C) through multiplication of 6 would equal 54, but, as stated before, this A does not equal D + 3:2, but rather G + 10:9 or C + 5:3. For Fricker, it is equal to the former, as follows:
a) C + P5'*' (3:2) = C (8) x 6 = 48, therefore, 48 stands for G. b) G (48) + M2 ( 10:9) = 48/1 + 10/9 = 480/9 reduced to 160/3, which cannot be reduced further, because it becomes irrational.
If he had calculated A from D + P5"', then it would simply have been D(9) x 6 = 54, which equals 162/3.
56 §20 20
Conclusio ex dictis, ad considerationem The conclusion, from what has been Mu- [22] sicae PSycholo^cam said about a psychological pertinens, anfanam exercere tres vires consideration of music, is that the mind In conferendis inter se singulis Tonis. exercises three powers in comparing individual tones among themselves.
Sub finem ergo hujus Analysées invenio, Before the end, therefore, of this analysis, animam, [22] quatenus ilia quidem a I find that the mind, as far as it is certainly Musica adftcitur, tres exserere vires, pro affected by music, exercises three forces, simplicitate numerorum primorum very active in relation to the simplicity of cardinalium 2,3, & 5 actuosissimas, the prime cardinal numbers 2,3, and Î ipsam vero, quasi unitatem, earundem The mind itself however, as if a unity, is esse Centrum, & hoc quidem breviter the center of those same forces, and repetendo Analysin ita probo: therefore I prove this, certainly, by briefly repeating the analysis;
1) ex §. 1-6. Anima innumeros in Musica I) From § 1-6. The mind compares the inter se comparât sonos, qualitate (ut innumerable sounds in music between videntur, sed revera quantitate tantum) themselves. They are diverse sounds in diverses, sola Multiplicatione ac quality (so they seem, but actually only by Divisione, &, quamvis eadem Tonos v, gr. quantity), by mere multiplication and & ‘*®/64 inter se confundat, agnoscit division, and, although the mind mixes tamen Tonorum vel concentum vel together the tones between themselves— dissensum ex originibus eorundem, sc. for example and ‘‘^ 54—nevertheless it numeris primis 2,3, & 5, ita ut hos quam recognizes either a consonance or certissime a se invicem distinguât, nec dissonance in the tones. It does this from ulla plane ratione inter se commisceat, sed the original numbers of those same ratios, potius eosdem numéros vel in maximis that is, by the prime numbers 2,3, and S. potentiis & productis suis, ubi sunt inter Thus, the result is that it distinguishes se commixti, secemat, & ad suas quemque those tones from themselves, one after classes referat. another, as specifically as possible. Nor clearly, does it mix together any ratio among itself, but rather, the mind separates those same numbers, even in very large powers or their products, where they are mixed together between themselves, and compares each one to their own class.
57 §21 §21
Fundamentum adfectuum Musicorum The foundation of the musical in basi harmonia situm. affections situated in the harmonic basis.
Facilius id ulterius possem deraonstrare ex It would be much easier for me to doctrina adfectuum Musicorum, si eam demonstrate from the doctrine of the quidem hie tradere vellem, cujus tamen musical affections, indeed—if I wanted to vestigia quaedam, immo fundamentum relate it—nevertheless, I will reveal omne, rei cupidis ostendam. Basis certain vestiges. Actually, I will relate the Musicae melodica: entire foundation, to those desirous of the matter. The melodic and harmonic basis of music:
melodica: mormum, fimplex, artificiolùm. I) 3- I, 3 ) 3 ) f. harmonica: nigi- (ecer- domU jucun- dum, nens, nans, dum.
Melody: Death Simplicity Art 1 3
i 4 Harmony: Coldness Separation Dominating Delight
Figure 19: Flicker’s application of the doctrine of the affections to the prime numbers with translation.
Quisquis ex hactenus traditis principiis Whoever, from these handed down contu- [23] lerit hanc generalem principles, compares this general synopsis adfectuum Musicorum synopsin a multa of the musical affections, by much experientia atque intrinseca numerorum experience, as well as the by the intrinsic analogia derivatam cum exemplis, analogy of the numbers, derived with veritatem ejus deprehendet. Veritatis examples, he will discover the truth of it. autem amantibus etiam hanc caelestium However, for those desirous of truth, I adfectuum cum terrestribus harmoniae will add even this table of harmony of the tabulam adjungo: heavenly affections with the earthly [affections]:
58 {De adfectibus Musicae ceolestis (Concerning the affections of the beatorum} blessings of heavenly music}
'frigw fccer* domi* jacan* occul- liis dam I nens, nuui dam, tnm. IT- I" 2 * 3* f* 7* terreftris. o- uniras radix. cat* 3* f * 7* i i “cœleftis. aoenst e d b ti. cnute- mijtfta- eKhilann- mcomwe- cum, le, ticumj_tiŒmum,^henfibiIe.
Coldness Separation Dominant Delight Hidden r 2‘ 3^ 5‘ 7® earthly.
r 2‘® 3’ 5^ 7' 11® heavenly. Silence Ecstasy Cruel Majestic Exhilarating Incomprehensible®*
Figure 20: Another application of the affections with translation.
Confer, summe Rever. D. Bengelii Compare this with the great Reverend D. explicatio Apocalypseos germanica in [Doctor] Bengel’s German explanation of Introduct. part. 3. in Progressione the Revelation in the Introduction, part 3. arithmetica periodorum cardinalium, The arithmetic progression of the cardinal comparanda cum Triade perfecta periods is compared with the major chord, (duraccord) seu potius Pentade coelesti, or rather with the heavenly fifth, likewise item in ratione Chroni ad Ævum p. 127. explained in the reasoning of Times Quae omnia devotus ac studiosus scrutator tofwards] Eternity, p. 127. All of which intelligere poterit. the faithful and studious investigator will be able to understand.
®* Henck presents a german version of these affections from a later work by Flicker, Mathematischer, aus der Natur der Music und der Zahlen hergenommender Beweifi vor die Gottlichkeit der Offenbarung Jesu Christi, 1751 (manuscript, see Johann Ludwig Flicker, “Mathematischer, aus der Natur der Music und der 2ühlen hergenommender BeweiB vor die Gdttlichkeit der Offenbarung Jesu Christi,’’ Ms. 21-1. Bibliothek des Evangelischen Stifts Tübingen, 1751). The presentation is almost identical; Irrdische (earthly): 1 = schweigend (silence); 2 = Trennung (separation); 3 = herrschend (reigning); 5 = angenehm (pleasing); 7 = radix (root) [Wurzel]; Himmlische (heavenly): 1 = schweigend (silence); 2 = entsetzend (horrified); 3 = tyrannisch (tyrannical); 5 = majestütisch (majestic); 7 = entzUckend (enchanting); 11 = radix (root).
59 §22 §22
niae vires animae activitate sunt Those powers of the mind are very diversissimae praesertlm In concentu diverse in activity, especially respecting plurium sonorum. the concord of many sounds.
2) Ex. §. 7-10. Diversa autem plane 2) From § 7-10. Clearly, however, the virtute & vi illos tres numéros vibrationes mind by a diverse strength and force sonorum audiendo, primum quasi ad apprehends those three numbers by omnem numerum indeterminatas, hearing the vibrations of sounds, first as if adprehendit, immo potius ex ipso suo indeterminate, according to every number. fundo evocat, & ad perceptas vibrationes No, rather, it calls up out of its own base adplicat anima. and applies them to vibrations that have been perceived.
(Hinc videmus, cur Quarta sit (Hence we see, why the fourth is a Dissonantia; cum enim propter numerum dissonance. For since, on account of the vibrationum sonus gravis sit simplicior number of vibrations, the lower sound is acuto, anima, consonantiam quaerens, & simpler than the higher sound. The mind, post rationem 1:2 alteram 2:3 adplicans, seeking a consonance, and joining another quod fere in toto reliquo Cantu [24] bene ratio 2:3 after the ratio 1:2, because successerat, offendit discongruam having generally advanced well in the rationem 3:4, quia : % = 8:9). entire remaining song, hits upon the discongruous ratio 3:4, because % = Z :965 .f
Scilicet anima binarium numerum multo Of course the mind holds the binary magis in potestate habet sua atque number by much more in its power as exercitio, quam temarium, & hunc, quam well as training, than the ternary number, quinarium; quod inde probo: In Organo and this more than five. I prove this Registra dantur plurima, e quibus unum thence: in an organ, there are many etiam compositum Mixtura adpellatur, registers, out of which furthermore one eundem semper sonum in Octavis puris, composite register is called a mixture, acutioribus & gravioribus referenda; always giving the same sound in pure Quintarum vero Registrum vix semel aut octaves, with reference to the high ones bis (si in Mixtura etiam continetur) audiri and low ones. However, in the fifth’s potest sine durissima aurium verberatione; register, it is hardly able to be heard once datur quidem etiam Tertiarum Registrum, or twice without a very rough beat of the Nasar dictum, valde acutum, ut celeritate ears (if it is also contained in the mixture). vibrationum anima numerando sublevetur, It happens certainly in the third’s register, ceterum tamen aures acerrime called the Nasard (French, organ stop], praestringens: very severely, so that the mind, by the
^ Going up the series; l/2:2/3 = 3:4, but 2/3:3/4 = 8/9, the next pair, creates a dissonance. See also section 23. Could he be at all suggesting this as a “natural law" for why the fourth is considerd a dissonance?
60 speed of the vibrations, is assisted with counting, nevertheless it stretches the ears terribly 66
atqui Octavae formantur & characterem Yet the octaves are formed, and they suum accipiunt a numéro 2, Quintae a 3, receive their character, by the number 2, Tertiae a S, quod observare licet in §. 1. & the fifths by 3, the thirds by 5, which aliis. Ergo potentias vel magnas binarii someone can observe in § 1, etc. anima facillime intelligit, temarii vix Therefore, the mind easily understands the duas, quinarii non ultra unam, v.gr. in powers or the large numbers of the binary, unico Tono anima quinquies 2 aeque the ternary with difficulty to two, of the cito in se multiplicat, ac 3 bis, & hanc 5th not beyond 1. For example, in the figuram: uncommon tone ^^/s, the mind quickly multiplies 2 five times^’ into itself just as, also 3 two times,®* and this figure:
a I eeque cito confiât, quam duas has: 2 & 1 • • • 1 [this it constructs equally as quickly as these two]®^
Figure 21: Graphic representation of how the mind compares quantities.
Idem vero sexcentis aliis exemplis However, if it was necessary, the same Musicis, quorum quaedam hinc illinc in could be proven in 600 other musical Analysi nostra dedimus, probari posset, si examples, of which we have given some opus esset. here, another there, in our analysis.
^ Based on Pythagorean tuning (the historically ideal tuning), the intervals from octaves to thirds would have increasing beats, respectively. Flicker uses this as a further demonstration for the rationale of the three prime numbers as the basis of intervals. Flicker’s knowledge of the organ was most likely greatly enhanced by his friendship with the organ builder, Hausdôrfer (see Roessle). " T h a t is, 2^:5. “ He is not referring to the 32/5 example here, but speaking hypothetically. " The mind will calculate 2 raised up to its highest power just as fast as the other primes are raised to lower powers; i.e., the mind can compute relatively complex ratios just as fast as simple ones by means of the prime numbers.
61 §23 §23
[25] Repraesentatur per figurant actus The mind’s action by which it animae, quo harmonlam Tonorum understands the harmony of the tones is Intelligit* [sic] represented through a figure.
Atque vero id ipsum, quod caput est, However, the chief matter is most maxime elucet ex comparatione apparent from the comparison of methods, modorum, quibus anima & Melodiam & by which methods the mind examines Harmoniam Musicae pervestigat. both the melody and harmony of music. Posteriorem itaque imaginationi sistimus And thus, we set up harmony for the his lineis: imagination in these lines:
a
Figure 22: Graphic representation of ratios on a string.
Concipe vibrationes sonorum ad iineam Imagine that the vibrations of sounds are rectam A B relatas, ut particulas linearum related to the straight line A B, as the sectarum a b & c d: erit, si anguli small parts of the divided lines a b and c lineolarum, a punctis in a & b item in c & d. It will be, that if segments of the d confluentibus concipiendarum, quae conceived lines ftom the assembled points lineam A B tranversim secent, sint on a and b, are equals, likewise on c and aequales, seu si lineolae secantes sint sibi d, which divide the transverse line A B, or paiallelae, sonus a b, ad sonum c d ut 3 ad if the divided lines are parallels to each 5: id quod ex numeris & lineis patet; unde other, the sound a b to the soundc dis as intelligitur modus, quo anima plures 3 is to 5. That is apparent from the sonos, etiamsi non eodem praecise numbers and lines. From this one temporis momento audiri incipiant, possit understands the manner by which the inter se conferre. mind is able to compare many sounds between themselves, even if they do not begin to be heard precisely at the same moment of time.^
™ That is, even though the beginning of each segment may not align, or is out of phase, the mind can still
62 Scilicet trium plagarum potentiam habet, Of course, the mind has the power of three seu ad triplicem se dimensionem regions, that is, the mind is able to extend extendere potest anima, ut rectas & itself to a triple measure, so that it is able transversas & demittentes se lineas una to follow, at the same time, the straight, sequi possit. Sed omnino etiam in hac transverse, and broken lines. But, mera singulorum sonorum collatione altogether even in this pure comparison of anima, ut dixi, non mere passive se habet, individual tones, the mind, as I stated, nam quaerit in singulis sonorum paribus does not merely passively hold itself. It onmes citius simpliciores rationes, donee quickly seeks in the equal individual parts graviores accipete cogitur, ex.gr. Itali of the sounds all the more simple ratios, nonnunquam in C dur Tonum finalem 1, until it is compelled to accept the lower ita cadunt: ones. For example, the Italians sometimes cadence in C major, the final tone I, thus:
(Explicatio Dissonantiae Quartae.} (An explanation of the dissonant fourth.}
Taft. I ex nods : tr . ^T|TiT|T;T • i T e x a gf t à( l ^ + flÿTexnof.4- g g .jiT ex c a x ex Fj^Texpuna. GJ punft. GJpun£l- CJ pun& ta & . I . taa.3. t a ^ 3 . ta & q .
Figure 23: Fricker’s representation of the explanation of the dissonant fourth.
extrapolate the relationship. This is akin to the Gestalt principle o f good continuity, wherein the eye can perceive relationships even when segments of a line (in a word or picture) are missing.
63 i
IV I 6M
Figure 24: Musical notation of Figure 23.
[26] Cum vero Bassus propter To be sure, since the bass is closer to the fortitudinem soni, quam naturaiiter habet, mind— because of the strength of sound, & tarditatem vibrationum animae sit which it has naturally—and because the propior, eique per minores suos numéros slowness of vibrations, and because the Discantum explicet, tactus 2‘‘*” [secundus] bass explains the soprano to the mind ita 1““'" [primum] excipiet, ut infimus in through its fewer numbers, the second I"” [primo] tactu Tonus ab altemationibus measure will relieve the first measure. The Tonorum ejusdem in Discantu saepe result is that the lowest tone in measure divisus sit, id quod jam non fit in one was divided several times by the subséquente 2 [secundi] tactus infimo; alternation of tones in the soprano, which does not happen to the subsequent lowest note of the second measure.
itaque ejus non saltem 3 partes, sed cum And thus not just with three parts, but also iis quoque 16 supremi Discantus Toni with 16 of the higher tones of the upper numerate debet anima, donee rationem voice,^' too, the mind ought to number, sonorum inspiciat: unde intelligitur nunc until it observes the ratio of the sounds: Quartae Dissonantia, de qua etiam supra whence it is understood now the §. 22. quam Mattheson contra veteres dissonance of the fourth, concerning propugnat in dem erofheten Orcheste, 3. which [dissonance] even above in ter [dritter] Theil oder Erofiiung Sect. 2. mentioned § 22, which, Mattheson fights Quartae blanditiae inscripta. against the ancients in dem erofneten Orchestre, 3 ter Theil oder Erofiiung [1713], Section 2, "Flatteries of the Fourth.”
His example applies not only for a simple, three part song, but also for a complex, multiple part song or piece—sixteen seems to be a randomly large number to make his point, since there is no reference to sixteen anywhere else. Literally, there are sixteen notes in the example, but not only in the upper voice.
64 Scilicet anima, a priori tactu in exeicitium Of course, the mind, having been lead adducta cito & sine multis numeris from the Rrst measure into the practice of colligendi rationes, nunc aiiud plane reckoning the ratios, both quickly and exercitium ingiedi debet, difficilius without a lot of numbers, now clearly it propter diutumitatem ac magnitudinem ought to begin another training, more operis. Ceterum Quartae Dissonantiam re difficult on account of the duration or vera tantum relativam, non absolutam, magnitude of the work. But the satis intelliget, qui ex traditis hactenus dissonance of the fourth, is in reality only principiis gustum modemum ab antiquo relative, not absolute. One who will have probe discreverit. Nonne enim in ferie properly distinguished the modem taste ex.gr. from the antique—from the hitherto handed down principles—will understand well enough. For will not the last point, in the series, for example,
3* 3’.i* 3*.2‘ 2*. 2' - I - i
Figure 25: An arithmetic series representation of the ratios of the fifth (ratios 1-4) and the fourth (ratio 5).
[27] ultimus terminus plane incongruus seem clearly incongruous to him, who will videbitur ei, qui seriem in suo ordine ac have gathered the series in its order or true vera numerorum & potentiarum progression of the numbers or powers?’^ progressione legerit?
§24 §24
Activitas animae In attendendo ad Further explanation of the mind’s Melodiam ulterius expilcatur. activities In attending to melody.
In Melodia non pari modo anima In melody, the mind does not seem to seek simplices requirre rationes videtur, quo in for simple ratios in the same manner, by Harmonia, sed potius componere vult eas which it does in harmony. Rather it atque augere, & continuando wishes to construct, as well as augment, aggregationes atque excrescentias those ratios, and be delighted by rationum simplicium in eodem continuing the aggregations and
^ The previous figure shows the fourth as more difficult because the bottom ratio contains three.
65 simplicissimo ac directo fluxu increasings of the simple ratios, in the provectarum delectari. same most simple and direct flow of the promotions.
Sic omnino in Melodiis Italicis Toni longe Thus, in every way, the tones in the Italian durantes perquam delectant, ubi videlicet melodies that last all the way through are anima methodo simplicissima suaeque extremely pleasing [i.e., the bass notes naturae atque indoli maxime propria under a florid melody]. Namely, the simplicissimas colligendi rationes mind, by the simplest method of its nature libertatem habet atque exercet. Sequitur and (more than anything) with respect to inde & perspicuum est, quanta sit animae its innate character, maintains and activitas in attendendo ad Melodiam & exercises the freedom of collecting the Musicum Tactum, & quomodo quasi per simplest ratios. It follows thence, and is se ipsa ac sponte ilium continuet, dum evident, how great is the activity of the soni ipsi non quidem vibrando, altemando mind in attending to melody and the tamen inter se & variando, cessant. Heic musical beat, and how, as it were, through enim anima ex innumeris fere unius itself, and also willingly, the mind longius durantis soni vibrationibus per continues that beat while the sounds accuratam numerationem illas praecise themselves cease. Certainly not with notât, quae majores Tactus partes secant. respect to vibrating, but with respect to alternating and varying among themselves. For in this, the mind observes exactly those ratios from the almost innumerable vibrations of one long sound, remaining through an accurate counting, which ratios divide the larger parts of the beat.
§25 §25
Tota operatic animae Musica in The mind’s entire musical operation is synthesi & figura Ezechielitica suggested in a synthesis and figure hy proponitur. Ezekiel.
Ex his concludo, animam, quatenus ea From this I conclude that the mind, as far cum Musica operatur, esse concipiendam as it is occupied with music, is to be fere, ut Ezechielis descriptio vitae; Cap. 1. conceived almost as Ezekiel’s description certe hac hypothesi stante explicari posse of life (Ezekiel, Chapter I). The effect of effectum Musicae in anima, atque ex ea music in the mind is certainly able to be totam derivari posse synthetica methodo explained by this firm hypothesis. And Musicam; [28] non item ex alia. Ex.gr. from it all music is able to be derived by Describantur circuli Concentrici radiis 2, the synthesis method—not otherwise from 3,4,5,6,7,8,9,10,15,16, &c. another method. For example, concentric circles are described by the radiuses 2,3, 4 ,5 ,6 ,7 , 8,9,10,15,16, and so on.
66 f 6 7* Cenfum ' Radii pro diftanda a centre dlmenfi. [Center: Radiuses measured according to the distance from the center.]
Figure 26; Representation of the radii from the center.
Inter illos vero peculiaribus nods However, among those circles, let three be disdnguantur tres, qui radiis numerorum disdnguished by peculiar marks, which 2,3 & 5 descripd sunt: are described by the radiuses of the numbers 2,3, and 5:
sic concipiantur per illos activae in tali Thus, let there be conceived through those figura ties vires gyratoriae, circulos ac numbers, three powers of gyrating acdvity spiras suas ita revolventes, ut proxima in such a figure. Their circles and quaeque centro rapidissime & in spirals^^ thus revolving, so that each remodssimum quemque circulum nearest power moves very rapidly by the muldplicatione radii sui exeat, ac deinceps center, and moves out to each most redeat: remote circle by multiplication of its radius, and returns successively:
Igitur 2 condnua vi agit centrifuga, Therefore two, by continuous centrifugal ceterasque etiam vires 3 & S ex orbids force, moves away from the center, and suis veris educit per muldplicadonem draws out the other powers—certainly suarum potendarum in illas; 3 idem facit, three and five—from their true orbits sed parum, immo podus in 2 reagit, through muldplicadon of their powers into praeserdm eo respectu, quod 2 ex centro those. Three does the same thing, but incomparabiliter fordus agit: less. No, on the contrary, it drives back into two, especially in this respect, that two drives incomparably stronger from the center:
73 Or. “coils.”
67 sed pro diversitate facultatis arithmeticae But according to the difference of the in subjectis actio & reactio conveniunt vel arithmetical ability in those numbers proprius centro simplici 1, vel propius which have been subjected, the action and unitati perfectam multipUcitatem reaction agree, either closer to the simple includenti 7®: center one, or closer to the enclosing unity 7°—the perfect multiplication: hinc S respectu totius rotae seu ceterarum Hence, with respect of the whole wheel, virium 2 & 3 potius reagere videbitur or of the other powers two and three, five versus centrum, quia semper restituit se in will seem to drive back towards the constantem sibi circulum, semperque center, because it always restores itself in iterum praesertim vi 2 ex eo fugatur, ut is a circle that is constant to itself, and constantia sua motum rotae in limitibus always is driven out of it a second time by suis, uti 2 in activitate, conservare the power two so that five seems to videatur. conserve the motion of the wheel by its steadiness in its limits, as two does in its activity.
Et sic dupliciter anima in numeris primis And thus the mind is doubly active in the minoribus 2 & 3 multo actuosior est, smaller prime numbers 2 and 3, much quam in majoribus 3 & 5, scilicet in more than in the larger numbers 3 and 5, agnoscenda Harmonia §. 22. & Melodia §. namely, in the recognizing of harmony of 24. Tonorum. In Harmonia ni- [29] the tones in § 22 and in the melody of § minim anima numerum 5 fere semper 24. Evidently, in harmony, the mind sees nudum atque ex ceteris productis the number 5 almost always bare,’"* as eminentem conspicit, nam in una Melodia well as standing out from the other id per se patet, cum c:e = 4:5, & cum products, for in one melody it is dissonantiae % = ’/lo, % = ®/i 5, &c. rarius accessible through itself, since c:e = 4:5, occurentes, statim in consonantias resolvi and since the dissonances % = ’/lo, % = debeant; in pluribus autem Melodiis ®/i5, and other rarely occurring ratios, concinentibus inferiores fere semper ought to be resolved immediately into numéros agnoscunt minores, ut iterum 5 consonances. However, in the greater in eo loco conspiciatur, ubi quasi number of melodies that are harmonious, simplicissimo modo ex unitate effluere musicians nearly always recognize the videtur, uti id observare licet in smaller numbers, so that again 5 is consonantiis supra §. 18. & 19. traditis. observed in that place, where, as it were, by the simplest manner, it seems to flow ficm unity, as can be observed in the given consonances in the above § 18 and 19.
Pari ratione numerum 3 fere non By the same reason, you will scarcely see conspexeris elevatum, nisi in dissonantiis. the number 3 raised, unless in the
Literally, “obvious” or “instant”
68 etenim in allegata §.19. serie ordinaria dissonances, because in the ordinary nudus adparet, quamvis etiam in series adduced in §19, three appears bare, consonante numéro 6 binario sit admixtus; although even in the consonant binary sed in modemo harmonicae compositionis number 6 it was mixed. But in the gustu cantuum spiritualium simplicissima modem style of harmonic composition of melodia incedentium sequelae the rousing spiritual songs, by the simplest consonantiraum fere semper in ratione 8:9 melody, the sequences of progressing adparent, ut supra §. 18. ostensum. consonances almost always occur in the Contra quantum binarius in Harmonia ratio 8:9, as pointed out in § 18 above. Tonorum valeat, vel ex solo adspectu On the other hand, how much the binary is baseos in 2^ satis elucescit, atque etiam worth in the harmony of the tones, is even demonstratio §. 22. tradita probat. sufficiently apparent whether from the bare aspect of the base 2^, and also proven from the demonstration given in § 22.
In Melodia autem anima ne vel semel satis However, the mind, in melody, is never commode 5 tactus potest colligere, ut able to collect 5 beats adequately enough, exemplum §. 17. id docet, cum contra as the example in § 17 shows.^^ On the vicies potius in ipsis partibus Tactuum contrary, 3 easily counts twenty times into triplorum principalibus facile 3 numerat, the very main parts of the triplets, and 2 & 2 indefmits, ne dicam infmitis, vicibus, into uncounted successions—lest 1 say partim in colligendis partibus Tactus into infinities—partly in reckoning main principalibus & minus principalibus parts of the beat, and less into all the main omnibus ac singulis ex innumeris parts or individual parts from the sonorum vibrationibus, partim in innumerable vibrations of the sounds, colligendis ipsis tactibus, & ad suas [30] partly in reckoning the beats themselves, propositiones referendis, pro acquirendo and comparing to their own proportions, sensu cantilenae totali ac pleno. for the acquiring a sense or fullness of the whole song. §26 §26
Praecepta Harmoniae Musicae in The precepts of musical harmony very synthesi tradita brevissima. briefly summarized.
Neque tamen etiam incongruum erit, rudi Nevertheless it will not be disagreeable, saltim penicillo ostendere modum, quo even with a rough brush to reveal the principia haec arithmetico-musica ex vera manner, by which these principles, by Musicae idea possint derivari, & musical arithmetic, can be derived from syntheticae proponi: the true idea of music, and can be displayed by a summary:
A reference to the asymmetrical phrase example. See p. 48.
69 1. Musica est Scientia dextra 1. Music is a science, by the skillful compositione Tonorum animum composition of tones, of delighting the delectandi variasque ejus adfectiones mind*, and of expressing its various exprimendi. affections.
2. Toni sunt rationes duorum sonorum: 2. Tones are the ratios of two sounds: a sonus scilicet est incredibilis & innumera sound is, of course, incredible and quasi multitudio vibrationum, aequabili innumerable as it were a multitude of motu sibi invicem succedentium; itaque vibrations, immediately succeeding to anima duos sonos percipiens, themselves by equal motion. Therefore quotiescunque una inter vibrationes soni the mind, perceiving two sounds, becomes A cum altera aliqua vibrationum soni B more attentive—no matter how many coincidit, fit attention ut crebriori recursu there are among the vibrations of sound talium duarum ex illis, quas habent duo A, one vibration coincides with another of soni, vibrationum, qui aequabilil fit motu, the many vibrations of sound B. With the quia ipsae singulorum sonorum result that by a more crowded feedback of vibrationes aequabiliter moventur, two such vibrations from those many investiget & inveniat, quot in spatio talis vibrations—which the two sounds recursus ex singulis sonis contineantur contain, and come about with equal vibrationes, seu quot vibrationes eodem motion, because the same vibrations of tempore unusquisque habeat sonus. Et sic individual sounds are moved equally—the Tonus est ratio singulorum duorum mind investigates and discovers how sonorum, v:c:l:2 &c. vid. fig. §. 25. many vibrations are contained in the space of such feedback from the individual sounds, or how many vibrations each sound holds at the same time. And thus a tone is a ratio of two individual sounds, v[oice]:c[ontinuo]: 1:2 and so on. See the figure in section 25.
3. Quoniam vero Musica praestatur 3. However, because music is presented compositione Tonorum, hinc Toni sunt by the composition of tones, fhom this, innumeri, singuli ergo citissimi & quasi tones are innumerable, individual momentanei; ergo, si velint esse animae therefore, very quick, and, as it were, intelligibles, ac delectabiles, non poterunt momentary. Therefore, if they might wish fere nisi simplicissimi [31] esse, ex.gr. to be understandable to the mind, and be 1:2,1:3,1:4,1:5, nec facile plures: delightful, they will hardly ever be able to Compositio autem illorum, cum debeat be so unless they are very simple—for esse magna atque adeo propter eorum example 1:2,1:3,1:4,1:5, and not easily paucitatem variantissima, pariter non more. Moreover, the composition of poterit esse nisi simplicissima, si quidem tones, since it ought to be long as well as velit esse animae perceptibilis & highly varied—indeed because of the
™ “Harmonic” and “Melodic,” respectively. ^ That is, the rules o f counterpoint.
70 delectabilis; debebit autem, cum Toni sint paucity of tones—in like manner, it will momentanei, esse & simultanea & not able to be so lest it is very simple; if successiva. De utraque specialius indeed, one wishes to be perceptible and agendum. pleasing to the mind. Moreover it ought to be both simultaneous and successive, 76 since the tones are momentary. And concerning both, a more special handling is called for.’^
4. Compositio Tonorum simultaneorum 4. The composition of simultaneous tones, seu Harmonia stricte dicta, erit vel or strict harmony as it is called, will either consona, prout rationes s. Toni ejus ex se be consonant or not—just as the ratios, invicem sunt explicabiles, vel non: Cum namely, the tones of the harmony are vero in numeris §. 2. adsum[p]tis 2,3,4, mutually arranged according to one 5, numerus 4 possit explicari ex numéro 2, another. However, since in the adopted nam 1:2 = 2:4, igitur numeri ex istis 2,3, numbers 2 ,3 ,4 ,5 (§ 2), the number 4 is 4, S; ope binarii, & item ope quatemarii able to be explained from the number 2, producti facile erunt ex binario per for instance 1:2 = 2:4, therefore the potentias sumendo intelligibiles, & inter numbers are from 2 ,3 ,4 ,5 . By the power se semper eandem servabunt rationem of two, and likewise by the power of four, simplicissimam 2,3,4,5; contra producta the products will easily be understandable numerorum 3 & S omnino constituent from the mentioned binary by powers. dissonantias, quae non valde debent And between themselves they will always cumulari, ne anima illis ita obruatur, ut preserve the most simple ratio 2,3,4, S. plane nullam harmoniam possit percipere. On the other hand the products of the numbers 3 and 5 entirely establish the dissonances, which ought not to accumulate greatly, lest the mind be overwhelmed by dissonances, so that no harmony can be perceived clearly.
71 3. [sic, S.] Erunt ergo hac ratione: 5) Therefore, they will be by this reckoning:
[Consonances:] Confbnanciæ A, B, c . D : 3, 4, f, 2 *. 4 > 6 , 8 , 10, 2 *. b. 8 , 12, i6, 20, 2*. C. l6y 24, 32, 40, 2 *. d. 3 2 , 48, 64, 80, a*, c. 9^) 1 2 8 , 16O; 6 (4
Figure 27: Flicker's table of consonances and dissonances.
Dissonantes numeri illis consonantiis, The dissonant numbers, arising from 3 inprimis numeris serierum A & C ex 3 & and 5, according to those consonant 5 oriundi: 15,9; dissonantissimi: 27,25; numbers, and the non-prime numbers of vid §. 17. Analys. & 45. the series A and C, are: 15,9; the most dissonant numbers are: 27,25 and 45; see the analysis of § 17.
[32] 6. Ergo pro simultaneis Tonis in 6) Therefore, let them be taken for Harmonia vera adsumantur, quotcunque & simultaneous tones in true harmony, qualescunque adsumere libitum fuerit ex however many and whatever kind should sequentibus numeris be agreeable to adopt from the sequential numbers—
72 Confonandbns; i, 3, 11, 1^ ,20, 24, 32, && quz alternare inter fc ita queant, ut alitais cx his 9 , l y , 2 7 ,4 9 nonnunquam mterccdae diflbnaturus. &«. Consonances: Which are thus able to alternate among themselves— so some of these: 9,15, etc.— sometimes intercede to cause a dissonance.
Figure 28: An example of a series of consonances interspersed with dissonances.
7. In successione Tonorum seu in Melodia 7) In a succession of tones, or in a attendendum est ad successiones & melody, one must pay attention to the singulorum & simultaneorum, & quidem successions both of individual tones and ad rationes succedentium ipsorum inter se of simultaneities, and certainly to the & ad rationes succedendi ipsas, seu ad ratios of the successions themselves diversa tempora, quibus sibi invicem between themselves, and to the ratios succedunt. themselves of the succession, or to diverse times, by which they succeed one another in turn.
8. Rationes sequentium se singulorum 8) The ratios of two individual tones duorum immediate possunt esse immediately following one another can be difficillimae & intricatissimae, modo & very difficult and very intricate. They are simultaneorum & successionis rationes obtained by the manner of both sint expeditae. Nam quid v:c. impedit simultaneities and succession. For successionem numeri 15, qua is numerum instance, what v[oice]:c[ontinuo] prevents 16 sequitur, in altera serie priorem ita the succession of the number 15, by which sequente? succession 15 follows the number 16, in a second series thus following the first?
4 > 8 ) 1 2 ) 1 6 ) 2 0 , 3 > If, 24,
Figure 29: A comparison of two series.
73 satis enim numeris iste IS, expilcatur ex For it is well enough that number 15 is serie sua, 3,6,9,12,15, quae est ut 1,2, explained according to its series, 3,6,9, 3,4,5; baud secus ac alter 16, ex serie 4, 12,15, which is as 1,2 ,3 ,4 ,5 , and not at 8,12,16,20, quae iterum se habet, ut 1,2, all diKerently than the second number 16, 3,4,5. which again contains itself from the series 4,8,12,16,20, as 1,2,3,4,
9. Sed hoc observandum, non posse facile 9) But this must be noted: series of many plurium succedentium series multum a se successions cannot easily be out of tune invicem abludere, nam ex.gr. [differ] from one another alternately by much, for example.
n m abludere, nam ex.gr. A B Serieshæc: 4, g, 10, 12, itaabalîera&hæç: 3, 6, 9 , i f , acecnadiTcrep^t: a , f , lo , 2,0, This series: is out of tune[differs] thus from another, and this series: differs from a third
Figure 30: A comparison of three series.
[33] hie 5 ab 8,20 a 12 & 16 non valde Here, 5 does not greatly differ from 8, nor abhorret, quamvis 9 a 10,15 a 16 Satis: 20 from 12 and 16, although 9 does differ quia rationes sequentium ferierum B non enough from 10, and 15 from 16: because sunt nisi repetitae priorum A. the ratios of the second series B are just repetitions of the first series A.
10. Rationes simultaneorum succedentes 10) Successive ratios of simultaneities aestimandae sunt ex numeris simplicibus must be reckoned from the simple simultaneorum, qui, ut statim §. 6. dictum numbers of the simultaneities, which, as inter se alternant; sic successiones firmly stated in section 6, alternate among numerorum themselves. Thus the successions of numbers
^ The bottom series can be explained to 15 by 3 times 1-5, excluding 24; it can also be explained to 15 by 3 + 3, etc.; the top series can be explained by 4 times 1-5, beginning with 4 in the series. Even though the 1-5 reduction does not match terms between the series, it does not matter, because the mind is satisfied in simply rinding this reduction.
74 A 8, lO, 12»i(»20, 30, 3%, B 12, If»18» a4 »30»36» C 4 > 8 > l 6y 20, 24» D 9> la»If»18» 36»4 f» E lO, If»20, 30, 40 »4 f» F la , *4 »3a»48, 64».
Figure 31: Several possible series generated from the three prime powers.
aestimantur A ex 2, B ex 3, C ex 2, D ex are judged, A from 2, B from 3, C from 2, 3, E ex 5, F ex 2 & sic deinceps. Ergo D from 3, E from 5, F from 2, and thus in rationes hae per se sunt valde simplices. order. Therefore, these ratios by themselves are very simple. 79
11. Rationes autem ipsarum 11) However, the ratios of the successionum, quae id vere constituant, successions themselves—which truly quod supra §. 11. Analys. melodiam establish that which I addressed above, in adpellavi, quamvis in duobus immediate section 11, “Melodic Analysis”—can be se invicem sequentibus Tonis possint esse very awkward, however much they are in valde inconcinnae, ut 9 & 4,3 ad 8, tamen two successive tones immediately ita debent coaptari, ut in praecipuis Tonis alternating themselves, as 9 and 4,3 to 8. semper non tantum simplices, sed plane Nevertheless they thus ought to be eaedem recurrant, ut prolixius §. 1 Î & 14. adjusted, so that not only the simple ratios Analys. Ostendi. return into particular tones, but clearly the same [complex] ones return, as I more broadly showed in the analysis in § 12 and 14.
12. Quod si nunc ea, quae in brevissima 12) But if now those things, which were hac synthesi dicta sunt, animo colligantur, stated in this concise summary, are facile patebit, omnes Musicae Tonos ex connected by the mind*, it will easily be basi 2 , 3 \ 5* debere fluere, illasque known, that all tones of music ought to rationes ex hac desum[p]tas, quae se flow firom the basis 2^, 3^, 5*. Those invicem non déprimant & in alias mutent, ratios obtained from this basis, which do verum debere constituere Musicae not compress themselves in turn, and promtuarium. change into others, truly ought to constitute the storehouse of music.^°
^ The issue of points 5-10 is that the mind takes any number of series and reduces them to 2,3,5—no matter how unmusical and out of order this might seem. ^ The ratios which do not adjust make-up the foundation intervals.
75 Nam si omnis compositio & coilectio [34] For, if every composition and collection rationum, rationalis, ut ita dicam, h e. of ratios is susceptible to reckoning—as perceptibilis animae per rationes, debet thus I stated, that is, perceptible to the fieri per regulam de Tri seu per mind by ratios—it ought to come about multipiicationem, talis autem per through a rule concerning “the three” multiplicationis speciem facta coilectio primes or by multiplication. In other pro elementis seu simplicibus supponit words, by the type of multiplication numéros primas, whereby the reckoning made replaces prime numbers for elemental or simple numbers. sequitur, talem compositionem It follows that such a composition and rationumque collectionem, quae collection of ratios, which always variantissimis omnibusque possibilibus produces ratios, by the most varied and all modis simplicissimas semper prodit possible simplest manners, according to rationes pro simplicibus componendis non the simple composings, should only supponere nisi numéros inter se primas, replace the prime numbers with h.e. Tonos illos §. 8. Anal, in repetitis themselves—that is, those tones octavarum intervalli iterato continuandos. continuing, in the analysis in §8, in the repetitions of the octaves by repetition of the interval.
§27 § 27
a Calcule magis remota. [A synthesis] by a more distant calculation.
Abstrahendo magis a Calculo ita posset Therefore, a synthesis can be established Synthesis institui; to a greater extent by an abstracting calculation;
1. Musica est Compositio Tonorum; sed 1. Music is the composition of tones; but dices: you say:
1) quid sunt Toni? Rx. Relationes s. 1) What are tones? Answer: Relations, of rationes sonorum; ergo per fractiones course, ratios of sounds; therefore, they numerorum exprimendae; 2) unde must be expressed through the fractions of oriuntur? Rx. Componunturex numbers; 2) From where do they rationibus; itaque revera nulla datur in originate? Answer: They are constructed Musica Compositio, nisi ipsorum fmm ratios; and thus actually no Tonorum, tanquam elementorum ex composition is given in music, except simplicibus invisibilibus: caeterum datur from the tones themselves, just as from saltim ordinata aggregatio Tonorum pro elements out of the invisible simplicities: componenda cantilena, ita instituenda, ut otherwise, an ordered collection of tones
76 omnia & aggregates singulos Tonos & is given for composing a song, begun in ordinem aggregadonis etiam per numéros such a way that one easily perceives facile percipiat. everything, both individual tone collections or a series of a collection, even according to numbers.
2. Toni se habent ad suam aggregationem 2. Tones relate to their collection, as uti substantia ad accidens, uti res percepta substance is to accidence, as a thing & perceptio ejus: Toni enim magis extra perceived to the perception of it: for tones nobis esse videntur, quam ipsorum seem to be more outside us, than their aggregatio; quia per se exsistunt. Hue collection, because they exist through pertinet Dni. Maupertuisii distinctio reaiis themselves. Pertaining to this point, the inter objectum & sensum. [25, sic 35] real distinction—according to Mr. Dicit is: Judicium, hoc est arbor, aut idea Maupertius—is between object and sense. de arbore emergit, quando sensus He asserts: The judgment, this is a tree, or frequentiores: sui in hoc loco & vidi the idea concerning a tree emerges when arborem, sum iterum in illo eodem loco, the senses are more frequent: I was in this & iterum video arborem; hinc place, and I saw a tree; I am, again, in that quotiescunque ero hoc in loco, videbo same place, and again I see a tree. Hence, arborem. Ita denique scilicet sensum as often as I will be in this place, I will see nostrum transferimus in ipsum objectum. a tree. Thus in short, naturally, we transfer our sense into that object.
3. Hinc sequitur, Tonos vim habere 3. Hence it follows, that tones have more majorem, quam anima nostra, quia power, than our ntind, because they take recipiunt sensum nostrum. Tonis in our sense. All our senses are divided tribuuntur omnes sensus nostri singulis; by individual tones. Therefore they are a sunt igitur magis compositum quid, quam thing more ordered than our senses, sensus noster. Unde sequitur, because it follows that a ruled collection aggregationem Tonorum regulam ordinis of tones of a succession holds more habere simpliciorem, quam quae est simplicity than something which is a rule Tonorum régula compositionis. of composition of tones.*
4. Posito igitur, Tonos esse composites: 4 .1 propose, therefore, that tones are quid erunt ipsorum simplicia, nisi humeri? ordered: what will their basic parts be, Et cum Toni sint rationes vibrationum in except shoulders?*^ And since tones are sonis variis inter se quoad celeritatem ratios of vibrations in sounds that vary comparatarum; ratio autem sit fractio; & between themselves as far as their speed fractionum ratio intelligenda nullam possit when compared; moreover since a ratio is requirere compositionem nisi per a fraction; and since of fractions, the ratio multiplcationem; multiplicationis denique to be understood is able to seek no
" The rules for specific compositions are necessarily more complex than the rules governing the fundamental cognitive construct. 82 Literally, “bones,” "structure,” or “supports.”
77 simplicia sint numeri primi: sequitur, composition except through Tonos omnes Musicos esse fractiones, multiplication; fînally since the quae emergunt per continuam simplicities of multiplication are prime multipiicationem & divisionem numbers: it follows, that all musical tones numerorum primorum, ordine are fractions, which emerge through simplicitatis se invicem subsequentium. continuous multiplication and division of the prime numbers, by order of the simplicity of the subsequent ones altemately with themselves.
5. Et cum aggregatio Tonorum pariter non S. And since the tone collections fît nisi ex rationibus s. fractionibus constet, together only from the ratios, that is, the saltim cum ilia ab ipsorum compositione fractions—at least with that distinction®^ singulorum Tonorum distinctione, ut illius which refers to the composition of régula sit simplicior, ideo pauciores individual tones—so it is that the rule is numeri primi ordine naturali sequentes more simple, to the extent that, fewer pertinent ad aggrega;[t]ionem [36] prime numbers—following by natural Tonoruum, quam ad singulorum order—pertain to the aggregate of the compositionem; caeterum & ilia fît per tones, than to the composition of multipiicationem istorum numerorum. individual tones. Still, even that comes about through the multiplication of those numbers.
6. Sed si ilia in utraque methodo adhibita 6. But if multiplication accepts no limits multiplicatio nullos accipiat limites, erit in both methods employed, it will be infinita; Musica vero infinitorum infînite. However, music of infinite tones Tonorum esset absurda, immo would be absurd; on the contrary, multiplicatio non longe debet progredi, ne multiplication should not progress too far, a simplicitatis & intelligentiae régula lest it depart from the rule of simplicity discedatur. Jam cincipe numéros tres and intelligence. Now take the three primos 2,3,5, ordine simplicissimos, pro simplest prime numbers 2 ,3 ,5 in order— ratione difficilioris ex simplicissima instead of the ratio of a more difficult unitate intelligentiae minus elevatos, sc. reckoning®^—having been less elevated 2®, 3^, 5* pro compositione Tonorum, & from the simplest unity of understanding, iterum pauciores 2°°, 3^ pro aggregatione that is, take 2^, 3^, 5* for the composition Tonorum, habes bases Musicae. of the tones, and again, take the lesser ones 2“ 3‘ for the aggregate of the tones, thereby you have the bases of music.®^
The distinction between the aggregate collection of tones and their use in a specific composition. “ That is, Euler’s method of raising 2 ,3 ,5 to very high powers. Flicker touches on an important point here. The musical basis represents the numbers used to derive ratios (pitches) used in actual pieces. The other numbers series, T and 3‘, which Pricker calls the “aggregate,” represents an even more fundamental cognitive construct. See discussion in Chapter 1.
78 {Conclusio.} Concipiat nunc, quicunque {Conclusion.} Let one now conceive, velit, aliam methodum explicandae whoever wishes, another method of Musicae; & conférât cum ea, &, quantum explaining music. Let him compare with vel ilia vel haec ad explicanda omnia this method, let him inquire how much sufRciat, inquirat. Satis sit ergo dictum; either that method or this suffices to the neque enim haec quisquam satis bene purpose of explaining everything. Let this intellexerit, non dicam dijudicaverit, qui dictum suffice: for neither will anyone ipse ad pervestigationem scientiae understand the things I have said, let alone Musicae arithmeticam tardus fuerit, cujus pass judgment on them, who himself will certe methodum & vias fideliter be slow according to the thorough ostendimus. mathematical investigation of the science of music, of which, certainly, we have faithfully shown the method and the paths.
79 [OETINGER: SECTION H]
§1
Ex Auctoris theoriae Musicae cogitatis From what has heen considered hy the fit applicatio propior ad animam. author of the Theoria Musica^ we make an application that is more closely related to the soul.
Haec sunt, quae peramicus mihi Dn. These are some things which the very dear Friker, Stipendiarius Tubingensis, Mr. Flicker, Fellow of Tubingen'—one Arithmeticae Musicae deditissimus, post most dedicated to the arithmetic of music, mutua mecum colloquia chartae mandavit, after mutual conferences with me in postquam multis observationibus Musicis writing—entrusted to me, after having & calculo operam impenderat. Quoniam devoted effort to many musical itaque in superioribus saepe ad Musicam observations and to calculation. And thus, provocavi, igitur applicabo nunc eadem because I often appealed to music in the [37] ad animam. Homo constat corpore above matters,^herefore I will make an ac anima: anima autem viribus gaudet application now likewise to the mind. diversis, inter quas quaedam sunt centrum Man is established by body as well as reliquarum. mind: however the mind rejoices in diverse powers, among which certain ones are the center of the others.
Centrum hoc redditur magis vel minus This center is retumed more or less pure. purum, prout ilia vel in corporea vel in Just as that power dwells, more or less, spirituali indole magis vel minus versatur, either in the innate flesh or in the innate & a spiritu sive praesentia Dei magis vel spirit, and is activated, more or less, by minus actuatur: est enim juxta Rom. Vm, the spirit or presence of God: for it is like 5. anima vel magis in spiritu, vel magis in Romans 8:5,^ the mind is either strong in came. Si est in Spiritu, tum separatur a the spirit, or strong in the flesh. If it is camalibus & habet in se quoddam strong in the spirit, then it is separated dominium, quod repellit sensationes from fleshly things and has in itself a camales; si est in came, tum sentit etiam certain control, because it repels fleshly camaliter. sensations. If it is strong in the flesh, then it experiences camality.
‘ Flicker received a scholarship to study theology and philosophy at Tubingen, which he did from 1747- 1749, after which he went to study with Oetinger. Roessle, p. 8. ^ See pp. I89ff., Oetinger’s Inquisitio. ' Romans 8:5 “For they that are after the flesh do mind the things of the flesh; but they that are after the Spirit the things of the Spirit.”
80 Anima quoad dominium a came separata The mind that is separated from the flesh, & divinae assueta praesentiae, operationes with ragard to its control, and accustomed Naturae & Gratiae in centro receptas ex to the presenceof God reflects those viribus manifestativis sui tanquam ex activities of nature and grace that it has speculo repercutit cum serena taken into its center, reflecting them from conscientiae suavitate. Sic ex fundo sui powers that manifest the mind itself just manifestationes cogitando vel coagitando as from a mirror, with a serene sweetness elicit, si tamen prius percharacteres of recognition. Thus the mind elicits creaturarum extemos excitatae fuerint. manifestations of itself by thinking and Sic Musica, sub consonantiis debitis stirring ideas out of its own foundation. percepta, partim ex fundo animae, partim Even if these manifestations are first per extemas aëris vibrationes illam tam stirred through extemal impressions of suaviter afficit. living things. Thus music, having been perceived under appropriate consonances, partly from the foundation of the mind, partly through the extemal vibrations of the air, affects the mind so pleasantly.
Etenim anima ab ipso Deo quoad vires And indeed the mind, with respect to the insitas & numéros potentiales in inbom forces and numerical powers, was harmoniam posita est, ita, ut omnes placed in harmony by God himself, thus, imagines rerum sint una quasi imago so that all images of objects are, as it naturam referons. In hac facile concentus were, one image referring to nature. In Musici exprimuntur tanquam in this, the concords of a musician are easily clavichordio intemo, & cum delectatione expressed as in an intemal clavichord, and percipiuntur. Delectatio ilia hominem they are perceived with delight. That naturalem ita occupât, ut non nisi delight thus occupies the natural man, so delectionibus aliorum sensuum superetur. that it is not surpassed except by the delights of the other senses.
Homo vero illuminatus centrum suum However, an enlightened man keeps his possidet liberum, ut impressiones extemas center free so that he might moderate ex facultate sensitiva in fa- [38] cultatem extemal impressions that reverberate from imaginativam & ex hac in illam the sensate faculty into the faculty of repercussas per delectionem puriorem & images (as well as from the latter into the intimiorem moderari, in subordinationem former) through enjoyment that is more reducere & pro sapientiae mensuris suo pure and more inward, and so that he loco ponere possit. might reduce them into subordinates and place them each in its own place according to the measures of wisdom.
81 §2 §2
Occurritur objection!, quasi anima An objection suggests itseif: because the esset materialis, quia ex viribus oiversis mind Is united by diverse powers, it [sic, diversis] coadunata. might be material, as it were.
Haec pro idea quadam general! possunt These things can be taken for a certain haberi. Ut jam distinctius id explicem, general Idea. I would like now to explain velim, ut vires animae concipiantur ut it more distinctly, so that the forces of the vires centrales ex centripetis & centrifugis mind are conceived as central forces compositae; concipiantur numeri Musici composed from centripetal and centrifugal primitivi 3 & 5 ibidem ut centrales, forces. The musical prime numbers 3 and motum gyratiorum a Deo impressum S are taken up at the same time as the constantissime servantes per 3 sphaeram central forces—the gyrating motion centripetarum 5 sphaeram centrifugarum having been Imprinted by God, most vel vice versa per tertiam 2 ampliantur, & constantly. The reserved numbers are in actione sustentantur per exitus & enlarged through the 3 sphere of the reditus continues. centripetal forces, the S sphere of the centrifugal forces or vice versa through the third power of 2, and they are sustained in the action by the continuous departure and return.
Nam vis superior 2 a centro I divinitas For the higher power 2, having been mota, ex intimo sui continua fulminatione divinely moved from the center 1, begets instantaneas edit operationes, sic, ut instantaneous operations from its inmost ceterae duae 3 & 5 vitam alant per continuous fulmination so that the superiorem illam 2. Unde vita naturaiiter operations nourish the life of the other indestructibilis oritur, nec potest dici two, 3 and 5, through that higher power 2. materialis, quia vires non sunt materiales: From whence, life arises naturally neque enim coadunatio virium infert indestructible, nor is it able to be called materialitatem; alias vita in Deo, a Spiritu material, because the forces are not Sancto indlssolubilis nominata Ebr. 7,16. material things. For neither does the ipsa materialis esset dicenda; quod est joining together of the forces infer absurdum. materiality. Otherwise, the life in God, called “indissoluble” by the holy spirit in Hebrews 7:16\ would be called material, which is absurd.
Vel ipsa virium motricium activarum Indeed, the very composition of the Leibnizii composite materialis esset active, moving powers of Leibniz would dicenda; quod iterum est absurdum: nam have to be called material, which, again, is
'* Hebrews 7; 16, “Who is made, not after the law of a carnal commandment, but after the power of an endless life”
82 ens unum simplex eodem tempore agere absurd: for one simple thing is unable to vel resistere nequit, nisi versus unam drive or oppose at the same time, except plagam; ens enim simplex conatum & against one region. Indeed the simple directionem tantum simplicem habet; ut thing has only a simple tendency and igitur directio versus diversas [39] plagas direction. Therefore, the result is that the emergat, debent vires esse compositae, true direction emerges into diverse vires hae compositae non dici possunt regions, the powers ought to be materiales: nam sunt materia priores, & composites^, these composite powers are haec regitur a viribus simplicioribus, not able to be called material: for they are quarum compositio materialismum infert prior to the material, and this power is nullum. directed by the simpler powers, of which powers the composition of the materials produces nothing.
Deus certe in S. Scriptura loquitur de God certainly talks in the sacred scriptures anima ut composite ex viribus centralibus; about the mind as being composed from hinc dicit, se plasmare vel formare central powers. Hence he states, that he spiritum in centro hominis Zach. 12,1 & shapes and forms the spirit in the center of Psalmo 33, IS. dicitur Deus plastice man—Zechariah 12:1 and Psalm 33:15^. formare corda vel intima hominum simul, God is said to form with clay^ the hearts h.e. continue actu coagmentare vires in or inmost parts of men similarly, that is, vitam, & in esse conservare centrum by a continuous act he joins the powers virium. An inde sequitur, animam vel into life, and is conserved in the center of centrum virium cum manifestatione sui the powers. Does it follow then, that the diffusiva esse materiale? nequaquam! Sed mind or center of the powers, with its pergendum est ad rem ipsam. Dictum est, diffuse manifestation, is material? By no vires 3 & S per tertiam fulminatoriam in means! One must proceed to the issue perenni actu sustentari. itself. It was stated that the powers 3 and 5 are sustained through a third fulmination in a continual action .
Profecto & Physica illam tertiam requirat; And certainly, physics requires that third quid enim est, quod directionem fulmination. For what is it, which thus ceterarum duarum in otu peripherico ita continuously changes the direction of the continue mutet, ut eadem respectu puncti other two in a peripheral motion, so that at unius s. centri constans maneat, scil. the same time, it remains constant with tangentialis in centrifuga, verticalis in respect to one point, namely, of the centripeta, nisi unitas centri illius educta center—that is, of the tangential in the
^ A reference to Leibniz’s monad. * That is, as opposed to monads. ^ 2kchariah 12:1, "The burden of the word of the LORD for Israel, saith the LORD, which stretcheth forth the heavens, and layeth the foundation of the earth, and formeth the spirit of man within him." Psalm 33:15, "He fashioneth their hearts alike; he considereth all their works.” ’ Job 33:6, "Behold, I am according to thy wish in God's stead: I also am formed out of the clay ” ’ That is, by the action o f the two force.
83 usque ad totam, quaqua patet, centrifugal force, and of the vertical in the peripheriam? centripetal force—except the unity of that center, having been led up all the way to the total periphery, whenever it opens • 10 up"
Quid est, quod centripetam vim sistat, What is it which stops the centripetal dum centrifuga punctum peripheriae in power, while the centrifugal force moves circulo promovet? Nam revera peripheria forward the point of the periphery in the est Summa tangentis, non radii neque circle? For actually, the periphery is the diagonalis duarum virium; alias enim highest of the tangent, not of the radius radius cum peripheria angulum 45° unum nor of the diagonal of the two forces. facere deberet, non rectum. Sed omnino Otherwise, the radius, with the periphery, Physici hac tertia vi opus non habent, quia ought to make one 45° angle, not a adsumunt motum circulaiem jam straight line. Physicists have no need of existentem ubique in natura, non demum this third force at all, because they assume gignendum, ne- [40] que unquam amplius that the circular motion now existing sistendum. everywhere in nature, is neither something which must be bom finally, nor something which must be established further.
Ceterum cum nullum peripheriae punctum But, since there is no point of the sit altero prius aut posterius, quippe ab eo periphery prior to or after any other point, nulla ratione distinguendum, haec vis, and obviously by no reckoning is there quam deinceps brevitatis causa any distinguishing from it, this force, excentricam appellabimus, ubique in which we will call successively, for the peripheria praesens & simultanea est. sake of brevity, outside-the-center, is Estque revera haec vis ipsa ilia conditio, everywhere present and simultaneous in qua Physici ponunt existera jam motum the periphery. And it is actually this very circularem. force that is the very condition whereby physicists now posit to exist in circular motion.
That is, the two force is the third fulmination which is able to be a constant force with respect to the center and a driving force against the other two (cenlrifugally, pushing them out to higher powers). By equating centrifugal with tangential and centripetal with vertical, he is trying to describe the “circle” actually as a sphere. To illustrate, 1 suggest the picture the globe: tangential (centrifugal) represents the horizontal, latitudinal surface, pushing the numbers out along the equator, if you will, and vertical (centripetal) represents the numbers falling back to the center longitudinally through the poles of the globe.
84 §3 §3
Specialior virium animae per Musicam The disposition of the mind’s powers is, affectio. through music [made] more specific.
Jam Specilaius determinandum, quomodo Now, it must be determined more facultates sensitivae & facultates specifically how the faculties of the sense imaginativae se habeant ad Musicam. and the faculties of the imagination are Musica in facultate sensitiva, naephaesch disposed with respect to music. Music in hebraeis dicta, innumeris sensationibus the sense faculty (called Nephesh in the numerorum, tres illas vires secundum Hebrew), by means of the innumerable singularum characteres & universarum sensations of the numbers, expresses the inter se proportiones certa mensura three forces according to both their temperandas, ad peculiares affectus individual characters and universal excitandos exprimit, quarum deinde proportions amongst themselves, to be colligationem & concentum facuitas tempered by a certain measure, for the sublimior imaginativa, Ruach, dicta Zach. purpose of exciting peculiar affections. 12,1. simili effigie ex se evocanda Of these, finally, the more sublime repraesentat. Praemittenda vero adhuc imagination faculty, the Ruach—as it is plura sunt de viribus & characteribus called in Zechariah 12:1," represents the numerorum. connection and harmony by a similar image evoking from itself. However, at this point more premises must be made concerning the strengths and characteristics of the numbers.
§4
Praemittitur consideratio de viribus & A consideration concerning the powers characteribus numerorum. and characteristics of the numbers is advanced.
Vires illae sunt simplicissimum in natura, Those powers are the simpleest thing in ita, ut totam hanc in omnibus effectibus nature, such that they direct and vivify suis regant & vivificent. Ergo in Musica, nature in all its effects. Therefore in quando quidem harmonia est obtinenda, music, whenever a harmony is to be naturam facuitas sensitiva sentiat oportet, obtained, the sense faculty should prout ex his producitur viribus velut perceive the nature, in so far as it is elementis suis. Plurimi igitur numeri a produced from these powers, as its own sensitiva sentiendi debent esse ita elements. Therefore, many numbers, comparati, ut facile & [41] per se sensed by the sense faculty ought to be intelligantur, nemine monente, ex tribus thus compared, so that they are
" Zechariah 12:1 “The burden of the word of the LORD for Israel, saith the LORD, which stretcheth forth the heavens, and layeth the foundation of the earth, and formeth the spirit of man within him.”
85 simplicissimis numerorum elementis understood easily and through themselves, simplicissima combinatione orti. on their own terms, having arisen from the three simplest elements of the numbers by the simpleest combinations.
Jam vero simplicissimum quidem However, now the simpleest type of combinationis genus est additio, sed per combination is addition; but through this hanc numeri omnes non nisi ex unitate all the numbers do not grow except by nascuntur, ita, ut in hoc genera non dentur unity, with the result that in this type the tria elementa. Nam & binarius & three elements are not furnished. For both temarius ex unitate addendo oriuntur. the binary and the ternary arise by adding Ergo alterum multiplicationis genus est from unity. Therefore, musical Musicum, in quo toni jam sunt compositi combination is of the second type in ex primus tribus; hi sunt 2,3 & S, qui which tones are composed from the three habent analogiam cum compositione primes. These are 2,3 and S, which are trium vitae virium, qui denarium efficiunt analogous with the composition of the in sua summa. three forces of life, which produce the denarium in its own sum.'
Qui numeri in eo quidem conveniunt, ut To which the numbers certainly come non secus ac reliqui omnes ex unitate together in it, so that not otherwise are all possint derivari per additionem, sed de the rest able to be derived through cetro & per se & in effectibus suis s. addition from unity. But concerning the virtutibus multum discrepant, scil. in vita others—both in themselves and in their exprimenda legem tenent, ut vis effects, namely, their forces—they simplicior ceterarum minus simplicium disagree greatly, namely, in expressing effectus omnes possibiles & pemicitate & life they preserve the law, that the efficacia superet, h.e. cum effectus unius simplest power surpasses all possible vis sint producta ex numéro ilia effects of the other less simple, both by respondente seu potentiae ejus & omnes speed and effect. That is since the effects plurium virium possibiles sint producta ex of the one power are products from that suarum ipsarum productis seu potentiis corresponding number or are powers of it, primorum numerorum facta, quisque and since all possible numbers of the numéros primus eousque [sic, eo usque] many forces are produced from the potest elevari, donee productum products of the numbers, or are powers
That is, you must add 1+1 =2; 2+1 = 3 The “denarium": 2+3+5 = 10, cf. decad. Oetinger is appealing to Greek numerology. The decad was used to represent the musical ratios 1:2,2:3,3:4. As a figure:
• 1 • • 2 • • • 3 • • • • 4 The total is ten. A similar Greek concept was the admiration for the numbers two and three which both add and multiply to six.
86 elevationum omnium ex ceteris majoribus made from the prime numbers, then every numeris primis factarum quantitate prime number is able to be elevated all the superet. way up to that degree, until it can surpass the product of every elevation made from the remaining larger prime numbers.
Sic 5 maximus inter tres virium Thus 5, the largest prime among the three characteres eievatur ad primam characteristics of the forces, is elevated to dignitatem, quippe quae jam nullam the first digit [1], which now, of course, elevationem, quam ex nulle majore surpasses no elevation, than which it holds numéro ante se habet, superet; 3 minor ad no larger number before it. The smaller 3 secundam 9 nam 3 minor S sed 9 major S is elevated to the second, 9, for 3 is denique 2 minimus ad [42] sextam 64 nam smaller than 5, but 9 is larger than S. 32 minor S 9 sed 64 major S 9 & haec Finally, the smallest 2, is elevated to the est basis tonorum musicorum 1“ 2^ 3^ 5* sixth, 64, for 32 is smaller than S x 9 but 7° in qua septenarius ille Divinus occulta 64 is larger than 5 x 9 , and this is the basis radix est. Atque ita etiam in ipsa Physica of the musical tones 1", 2®, 3 \ 5', 7° in effectus virium aestimantur per producta which is that divine, hidden, septenarium eorum omnium, quae per vires root. And thus even in physics itself, the praestantur. effects of the forces are judged through the products of all of those numbers that are exhibited through the powers.
§5 §5
Applicatur id ulterius ad rotam vitae. This [idea] is applied to the wheel of life.
Atque ex his elevationibus distinctius The distinct characteristics of the powers adhuc ipsarum virium characteres themselves can be perceived (as well as distinctivi possunt percipi, sin illae from these elevations, with more precision concipiantur per motum circularem se at this point), if those characteristics are exerere, ubi unitas ipsa centrum circuli thought to exert themselves through the constituât. circular motion where unity itself establishes the center of the circle.
Sic enim 5 repraesentabit elementum seu For thus, 5 will represent the element or differentiale vis tangentialis seu difference of the tangential or centrifugal centrifugae, temarius in 9 elevatus force, the ternary, having been elevated duarum illius deflexionum: Sive 3 & 5 into 9, will represent the element of the conficient vires contrarias 2 vero two modifications of that power; or, if 3 decussatoriam fulminantem ex reditu and 5 will complete the contrary powers, altemo ab excentricitate versus centrum in however, 2 will complete the intersecting sphaeris 3 & S. fulmination from a second return from
87 outside-the-center towards the center in the spheres 3 and S.
In omnibus vitae speciebus per hunc In every appearance of life through the reditum hujus excentricae vis sit return of this eccentric power, let there be intersectio virium 3 & 5 & dein separatio an intersection and then a separation of s. praecipitatio crassiorum atmorum molis the powers 3 and S. In other words, the & elevatio atomorum puriorum, hoc est, falling of the denser atoms of the mass reductio virium vitalium, quas Pythagoras and the elevation of the freer atoms, is fontem aetemae virtutis sub quatemarii this: the reduction of the vital powers, titulo, nam 4 est ilia potentia numeri 2, which powers Pythagoras calls the quae ceteros duos in statu, quo émanant fountain of eternal virtue under the title ex unitate, combinat, 3,4, S. appellat, in “quaternary”—for 4 is that power of the rotam vitae insimul quoque coadunatio number 2, which combines the two others atomorum molis cum viribus vitalibus. in position, by which 3,4, and 5 emanate from u n ity .L e t there be, simultaneously, a coming together of the atoms of the mass with the vital forces into life’s wheel.
Posset aliquis heic opponere: Si 3 & S One might be able to object here: if 3 and sunt vires illae contariae in vitae rota tum 5 are those contrary forces in the wheel of elevata 3 etiam altera deberet elevari life, then 3, having been elevated, ought to statumque mutare, quod in numeris nequit be elevated by another and change status, fieri, qui elevationem in 25 non admittit. because it is not able to come about in Rx. Rota [43] ilia vitae debet esse limitata numbers, which does not admit elevation pro capacitate cujusvis creaturae; sed into 25. Answer: That wheel of life ought limitatio per mutabiles vires, quae mox to be limited by the capacity of any given cum primo, mox in secundo elevationis creature. But the limitation through the gradu currerent, non esset satis certa & changeable forces [2 and 3]—which eievatur quidem numerus 5, per forces would soon run with the first, then multiplicationem ex numéro 3 in 15 & 45, into the second step of the elevation, sed non sua propria vi, quae ab occulta would not be suffîciently fixed, and the radice coelesti 7®, juxta reg. §. 7. number 5 is certainly elevated, through analyseos, coercetur, in prima saltem multiplication from the number 3 into 15 potentia. and 45, but not by its own power, which by the hidden, heavenly root 7®, like the rule of analysis in §7, is restrained, even in the first power.
" The number four acts a “gap-Bll” in the series. Compare this with Part I, section 26, paragraphs 4, S.
88 Itaque vis ignea 3 illam constantem & And thus the fîery power 3 attacks that placidam 5 oppugnat, ut nunc sursum, constant and placid 5, so that now nunc deorsum earn incitet ad devia upwards now downwards it urges it cupienda, & tamen ultra limites non nimis towards contrary desires and nevertheless excurrat. does not run too far outside the limits.
Excentrica vero 2, omnino fulminatoria Two, however, must be conceived as quippe summa pemicitate excurrens & lightening-like, racing out with utmost rediens, si non uno momento, certe tamen speed, if not in a single moment, then tempore aequali revolutioni potentiae certainly in time to the revolution of the simplici S‘ ab una usque ad sextam simple power 5*. Two will be adopted— dignitatem evecta & depressa concipienda raised and lowered—from one all the way est, id vero non nisi per instantaneam to the sixth digit. However it is not able ipsius puncti intimi s. centri alias to be understood unless through the immobilis usque ad peripheriam & ultra instantiation [instant?] of the inmost point extra positionem & recursum itself, that is, the other immovables of the comprehendi potest. Certe ex hujus vis 2* center up to the periphery both beyond the usque ad 2^ h.e. ex dignitatibus sex outside position and beyond the return. apparet, quomodo possit dici decussatoria Certainly, of this force 2‘ up to 2®, that is, inter duas reliquas. from six digits, it is apparent how it [two] can be called the intersection between the other two powers.'® §6 §6
Tacta superius §. 3. generallor virium A more general disposition of the animae per Musicam affectio specialius mind’s powers (mentioned above in §3) explicatur. is, through music, explained more specially.
Haec jam omnia, quomodo sentit animae How does the sensitive power of the sensitiva vis, naephaesch, quomodo mind, the naephaesch,^ sense all these imaginatur conceptiva vis, Ruach dicta? things? How does the conceiving force, ad prius respondeo, in Musica unitates called the ruach,^^ imagine them? I separatae innumerae continuo efficiuntur respond to the first: in music, tonis, quarum semper, quae se invicem innumerable, separate unities are subsequuntur, per tempus aliquod, sunt continuously produced in the tones. plane similes, quam- [44] vis, quae se These that adways follow one another in comitentur, possint discrepare. Ita tum, through some time, are clearly oriuntur conceptus relationum numerorum similar, while those that travel in tandem proportionum harmoniae. Concipe tibi may differ. Thus arise conceptions of the per lineolas longiores, aeris vibrationes proportional relations of the numbers of
That is, because two surpasses them. This is the Hebrew word for “soul.” This is the Hebrew word for "spirit.”
89 lentas, per lineolas succedentes magis the harmony. Conceive this for yourself curtas, vibrationes breviores, per lineolas by means of longer lines, slow vibrations adhuc minutiores, iterum breviores of the air, by means of succeeding lines vibrationes, tandem per minutissimas, much shorter, briefer vibrations, by means vibrationes brevissimas. of still smaller lines, again briefer vibrations, Anally, by means of very short lines, very brief vibrations.
Quem ad modum igitur lineolae duorum Therefore, how the separate lines of the sonorum separatae (nam numerus est two sounds (for the number is a discrete quantitas discreta) non nunquam eundam amount) sometimes recognize the same terminum & principii & finis, licet, boundary of both the beginning and end of magnitudine discrepent, agnoscunt, ita the lines, even though they differ by size, saepe repetendo eundem terminum orietur thus, frequently by returning the same relatio lineolarum idem spatium boundary, a relation of the lines occupantium e.g. occupying the same space will arise, for example.
A4 B4
3 . 5 .
Figure 32; Graphic representation of the ratios on a string.
seu duorum sonorum, quae est tonus or a relation of the two sounds will arise, Musicus s. intervallum heic numeris 3 & 4 which is a musical tone, namely, the exprimendum, quod quidem non primo interval to be expressed here by the loco Lit. A. cognoscitur, sed post iteratas numbers 3 and 4, which certainly is not sectionum conjunctiones neque ex eo, recognized by the first position letter A, quod soni alicujus toni non ita praecise but after the repeated conjunctions of the eodem momento in aërem edantur, aliud sections it is recognized, and not from the sequitur, quam sectiones dari & verticales fact that the sounds of any tone are not I I &obliquas v quascunque sed sibi precisely at the same moment emitted into parallelas. Nam in his rebus omnibus the air, rather one follows the other, with sensus ad naturam tenestrem pertinentes respect to which, divisions are given both magis limitati sunt, quam imaginatio ex vertical I I and oblique, v which ever coelestibus participans, ut momentum non ones, but parallel to themselves. For in all satis obsevare possimus. these things, the senses pertaining to the
90 earthly nature are more limited, than the imagination partaking of the heavens, so that we are not able to observe the motion well enough.
§7 § 7
Conclusio. Conclusion.
Cum igitur omnes in Musica oc- [45] Therefore, all tones occurring in music are currentes Toni numeris exprimantur, non expressed in numbers, entirely from the nisi ex elementis 2,3,5, productis nec elements 2,3,5, nor beyond the products ultra dignitatum, quas ilia habent, limites of these digits, which digits they hold, sc. 2^, 3 \ 5% ductis: effigies quam inde namely, 2^, 3 \ 5'. The image which the lecipit Ruach s. vis conceptiva, abstrahens Ruach, obviously the conceiving force, quodam modo a multiplicitate ipsa, in qua receives, abstracting in a certain manner, numeri primi in sensationibus rq by multiplication itself, in which the naephaesch s. sensitivae apparent, quae prime numbers appear in the sensations (effigies) forte commercium inter Ruach the Naephaesch, of course, of the & Naephaesch sustentât & laetitiam ex sensitivity, which (image), perhaps, Musica oriundam producit, est ilia ipsa sustains the exchange between the Ruach basis Musica I” 2*3^5' 7° ubi vires per and Naephaesch and produces the joy indices dignitatum in sua vivacitate, qua arising from music. It is that very musical vitam regunt, repraesentantur. basis, r , 2®, 3 \ 5% 7® where the forces are represented through the indices of the digits in their natural vigor by which they guide life.
§8 §8
Ad facUiorem superiorum conceptum For an easier conception of the Electricitas & Neutoni optica In preceding [matters], electricity and subsidium vocatur. Newton’s Optics are summoned In support
Haec omnia nimis abstracta habebuntur, All these things will be considered too quia non omnia, quae Natura supponit, abstract, because not everything, which substrata sunt huic Analysi. Age igitur, nature supports, is put at one’s service in concipe tibi ex Philosophia Electrica & ex this analysis. Therefore, conceive for Neutoni demonstratis hypothesin yourself from the Electric Philosophy and quandam palpabilem, quae mihi videtur in from the demonstrations of Newton a omni vita supponi debere ante numéros & certain palpable hypothesis, which seems ante conceptus harmoniae. to me in all life o u ^ t to be supported in view of the numbers and the conception of harmony.
91 Constat inter Physicos materiam It is agreed among physicists that electricitatis habendam pro atmosphaera electricity’s material must be considered corporum, plane distincta a partibus as an atmosphere of bodies clearly distinct coporum solidis, si vorticem appellare from the soUd parts of the bodies (if you mavis, non renuo. Jam demonstravit prefer to call it the vortex, I do not object). Neutonus, causam, cur lux a coporibus Now, Newton demonstrated, that the repercutitur, non esse, quod lux ad solida reason why light reflects from bodies is allidat corpora. not because light dashes against a solid body.
Describit expérimenta in Libr. H. opticae He describes the experiment in Book II of p. 224. Inde concludit, allabentes radios Optics, p. 224. Thence, he concludes that fortasse per vim quandam repelli, quae se falling rays are repelled, perhaps by a aequabiliter effundat per corpus ad certain power, that equally sends itself out superficiem, qua fiat, ut corpus agat in through the body to the surface, so that it radios, sine radiorum immediate attactu comes about, that the body acts on the [46] in corpus ipsum. rays, without immediate action of the rays on the body itself.
Porro dicit Libr. lU. per modum Furthermore, he states in Book m '^ in the quaestionis, an non corpora per spatium form of a question, whether or not bodies interlabens in lucem agant & per hoc act on light, through the interlying space, radios inflectant; porro, annon radii, ad and through this bend the rays? corpus directi, priusqam illud attingant, Furthermore, he asks whether or not the jam jam incipiant curvari; in quaestione rays, directed at a body, begin to be tertia dicit, corpus diaphanum agere per curved little by little, before they arrive spatium inteijectum in radios ad there? In the third question he states that inflexionem, refractionem & reflexionem the body acts invisibly, through the & radios, movere, & calefacere per interlying space, on the rays moving them spatium interlabens partes corporis & to inflection, both refraction and hanc actionem & reactionem per tale reflection, heating parts of the body spatium, videri analogam virtuti through the interlying space, and this attractivae. action and reaction through such space seems proportional to the strength of the attraction.
Newton, p. 339ff. Also sec Dennis Sepper, Newton’s Optical Writings: A Guided Study. New Brunswick: Rutgers University Press, 1994.
92 §9 §9
Novae quaestiones. New questions.
Liceat jam mihi quaerere, annon in Now it is permitted for me to inquire Musicae & coicrum & saponun & whether, in the perception of music, odorum perceptione fiat sensus mediante colors, tastes, and odors, it happens that tali atmosphaera, cum qua vis quaedam the sense is connected mediately by such subtilior animae spirituum transmissiva, an atmosphere,'^ with which a certain imagines recipiens & in sphaeram subtler power transmits the spirit to the activitatis sensuum trahens, sit connexa? mind, receiving and drawing images into the sphere of activity of the senses?
Si corpus habet vorticem electricum, cum If the body has an electrical vortex, with quo sese lux & sonus commiscet, which light and sound mix themselves absurdum ne erit animae tribuere together, will it not be absurd to assign sphaeram talem activitatis per quam such a sphere of activity to the mind, Baconis transmissio virtutum through which Bacon’s transmission of immateriatarum & receptio earundem ab immaterial powers, and the reception of aliis accidat? Nonne haec etiam ad them, occurs apart from the others? Are originem pulsus s. systoles & diastoles not even these able to come about for the cordis explicandam facere possent? purpose of explaining the origin of the pulse, that is, the systolic and diastolic pulses of the heart?
Pulsus sanguinis altemans causam habere The alternating pulse of the blood ought to debet in simili altematione positam. Pone have a cause set in a similar alternation. vitam animae per circulum s. vorticem Suppose that the life of the mind through igneum spiritualem numéro 3 the circle (the fiery, spiritual vortex respondentem & per alium vorticem corresponding to the number 3 and gyratorium placidioris indolis numéro S through another gyrating vortex of a more respondentem rotari pone potentias ex gentle innate character, corresponding to centro per vorticem exeun- [47] tes & the number S) is rotated behind the redeuntes, annon interfectio vorticis per powers exiting and returning from the excentricam 2 exeuntem & redeuntem center through the vortex (whether or not altemationem effîciet, altemationis pulsus the destruction of the vortex through two effectricem? exiting and returning affects the alternation)—is the pulse of the alternation effected?
"A n electric atmosphere (see section 8).
93 §10 §10
De sensuum reliquorum analogia & de Concerning the resemblance of the sensus Musici cum Morali remaining senses and the comparison of comparatione. the musical sense with the moral sense.
Haec licet nondum sint omnibus ad Although these matters are not yet certitudinem requisitis instructa, tamen ad established in all we have sought to a Ezechielis rotas apprime quadrant illasque point of certainty, nevertheless according illustrant. Quoties a Phylosophis to Ezekiel’s wheels they especially hypotheses concipiuntur, donec per ea, complete and illuminate those quae inde sequuntur & cum aliis requirements. How often are hypotheses veritatibus concordant, verifîcetur ipsa adopted by philosophers, until through hypothesis? Sine hoc artificio ars those things that follow from them and inveniendi nulla esset. niustr. Dominus agree with other truths, they are verified de Maupertuis talia supra limites by the same hypothesis? Without this intellectus nostri autumat, sed tamen method, there would be no art of licebit illius more, etiam ex Musicis aeque discovery. The illustrious Lord ac ex Linguis conjectare ad Naturam Maupertuis, asserts that such things animae, ad sensum communem, imo ad exceed the limits of our intellect, but singulas sensationes. nevertheless will be allowed in the manner of that man to conjecture from music (and also from languages) on the nature of the mind, on the sensus communis, indeed, on the individual sensations.
Colores Dominus Pater Castell duodecim, Lord Father Castel^® discovered not only non tantum septem more Neutoni, invenit; the 7 colors in the manner of Newton, but sed colores fortasse ex tribus primitivis twelve colors. However, perhaps colors viribus lucis a Deo sunt compositi, ut sint are composed by God from the three primo tres vires, dein septem, tandem primary light powers, as are the three duodecim, ad typum tonorum Musicorum. prime forces, then seven, finally twelve, Hinc ipse Castell ante Krûgerum according to the type of musical tones. construxit clavichordium colorum. Sic From here, Castell himself, before gustus & odoratus eandem fortasse Krugerum, constructed a clavichord of compositionem ex viribus primitivis in se colors. Thus taste and odor, perhaps, have habet. in themselves the same composition from the three prime forces.
^ Louis-Bertrand Castel (1688-1757). A Jesuit priest, philosopher, and mathemtician who built a “color” clavichord in 1734, which produced various light colors (by means of colored paper over candles) when the different keys were pressed.
94 Austerum, amaram & dulce sunt fortasse Sharp, bitter, and sweet are, perhaps, the primitiva gustus, unde reliqi sapotes sunt three primary tastes, from whence the compositi. Anima igitur, ut habet remaining tastes are composed. Musicae basin, sic & colorum, odorum & Therefore, the mind itself, as it holds the gustus basin in se ipsa, non tantum in basis of music, even thus holds the basis sensoriis habet. of colors, of odors, and the basis of taste, in itself, not so much in the senses.
In sensu quodam [48] communi Physico The mind, perhaps similarly, holds these habet haec anima fortasse simul, quem ad things in a certain, common, physical modum lux colores habet in se, sed sense, as light holds the colors in itself^', sensoriis opus ad peculiarem but in the senses the task is according to a modificationem. Datur sensus communis peculiar modification. The common, Physicus, per quem Deus providit physical sense is given, through which hominibus, ut sine longa Analysi judicare God provides for men, that without a possint de eo, quod humanae convenit tedious analysis they are able to judge naturae. Multo magis stautendum est, according to it, because it agrees with Deum providisse homini per sensum human nature. Still more it must be communem Moralem, unde, si established, that God provides man with a rectitudinem cordis sequitur homo, nullis common moral sense, from which, if a aliunde regulis opus habet, nisi ut bene man follows the rightness of the heart, he utatur occasionibus oblatis. has no need of other means from other sources, unless to make good use of opportunities offered.
Sequatur homo, quo ducit instinctus Let a man follow where the instinct of pudoris, timoris, admirationis, tum shame leads, and of fear, of wonder. Then timorem & praesentiam Dei persentiscere he strives to sense the fear and presence of nitatur; & tum omnes instinctus boni per God, and then all good instincts will be exercitium regularem sapientiae, omnibus elevated (by degrees) through the regular obviae, in gradus elevabuntur. Experietur exercise of wisdom, obvious to all. A homo tales instinctus a Deo ita evehi, ut man will prove such instincts to be thus superioritatem illorum, lege Dei vivifîca lifted up by God, as superior of those, fultam, distinguere possit a se ipso; & tum supported by a law; having been revived sensoria secmdum Ebr. S, 14. exercitata of God, he is able to distinguish by the habebit ad boni & mali discietionem. very thing itself. And then the sense,
cf. Newton ^ Hebrews 5:14 “But strong meat belongeth to them that are of full age, even those who by reason of use have their senses exercised to discern both good and evil.”
95 according to Hebrews 5:14, will have exercise to discern good and evil/^
96 [FRICKER: SECTION DI]
[49] Pars secunda theoriae musicae Fart Two of the THEORY of MUSIC: physica seu psychologica, sed saltim Physical or Aychological, hut in any conjecturaria. case, Conjectural.
§1 §1
Occasio secundae hujus partis Theoriae The reason for the second part of the Musicae. Theory of Music.
Cum ego de ipso animae & corporis nexu Although I had never considered the ac mutuo influxu in negatio Musicae connection of the mind and the body and nunquam cogitassem, nec de peculiari their mutual flow in the subject of music, utriusque statu ac dispositione sollicitas and although I had not concerned myself essem, nunc sane Analysin hanc Theoriae about the unique condition and Musicae fini turns eram, nisi & Auctoris arrangement of the two, now certainly, I hujus de Sensu communi scripti consilio would have been about to fînish this adductus & ejusdem conjectura analysis of music, had I not been psychologica §.31. adjutus, etiam persuaded by reason of what this author paululum physica seu realia animae has written concerning the sensus Musicae hactenus descriptae vestigia communis, and been compelled by the scrutari, atque in iis opinionem meam & psychological theory of this same author imaginationem problematis ad instar in § 31.^*" Now too, I ought to examine proferre deberem. Quod igitur deinceps somewhat the physical or actual vestiges trado, id, licet brevitatis causa thetice & of the description, thus far, of the mind of positive atque absolute dicatur, plane non music, and, in regard to these problems, I ita intelligendum esse volo, sed omnino ought to express my opinion and omnium judicio relinquo. imagination according to the standard of the issues presented. Therefore, that which I am passing on next, this theory, although it is being stated for the sake of brevity with a heavy-handed, positive, as well as absolute manner, clearly, I do not wish to be understood in this way, but I leave it entirely for the judgment of all.
' Flicker is referring to a passage in the Inquisido where Oetinger speculates about the connection between the mind and the body and the role of the musical numbers (ratios) in perception.
97 §2 §2
Anima habet principium fixiim seu The mind regards the fixed origin or centrum Ideanim; res extemae non center of ideas; this is not so with Item. external matters.
In omni rerum natura, extema certe & In the whole nature of matters, certainly corporea, locum obtinet fundament!, ex external and bodily, the order of the que discrimina fieri possint, naturalis natural numbers 1,2 ,3 ,4 ,5 ,6 ,7 ,8 , etc., numerorum ordo 1,2 ,3 ,4 ,5 ,6 ,7 , 8, &c. occupies the place of foundation, from quem ad primum Arithmeticae genus, which distinctions can be made. I ascribe rationes spectans arithmeticas, h.e. meras this order to the first type of arithmetic, differentias quantitatum, uti secundum considering them arithmetical ratios, that rationes veras & geometricas, refero. is, simple differences of quantities, just as I refer to the second type, considering them actual and geometrical ratios.
Saltem in hoc differt anima a natura Even in this, the mind differs from the terrestri, quod in ilia unitas non ubique earthly nature, because in the earthly simul possit figi, sed per [SO] tempus nature, unity cannot be established aliquod centrum intelligentiae seu everywhere simultaneously, but through conscientiae heic praecise nec alibi time some center of intelligence or deprehendatur, ad quod ceterae ideae knowledge is discovered in this place referuntur; quapropter Philosophi animae exactly and not elsewhere, to which other simplicitatem seu unitatem adeo urgent ideas are compared. Wherefore contra materialistas, & recte; quamvis philosophers of the mind urge simplicity etiam centrum illud leniter videater or unity against the materialists, and mover! in general! totius vitae rotatione. rightly so. However, even that center seems to move mildly, in the general rotation of all of life.
Cum enim conscientia summam For, since consciousness requires, necessario animae simplicitatem requirat, necessarily, the greatest simplicity of non poterit idea ad conscientiam pertinens mind, an idea pertaining to consciousness ad ceteras ei vel propinquas vel will not be able to retain itself with remotiores se habere ut x ad y + v + z; sed respect to other ideas, either near ones to ut a ad X + y + z, imo potius ut 1 ad a + b it or more remote ones to it, as X = + c, h.e. ut unitas ipsa ad ceteros numéros, Y+V+Z; but as A = X+Y+Z, on the sic ad reliquas referri debebit ideas, & contrary, preferably as 1 = A+B+C, that is illarum esse centrum. as unity itself is in relation to the other numbers. Thus, it ought to be compared to the remaining ideas, and of those to be the center.^
^ The mind (sensus communis) acts as the repository of ideas. Consciousness arises not as simple, direct,
98 §3 §3
Cerebrum sensuale quid? How Is the brain endowed with sensation?
Intelligitur inde facile, quomodo per And so, it is easily understood, how a diversorum sonorum vibrationes, certain material representation of high and celeritate h.e. numéro ad idem tempus low tones arises in the brain. It is through relato diversas, & cerebro per extemam the vibrations of different sounds— materiae subtilioris vim repetitis vicibus, quickly, that is: by a numerical relation of cum unus sonus innumeris quasi the diverse vibrations calculated at the vibrationibus constet, impressas, oriatur in same time; by the brain, through the cerebro aliqua Tonorum acutiorum & external power of a more delicate matter graviorum materialis repraesentatio. by repeatedly changing impressions (since one sound corresponds as if with innumerable vibrations).
Cum vero diversitas, ut §. antec. dictum, When however, as mentioned in the omnis in extema natura aestimetur ex preceeding section, everything in external differentiis seu numeris naturaii ordine nature is judged from the differences or progredientibus, imago ex duorum numbers progressing in their natural diversorum sonomm impressione facta order, the image, having been made from non potest non esse repraesentatio the impression of two different sounds, distantiae seu intervalli alicujus. Hinc cannot help but be a representation of perceptionem intemam numerorum & some distance or interval. Hence, I assert intuitivam animae immaterialis comitatur that with respect to the internal perception in subtilissimo materiaii dimensio of numbers and the intuition of the graduum seu intervallorum ope cujusdam immaterial mind, the measurement is reguiae; atque hanc animae vim metiendi accompanied in accordance with the most merasque differentias inter se conferendi delicate matter of steps or intervals by the in cerebro sensuali peragi dico. power of a certain rule. And I assert that the mind’s power of measuring and comparing the simple differences between themselves is completed in the brain-sense.
obvious relationships, such as X = Y+V+Z, which is too specific, but as more remote relationships, such as A = X+Y+Z, which stands for a more abstract representation. Unity, or 1 = A+B+C, represents the most abstract relationship. This enables one to conceive of multiple ideas that are not dependent on a specific representation.
99 §4 §4
[51] De intervallis Musicis veterum Concerning the musical intervals, praecipue. especially those of the ancients.
Quid ergo mirum, si ad hunc fere diem What, therefore, is the remarkable thing, usque Musici semper harmoniam derivare if, in general, up to this day musicians volerunt ex doctrina intervallorum? continuously and always wished to derive Videlicet ex vero Musicae sensu harmony from the doctrine of intervals? subintellexerunt, animam Tonos non Clearly they realized from the true sense cumulatim atque acervatim, sed sigillatim of music, that the mind does not perceive percipere, singulos cum singulis conferre, tones heaped-up and without order, but dijudicare, id quod & in priori parte satis impressionistically, comparing and ostensum, ubi vero & hoc simul vidimus, judging each one with each one rationes Tonorum esse geometricas, non successively (that which was shown arithmeticas; sufficiently in the fîrst part and where, however, at the same time we saw this— that the ratios of tones are geometrical, not arithmetical).^ sed decepti deinde ab idea Toni tanquam intervalli, non raüonis Geometricae, But thereupon, having been deceived by duorum sonorum, in cerebro sensuali the idea of a tone as an interval of two materi aliter expressa, fundamentum sounds, and not their geometrical ratio harmoniae ipsius semper in intervallis seu expressed in a material in the sensate distantiis quaesiverunt, id quod unicum brain, they always sought the foundation fere, at insigne fuit impedimentum of harmony itself in the intervals or scientiae harmonicae apud veteres certe. distances, that which was nearly unique. Ex. gr. légat aliquis Baconis Hist. Nat. But, it was a well-known impediment for Cent. n. sine praejudicio suae opinionis, the science of harmony among the & deprehendet, quam necessario etiam ancients. For example, let anyone read vera Tonorum ex sensu dijudicatio ideam Bacon’s Natural History without distantiae secum ferat. prejudice of opinion, and he will discover how by necessity the true judgment of the tones from sense perception brings with it the idea of distance.
^ His use of the terms geometrical and arithmetical, in this context, refer to the ratios being calculated by multiplication and division, as opposed to addition and subtraction, respectively.
100 §5 §5
De intervallo k u t E%OKnv, sc. de gradu Concerning the most eminent ancient^ seu Tono ipso. interval, namely, the step or tone itself.
Ex illis ergo intervallis seu distantiis varii Therefore, from those intervals or generis infmitis una est praecipua. quam distances of various infinite types, one is secundum excellentiam Tonum, hodie particular, which is the second excellent Secundam. appellant, ratione 8:9 = 2^:3" tone, today they call it the Second. It exprimendam; nam praecipue major should be expressed by the ratio 8:9 = intelligitur Tonus, non minor 9:10. Cujus 2^:3". for. actually, the major tone is ideam adeo naturalem esse animae probat understood, not the minor 9:10. experientia. & idem libellas citatus. ut ei Experience shows that the idea of the tone quasi connata videatur. & a Deo is so natural to the mind (and the same concreata. tanquam [52] mensura aliqua book is cited) that the idea seems, as it ipsi animae proportionatissima. qua were, bom with it and created with it by dimetiri ac dijudicare possit omnia cetera God just as if some measure were most intervalla. Inde factum, ut veteres proportionate to the mind, by which it is suavitatem Octavaetribuerent septenario. able to measure and judge all other quippe post quem ilia recurrat. intervals. Thereupon it happened that, the ancients attributed the agreeableness of the octave to the septenario. and after that, of course, it repeats.
§6 §6
Anima in Tono tertio Arithmeticae With respect to the tone, the mind uses genere, scil. elevatione ad potentias the third type of arithmetic, that is, utitur. raising to powers.
Cum vero anima musica in numeris. quos However, since ( I ) the musical mind in basis Musica 2*. 3". 5* admittit. in the matter of numbers (which numbers the simultaneis Tonis primo & secundo musical foundation admits as 2^. 3‘. S'). in Arithmeticae genere in successivis cum accordance with simultaneous tones, uses maxima activitate utatur. id quod in priori the first and second type of arithmetic, in parte Analyseos satis demonstratum est; successive numbers with the greatest cum ea porro potentias binarii majores activity (that is what was sufficiently quam citissime intelligat. & eadem opera demonstrated in the first part of the numerum 64 velut ex 2 oriundum. qua 5. analysis); and liirthermore. since (2) the quod certe non potest aliter fieri in ideis mind understands the larger of two symbolicis nisi substitutione signi alicujus powers as quickly as possible, and ex. gr. 2^ in locum signi 2 2 2 2 2 2 likewise by the procedure it understands h.e. usu tertii Arithmeticae generis: the number 64 as arising from 2,
* Greek "ancient.” by implication, “prominent" or "revered.”
101 [multiplied] S more times, which certainly cannot otherwise to come about in the symbolic ideas unless by the substitution of some sign (for example ' t in place of the sign 2 2 2 2 2 2, that is, by use of the third type of arithmetic); cum denique ipse Tonus 8:9 = 2^:3^ and finally, since (3) this tone is most animae sit familiarissimus & quasi familiar to the mind—8:9 = 2^:3^—is so notissimus: maximi momenti conjectura familiar and, so to speak, most well- est, animam etiam tertio Arithmeticae known: the inference is of the greatest genere sc. elevatione ad potentias, importance, that is, the mind also uses the quantum basis Musica id quidem third type of arithmetic (that is, by the concedit, uti, Tonumque non ex numeris 8 elevation to powers), certainly as much as & 9 qua talibus, nec qua productis 2 2 2 the musical basis admits it, and it & 3 3, sed qua potentiis 2^ & 3^ estimates the tone not from such numbers aestimare. as 8 and 9, nor by the products of 2 2 2 and 3 3, but as powers 2^:3^.
§7 §7
& Ita quidem, ut Tonus vere possit did and It Is Indeed such that the tone can Intervallorum mensura, truly he called the measure of the Intervals,
Ex hac certe hypothesi traditam Toni Certainly, from this hypothesis we find ideam invenimus; nam cum mensurae idea the traditional idea of the tone; but now, involvat ex una parte diversitatem, ex since the idea of the measure involves a altera homogeneitatem: unitas non satis difference from one part, a similarity from erit apta ad [S3] construendam mensuram another [part]: unity will not be suitable ex tertio Arithmeticae genere; potius talis enough for construing the measure which requiritur numerus, qui certe in secundo must be arranged from the third type of genere mutationem possit inducere. arithmetic; rather such a number is required, which certainly in the second type is able to induce a mutation.
Atqui la = a, 1* = 1^ = 1^ = i“, sic nec ex Notwithstanding, la = a , l ‘ = 1^ = 1^ = numeris 2% 3 \ satis cognoscitur usus tertii 1", thus, not even from the numbers 2% 3* generis, quia 2* = 2 1,3* =3 1; contra is the use of the third type sufficiently vero ratio 2^:3^ & diversitatem satis recognized, because 2' = 2 1,3' = 3 1; indicat veris numerorum elevationibus, & on the other hand, however, the ratio 2^:3^ homogeneitatem mutuo & potentiarum in indicates well enough both the difference exponendbus & radicum in numeris from the true elevations of numbers, and eorum consensu.
102 the similarity by a mutual agreement of both powers in exponents and of roots in their numbers—
fcil. a r c
Figure 33: Inversion of bases and exponents in the ratio 8:9, expressed as 2^: 3^.
Cum vero ultra 3 in Musica non detur However, since in music a power is not potentia praeter illas binarii, quae merae granted beyond 3^ beyond those of sunt relationes ad 2 \ non tales, quae novi binariam (which are pure relations to 2^, quid inferre possint, & S', quae non est not such ones, which are able to infer vera elevatio: patet in tertio genere something new), and 5% which is not a rationem 2^:3 esse veri nominis unicam, true elevation; it is obvious in the third atque adeo, propter excellentiam & suam type that the ratio 2^:3^ is truly called & generis sui, mensuram animae a Deo unique, and exceedingly so according to collatam. its own excellence and that of its type, the measure of the mind brought together by God.
& ut tanta in eo sit animae activitas, and the activity of the mind is so great quanta ad conununicationem cum as is required for conununication with cerebro intellectuali requiritur. the reasoning function of the hrain.
Haec Toni idea est clara, non proxime This idea of the tone is clear, not only tantum, ut ceterorum intervallorum, de approximately as of the rest of the quibus deinceps, sum[p]ta sed praecise; intervals, from which tone they are taken est vividissima, &, ut verbo dicam, solida successively, but exactly; it is most lively, seu corporea; tanta enim in percipienda and, as I might say with a word, solid or ilia ratione 8:9 est vis atque activitas bodily. For so great is the power and animae, ut ea ad peripheriam usque suam activity of the mind in perceiving that & in ipsum sensuale cerebrum pénétrât, ratio 8:9, that it pentrates all the way to its [54] quapropter & homo inter omnes hanc periphery and into its brain-sense, and facillime ex fundo suo profert rationem, & wherefore a man easily produces this 103 quemadmodum unumquodque ulterius ratio, among all [the ratios], from its Arithmeticae genus inrinities alterum se authority, and how each and every higher inferius comprehendit. type of arithmetic comprehends another lower [type] itself to infinity. ita perultimum hoc Arithmeticae genus, Thus, this second to last type of arithmetic cujus fere nondum est capax anima, (of which [type] the mind is almost unicam saltim in illo rationem 2^:3^ incapable), at any rate, recognizing in that agnoscens, sed omni tamen nisu in illud unique ratio 2^:3 , but nevertheless, quippe in sui ipsius elevationem tendens, extending with all effort into that type ab inferioribus ordine generibus pergendo (naturally, into the elevation of its very omnem suam vim ac virtutem in his self), by proceeding from the inferior paululum dissipatam in eadem ilia ratione types (in order), it collects all its own colligit, atque ita exserit, ut per eam, quae force and strength scattered in these velut paries est inter animam & corpus, [types] in the very same ratio, and thus it quasi actuet & moveat corpus suum stretches out, so that through it [the ratio], simplicissima. which is as a wall between the mind and the body, [this] simplest ratio activates and moves, as it were, its body.
§9 §9
distinctio ideanim in lineares, The division of ideas into linear, superflciales & corporeas. surface, and corporeal.
Nimirum ideae, quae dantur in anima, Evidently ideas, which are furnished in habent suas inter se rationes, atque ex the mind, have their ratios between harum methodo seu natura aestimatae sunt themselves, and from the method or vel lineares, vel superflciales, vel solidae nature of these ideas, they are judged [as] seu corporeae: In istis rationes ex primo either linear or surface or solid (or bodily) Arithmeticae genere, Additionis & [ideas]. In those [linear ideas], the ratios Subtractionis, in illis ex secundo are selected from the first type of Multiplcationis & Divisionis, in his ex arithmetic, addition and subtraction; in tertio elevationis ad potentias & radicum those [surface ideas], [the ratios are extractionis desumuntur. selected] fiom the second [type], multiplication and division; in these [solid ideas], [the ratios are selected] from the third [type], elevations to powers and extractions of roots.
Pro primo genere suppeditat basis nostra The musical basis supplies the ratios for Musica rationes: 1:2,2:3,3:4,4:5,5:6, the first type: 1:2,2:3,3:4,4:5,5:6,8:9, 8:9,9:10,15:16, quia 1:2 = 1:1+1,2:3 = 9:10,15:16, because 1:2 = 1:1+1,2:3 = 2:2+1 &c. pro secundo in unaquaque 2:2+1, etc. For the second [type] [the Octava illos 15 tonos; pro tertio vero, uti musical basis supplies] those 15 tones in
104 jam dictum, non nisi 2^:3\ quamvis each and every octave. However, for the potentiae 1”, & l \ 2 \ 2 \ &c. 3% 5', 3 5 third type, as already stated, not except by &c. medium quid inter tertium & 2^:3^, although the powers 1", and 2% 2 \ secundum inférant genus, & ideam hanc, 2^*, etc., and 3% 5‘, 3.5, etc., bring in a latere in mensura 8:9 totum aliquod middle type which is between the third novum genus. and second type, and this idea, hides in the measure 8:9, some entirely new type.
Ceterum uti superius & sequens quodque Thus, as each greater and successive type genus antecedens in se plane clearly includes the preceeding type comprehendit, ita & ideae superflciales within itself, and surface ideas include includunt lineares, & solidae tam lineares linear [ideas], and solid ideas include quam superflciales. linear ideas as much as [they include] surface ideas.
§10
de diversis intellectus & imaginatlonis Concerning the diverse operations of sensatae In cerebro operationlbus, the intellect and the sensate ordinem numerorum 1, 2,3,4, &c. imagination, following the order of the sequentibus. numbers 1, 2,3,4, etc..
Sedes materialis, cujus partium motus The material seat, of which the motion of actionem animae comitatur, in usu its parts accompanies the action of the facultatis primi generis Arithmeticae, mind—in the use of the skill of the first quam rationem voco, facultatem scil. type of arithmetic, which I call a ratio, the abstrahendi, putem enim abstractionem ab ability of course to abstract, for I think anima fieri, progrediendo saltim in ordine abstraction to come about by the mind, in numerorum naturaii 1 ,2 ,3 ,4 ,5 ,6 , &c. any event by proceeding in the natural sine dubio est cerebrum, seu locus, ubi order of the numbers 1 ,2 ,3 ,4 ,5 ,6 , etc.— maxima moles materiae subtilis without doubt it [the seat] is the brain, or reservatur, & motus fit ita, ut per tempus location, where the greatest bulk of aliquod una pars eius materiae minima pro delicate material is reserved. Thus it centro adsumatur, seu potius, ut in eo comes about that the motion, through unum punctum aliquod physicum sit some time, some single, smallest part of terminus, a quo motus & praecipua in its material is taken as the center, or linea directionis puncta fluant: rather, that in it one single, physical thing becomes a terminus, from which the motion and the particular points flow on a line of direction:
' See Sections 1,6,8.
105 1^ ; 4 f < 7 89*®.
Figure 34: Graphic representation of simple arithmetic progression conceived as points on a line.
quo pacto haec facuitas satis distinguitur a In this way this faculty is distinguished facultate tertii generis, quam adpellare well enough from the third type of non aliter novi, quam imaginationem arithmetic, which I knew not to call sensatam, quae per ductus organorum in otherwise than the imagination-feeling, cerebrum ordinarios impressiones recipit, which receives ordinary impressions easque aestimat pro distantia ab ostio through the duct of organs in the brain. singulorum ductuum organorum, ac pro And it judges according to the distance recti tudine vel obliquitate directionis, from the entrance of the individual organ quam impressio ex ductu per oftium in ducts, and according to the straightness or cerebrum sequitur, quamvis ceterum ea obliqueness of the direction, which facuitas pariter naturaii numerorum ordine [distance] the impression follows from the incedat. duct through the entrance into the brain, although otherwise [still], the faculty moves it equally by the natural order of numbers.
§11 §11
[56] de animo in corde residente & Concerning the mind* residing in the secundum Arithmeticae genus heart and following the second type of sequente. arithmetic.
Sed omninio sedes facultatis secundi But in general the seat of the faculty of the Arithmeticae generis, quam ipsum second type of arithmetic, which I would vocaverim animum, est cor, quod per call the mind* itself, is the heart and subtilissimum aliquem spiritum occupies unity[The seat] holds together (materialem an immaterialem? posterius very closely with the brain, and drives sane crediderim, ac virtutem seu vim into distances, the greatest, incredible aliquam divinam esse putaverim), quem distances of ratios through some very vita ejusque rota quippe continuo agitata subtle spirit—which life and its necessario supponit, cum cerebro continuously moving wheel necessarily arctissime cohaeret, & in distantias agit, supports. ([Is the spirit] material or rationumque summas incredibiles. immaterial? The latter, certainly, I would
’ Or "occupies one place.”
106 diversissimas, intellectui certe atque believe, in fact, I might suppose it to be iraaginationi sensatae ita visas, occulto some divine virtue or power.) The tamen vinculo multiplicationis scil. distances having been seen by the numerorum primorum cohaerentes, una intellect, certainly, and the imagination capit. sense thus, nevertheless, are joined by a hidden chain of multiplication, of course, of the prime numbers.
Atque ita intellectus circulum incipiens And thus the intellect begins the circle by ratiociniis suis, animus mediante cordis its ratios; the mind*, measuring the motu promovens sensibus seu potius motion of the heart, advances it by the sensis atque adfectibus suis, denique senses, or rather by perceptions as well as imaginatio sensata eundem fmiens ac their affections; finally, the imagination terminans symbolica sua cognitione, sense ends the circle by its symbolic idearum longitudinem, latitudinem & cognition. These three—the intellect, the profunditatem seu soliditatem constituunt. mind, and the imagination—establish the length, width, and depth (or solidity) of ideas.’
§12 §12
Ad imaginationem sensatam ejusque Human speech, stemming from the primum Arithmeticae genus a tertio third [type], is returned to the sensate productuum refertur loqueia hominis. inugination and its first type of arithmetic.
Sive potius dixeris, animum sensa Or rather, you might prefer to say, that the transferre sua immediate ad loquelam mind conveys its perceptions immediately eiusque organa, sc. guttur, palatum, to speech and its organs—that is, the linguam, dentes, labia, quibus animus throat, palate, tongue, teeth, lips—by maxime est praesens: tamen & hoc non which things the mind* is very present. poterit fieri nisi per tertium genus Nevertheless, even this [situation] is not Arithmeticae ejusque revolutionem ad able to happen unless through the third primum genus, novo modo sc. in materia type of arithmetic and its revolving back seu corpore in- [57] stituendum; atque to the first type, by a new manner, that is, idem ita Get, ut continuus influxus seu by being established in material or flesh. consonantia materiae, h.e. non ad Thus this very thing will happen, such that adcuratam matheseos, quam intellectus a continuous influx or consonanance of sequitur, praecisionem ejusque normam material obtains, that is, [a consonance] dijudicanda inter impressiones ab extra which must be judged—among the cerebro & ab intra organis loquelae factas impressions having been made from obtineat. outside the brain and from within the organs of speech—not according to a
^ That is, linear, surface, and solid ideas.
107 precision of reckoning, which the intellect follows, [but according to] a shortened [reckoning] and its standard.
§13 §13
De cohaerentia facultatum harum Concerning the connection of these imperscrutabili. inscrutable faculties.
Poterit ergo per anatomiam investigari, Therefore, it will be possible, through quomodo cohaeret cerebrum cum loquelae anatomy, to investigate, how the brain organis, atque ex traditis principiis & holds together with the organs of speech, special! ex.gr. tractationis Musicae vel and to discover that they agree in some usus linguarum, &c. adplicatione nonnihil measure, by application of traditional congru! in venir!: quo vero pacto organa principles and especially—for example— haec cum animo, animus cum intellectu [those ] of the treatment of music or of the seu ratione, ratio cum imptessionibus in use of languages, and so on. However, cerebrum ab extra factis jungantur, vix how these organs are joined with the quisquam investigaverit: mind*, the mind* with the intellect or reason, reason with impressions having been made in the brain from outside— scarcely anyone has invesitgated.*
Ex tradita vero hypothesi patet, nexum However, from the traditional hypothesis esse ilium, quem habent inter se genera it is well known, that the connection is Arithmeticae tria, & ita quidem, ut tertium that, which the three types of arithmetic genus per unicam suam rationem 2^:3^ have between themselves, and it is revolvatur iterum in primum, nam 9 -8 = certainly, such that the third type is 1; qui sane nexus Arithmeticus a priori returned—through its one ratio 2^:3^—a nequit investigari, & a Deo toti naturae second time into the first [type], for absoluta insitus videtur esse necessitate. instance 9-8=1; obviously this arithmetical connection cannot be discovered by the previous [type]. Furthermore, in the whole of nature, the complete [ratio] seems to be, by absolute necessity, innate by God.
' He is admitting, for a second time, to speculation.
108 §14 §14
Mutua actio & reactio facultatum The mutual action and reaction of these hanim in cycle. faculties in the cycle.
Quod vero §. 11. diximus, ab ratione However, what we stated in section 11, incipere ciiculum, id ita est intelligendum, namely, that the circle begins by ratio ut ponatur anima in hoc casu activitatem [reason], should be understood thus, that suam in corpus educere: contra quatenus the mind is supposed, in this event, to passive se habet ilia discitque, atque ab draw out its activity into the body. On the extra impressiones seu ideas recipit, other hand, in so far as the mind holds eatenus potius [58] ita currit vitae rota, ut itself, and leams, as well as accepts res extemae per organa sensuum in impressions or ideas from outside, rather, cerebro, per hoc in organis loquelae, per to this extent, thus the wheel of life runs; haec in corde, per hoc in cerebro iterum so that external matters [run] through the sed intellectuali vestigia relinquant sua, organs of sense into the brain, through this ubi omnino in posterioribus casibus atque into the organs of speech, through these etiam ipsa idearum receptione subinde [organs] into the heart, through this into magis activam se praebet anima suamque the brain a second time, but, [now] as an non rerum methodum sequitur. intellectual [organ]. They leave their vestiges behind, where entirely in later cases as well as even by a reception of ideas, suddenly the mind shows itself more active and follows its method, not [that] of the external.
§15 §15
De nimia philosophoram continua Concerning the excessive, continuous abstractione cyclum vitae turbante & abstraction of the phiiosopbers ab eodem turbata. confusing the cycle of life and likewise having been confused by it.
Neque vero negaverim, brevius posse Certainly, I would not have denied that a absolvi cyclum, & dare etiam semitam v. shorter cycle could complete, and provide gr. per cerebrum sensuale statim in a path through the brain sense intellectuale, vel tantum vix paululum immediately into the intellect.^ For intercedente animi actu; instance, [completing] such a small [path] barely by an interceding act of the mind*.
eamque philosophes nostros And I would assert that our metaphysical Metaphysicos potissimum terere fere philosophers, especially, trivialize that dixerim, abstrahendo nimium a path by abstracting too much from the
’ That is, bypassing language. This references the description given in the preceding section.
109 sensationibus, imo rejiciendo etiam sensations—no—by rejecting the subtle subtilem cognitionis omnis sensum a sense of all cognition animated by the corde animatum, qua fit, ut centrum heart. By which it happens, that some aliquod in cerebro pro formandis ideis center in the brain (which forms intellectuaiibus incertum & vagum nunc intellectual ideas)—now uncertain and in hoc, nunc in illo ostio ductus organici wandering in this, now in that, opening of alicujus in cerebrum ferentis, & nunc hac some organ duct leading into the brain— nunc alia directione, qua impressio quasi through which an impression as if seized adfectata (forcée) ferri posset, contra (forced), is able to be produced. On the ipsum animi ex corde nisum, centrum contrary, the very exertion of the mind illud intellectuale etiam insciis & invitis from the heart, that center pertaining to hominibus sensim dirigentem ac lente the intellect—even to ignorant and promoventem, non satis tuto stabiliant. unwilling people—gradually directing and slowly, thus the philosophers do not establish this securely enough.
§16 §16
De via fidei ex cerebro sensual! in cor Concerning the way of faith, leading ducente. from the sensual brain into the heart.
Pari modo dixerim, immediatum dari I asserted equally, that an immediate duct ductum ex cerebro sensuali seu potius is furnished from the brain sense, or [59] imaginativo in cor ipsum, sine rather, from the imagination, directly into intercessione organorum loquelae, the heart, without intercession of the hancque fortasse fidei esse viam, quippe organs of speech. Furthermore, that this quam Paulus ex auditu prodire pronunciat. speech, perhaps, is the path of faith, naturally, which [the Apostle] Paul proclaims to proceed from hearing. 10
Attamen cum ad id requiratur However, since intelligence of discourse intelligentia sermonis seu linguae, qua or language is required for this (whereby Evangelium nobis praedicatur, non poterit the Gospel is proclaimed to us), it is not fere non aliqua organorum loquelae able to not intercede by some operation of operatio nisusque in illis excitatio the organs of speech and by an excitation intercedere: nisi velimus dicere, of exertion in them. Unless, we should perceptionem linguae alio modo fieri in say, that the perception of speech occurs corpore, quam expressionem; scil. cum in by another manner in the body, other than perceptione anima se passive habeat, expression. Of course, since the mind lingua sine dubio seorsim tota jam residet holds itself passively in perception.
10 Romans 10:17, “So then faith cometh by hearing, and hearing by the word of God."
110 in cerebro sensuali, atque eadem propter language, without a doubt, resides in the usum suum etiam in organis loquelae sensual brain apart from the whole, and seorsim. likewise—on account of its use—in the organs of speech.
§17 §17
De via fidei divinae spuria & vera. Concerning the false and true way of divine faith.
Viam autem fidei divinae ex cerebro Moreover, the way of divine faith, leading sensuali per cerebrum intellectuale in cor from the sensual brain through the brain ducentem, qua in dicto Paulino inter (intellectually), into the heart (which, in auditum & fidem ratiocinium ponitur, esse what was stated by Paul), is placed among spuriam, testabitur unusquisque regenitus the hearing and reasoning faith, is false." suo exemplo suaque experientia: quamvis Each and every “bom again” person will ad illam fidem non primum quidem testify to this by their example and excitandam, sed potius plantandam & experience. Although with respect to that radicis instar generandam cum longior faith, [language] is not primarily for usus ac volutatio verbi divini auditi, sed in “stirring up,” but rather for planting and memoria saltim, non in facultate generating the equivalent of a root, not abstractiva & ratiocinativa, memoria vero only when a longer engagement and ad cerebrum sensuale pertinet, turn mulling over of the hearing of the divine submissio cordis, tanquam vasis word is required,'** but—in the memory at coelestem doctrinam per ductum cerebri least (not in the abstract and reasoning sensualis hausturi requiratur. faculty) truly the memory pertains to the sensual brain—but also the submission of the heart (as it were of the vessel) is required for the absorbing of the heavenly doctrine through the duct of the sensual brain.
" That is, the path which would bypass language, because it would bypass the God given means whereby one hears the Gospel. 2 Timothy 1:6, “Wherefore I put thee in remembrance that thou stir up the gift of God, which is in thee by the putting on of my hands.” 2 Peter 1:13, “Yea, I think it meet, as long as I am in this tabernacle, to stir you up by putting you in remembrance.” 2 Peter 3:1, “This second epistle, beloved, I now write unto you; in both which I stir up your pure minds by way of remembrance.” [Emphasis added] 1 Corinthians 3:6-8, “I have planted, Apollos watered; but God gave the increase. So then neither is he that planteth any thing, neither he that watereth; but God that giveth the increase. Now he that planteth and he that watereth are one: and every man shall receive his own reward according to his own labour.” [Emphasis added] He is referring to Pietist theology, and the steps a believer goes through: rebirth, etc.
i l l §18 §18
Descriptio cognitionis verae. A description of the true cognition.
Cetemm ne quis existimet, philosophonim However, one should not think that the cogni- [60] tionem crebra acquisitam strained cognition of the philosophers, by abstractione praestare cognitione vulgi, constant abstraction, surpasses the quod brevior sit via a cerebro sensuali ad cognition of the common people, because intellectuale, quod fortasse plane idem est it is a briefer way from the brain sense to atque unum, & saltim in motibus partium the intellect. The intellect [quod\ (perhaps eorundemque directionibus ac completely one and the same [as the brain celeritatibns [sic, bus] discrimen admittit, sense]) admits a separation—at least in quam a cerebro sensuali per cor in the motions of the same parts with respect intellectuale, notandum est, cyclum vitae to their directions and also their speeds. rapidissimo motu gyrari in circulo non in [This separated] way [quam = via]** from linea recta, vimque divinam, quae cor the brain sense through the heart into the tanquam centrum respicit, non cor saltim intellect, should be observed. A cycle of & imaginationem sensatam, sed & ipsam life with a very rapid gyrating motion in a loquelam atque intellectum actuare ita circle—not a straight line—and a divine quidem, ut nec hie pedem possit force, which looks back at the heart (that promovere ex numéro 1 in 2 & deinceps, is, the center). And [looks back at] not nisi ope generalis totius animi & vitae just the heart and imagination sense, but revolutionis. even speech itself (as well as the intellect). Thus indeed, [the cycle] acts, so that neither is this [one] able to move [his] foot from number 1 to 2 successively, unless by the power of the complete, common mind* and [of the] revolution of life.
§19 §19
De statu contemplationis & Concerning the condition of divine Uluminationis divinae. contemplation and enlightenment
Vera igitur cognitio non in linea cycli Therefore, true cognition proceeds not in vitae tangentiali abstractiva sed in directa an abstract line of the tangential cycle of ex centra progreditur, unde intelligitur life, but [proceeds] in a direct [line] hrom sensum animi, qui centrum habet in corde, the center, from whence it is understood esse lucemam Dei, qui si eo sit elevatus, that the sense of the mind*, which has ut homo etiam in tertio Arithmeticae [its] center in the heart, is the light of God, genere novas invenire possit rationes, which if it has been elevated by it, so that mensuramque 2^:3^ distincte intelligere, & a man—even in the third type of
" He is referring to the separation between the abstract way and the common sense way.
112 quasi cum aliis possibilibus mensuris, mathematics—is able to discover new ex.gr. 3^:5^, 2^:5^ conferre & mensurare, ratios, and clearly understands the videtur sistere statum Uluminationis measure 2^:3^ and, as it were, compares divinae. and measures with other possible measures, e.g. 3*:5^, 2^:5 , [so that] it seems to establish the position of divine enlightenment.
§20 §20
Conclusio. Conclusion.
Denique mentio dupilcis cerebri non adeo Finally, making mention of the twofold peregrina erit auribus anatomiae cultorum, brain will not indeed seem strange to the qui sollicite cerebrum a cerebello ears of the supporters of anatomy, who distinguunt, neque talis aliqua animae anxiously separate the brain from the cum [61] corpore, qualem descripsimus, cerebellum. Neither [will] some such communicatio, cum satis intelligant, etiam communication'^ of the mind with the unius in corpore musculi hanc vel illam body (such as we described) [seem directionem motus, cum per se plures strange],'^ since they understand well darentur, non posse nisi a praesentissimae enough, that even [for] the motions of one toti corpori animae impressione, quae non little muscle to proceed in the body in this ex communi aliquo totius corporis centro or that direction (since several muscles are derivari queat, proficisci. In hoc certe given for a motion), this is not possible argumento quippe problematico praestat except by an impression of a very present in exsistentibus imaginari, quam in mind [given] to the whole body. This abstractis rationicari. impression [quae] cannot be derived from some common center of the entire body. Certainly, in this problematic argument, of course, it is better to imagine in actualities, than to reason in abstractions.'^
talis aliqua...communicatio. Repetition of peregrina erit. " That is, think literally about the process.
113 [OETINGER: SECTION IV]
Accipe, Lector benevole, partem alteram Accept, dear reader, part two of the Analysées Musicae a Dn. Frickero post analysis of music written down by Mr. lectum de sensu communi librum Pricker after having read [my] book exaratam. Dabo & hie notas quasdam. concerning the .renfuj commwnzf .' And I will furnish here some notes.
Ad§l On §1
Psychologiae Musicae requlsita It is asserted that a full examination of nondum plena esse, asseritur. the psychology of music does not yet exist.
Animam in prioribus ut sphaeram ex [Flicker states that], in the previous potentiis secundum numéros triplices [sections], it is [a] very difficult musicos 2^ 3^ s' coadunatam, & jam hie conjecture, not springing from fantasy, but in binario quodam 8:9 seu 2^:3^ omnem having been founded from the ratios, that multiplicitatem potentiarum terminantem the mind, having been united, can be considerari posse, conjectura est valde considered as a sphere—as a result—of ardua, non ex phantasia propullulans, sed the powers according to the three musical rationibus fundata. Posset auctor hanc numbers' t 3^ 5‘, and now here in a hypothesin pluribus comprobare datis. certain binary 8:9 or 2^:3\ terminating every multiplication of the powers. The author could prove this hypothesis with many given [examples].
Sed sufficit hac vice per numéros Musicos But it is suffîcient, in his tum,^ to have hoc tentamen fecisse. Etenim bene novi, made this attempt through the musical quanta paupertas datorum vexet numbers. And indeed, I knew well, how philosophos in investigatione animae. great the lack of details [data] vexes Deberent enim a sensoriis visus, auditus, philosophers in their investigation of the gustus, olfactus, nec non tactus incipere, mind. For they ought to begin from the nec non a [62] consideratione linguarum senses of sight, hearing, taste, smell, and & earum nexu cum anima deberent animi certainly touch, and certainly from a & animae differentias ab antiquis agnitas consideration of languages, and of those, revocare; their connection with the mind; they ought
' Compare this with Pricker’s statement in Part 2, Section 1, where he alludes to the fact that Part 2 was written after having read Oetinger’s book. ‘ Literally, “with this succession. ”
114 to revive the distinctions from the ancient knowledges of the mind* and of the soul/
{Opus esset pleniore elementorum (This task should be an investigation corporeorum, potentiarum, vitalium,} more of elements, bodies, powers, vitalities,}
deberent elementorum extemorum aëris, They should compare the more subtle aetheris, ignis, luminis, mated ae character of external elements'* of the air, magneticae, aquae, terae &c. habitum ether, light, magnetic material, water, subtilius conferre cum sensoriis, cumque earth, etc., with the senses^, and of those eorum membranis & cum fluidis in illis with membranes and with fluids in those contentis aëreis, aethereis, igneis, lucidis, continuous airs, ethers, fires, lights, magneticis, aqueis, terreis; magnetics, waters, earths.^
deberent determinare aliquo modo, They ought to determine by some manner, quomodo anima in sua intrinseca how the mind in its intrinsic constitution constitutione différât ab his compositis, in differs from these compositions, in which quo superet natura animae has naturas the nature of the mind surpasses these elementares, quarum motus tremulatorius natural elements, of which the trembling semper se habet quadratice ut diametri; motion always holds itself squarely as the deberent determinare differentias diameter. They ought to determine the undulationis a tremulatione, & quomodo differences of the undulations from the se mutuo habeant undulationes ad trembling and how the undulations hold tremulationes. themselves mutually in relation to the tremblings.
Undulatio est qualis in aqua conspicitur The undulation^ is such a kind as observed dum in citculos abit, & qualis in materia in water as it goes forth in circles, and is sono affecta esse colligitur, quae movet such a kind as is obtained in a material superficiem integram. Tremulatio vero affected by sound, which moves the whole est uniuscuj usque particulae cum vicinis surface. However, the tremor® is a concussio reciproca, ubi localis motus non concussion moving back and forth of each apparet ut in undulatione. Nam cum and every partficle] with neighboring tremulat aer a sono, undulat aether, quae [particles], where the location of the undulatio facit tremulationem in motion is not apparent as in the undulation membranis nostri auditus, ubi sonus [wave]. For when the air vibrates by a aethereus pénétrât ipsas membranas. sound, the ether undulates, which
^ This is the only time that Oetinger makes use of the masculine, animus, and the feminine anima appears with it twice. * That is, physical reality. ^ That is, sensorial elements. ‘ That is, the “subtle,” discrete components that make-up “continuous” phenomenon such as air, etc. A sine or transverse wave. ' A longitudinal or compression wave.
115 quemadmodum alias saxa, ferra, vitrum, undulation makes a tremor in the per quae aeri nullus est transitas. membranes of our ears, where the etherial sound penetrates those same membranes, just as other [materials] stones, iron tools [weapons], woad [lit, vitrium; plants, wood], through which [things], nothing is transmitted to the air.^
Pari modo tremulante aethere undulare In like manner, light is able to undulate by potest lux, et per totum coelum undulare the trembling ether, and the rays from the possunt radii a sole. An ipsa anima sun are able to undulate through the entire undulationem gyratoriam in fluidis & sky. Whether the mind itself should limit tremulationem in mem- [63] branis ab [or] modify the gyrating undulation in extra concitatam limitet, modificet, in fluids and the tremor in the membranes, totalitatem perceptionis revocet, nescimus, stirred from outside, [or] check them in & an id per intelligentias arithmeticas the totality of perception, we do not know, praestet, adhuc investigamus. and whether it exhibits it in arithmetical understandings, we are still investigating.
Deberent determiare an sensationes sint They ought to determine whether the merae tremulationes in membranis, an sensations are mere tremors in the aliquid superveniat altioris naturae, & membranes, or whether something else is quid sit illud subtilissimum, added on top of a higher nature, and what vivacissimum, actuosissimum ideale, that something of a most delicate, lively, quod tremulationem in membranis, sive active idea is, which thus modifies, ad visum, sive ad auditum, sive ad gustum according to the totality of [the] ita modificet ad totalitatem perceptionis; perception, the tremor in the membranes, nam ex meris undulationibus & either with reference to sight, or to tremulationibus nulla résultat perceptio hearing, or to taste. For from the mere totalis. undulations and tremors no perception of the whole results.
{nec non structurae mechanicae (than the mechanical structure of the sensoriorum indagatione.} senses.} 10
Novimus auditum ab undulatione aëris We know that the hearing arises from the oriri, quia aures mechanice ad ejus undulation of the air, because the ears are receptionem formatae sunt, id quod mechanically shaped for the reception of
’ Oetinger is making a distinction between observable and unobservable waves. One might be tempted to think he abstrusely referring to the distinction between transverse and longitudinal waves, but that would be incorrect. He is referring to the transmission of observable sound waves into the ear membranes and the brain, where they are no longer observable. As he states, the observable sound wave also sets the invisible ether in motion, and it is the ether which actually transmits the sound to our ear membranes. It is ironic that this marginal reference (which is a continuation of the previous), and disparages looking into the “mechanical structure” of the senses, immediately precedes an anatomical description of the ear.
116 summus Philosophus Swedenborg ita it, that which the great philosopher describit:HAcrnkir Swedenborg describes: *
[“] Membrana nervea cartilagini auris The nerve membrane attaches directly to immediate adhaeret. Figura auris sulcata the cartilage of the ear. The shape of the est, plicata est, helix & antihelix, tragus et ear is a folded, bent, helix and anti-helix, anti tragus et concha sic formata sunt ut tragus and anti-tragus and the concha thus tremor soni distincte versus concham et having been formed so that a tremor of versus tympanum deferatur. Membrana sound is distinctly brought against the tympani convexa, obliqua, malleo juncta, shell and against the tympanum. The non tensa sed laxa ad sonos articulates membrane of the tympanum is convex, recipiendos & ulterius transferendos; oblique, connected by a mallet, not tense, but lax for the sake of receiving articulate sounds and transferring them further. manubrium mallei adhaeret a superiori The mallet's handle attaches from the margine usque ad centrum, quod illam upper edge all the way to the center. The marginem ita disponit, ut omnem center thus disposes that edge in such a varietatem in tonis recipiat, quos distincte way that it receives the entire variation in non posset recipere, nisi manubrium radii tones that it could not receive distinctly aut semidiametri instar ei se adfixisset, ita had not the handle affixed itself to it in a enim nullus tonus alibi membranam measure equivalent to the radius or tremulam potest reddere, quam 'semidiameter'. For no other tone can convenienter distantiae a radio suo seu make the membrane vibrate from any manubrio unde tot distantiae in membrana other source, other than that of a suitable [64] auxilio manubrii formatae visuntur, distance from its own radius or handle quot usquam dari varietates in sonis from which as many distances are seen possunt: at vero si nuda sine distinctione fashioned in the membrane, with the help of the handle, as are the varieties in per aliquod mallei crus foret appensa, sounds that are capable of being produced. aliquid distincte tremulum in se vix But if it (the membrane), uncovered, were admitteret: appended indiscriminately across a certain shank of the handle, it could hardly admit any distinct vibration in itself.
Furthermore, a string is present, running adest etiam chorda membranam through the membrane, so that with help, percurrens, ut ejus ope scire possit anima the mind is able to know the direction of directrix, num rite & ad omnem it, and whether or not, rightly, the tension peripheriam tensa sit vel non; statim ut is to the entire periphery. Immediately, as minus ab una parte quam ab altera laxior a result (less from one part than from
' ' What follows is an extensive quote from one of Emmanuel Swedenborg’s numerous treatises on the soul’s coiuiection with the body. I have been unable to locate the exact passage and source, but it is not critical to this treatise. There are similar passages located in “The soul or rational psychology ” and “The economy of the animal kingdom.’’
117 aut tensior sit, admonet nervulus ab usu et another) it is looser or tenser. The little exercitio quasi infonnatus, & tandem nerve (having been formed, as it were, by sponte ex habitu, nesciente homine, quod use and exercise, and finally voluntarily scilicet aliter tendendus sit; fiom habit, unknown by a man) stimulates, because, of course, it has been stretched differently. unde duo musculi dati sunt malleo, unus From whence, two muscles are given with qui manubrium torquere possit & paullum the hammer, one which is able to twist the circumflectere, ut sic, si ab uno latere handle and able to bend a little, so that the minus laxa aut tensa foret membrana membrane, if from one side is less lax or quam ab altera, statim per exiguam tense than from another, it is flexuram faciat, ut ab utroque latere accomplished immediately through the manubrii tensio ejus sit adaequata; additus small flexing, so that from both sides of etiam est alter musculus, qui actionem the handle the tension of it is adequate. aliam in malleum etiam exerceat, For there is another added muscle, which membranam elevando, si infra vel in even exercises another action in the inferiori membranae circulo sit restituenda hammer, elevating the membrane, if tensio; below or in a lower circle of the membrane the tension is restored. verbo motus musculorum talis est, ut In a word, the motion of the muscles is so illorum ope ad omnem directionem aeque great, that by the power of those, the intendi & laxari possit tympanum, tympanic membrane is able to be admonente subtili chorda; ejus figuram stretched and relaxed equally in every concavam praetereamus, quia obvium est, direction, prompting the delicate string. tremores per circulos et curvaturas ut sui We bypass the concave shape of it, dissipatione convenientiores fluere velle. because it is obvious, that the tremors wish to flow through the circles and curvatures as their own [circles] by a more harmonic dissipation.
Sunt jam tria ossicula. Malleus, qui Now, there are three bones, the hammer articulatur, prout alia ossa, cum incude, [malleus], which is divided, just as the ligaturque per tendines, ut non modo other bones, with the anvi] [incude], and musculis suis obedire possit, sed etiam connected through the tendons, so that not cuicunque tremori suae membranae: Incus only are they able to submit to its muscles, vero stapedi adjungitur per exiguum but even to each and every tremor of the ossiculum, quod quartum vocatur, uno membrane. However, the anvil is crure per [65] tendnem ligato & in attached to the stirrup [stapedi] through a foveola detento: small bone, which is called the fourth [quartum], with one leg fastened [to] a tendon and detained in a socket.
118 ipse stapes bina crura aequaiissima The stirrup holds very equally with two possidet, quae non mode excavata sunt, legs (which are not only hollowed out, but sed per se quoque flexibilia & elastica, through themselves also [are] flexible and perque limbum circularem pariter cavum elastic), and through the circle[ular] incumbit fenestrae ovaii; transfertur sic border together presses on a cavity of the mire tremor a membrana tympani ad oval window. TTius, the tremor is merely fenestram hanc ovalem; per ligaturas transferred by the tympanic membrane to ossiculorum adest momento idem tremor this oval window. Through the ligaments in fenestra, qui per membranam receptus of the bones the same tremor arrives in the est; window, which is received through the membrane. ipse stapes ad eosdem nictus se illico The stirrup moves itself immediately movet, ad quos motum est manabrium according to the twitchings [nicms], [sic, manubrium] tympani; ipsa flexibilitas according to which [twitchings] is the in stapedis chelis aut uncis efficit, ut motion of handle of the tympanic aptior sit fenestram ovalem in similes [membrane]. This very flexibility nictus movendi; ipsa cavitas in stapedis produces [twitchings] in the clawed or collo vel capite, cruribus & limbo, ut hooked stirrup, so that it is more tremor etiam intus sentiatur, sive inclusus appropriately moving the oval window in sit aër sive aether, qui sic quoque a similar twitching. The cavity in the communicari possit cum fenestra & fere stirrup [moves] by the neck or head, by instanti cum labyrinthe; the legs or borders, so that the tremor is sensed even inside. Either it is included in the air or the ether, which thus also is able to be communicated with the window and almost instantly with the labyrinth. ipsa fenestra per tales modules se etiam The opening itself stretches and relaxes tendit & laxat, prout se stapes elevat vel itself also through such small measures, deprimit, & facit, ut tremor aëris inclusi & just as the stirrup lifts or lowers itself. per sues ductus semper praesentis possit And it brings it about that the tremor of air recipi: sicque multipliciter se in fenestram that is closed off and always present a membrana tympani dérivât tremor, & throughout its ducts is able to be received. omnia unanimiter conspirant in ejusdem And thus the tremor channels itself off tremoris receptionem: utque haec in suo from the tympanic membrane into the statu teneantur integra & aequilibrata, opening in many different ways, and the stapedi etiam musculus additus est, qui se whole (apparatus) conspires in unison to ad nutus praedictorum muscuolorum in receive that same tremor. And, in order malleo aptare noverit. that these things be maintained at their own degree, whole and balanced, a muscle is also added to the stirrup that knows how to fit itself to the urgings of the muscles in the hammer mentioned above.
119 In labyrinthe occurrunt innumera notatu In the labyrinth, innumerable noteworthy dignissima: occurrunt ibi tres canales, things occur. There are three canals, semicirculares dicti, qui tubarum seu called semicircular [canals], which are comuum instar fonnati per quinque formed in the image of tubes or horns, orificia patent, duobus in unum foramen accessible through five openings, coming coëuntibus, in quorum cavitate idem together in two [tubes] into one opening. elemen- [66] tare adest, quod in ipso In the cavity of which the same [opening] labyrinthe, quo pleni & quasi distenti appears to be rudimentary, because in the cubant; investiti sunt membrana tenui, & same labyrinth, by which they lie full and extrinsecus substantia porosa, hinc & inde as it were spread out. They are invested allegati ossi; with a delicate membrane, and on the outside a porous substance, here and there, they have been layed before the bone.
unde tremor jam a fenestra ovali illuc From here the tremor is now instantly translatus per parietes membranaceos present, transferred to this spot from the labyrinth! perque aërem intermedium in oval opening through the membrane walls unoquoque canali momento adest, and through the air that intervenes in each figiturque & incipit quilibet in sua canal. And each one (whatever) becomes distantia ab orificiis, serpitque utrinque fîxed and begins in its own distance from [sic, utrimque] versus orificia; utque hoc the openings, and it winds from both sides sentiat anima, nervulus ex pari quinto towards the openings. And in order that molli percurrit omnes, qui conscius soni the mind sense this, a small nerve runs articulati & ad hanc vel illam distantiam evenly through them all from the fifth soft & consequenter celeritatem moti, defert nerve, which is aware of articulate ilium per nervum totum auditorium, & sic sound.According to this or that distance in cerebrum, medullam oblongatam, ad and following the speed of the motion, it ipsam animam. carries that sound down along the entire auditory nerve, and thus into the cerebrum, to the medulla oblongata, then to the mind itself.
Ut adhuc articulatius percipiatur sonus, The result, thus far, is that the articulated addita etiam cochlea ad omnem sound is perceived, indeed, by the mechanismum harmonicum formata, attached cochlea (which has been formed serpentino modo se circum suum nucleum to every harmonic mechanism, in a snake flectens, circulus spiralis ab una parte like manner, bending itself around its latior est, ab altero per gradus angustior, nucleus; the spiral circle is broader on one adeo ut nihil in sono dari possit, quod non side, and more narrow, by degrees, on the habere queat suam distantiam aut other [side]). To the end that nothing is centrum, ubi incipiat, & ab inde ad able to be produced in a sound, in so far as
He is referring to the brain perceiving the impulse according to its distance from the window. Compare to Pricker's passage in Section ten, p. 106.
120 extremitatem pergat, omnes enim it is unable to hold its distance or center dimensiones adsunt: [of the cochlea], whereby it begins, and from whence it proceeds to the extreme [end, of the cochlea]: for, every dimension exists.'^
pariter ut sonus altior & altior fiat, & At the same time, the sound becomes progrediendo a termino ad terminum higher and higher, and by progressing versus fenestram rotundam intendatur: from end to end it [the sound] is exerted ilium etiam in finem est canalis hie toward the oval window: likewise, diversae crassitiei, circa nucleum [toward] the oval window [ilium], in the tenuissimus, ut sic omnem gradum end, this canal is diverse in density, very tremoris recipiendi capax sit: dividitur delicate around the nucleus, so that it is etiam canalis per tenuem lamellam capable of receiving every degree of the osseam & membranaceam a summo ad tremor. Also, the canal is divided through imum, ut membranaceum ejus eo melius a thin plate bone and a membrane from tremiscere possit, quum alligatum si high to low, so that its membrane is able laminae tenui, durae & fragili, & ab uno to tremble better by it, since it is attached [67] fine sequatur ad alterum; utque to a plate, hard and fragile, and it goes tremor eo distinctius veniat ad fînem, ideo from one end to the other. And so the etiam ei data est apertura versus tremor comes, therefore, more distinctly vestibulum, ut idem effectus in cochlea, to the end. Likewise, for that reason, the qui in tubis aut canalibus praedictis opening was produced for it toward the existât: vestibula, that results in the same effect in the cochlea, which exists in the afforementioned tubes or canals.
& quia nullus tonus fere alteri similis est, And because no tone is exactly the same sed datur indefinita varietas, hinc cochlea as another, but is given indefinite variety, formata est cum omni varietate qua the cochlea was formed with every variety formam, crassitiem, amplitudinem & in so far as form, density, amplitude and plura, tremor etiam amat gyros spirales, many more [things]. Actually, a tremor circulares, & hyperbolicos, hinc nihil dari has a propensity [amat] for circular potest in sono, quod hie non locum suum spirals, circles, and hyperboles. Hence, sortiri queat. nothing is able to be perceived [dari] in sound, which, here, is not able to be received in its place.
He is describing, essentially, the place theory of hearing, i.e., that sounds are judged by activating a nerve a certain distance from the oval window on the basilar membrance in the cochlea. *'* That is, it matchs the shape of the cochlea, and thus be perceived. This is somewhat different from the place theory, which does not rely on sympathetic vibration per se, but stimulation of the cilia on the basilar membrane.
121 Sunt etiam nervi innumeris in locis Furthermore, the nerves cross the nucleus nucleum & laminam transeuntes, ut illi and plate in innumerable places, so that varietates dictas in se possint recipere, they are able lo receive the designated man nullus sensus est sine nervis, & varieties, for there is no sensation without nullus in nervis nisi per motum, hinc nerves, and there is no sensation in the omnis tremor in cochlea & membranis nerves except through motion. Hence, ejus statim sentitur similis in nervis, qui every tremor in the cochlea and in its ideo tam multis in locis per cochleae membranes is immediately sensed in interiora se dispandunt; similar nerves, which therefore, to such a degree, spread themselves out in many places through the interior of the cochlea. sitque omnis motus eo sensibilior, quum And it happens that every motion, filamenta nervorum cochleam percurrentia therefore, [is] more perceptible, since the secundum longitudinem ejus serpant, & round opening around which the nerve quidem versus fenestram rotundam, circa itself enters, such that the tremor, as it quam ipse nervus intrat, adeo ut tremor follows the filaments into the thicker and filamenta sequens in crassiorem & still thicker part of the cochlea, seems to crassiorem ejus partem se conferre collect (i.e. compress) itself, so that videatur, ut eo distinctius & sensibilius thereby it might, once received by the per totum nervum auditorium exceptus auditory nerve, be carried along its entire deferatur. length in a more distinct and sensible fashion.
Hie iterum vides, qualis sit mechanismus Here again, you see, what the mechanism naturae, ut modo tremor in aëre deferatur of nature would be (as the tremor in the versus interiora, imo versus ipsam air is carried down toward the interior, on animam, si rationale quid in sono sit, quod the contrary, toward the mind itself), if per articulationes & voces fit, & dein per reasoning were something in the sound— earum cempositiones [sic, compositiones] by means of articulations and voices, and altius & altius; hie vides non modo finally, by means of higher and higher Mechanismum, sed etiam tot ejus partes compositions of them. Here you see not ad unum [68] eundemque finem merely a mechanism, but actually so many tendentes, ut audiat homo, ut delitias of its parts extending to one and the same mundi cum ratione percipiat, ut inde sciat, end, so that a man hears, so that he quod qua animam vivat, ut sic miscere perceives the delights of the world with sciat delitias mundi cum delitiis animae, reason, so that he knows, what, by which, & animae cum delitiis ex mundo adhuc the mind lives, so that thus he knows how subtiliori; tandem cum Deo, in quern sic to stir up the delights of the world with the desinet omne tanquam in unicum Rnem." delights of the mind, and of the mind with [End of Swedenborg quote] the delights from a world hitherto more subtle. In the end, with God, in whom, thus he will finish everything, so to speak, in one end [point, goal].”
122 Excusabit Lector prolixitatem notis hisce The reader will excuse the disprcpcrtionatam; Sedquoniam disproportionate length of this quote, but Swedenborgii liber non in omnium because Swedenborg’s book is not in the manibus est, & motus undulationis hands of all, and the motion of the sonorum ex mechanica hac structura undulations of sounds can be collected so optime colligi potest, integram well from this mechanical structure,'^ I demonstrationem adposui. have set before you the entire demostration.
Habes, itaque Lector! mechanicam And thus you have it, reader! The deferendorum tonorum ad animam mechanism of conveying tones to the conformationem in aure, unde motus mind’s conformation in the ear, from undulatiorius sonorum colligitur. Jam whence the undulating motion of the regredior ad priora. Opus est porro, ut sounds is collected. Now I return to the Philosphi Psychologiam ex sensoriis previous arguments. It should be locupletaturi structurae hujus mechanicae necessary furthermore, that philosophers, habitum ad exteriores aëris & aetheris about to enrich the psychology from the variationes & vibrationes & ad spirtualem senses, more deeply penetrate the vocis articulationem altius penetrarent. condition of this mechanical structure according to the exterior variations and vibrations of the air of the ether and according to the spiritual articulation of the voice.
Opus esset, ut cosmologica principia in It should be necessary, that cosmological subsidium vocarentur talia, quibus principles—as such things are called into ostenditur, esse non quidem eandem support, and by which it is shown—be quanti tatem virium motricium in universo, not, certainly, the same magnitude of quum quies & motus accidentalia sint forces of motions in the universe, since tantum sed materiam passivam in continua rest and motion are accidence, but only subjectione ad potentias créatrices the passive material should be placed in positam esse. continuous subjection to the creature powers.
Has potentias nosse ut fontem motus & [If we were now able] to recognize'® these quietis, ut uniformem mutationum powers as the source of motion and rest, regulam, etiam sensibus, si quidem as the uniform rule of changes, even for intemoscere satis possemus uniformia & the senses; if in fact we were now able to simplicia, obviam, magni esset momenti, distinguish well enough between things e.g. si possent ignes illi multiplies, aquae uniform and things simple, this would clearly be of much importance, for
Even though he disparages too much investigation of the mechanical structure, he still believes it has merit. “Known”, nosse, contraction for novisse.
123 multiplices in regionibus polaribus example, if those multiple rites-- Domino de Mauper- [69] tuis cognitae recognized by Lord Maupertius as melius explicari; multiple waters in the polar regions-could be better explained.
cur in locis frigidis adeo formosi Why, then, in cold places, indeed, [would] appareant ignes; si possent vibrationum & beautiful fires appear, if they could be tremulationum ignis, aetheris, aëris & investigated (as attraction is) as the origin elasticae tensionis origo, ut attractionis of vibrations and tremors of rite, ether, investigari; si nexus cum Sole & Luna air, and elastic tension? If a connection ostendi, ut per hoc microcosmica were revealed with the sun and moon, as praecipua in homine, ad similitudinem through this particular microcosm in man, Sapientiae Salomoneae, constate possint, according to the similitude of the wisdom ut sic animae veram indolem cum suis of Solomon, they should be able to indumentis, vehiculis & organis, ut sunt correspond, as thus the true innate cor, cerebrum & aliae camerae corporis, character of the mind with its garments, determinatius sciremus. Sed quam longe vehicles, and organs, as are the heart, absumus abhinc! brain, and other chambers of the body, we know are more determined. But so far we are lacking from this standpoint!
Totum semper desideratur in fragmentis The whole is always desired in the cognitionis humanae. Igitur sane riagments of human comition. Therefore Hippocratis tres ignes cum Musica certainly the three fires ^ of Hippocrates, comparati non sunt absurdi. Multo minus having been compared with music, are not supervacuum videri debet ad animam absurd.'^ Much less it ought to seem intacta Musicae principia applicate; Et si unecessary to the mind to apply the intact sublimioris Chemiae data adderentur, principles of music. And if the data of the sperari posset per numéros plura posse more sublime chemistry^" are added, it concludi ad modum activitatis in anima could be hoped that through many declarandum, praecipue ad Sensus numbers it could be deduced for the communis, seu reductionis multipli ad purpose of declaring the manner of mensuram simplicis Philosophicam activities in the mind, especially in analysin. relation to the sensus communis, or of a multiple reduction according to the analytical philosophical measure of simplicity.
Refers to the Northern Lights. See footnote on page p. 148 for more detail. Oetinger mentioned these before in the main portion of the treatise on page 190, but still does not elaborate. ” A reference to the three forces (2,3, S) proposed earlier. 20 That is, alchemistry.
124 (Quae cum non plena sit, tantisper (Since this is not complete, for the consuliquoque debent principia Musica.} present, they ought to, likewise, consider the musical principal.}
Simus itaque contenti hactenus generaliori And thus we are content to this point with hac conjectura musica donee plura & this more general musical conjecture, until magis coagmentata se invicem confirment many and greater required connections requisita. could be reciprocally confirmed with themselves. Ad §2 On §2
Per principia Musica infertur Through the musical principie, the compositio potentiarum in anima, quae connecting of the facuities in the mind tamen nihil detrahit ejus is brought forward, which nevertheiess inunaterialitati. detracts nothing from its immateriality.
Principium animae, unde ideae cum The beginning of the mind, from whence conscientia nascuntur, esse quid ideas with conscience are bom, is immateriale, afferit auctor. Sed eo vix something immaterial, asserts Fricker aiiquis naturali indagine pertinget, ut [auctor]. But to that end, it scarcely centri cum circumferentia actiones & [70] extends to some surrounding nature, so reactiones exploret. that it explores the actions and reactions of a center with a circumference.
Actio centralis est respective immaterialis. The action of the center is with respect to Reactio autem videtur fluida, spiritus the immaterial. The reaction, however, animales dicta, seu ut Swedenborgio seems fluid, called animal spirits, or as it placet, mechanicas subordinationes a pleases Swedenborg, the subordinate meningibus ad intimos cerebri maeandros mechanisms seek from the meningibus^.21 requirere. Ipsi Leibnitiani nullam animam to the innermost winding paths of the sine schemate corporeo concipiunt; dicut brain. Lebnizians themselves conceive no enim, ubi datur sensus, ibi datur corpus. mind without a bodily scheme, for they say that where the sense is given, there the body is given.
Typus limitationis cogitationum ipsis est a The type of delimiting of thoughts in these Schemate corporeo in ipso statu post very bodies is from the bodily schema in mortem. Igitur certe Schema corporeum the same rank after death. Therefore the concurrit ad cogitationes. Sine hoc enim bodily scheme certainly runs in tandem omnitudo virtutis repraesentativae ab intra with thoughts [i.e. harmonizes with them]. prodeuntis non posset modificari. Non For without this [bodily schema] the igitur succenseri potest vel mihi vel Dno. entirety of the representational power that Frickero, quando animam immaterialem. proceeds from within could not be
Membranes covering the brain and spine.
125 immortalem, indestructibilem asserentes modified. Therefore one cannot be angry corporeum quid ad cogitationes either with me or with Mr. Fricker, since solidandas assumimus. we claim that the mind^^ is immaterial, immortal, indestructable, and we assert that the body is for the purpose of soldifying the thoughts.
{praesertim cum de anima nondum eo (Especially because, concerning the mind, progressi simus, quo Robert Hooke, cujus we have not progressed to that degree, for sententia allegatur.} which reason, the opinion of Robert Hooke is selected.}
Adscribam hie Celebris Mathematici I insert here the celebrated mathematician Robert Hoocke sententiam de anima, qui Robert Hooke’s opinion concerning the longe determinatius quam nos loqui audet mind, who ventured to speak much more de anima. Statuit ille, esse aliquam mentis determined than we concerning the mind. in cerebro sedem, ad quam omnes He establishes that, it is some seat of the sensuum impressiones deferantur; hoc mind in the brain, to which all idearum repositorium variam continere impressions of the senses are brought. materiam in usum animae, cui This repository of ideas contains diverse elementorum idearum tribuit nomen, material in the service of the mind, to eamque quintuplicem esse secundum which, of the elemental ideas, it assigns a quinque sensus; elementa ideis visus name, and it is a multiplied according to adaptata esse velut Phosphorum [sic, the five senses.^ Elements adapted to the Philosophonim] Bononianum vel ideas of sight are just as of the Balduinum; alia alter. philosophers Bononianum or Balduinum, in different ways.
Certam esse capacitatis Sphaeram circa A certain sphere of capacity exists around sedem mentis, quod centrum vocat, intra the seat of the mind, which he calls the quam ideae formentur, recipian- [71] tur, center, within which ideas are formed, retineantur; etsi anima ipsa sit immateriale received, retained. Even if the mind itself quid, tamen ideas esse corporeas & is something immaterial, nevertheless quamlibet determinatam habere figuram, ideas are bodily and, as much as you determinatum motus gradum, please, have a determined figure, a consequentur duas non posse una esse in determined step of motion. Consequently, eodem spatio; animam singulis momentis the two will not be able to be one in the aliquam harum idearum partim virtute same space. Any mind, in individual immediata, partim ope sensuum producere moments of these ideas, partly by & repositorio inserere, quod omnibus immediate strength, partly by the power of capiendis suffîcere calculo evincit; bas the senses, produces and inserts [ideas]
^ “Soul” is possible here since he refers to it as “immortal.” ^ 1 believe this is his way of stating that it acts as the sensus communis, “sixth sense,” coordinating organ for the other five.
126 ideas formari in centro versus into the repository, which by means of peripheriam, actionem animae eas capturing all [ideas], it suffices to conquer formantis esse id quod attentionem by calculation. These ideas are formed in vocamus; the center [and moving out] toward the periphery, the action of the mind forming these is that which we call attention.
caeterum semel formatas formam & Otherwise, once the forms, by the passage motum successu temporis amittere posse, of time, are able to lose the form and alterari, deleri; ut autem percipiatur, id motion, by another they are able to be deberi animae radiationi, solis instar erased. Moreover, with the result that, as continue radiantis, dum scilicet in spiritus it is perceived, it ought to be to the circumfluos agat, percipi nempe ab anima shining of the mind, a single image tanquam resistentiam & reactionem. continuously shining, until of course it goes into the surrounding spirits. It is perceived, of course, by the mind as much as resistence and reaction.
Ad §3 On §3
De cerebri sensualis necessitate 1. ad Concerning the necessity of the sensual mensuram Tonorum. 2. ad motum in brain with respect to 1) the proportion cerebro. 3. ad intelligentiam motus of tones, 2) the movement in the hrain, harmonici. and 3) the understanding of harmonic movement
In hoc paragrapho cerebrum sensuale In this paragraph the brain sense is consideratur ut materiale instrumentum, considered as a material instrument, per quod ipsi Toni harmoniam efficientes through which the tones themselves are percipiuntur sub characteribus peceived as affecting the harmony under substantiarum materialibus. Feci the characteristic materials of the tentamen pro solutione Problematis substances. I made an attempt at a Berolinensis de fluido nerveo, ubi aliqua solution of Berolin’s^^ problem hue facientia possunt legi; Sed cum ne concerning the fluid nerve, where ipsa quidem turgescentia musculorum per somehow to this point, things created are fluida possit mere mechanice explicari able to be read. But since not even the (nam electricitas ipsa, per quam muscolus swellings of the muscles themselves tergescere constat, nondum mechanice through the fluid can be explained merely discussa est): multo minus illud ce- [72] mechanicly (for the electricity itself, brum [sic, cerebrum] sensuale mere through which the muscles agree to swell, constabit structura mechanica. was not yet dissipated mechanically), then
^ Possibly a reference to Daniel Bernoulli’s work on fluid mechanics, Hvdrodvnamica. (1738). See Daniel Bernoulli, Hvdrodvnamica. sive. De viribus et motibus fluidorum commentarii. Argentorati; Sumptibus Johannis Reinholdi Dulseckeri, typis Job. Henr. Decked, Typographi Basiliensis, 1738.
127 much less will the brain agree with mechanical structure merely sensually.
Nondum enim certum est, quod in For it is not yet certain, what was minimis suis natura mechanice & determined by the mechanical and Geometrice fînita sit, vel quod regulae Geometrical nature in its smallest [parts], motus simpliciomm corpusculorum sint or what rules of motion of the simpler prout regulae motus majorum. Ne ipse corpuscles are just as the rules of motion quidem ignis potest explicari mere of the larger [parts]. Not even fire itself mechanice. Quidquid vero sit de hoc, can be explained merely mechanically. repetendum est ex superioribus, animam However, whatever it is in this respect, necessario singula numerorum paria seu that it is repeating from higher rationes praesentissima vi sentire & [principles], the mind necessarily feels agnoscere. Jam vero animae cognitio in and recognizes individual pairs of hoc aevo characteribus symbolicis indiget, numbers or ratios by a most effective ad quos requiritur aliqua materiae power. Now, however, the cognition of subtilioris in cerebro mutatio, ubi the mind in this time requires symbolic vibrationes aëris cum reprasentationibus characters, according to which is required, motuum non abstracte sed physice in the brain, some change of more delicate existentium coaequantur. material, where vibrations of the air are regarded as equals of the existing ones with representations of motion—not abstractly, but physically.
Haec varia in cerebro mutatio secum A mutation conveys these variations into cognitionem quandam characterum vehit, the brain according to a certain cognition per quos vibrationes aëris musicae of signs, through which the vibrations of similem faciunt repraesentationem, qui the air make a similar representation of characteres videri possent, si quis music, which signs could be seen, if microscopice in cerebrum alicujus ad someone were able to inspect Musicam attenti introspicere posset. microscopically into the brain of another Assumo ab experientia intensitatem & attending to the music.^ I claim from gravitatem vibrationum & tonorum, nec experience that the intensity and weight of non distinctionem inter plures concinentes the vibrations and of the tones, and per Altum, per Bassum, Tenorem, certainly the distinction among the many Discantum, diversasque melodias consonances through the alto, through the singulatim observari ab anima. bass, tenor, and soprano, and diverse melodies are observed by the mind separately.
^ He he is positing a symbolic cognition of the mind, as if the symbols are literally present and observable in the brain.
128 Ex Bacone etiam discimus, intervallorum, Even from Bacon we learn that of the quibus ob gravitatis diversitatem toni a se intervals—with respect to which they invicem distant, specialem & mutally differ from themselves because of individualem, etsi non accuratissimam the diversity of the weight of the tones—a cognitionem in anima obtinere. Jam beauty and uniqueness obtains, even if [it observatio incrementorum & is] not an accurate cognition in the mind. decrementorum hannoniae non pendet a Now an observation of the increments or Tonorum distantia, sed ab intemo decrements of the harmony does not essentiali arithmetico numerorum, [73] depend on the distance of the tones, but on quibus dimetimur Tonorum distantias, the essential internal arithmetic of the consensu vel dissensu. numbers, by which [numbers] I measure the distances of the tones, by consonance or dissonance. sed tamen, cum harmonia ipsa sit Nevertheless, since harmony itself is invisibile quid, animae saltem cognitum, something invisible, at any rate, a & cum Tonis transitorium, nisi quatenus recognition of the mind, and since with vestigia in anima abstracta, ut ita dicam, respect to the tones [it is something] & spiritualia relinquit, ipsi Toni, transitory (except as far as the abstract harmoniam expressuri, mediante corpore vestiges in the mind), thus, the result is percepti, per hoc instrumentum & that I assert (and leave behind spiritual vasculum quasi quoddam ab anima non [matters]), that the tones themselves— possunt hauriri, nisi sub characteribus about to express the harmony, [being] intervallorum & distantiarum perceptions by a mediating body—cannot materialibus. be absorbed, as it were certain, by the mind through this instrument and small vessel, except under the material signs of the intervals and distances.
Quodsi jam concedamus, singulos Tonos But if we should now concede, that ab anima ejusque facultate quasi materiali individual tones are measured by the seu in materia, tanquam sua offîcina, mind, and by its faculty—as it were by operatura, instrumento quasi quodam seu matter or in matter, just as if by its offîce, pertica mensurari, & numerum mensurae by its operation, by its instrument, as if by postea demum ab hac animae rationali & something [definite] or by a measuring spirituali tradi, tria necessario admittere rod, and that the number of the measure is debemus: delivered, hereafter finally, by this rationality and spirituality of the mind— we would have to admit three [things] necessarily:
1) certam quandam mensuram seu, ut dixi, 1) some certain measure or, as I have perticam, ad quam Toni exigantur, 2) stated, measuring rod, according to which notabilem in cerebro mutationem seu the tones are examined; 2) a notable motum seu lineolam, qua ille mutation in the brain, either a motion or
129 repraesentetur, imo fiat; 3) cognitionem small line, by which that tone might be distantiae, in suo corpore obtinentis, represented; 3) an accurate cognition of accuratam, qua motum ilium anima obtaining distance, in its body, by which intelligere possit. Ad haec omnia the mind is able to understand that cerebrum sensuale supramechanica motion. According to all these things, the quadam vi insita debet esse brain ought to be controlled by a certain contemperatum, h.e. Monochordum supermechanical innate power sense, that quoddam characteribus distinctum debet is, a certain distinct monochord with esse in ipso cerebro sensuali. respect to the signs ought to be in the very brain sense.
Etsi vero contradicere videatur sibi Even if, however, it seems to contradict aliqualiter, esse in cerebro sensuali itself to some extent, it is in the brain distantiarum notas symbolicas, & tamen sense that the symbolic signs of the harmoniae intuitivae sensum non ex distances, and still the sense of the distantiis emergere, tamen huic difficultati intuitive harmony, emerges, not from the sic occurritur, dicendo, ex una parte distances. Nevertheless, thus it occurs animam in cerebro sensuali, tanquam in with this difficulty, in saying that, on the extenso per distan- [74] tias jam distincto one hand, the mind is everywhere present esse ubique praesentem, & (Ustantias a se in the brain sense, as it were, in extending ipsa metiri a centro sui ipsius, sicque through distances, it is now everywhere harmoniam per calculationem sibi present by separation.^^ And the distances propriam ju^care ad mensuram 2^:3^; are measured from each other from the center of themselves, and thus the harmony through a calculation unique to itself Judges according to the measure 2^3l ex altera parte quum dentur in cerebro On the other hand, since the distances are sensuali distantiae per V9 unius vibrationis given in the brain sense through V 9 of the soni, seu quum justae Tonorum distantiae whole vibration of the sounds, or since the velut in monochordo jam signatae sint in just distances of the tones [are given], as it illo, dicendum, cerebrum sensuale were, on the monochord, [which] are now numéros primos, scilicet summas signs on that [monochord], I assert that, Tonorum uniuscujusque intervalli the brain sense expresses the prime exprimere ex tactu clavichordii jam sibi numbers, of course the highest [numbers] insiti, & non ex repetita ilia distantia V9. of the tones and of each and every interval, according to the clavichord touch, now innate to themselves, and not from that repeated distance V9.
^ I believe he is refering to an earlier issue—the mind being located at one point, and yet able to affect things at a distance. This touches on Gadamer’s point about Oetinger’s use of “action at a distance,” based on Newton’s theory o f gravity.
130 Ad §4 and S On § 4 and 5
Ex differentia simpiici intervallorum Harmony is not judged from the simple harmonia non Judicatur, sed ex difference of intervals, hut from the compositis rationibus. composite ratios.
Venilamius ex distantia judicavit Vemlamius^’ judges the harmony from harmoniam ita: the distance thus:
I I f z 3 3t 4 t 4 f fi 6 * H f V f i a \ i f T& i C C is D D is E F Fis G Gis A B H C Seamd. Tcrt. Quait Quint. Sen. Snt. Sept. CA. . . . «*»• "«/•
Figure 35: Francis Bacon’s representation of the muscial ratios.
At ex distantia falso judicatur harmonia; And the harmony is judged falsely on the Hoc tantum ex distantia percipitur, ut basis of the distance, namely, that the tone Tonus sit acutus aut gravis, & quanto sit is high or low, and by that amount one acutior Tonus unus altero. Hanc tone is higher than another. Verulamius distantiam Venilamius semper metitur ex always measures this distance from the ilia Toni idea, quod is sit intervallum inter former idea of a tone, which is the interval C & D ; atqui C ad D est uti 8 ad 9 quod ex between C and D. However C to D is as 8 monochordo apparet, assumendo chordam to 9 which is apparent from the plane liberam pro 9, quando subtenditur in monochord, by assuming the free [not 79, vel nona sui parte, tum 8 reliquae fretted] string clearly as 9, when it is partes sonantes sunt Secunda; stretched in 1/9, or a ninth part of it, then 8 remaining sounding parts are a second.
Ita Tertia, Quarta, Quinta &c. in Thus a third, fourth, fifth, and so on ought monochordo etiam observare debet to be observed even in the monochord in propriis manibus & oculis, ut vibra- [75] your own hands and eyes so that the tionum proportio fundet notiones primas proportion of the vibrations establishes the in Musica, alias sine his nihil intelligitur prime notions [ideas] in music. In other in tota hac Analysi Musica. Caeterum words without these nothing is understood vibrationes Toni acutioris D esse ad in this entire musical analysis. Otherwise,
Refers to Francis Bacon, see frontpiece of Novum Oreanum. See Francis Bacon, Novum organum. English. The new organon. Eds. Lisa Jardine and Michael Silverthome. New York: Cambridge University Press, 2000.
131 vibrationes gravions C eodem tempore the vibrations of the higher tone D are in absolvendas, ut 9 ad 8, dudum relation to the vibrations of the lower tone mathematice certissimum fecit Eulerus. C, acquitted at the same time, as 9 to 8, Sint ergo vibrationes soni long since demonstrated mathematiclly by Euler with the greatest certitude. Therefore, the vibrations of a sound are
c D
faciet ergo D impressionem ex ductu therefore D makes an impression from the nerveo in cerebrum tantam nerve duct into the brain of such a size
I a 3 4 f ^ 7 8;
& C tantam: and C of such a size
Figure 36: Graphic representation of the 9:8 ratio as a division of a string; then the individual eight and nine parts of the string, respectively.
differentia harum impressionum Vg lineae The difference of these impressions sets ultimae sistet in cerebro ideam gradus up the idea V, of the highest line in the Toni (dicunt enim sonos C & D Tono brain of a step of a tone (for they say that différé) seu Secundae. Jam edantur una the sounds C and D differ by a tone) or of soni CEO, tum C edet vibrationes tales a second. Now at the same time that the sounds CEO are emited, C will emit vibrations like
132 o I 2 3 4 f ^
Etales Elike
o I 2 3 4 f
& G tales. and G like
0 12 3 4
Figure 37: Graphic representation of the individual notes in the chord C - E - G as parts of a string.
sedrelataeadprioremToni ideamC &E But [by] the relationships to the previous different 2 gradibus, E & G dimidio & idea of a tone, C and E differ by 2 steps, E uno (quod in Clavichordio facile est and G by one and a half (which in the videre); erit ergo C ad E bis ut 8 ad 9; h.e. clavichord is easily seen). Therefore C to ut 8 8 ad 9 9 seu 64 ad 81; h.e. fere ut E will be twice as 8 to 9, that is as 88 to 64 ad 81; h.e. fere ut 64:80 = 4:5; cum 99 or 64 to 81. That is almost as 64:80 = enim Vg quae est mensura, qua Tonus 4:5. Since Vg, which is the measure, by indicatur, sit parvum quid, poterit Vg, which the tone is indicated, it is somewhat plane negligi, itaque anima ipsa pro ^/g, smaller, Vg, clearly will be able to be adsumit / s , fractionem seu rationem nunc neglected, and thus the mind itself ex corde ejusque rationibus geometricis assumes % for ^/g,, judging the fraction dijudicandam. or ratio now from the string and its geometrical ratios.
Quatenus ergo soni vel acutioies sunt vel Therefore, in so far as the sounds are graviores, eatenus anima sal- [76] tim either higher or lower, to such a degree inquirit rationem tanquam distantiarum the mind of course seeks a ratio, as it seu intervallorum ope intervalli C-D unius were, of measuring the distances or Toni dimetiendorum, quatenus vero in intervals by the power of one tone interval inventa hoc modo ratione valde composita C-D. However, in so far as in inventing, aliquid negligere licet, eatenus ilia ratio by this manner, a ratio greatly composed, transit & evadit in aliam, per secundum it is permitted to some degree to neglect, Arithmeticae genus Geometriae scil. to such a degree, that ratio which changes deinde a corde considerandam. and escapes into another [ratio], by considering the second type of arithmetical geometry, of course, then by the heart.
133 Impedimentum est omnino maximum, si The impediment is entirely great, if I wish harmoniam ex ilia mensurandi ratione, to judge the harmony from that measuring quam cetebro sensuali tribuo, velim ratio, which I attribute to the brain dijudicare, e.gr. Si dicam, Octavaest sense—for example, if I should say, “The consonantissima, quia ex Septenario octave is the most consonant, because it prodit; imo exemple possunt esse omnia comes from the Septanario.” Indeed, ilia Baconis loca: every passage of Bacon can be [explained] by a last example:
“In concentibus musicis, qui perfecti aut “In musical harmonies, which are called semiperfecti vocantur inter unitatem & perfect or semi-perfect between the unity diapasonta. Quinta omnium perfectissima and the octave, the fifth stands out as the exsistit, deinde Tertia, Sexta magis tetrica, most perfect, next the third, to a greater & judicio vetenun, ut & meo aliorumque extent the rough sixth (even by judgment Quarta, quam diatessaron vocant.— in of the ancients, and as even by me and of dissonantiis Secunda & Septima omnium others), and the fourth, which they call the maxime auribus ingrata occurrit, quarum diatessaron.^^ Among dissonances the prior unisonum proxime antecedit, second and seventh are the most posterior vero diapasonti proxime in unpieasing of all to the ears, of which the descentu succedit; unde apparet, quod former closely precedes the unison, but harmonia justam notarum distantiam the latter comes soon after the octave in requirat.” descending.^^ From this it is obvious that harmony demands a just distance between notes.
Jam quis legem hanc ex Bacone pro justa Now let one explain this law from Bacon notarum distantia explicet, ut a priori about the proper distance of the notes, so demonstret has propositiones: that it demonstrates a priori these propositions:
1) unius Semitonii distantia, ex.gr. C ad 1) The distance of one semitone, for Cis, H ad C pessime audit. example C to C#, B to C sounds bad. 2) unius Toni intervallum dissonat satis, C 2) The interval of one tone is dissonant adD. enough, C to D. 3) unius & dimidii Toni molliter se 3) Of one and a half tone[s] sneaks in insinuât, C-Dis. [77] itself [as] minor, C D#. 4) 2 Tonorum jucunde pénétrât aurem; C- 4) Two of the tones penetrates the ear E. pleasantly, C-E. 5) 2 V2 Toni interv. tetrice, tamen 5) Two and a half of a tone [is] rough, consonat; C-F. nevertheless it is consonant; C-F. 6) 3 Tonorum intervallum pessime 6) The interval of three tones beats the ear verberat aurem; C-Fis. badly; C-F#.
“ Possible ellipsis? ^ Descending down the scale.
134 7) 3 Vz Toni gratissima arte ei imperat. C- 7) The most pleasing of three and a half G. tone governs that art. C-G.
{Quae Verulamius ex distantia (Of which Verulamius judges from the intervallorum judicat, ea resoivuntur ex distance of the intervals, they are resolved mensura 8/9 multiplicata in rationes from the measure 8/9, having been compositas.} multiplied in the composite ratios.}
Atque haec Dn. Frickerus facillime And Doctor Pricker easily explains these explicat ex ilia de mensura % sententia, things from that thought concerning the hac ratione; Praemittuntur regulae measure %, by this reason: these general generates: rules are advanced:
I. Tonus est 2^:3^omniumque I. The tone 2^:3^ is the measure of all of intervallorum mensura, qua cerebrum the intervals which the brain sense makes sensuale utitur. use of. n. Cerebrum sensuale non agit nisi per Q. The brain sense does not function tertium Arithmeticae genus, scil. elevando unless through the third type of arithmetic, & extrahendo; quod fit, dum indices seu of course, elevating and extracting, which exponentes dignitatum multiplicantur & happens as long as the indices or dividuntur. exponents of the digits are being multiplied or divided. in. Auditus cerebro sensuali non affert m . Hearing by the brain sense does not nisi differentias, itaque cerebrum sensuale convey anything except differences, and iis numeris indices multiplicat & dividit, thus the brain sense multiplies and divides qui ex primo Arithmeticae genere & those numerical indices which flow from ordine 1 ,2 ,3 ,4 ,5 ,6 , etc. fluunt. the first type and order of arithmetic 1,2, 3,4,5,6, and so on.
IV. Rationem irregularem cerebrum IV. The brain sense easily permits the sensuale facile flecti patitur ad regularem, irregular ratio to be bent to the regular non multum differentem, quae deinde ratio, not differing much, which then cordi seu animae immateriali ad submits to the heart or to the immaterial harmoniam agnoscendam inservit. mind for the purpose of recognizing the harmony.
Nunc ad propositiones: Now to the propositions:
1) de C-Cis. Hoc intervallum est Vz 1) Concerning C-C#. This interval is a Tonus, he. 2^^:3^ = 2^^:3‘ = circa **/i 5 = semitone, that is, 2^^:3^ = 2^^:3* = about circ. = rationi satis compositae, ergo *‘*/i 5= about *Vi6 = close enough to the dissonanti. composite ratio, therefore [equal] to a dissonance.
135 2) de C-D, ipso Tone idem antea constat, 2) Concerning C-D, it corresponds with nam 8 & 9 sunt numeri pro nostro the tone itself, for 8 and 9 are not simple harmoniae sensu non simplices. [78] numbers according to our sense of the harmony.
3) de C-Dis, 1V2 = V2 Ton. = = 3) Concerning C-D#, 1V2 = % tones = circ. % = rationi simplici consonanti. 23.3/2.g23/2 _ about % = to a simple consonant ratio.
4) de C-E = 2 Tonis = 2^^:3^^ = ‘^/g, = 4) Concerning C-E = 2 tones 2^^:3^ ^ = circ. % = rationi consonanti. ^/gi = about /s = to a consonant ratio.
5) de C-F = 2 V2 Ton. = 5 = 2 ^ ^:3 ^^ = 5) Concerning C-F = 2 V2 tone = 5 = circ. % consonantiae, nam de Dissonantia 23 _ jQ about %, a consonance, Quartae §. 23. Analys. dictum. for concerning the dissonant fourth, it was discussed in the analysis, par. 23.
6) de C-Fis = 3 Ton. = 2^ ^3^^ = ^'^729 = 6) Concerning C-F# = 3 tones 2^^:3^^ = circ. - rationi maxime compositae. ^'^/729 = about ‘*^64 = to the most composite ratio.
7) de C-G = 3 V2 Ton. = % Ton. = 7) Concerning C-G = 3 V 2 tones = tones 237/2:32-7/2 _ 2/j _ rationi = = about = to the most simplicissmae praeter Octavam. simple ratio beside the octave.
Observandum hie, quomodo numeri ex Observing this, how the numbers from the primo genere Arithmeticae, in primo first type of arithmetic, obtaining statim sensu corporis seu cerebri immediately in the first sense of the body obtinentes, desum[p]ti adhibeantur or the brain, the selected [numbers] are immediate ad tertium genus elevationum, employed immediately according to the deinde computati, nec valde praecise third type of elevation, then [once] sum[p]ti, constituant rationes Geometricas computed (not begun precisely), they ab anima ex generali Euleri régula establish the geometrical ratios by the dijudicandas. mind, judging from Euler’s general rule.
Heic ergo numeri summas Tonorum in Here, therefore, are the highest numbers intervallis comprehensas indicantes 1V 2 = of the tones indicated in comprehensive V2, 2 ,2 V2 = V2, 3 ,3 V2 = V2, non poterant intervals: 1V2 = 2 ,2 V2 = k, 3,3 V2 = aliter adhiberi ad explicandam %. They could not otherwise be harmoniam, quam ducti prius in indices employed for the purpose of explaining potentiarum mensurae % = 2^:3^. Ergo hie harmony, than by the previously
136 est versus animae in corpore calculus: considered numbers in the indices of the nam calculus sine corpore est merus powers of the measure Vg = 2^:3^.^^°^ harmoniae sensus virium quasi jam Therefore this is the true reckoning of the calculatarum. mind in the body: for reckoning without the body is a mere sense of the harmony, as it were, actually, of the calculating powers.
Dantur distantiae in cerebro sensuali, Vg The distances are given in the brain sense, unius vibrationis soni C est id, quo Tonus Vg of one vibration of the sound C is that, notabilis fit; sed cum 2^:3^ talis sit by which the tone becomes a sign.^' But mensura, cujus exponentes seu indices since 2^:3^ is the measure of such, of tantum variabiles sunt, [79] quod tertiam which the exponents or indices are only Arithmeticae operationem requirit variables (because the third operation of maxime concentratam; non potest arithmetic requires the greatest calculatio modo facta nisi animae ipsi concentration), the calculation cannot tribui, ejusque vi immateriali, happen presently unless it is assigned to simplicissimae, actuosissimae. the mind itself, and of it to an immaterial, most simple, most active power.
Saltim dicendum est, cerebrum sensuale Of course it ought to be stated that the numéros primos, scilicet summas brain sense reveals the prime numbers, Tonorum uniuscujusque intervalli naturally, the highest numbers of the tones expromere, sed non ex repetita ilia and of each and every interval, but not distantia Vg, quin potius ex tactu from that repeated distance 1/9, but rather clavichordii jam sibi insiti; scilicet justas from the touch of the clavichord now Tonorum distantias velut in monochordo innate to itself. Of course, the just esse dimensas in cerebro sensuali per distances of the tones, just as in the praecipua lineae alicujus puncta; nam si monochord, are measures in the brain dicam, illam partem Vg esse mensuram sense through the particular points of Toni, ex.gr. & sic omnium, tum cogor some line. For if I should say that l/9th statuere mensuram illam esse variabilem, part is the measure of a tone, for example, & in valde acutis Tonis decrescere, in and thus of all [tones], then I would be gravibus augeri, quod definitioni compelled to establish that the measure is mensurae répugnât, quae ubique videtur variable, and decreased in very high tones, necessario esse eadem & sibi constans, increased in very low tones, which is nam inter c-d tantum dimidium ejus opposed to the definition of a measure, intercedit, quod inter C-D. which everywhere seems necessarily to be
^ I am taking this as a comparative, even though the syntax is not explicit. He is contrasting the true measure (8:9) with the “false” numbers (see first example in this section). Referring back to his symbolic cognition.
137 the same and constant to itself, for between c-d hardly half of it intercedes, as far as between C-D.^^
Ad §6 On § 6
Quatuor ilia momenta in unum Those four influences are reduced into revocantur. one.
Quatuor ilia momenta comprehendi Those four influences are able to be possunt hac figura: comprehended by this figure:
cer.inr. drc. cordis. cetftoC j,.a 1 , f v.li 4 64- 1 a'"' 1 4 3 1 .. 1 , Î 1 i A—• 1 3 2' ' 1 8 1 nwmor. cot
Figure 38: Table representing the physiological position of the various ratios.
[80] Spatium A est basis cerebri sensualis, The space A is the basis of the brain in qua mensura 8:9 = 2^:3^ dupliciter sense, in which the measure 8:9 = 2^:3^, expressa residet; ex hac non datur having been expressed twice, resides. transitas in basin cerebri intellectualis B, From here, the passage into the basin of nisi forte ita, ut non valores potentiarum the brain intellect B is not given, unless 2^, 3^, qui sunt 8,9, sed indices saltim perhaps, so that not the values of the separati in numeris 2 & 3 exprimantur. In powers 2^, 3^, which are 8,9, but separate spatio cerebri sensualis C 4 & 5 sunt indices are expressed in the numbes 2 and numeri ab extra recepti, qui secundum 3.^*“ ^' In the space of the brain sense C, communem mensuram 2 :3^expressi 4 and S are numbers received from mutantur in 2^^ & 3^ ^ hie ergo in B regnat outside, which, having been expressed
He is trying to account for the discrepancies between Just and mean-tone tuning. As he has stated before, 8 and 9 taken not as actual numbers, but taken as the indices of these numbers expressed as powers, i.e., 2 and 3.
138 numéros 2, potentiarom illarom 2^ & 3^ according to the common measure 2^:3^, indices mutans; are changed into in 2^^ and Here, therefore, in B the number 2 reigns, changing the indices of those powers Ÿ and3^
numeri vero 2^^ & 3^ ^ se ipsos in spatio D However, the numbers 2^^ and 3j2.2 '^ change . mutant in 81 & 64, nam 2^ =64, & 2^^ = the same [numbers] themselves in the 81; denique neglecta in hoc ultimo unitate space D into 81 and 64, for 2^^ = 64, and se resolvunt in S & 4, nam 80:64 = 5:4, & 2 =81. Finally having been ignored in se rite componunt in lineam rectam & this ultimate unity they resolve themselves radiantem, in qua intellectus a corde into 5 and 4, for 80:64 = 5:4, and illustrator per primum Arithmeticae genus accordingly they will compose themselves 1,2,3,4, etc. Hoc non obstante animae in a straight line and ray, in which the vis in sensu harmoniae se exserens intellect is enlightened by the heart circulum a ducit in circulum b altioris through the first type of arithmetic 1,2,3, ejusdem binarii potentiae. Si ex A in B 4, and so on. Since this [type] does not immediatus daretur transitus, quem oppose the mind, a power extending into coelestis Musica admittit, cognitio the sense of harmony itself leads a circle intellectualis & sensualis coinciderent. “a” into the circle “b” of a deeper power of the same binary. If the passage were immediately given from A into B, which the music of heaven admits, then the cognition of the intellect and of the sense would coincide.
Jam ad reliqua a Par. 8-20. nil habeo, Now to the remaining [thoughts] from §8- quod addam, quam hoc, ut rogem lectores 20. [About the remainder], I have nothing suspicionem materialismi abhorrentes, ne which I might add, [other] than this, that I ex terminis meris nos judicent, nec nisi invite the readers abhorring suspicion of rebus omnibus probe ponderatis de nostra the materialist, lest they judge us from sententia statuant. bare conclusions, nor lest they not establish all matters having been weighed properly according to our opinion.
(Ideae lineares, superfîciales & corporeae (Ideas can otherwise be denominated as aliter denominari possent.} linear, surface, and solid.}
Distinguuntur ideae in lineares, Ideas are divided into linear, surface, and superfîciales & corporeas. Possent aliis solid. They can be divided into other terminis distingui in abstractas, sui diffu- abstract limts, into their diffusions and [81] sivas & sui progenerativas. into their productions. We admit in the Admittimus in anima compositionem mind a composition of powers, but this
^ This refers to 4:5, as discussed earlier, being changed to 64:80 and compared to 64:81.
139 virium, sed haec nullum infert [power] infers nothing material. God materialismum; Deus enim vi sua provided deep within, by his own creative créatrice simplicitati & indestrucibilitati power, a simple and indestructable mind. animae penitus prospexit.
(Anima non est simplex absolutum ut (The mind is not perfectly simple as God Deus ipse; sed respectivum, quod himself, but respectively simple, which demonstrator 1. ex ejus finitudine} will be demonstrated, 1. from its finiteness.}
Anima est finitum quid, hinc non potest The mind is something rinite, hence it esse ens simplex tale, quale est Deus cannot be existing as such a simple thing infinitus. Si anima est taie simplex, quo as is the infinite God. If the mind is so Simplicius non datur, tum neque finita vel simple, by which a simpler thing is not gradibus limitata dici potest. Ipsa vis given, then neither can a finite or gradual repraesentativa esset immutabilis, omnis limit be asserted. The very representative animae divisio esset annihilatio. Nam si power is immutable, every division of the nulla compositio ingreditur essentiam mind is annhilated. For if no composition ejus, tum omnia quae in ilia sunt, simul undertakes the essence of it, then all the sunt, omnia sunt instananea. things which are in that are simultaneous, all are instantaneous. 35
Sicut ortus & intertius ejus concipi debet In the same way, the birth and death of it instantaneus; sic etiam momenta aut ought to be conceived as instantaneous. gradus in ilia debent esse aliquid Thus even movements or steps in those instantaneum. Jam vero certum est ought to be something instantaneous. animam esse finitum quid. Si hoc, non Now, however, it is certain that the mind potest esse sine quali, sine modo, sine is something finite. If this is so, then it mutabili. Si hoc, non amplius est cannot be without quality, without instantaneum, neque quoad ortum, neque measure, without changableness. If this is quoad interitum neque quoad momenta & so, then it is not more instantaneous, gradus, & sic non potest non habere neither as far as birth, nor as far as death régulas & modos successivos tum intra nor as far as movements and steps, and tum extra se. Sed quid juvant sola haec thus it cannot have rules and successive ratiocinia? measures first within, then without itself. But what assists these individual reasonings?
His argument is addressing, I believe, Leibniz’s monads, which are the most simple, complete, entity which comprise matter. There are also points reminiscent of Anselm’s argument (“that which nothing more perfect can be conceived ”).
140 {2. ex sacra Scriptura.} {2. from the sacred Scriptures.}
Jesus Christus nil pro nunciavit, iogw, Jesus Christ declared nothing on behalf of summae rationi non consonum. Dicit is the Word [that is] not harmonious with the Deum posse interimere non solum corpus highest reason. He states that God is able sed & animam in gehenna. Non potest to destroy not only the body but also the interfici anima ut corpus, habet igitur spirit in Gehenna.^^ The soul is not able immortalitatem at non ex intrinseca to be killed as [is] the body, therefore it natura. sed ex operatione Dei. Deus solus has immortality while not from intrinsic secundum sacram scripturam hàbet nature, but from the operation of God. afqarsian [82] & aqanasian intrinsecam, God alone, according to the sacred cujus nécessitas existendi sola infert vitam scripture, has intrinsic incorriiptihility and indissolubilem zwhn acataluton Ebr. 7. immoruiliiy. of which the only necessity of existence produces an indissoluable life— inUestnictahle life (Hebrews 7) 37
Reliquae creaturae omnes & quidquid All other remaining creatures and habitum vitarum habet, whatsoever holds a dwelling of lives, does indestructibilitatem non habent in se, sed not hold indestructability in themselves, in radice sua Deo. possunt alterari, but in its foundation by God. is able to be possunt solutionem pati. & tamen. quia in altered, is able to undergo a loosing, and Deo radicem existentiae habent, manent nevertheless, because they hold the indestructibiles, sed non immutabiles. nisi foundation of existence in God. they id fiat per gratiam. remain indestructable. but not immutable, unless it becomes so through grace.
{3. etiam ex principiis musicis.} {3. even from musical principles.}
His jam praesuppositis in lanta rerum Now with respect to these presuppositions immaterialium ignorantia possumus in such ignorance of immaterial matters, quaedam ex principiis musicis de animae we are able to acquire something from the natura colligere. multiplicationem musical principles concerning the nature quandam numericam contigere in anima, of the mind, to take hold of a certain in harmonia numéros valde crescere. ut numerical multiplication in the mind, to subito numerus 2 in potentiam 64 greatly attain numbers in harmony, so that elevetur, & vicissim 64 ad radicem 2 suddenly, the number 2 is elevated into revolvatur. the power 64. and in tum. 64 is resolved to the root 2.
Matthew 10:28, "And fear not them which kill the body, but are not able to kill the soul: but rather fear him which is able to destroy both soul and body in hell." Hebrews 7:16. "Who has become such not on the basis of a law of physical requirement, but according to the power of an indestructible life."
141 Possumus colligere confonniter Sacrae We are able to obtain conformity to the Scripturae animam esse in centro cor sacred scriptures that the soul is in what dicto, & hinc in cerebrum exserere has been called the “center” with respect potentias. Portasse potentiae ulterius imo to the heart, and hence exerts powers into plane extra corpus porriguntur certa lege. the brain. PeAaps the powers are Etsi nondum perfecte, fluida, nervos, extended by a further fîxed principle structuram mechanicam auris, cor, beyond, no more clearly, outside the body. cerebrum, animam ipsam quoad Psychen Even if not yet perfectly, we are able to & quoad pneuma possimus coordinare, ut set in order fluids, nerves, the mechanical videamus quomodo unum ab alio structure of the ear, the heart, brain, the dependeat, tamen sufficit hanc sententiam mind itself (as far as the psyche) and as cum conceptibus sacris multo melius far as the spirit, so that we see how one concordare, quam ullam aliam. depends on another; nevertheless it is sufRcient that this sentiment harmonizes much better with sacred concepts than any other.
Paulus verbo omnipraesenti tribuit Paul, by the all-present word, attributes divisionem Psyches & spiritus, nervorum the division of psyche and spirit, of nerves & fluidorum cerebri, a'rmw/n kai and fluids of the brain, to harmony and muelw/n, etsi non explicarit, quo pacto melody, even if he does not explain by hoc fiat. Ex Musicis porro discimus esse what manner this happens.^^ From in anima mensuram [83] quandam musicians, furthermore, we learn that a tonorum & harmoniae, in quem omnis certain measure of the tones and harmony multiplicitas trium Arithmeticae generum is in the mind, in which is resolved all resolvitur; multiplications of the three types of arithmetic.
{Fit regressus ad demonstrationem (We return to a demonstration of the necessitatis sensus communis ut in Musica necessity of the sensus communis', as it is sic in moralibus} in music, thus it is in morals.}
& talem quoque mensuram quoad res And also we take pains to demonstrate divinas & morales in anima & in nexu that such a measure by each [type], with animae cum corpore latere per integrum respect to divine matters and morals in the hunc librum ostendere allaboravimus. mind and in the connection of the mind with the body, works through this whole book.
Indeed, Paul does not even use the word harmony, and the reference Oetinger is citing cannot be construed in any way as a statement about the human psyche. Ephesians S: 19, “Speaking to one another in psalms and hymns and spiritual songs, singing and making melody with your heart to the Lord.”
142 Continua multipli harmonici in simplum A continuous reduction of a multiple reductio effîcit ut vel rusticus quid harmonies into something simple brings it consonet & quid dissonet statim percipiat about so that even a peasant immediately sine Syllogismis. Sic & simplicissimi, perceives something consonant and utilissimi & maxime necessarii indices something dissonant without syllogisms.^^ quosdam stabiles, per omne genus And thus we rightly assert that certain, scientiarum, sine Syllogismis animae stable, most simple, most useful, and most obversari jure asserimus. necessary indices, through every type of the sciences, are observed by the mind without syllogisms.
{Sensus communis elevatus Sapientiae (The sensus communis analogously Salomoneae analogus-) elevates the wisdom of Solomon.}
Sensus hic, si est summe elevatus per This sense, if it is highly elevated through gratiam aliquid Sapientiae Salomoneae grace somewhat analogous to the wisdom analogum, in Sacris Tuschiah dictum of Solomon, brings about what has been efficit. Scientiae semper fuere in mundo, stated in the sacred Tuschiah.^ The sed mox in inciemento mox in sciences have always been in the world, decremento constitutae. Quis fixum hic but have been built soon in an increase, dabit punctum ut mutabilia next in a decrease. Who will furnish here Philosophorum Systemata aestimare a fixed point so that the changing system possit & ad illud mensurae? Nonne of the philosophers is able to judge and consonum est rationi, si, quod in Literis ad measure according to that point? Is it not Sapientes indicatum est, hic iterum the consonance of reason, if, what has repetamus, Deum dedisse Salomon! talem been indicated in the letters according to Scientiae rupem, quae sub fluctibus the wise ones, we repeat here again, that temporum stetit immota. God gave Solomon such a rock of wisdom, which under the turbulences of the times stands immovable.
Mire triumphavit Leibnitius inventis suis Leibniz triumphed remarkably in his dynamicis; Sed jam arietat ilia & mire discovered dynamics. But now Lord conquassat Dom. de Maupertuis in suis Maupertius collides and remarkably Essais de Cosmologie. Acutus ille Vir unsettles the former idea in his Essays nititur ad simplicem quandam existentiae Concerning Cosmology. That wise man Dei demonstrationem, magis obviam advances according to a certain simple sensibus quam quae hucusque [hujus?] demonstration of the existence of God, data est. Nititur ad commune [84] quid, more against the senses than what was quod jam cemitur in mundo, & ad quod given thus far. He advances according to instinctu quodam trahimur omnes, sed something common, which is now quod scientifice non satis discemitur ob distinguished in the world, and according
A deprecation of the Scholastic method. ^ Most likely a Jewish mystic book related to the Kabbalah, in which Oetinger was interested.
143 Babelicam linguarum Philosophicarum to which by a certain instinct we are all confusionem. Ipsa Geometrica & derived, but which scientifically is not Arithmetica nondum ad rationes ere discerned satisfactorily on account of the metaphysicas revocata est. Babylonian confusion of the philosophical languages. Geometry and Arithmetic themselves have not yet been revived according to the truly metaphyscial ratios.
Credo simplicissima & utilissima nobilis I believe that the simplest and most useful figuramm situum & numerorum scientiae, of the familiar structures of figures and of quibus per sensum communem illico the science of numbers, into which we applauderemus, nondum esse in might immediately strike by the sensus propatulo. n i a vero Salomoni per sensum communis, are not yet in the open. elevatum fuisse notissima, However, that best-known [wisdom] of persuasissimum habeo. Arithmeticam Solomon having existed through the novenariam antiquis fuisse notam, elevated sense, I consider [to be the] most monumenta vetustissima produnt, persuasive. The new well-known Revocabitur ilia & suae simplicitati mathematics existed in antiquity; they restituetur nostris temporibus, quibus finis produced very ancient monuments. That redit in initium. [arithmetic] will be revived and restored to its smplicity within our times, within which the end returns to the beginning.
{Fit comparatio Matheseos Physicae & {A comparison of the studies of our Metaphysicae nostri aevi cum Sapientia ancient physics and metaphysics with the Salomonea, de qua dictum est fusius in wisdom of Solomon, concerning which it praesatione} is said that it is widely disseminated in the present.*” }
Instituam brevem comparationem I will prepare a brief more special Specialiotem Scientiarum modemarum comparison of the modem sciences with cum Sapientia Solomonis, mathematicae, the wisdom of Solomon, of mathematics, physicae, metaphysicae. physics, and metaphysics.
Primo de Mathesi nostrorum temporum ita First, I thus argue concerning the arguo. Per calculum differentialem hodie mathematics of our times. Through ex consideratione infînite parvorum differential calculus, today, from a eruitur ratio fînitorum. Infinite parva consideration of infinite little things, the habent rationem inter se, hinc non sunt ratio of limits is elicited. Infinitely small nihila. Differt nihilum & illarum ratio things have a ratio among themselves, fictarum quantitatum. Si nihilum est hence they are not nothing. A difference nihilum, ratio nihili non est nihilum. Si exists between nothing and the ratio of
That is, it is common knowledge today, which would tie-in with his notion of the sensus communis (common sense).
144 sciiemus figuras ordine, quo ex invisibili those false quantities. If nothing is exeunt, tum his fictionibus non haberemus nothing, the ratio of nothing is not opus, sed intuitus merus esset nothing. If we knew the figures in order, demonstratio. by which they go forth from the invisible, then we would have no need for these fictions, but this demonstration is a mere intuition.
Datur itaque perfectior Mathesis nostra, And thus our more perfect mathematics is dantur incognita adhuc Geometriae offered, the hitherto unknown principles principia, quibus quasi intuitu uno sensus of geometry are offered, by which, as it com- [85] munis elevatoris, supra were, by one intuition of the more consuetam mensuram hujus aetatis, sine elevated sensus communis, they can be ulla demonstratione maxima Theoremata examined above the accepted measure of perspici possint. Ad haec nititur calculus this age, without any greater theoretical differentialis; sed illis inventis hie plane demonstration. According to these, the esset inutilis. Principia ilia videntur nobis differential calculus advances, but with constare saltem in ordine figurarum, quo respect to these discoveries, this is clearly una alteram excellens praecedere debet, useless. Those principles seem to us to eumque ordinem Cluverus videtur docere correspond at least in the order of figures, voluisse, secutus punctorum numerum, where one excelling another ought to quo unaquaeque curva determinatur. surpass, Cluverus wishes to teach that it is in following the number of points, whereby each and every curve is determined.
Credo calculum differentialem, inventum I believe that the differential calculus (a in suo genere mirum, qua brevi remarkable invention in its kind, by calculatione generalissima inveniuntur which, with a small calculation, the most problemata, in aliam methodum general problems are discovered), can be generaliorem posse transferri, qua fere transferred into another more general nihil computetur, & quae se habeat ad method, by which almost nothing is ilium, ut iste ad communem Algebram, & computed.** This method holds itself to ut haec ad veterem; Sed novam illam that [method], as that [method holds methodum in eo omnibus multo fore itself] to the commen a]gebra, and as this excellentiorem, quod rursus pertingat ad [Algebra holds itself] to the ancient id, quod praeclarum fuit in veterarum [method]. But that new method, methodo, nimirum ut intuitiva maxime therefore, will be much more excellent in cognitione uterentur. everything, because it extends backwards to that, which is in the method of the
The original font becomes smaller at this point; most likely because the printer had runout of the previous size type. I belive this is a reference to Johann Bernoulli who worked on the calculus o f curves. ** Ashe states next, it intuits rather than computes.
145 noble ancients, without a doubt, so that it uses the greatest intuitive cognition.
Scivit Salomo simplicem hanc Mathesin, Solomon knew this simple mathematics, quoniam cognitio ejus fuit maxime because the cognition of it was very intuitiva, omnes disciplinas in uno intuitive, containing all disciplines in one. continens. Ignoravit fortasse ut He disregarded, perhaps, differential Archimedes calculum differentialem. qui calculus, as [did] Archimedes, who. comparative ad veterum Sapientiam est compared to the ancient wisdom, is a hominis quasi claudi fulcrum, an ftilcrum of a wavering man. Therefore, propterea cognitio Salomonis postponenda either the cognition of Solomon is est imperfectis aevi nostri scientiis? an considered secondary to the imperfect Salomonis Logica ex simplici ilia Mathesi sciences of our age. or the logic of fluens. fuit imperfectior nostra? plane Solomon, flowing from that simple non: Multis parasangis praeivit nostram. mathematics, was more imperfect than ours? Clearly not: many leagues^^ preceded"*^ us:
Secundo quod attinet Physicam nostri second, because it holds onto the physics aevi. Proh! quam ilia est imperfecta. of our age. Oh! that which is imperfect. Leibnizius adversatur Neutono. Neutono Leibniz opposes Newton. Euler opposes adversatur Eulerus. Eulerus Theoriam Newton.^ Euler explains Newton's Colorum Neutoni ex aliis plane principiis theory of colors from different clear explicat. Demonstrat lucem non emenare principles. He demonstrates that light ex sole, sed materiam ignis, lucis. does not eminate from the sun. rather, the colorum, jam ubique adesse matter of tire, of light, of colors, exists quemadmodum materia sonorum. Motus now everywhere, just as the material of tremulatorius seu vibratorius particularum sounds. The more trembling or vibrating aeris quae sonum producunt. ipsi est motion of the particles of air which celerrimus. Aer. ut fluidum elasticum. produce the sound, the faster it is with condensatur & rare fit alternative. respect to itself. Air. as elastic fluid, is Celeritas qua compressiones se mutuo condensed and rarefied, alternatively. The insequuntur. pendet ab elasticitate & speed by which mutual compressions condensatione conjunctim. nam est come after themselves, depends conjointly proportionalis radici quadratae elasticitatis on the elasticity and density, for [the divisae per densitatem. speed] is proportional to the square root of the elasticity, divided by the density.
45 Parasangis—Greek for the Persian farsang, a distance measure equivalent to thirty stadia. ' Praeivit, or praeevit. I am taking adversatur as the active of the deponent adversor and not the passive of adverso.
146 Una compressio non producit sonum, nam One compression does not produce a una si non ex- [86] cipitur a sequentibus sound, for one, if not followed by non efficit vibrationes in aeie, debent successive ones will not cause vibrations reiteratae compressiones efficere sonum. in the air; repeated compressions are Quaelibet particula aeris debet habere necessary to cause a sound. The particles motum vibratorium, ita ut motu altemo sit of the air ought to have a motion of rarefacta & compressa, & ut sensorium vibrations, so that by alternate motion auditus certo tempore recipiat there exists a rarefaction and compression, determinatam quantitatem compressionum and so that the hearing sense within a ex raefactione. Pendet abhinc differentiae certain time receives a determined soni gravis & acuti. Et talem in igne & quantity of compressions from a luce proportionem compressionum rarefaction. It depends, therefore, on the demonstrat, & colores esse dicit di^erentiation of the low sound and high reflexiones ex superficie particulis magis sound. It demonstrates such proportion of vel minus elasticis praedita. compressions in fire and light, and he asserts colors to be reflections from a surface furnished with more or less elastic particles.
Hinc corpora nigra dicit composita ex Hence he asserts that the black body is minus elasticis, qui absorbent radios sine composed from less elastic [particles], reflexione, corpora alba dicit composita which absorb the rays without reflection. ex particulis elasticis, quae se restituant ex He asserts that the white body is impressionibus indeterminate acceptis composed from elastic particles, which tanta vi ut in aethere proximo novos restore themselves from the radios producant, sic ut compressiones & indetermintately accepted impressions by impressiones se restituentes sint aequales. such power so that they produce new rays Corpora rubra sunt similia chordis in the proximate ether, so that the determinate tensis determinato numéro compressions and impressions restoring vibrationum ad consonantiam movendis, themselves are equals. Red bodies are adeo ut determinatus numerus similar to specifically [determinate] vibrationum lucidarum determinato stretched strings moving by a determined tempore illabatur in oculos. number of vibrations according to a consonance, exactly as the determined number of vibrations of light flowing into the eye by a determined time.
Vides Lector, Dlustrem Eulerum naturam You see, my reader, that the illustrious plane aliis oculis intueri, adeo ut attractio Euler looks upon nature with clearly other illi quasi evanescat. Sic & Dlustris de eyes, such that my attraction to him, as it Maupertuis in Essays de Cosmologie dicit were, disappears. And thus also does the famous Marpertuis say in his Essays on
147 se in Lapponia mille ignés mille modis Cosmology that he saw in Lapponia^* a motos vidisse; Elateres ultimos naturae thousand fires stirred in a thousand ways. desperat inventum iri. adeo ut quaestiones He despairs of discovering nature's Hiobo divinitus motas videatur ante ultimate 'drivers,'^^ to the very extent that oculos habuisse: In simplicissimis objectis he seems to have held those questions Dei vestigia se prodere assent, in primis divinely stirred by Hiobus“ right in front legibus Naturae praescriptis. quas tamen of his eyes. In the most simple objects of leges posse resultans quid esse ait ex God. vestiges assert to produce Natura & essentia corporum. themselves, in the first laws having been prescribed of nature, which laws nevertheless he affirms are able to be something resulting from nature and the essence of bodies.
Fatetur itaque Eulerus & Dn. de And thus he'' admits that Euler and Maupertuis ignorare se simplicissima & Maupertius themselves ignore the most primitiva naturae. Primam vibrationem simple and primitive [matters] of nature. elasticam, fontem caeterarum explicare. The first elastic vibration explains the multas difïicultates créât Eulero. Assumit source of the others, and this creates many incognitum scilicet motum vibratorium ut difficulties for Euler. He assumes, of cognitum, & ex elasticitate & densitate course, an unknown motion of vibrations hujus motus incognitum motum magis as cognition, and from the elasticity and déterminât. Brevibus: Compressio & density of it the unknown motion rarefactio in hac Theoria primae quasi determines to a greater extent the motion. sunt apparentiae. sed credo attractionem. In short: The compression and rarefaction gyrationem, aliosque motus primitivos his in this theory are. as it were, first esse socios. appearances, but 1 believe that the attraction, gyration, and other primitive motions are allies with these.
Jam supponamus Salomonem tam Now we suppose that Solomon’s rather ingeniosum calculum, cujus Neutonus, ingenious calculation, of which Newton Eulerus specimina dedere haud scivisse; and Euler have given proofs, was by no scivisse autem omnes illas Naturae primas means understood. On the other hand, all potentias compressionis, rarefactionis, of those prime powers of the nature of attractionis, gyrationis. [87] scivisse haec compressions, rarefaction, attraction, in ignibus tum consumentibus tum gyration were understood. It was nutrientibus. prout diversimode mixtae & understood in these fires, first in multiplicatae sunt; sci. visse [sic. consuming, then in nourishing, just as
Lapland: I believe he is referring to a northern region of Finland, near the Artie Circle, where one might see the Northern Lights. Literally "charioteers," Greek. ^ Possibly "Jehovah." the Hebrew word for God. Refering to Pricker.
148 multiplicatae sunt; sci. visse [sic, they are diversely mixtures and sc/vme] quomodo ignis simplex induat se multiplications. It was understood how a sub aquis tegumento ignis multiplicis simple fire covers itself under waters by creati; scivisse multiplicis reductionem in covering of a layered, changing fire. The simplex; reduction of multiples into [something] simple was understood. idque ex intuitiva tum simplicissimorum And it [is known] from the intuition of tum uniformiter compositorum & inde firstly the most simple [things], then resultantium legum cognitione; quae illi uniformily of compositions, and nescivere [nescire] scivisse ita, ut totam thereupon by knowledge of the resulting Naturae harmoniam in herbis. arboribus. laws. Which [laws] to them who did not animalibus & mineraiibus uniformiter know, were known,’" thus, so that he comparare noverit ex uno fonte; scivisse [Solomon] knew to compare the entire figurarum salium diversitates harmonicas, harmony of nature in grasses, trees, in quo teste Isaaco Hollando ex Respurio animals and minerals uniformly from one Cabalae Veterum rationalis explicatio source. The diverse harmonies of the constitit. Nonne dicendum est praestitisse figures of salt were known, in which the talem scientiam omnibus inventis nostris. explanation of the ratios agrees with the wittness of Issac Holland from the Rejeciion^^ o f (he Cahalla o f the Ancients. Was he not declaring to be exhibited such science in all ourdiscoveries[?]. [Yes.]
Tertio quod attinet ad Metaphysicam, plus Third, which pertains to metaphysics. 1 satis jam de ilia dixi. Nolo plura de ilia have declared more than enough now fari. Seculum nostrum in his est valde concerning that. 1 refuse to say more delicatum. Si vero priora se habent, ut concerning those things. Our secular dixi; si scientiam figurarum. ignium. charm in these things is great. If however, harmoniae corporum, inprimis the preceding things hold themselves, as 1 microcosmicae hominis tenuit; have said—if the science of figures, fires, metaphysicam certe millies bodies of harmony, held in the first men praestantiorem tenuit nostra. Metaphysica of the microcosm—then our metaphysics enim est scientia certa eorum quae ex holds certainly a thousand times more Physica resultant & in suum initium greatly. For metaphysics is a certain redeunt, ita ut t ô resultans principiatorum science of those [things] which result
" I believe he is referring to two passages from the book of R o m a n s : Romans 1:19-20. "Because that which is known about God is evident within them; for God made it evident to them. For since the creation of the world His invisible attributes. His eternal power and divine nature, have been clearly seen, being understood through what has been made.” Romans 2:14-15. "For when Gentiles who do not have the Law’ do instinctively the things o f the Law. these, not having the Law. are a law to themselves, in that they’ show the work o f the Law written in their hearts. their conscience bearing witness.” " Refers to Isaac Beeckman (1588-1637). ’ ■* Respurio = respuo.
149 arguat certitudinem principiorum. imo from physics W return into its beginning, recurrat circulariter in ipsa summa so that thing,^^ resulting, proves the principia invisibilia. certitude of the beginnings of the principles, more correctly, it returns circularly in its very own invisible principles.
Si nihil aliud quam linguarum If it were nothing other than the confusion Philosophicarum conftisio, ut Cartesii, of the philosophical languages, as Gassendi. Malebranci. Leibnitii & Decartes. Gassend. Malenbranc. Leibnitz, perpetuae altercationes inter illos de rebus and the perpetual altercations between Metaphysicis esset. de qua Illustris those [men] concerning metaphysical Maupertuis in supra translatis legendus. matters— concerning which the illustrious modestius de Metaphysica veterum Maupertius has been collected into the Hebraeorum statueremus. quippe qui tanta above translations'*—then we would se confusione non polluerunt. establish a more restrained [philosophy] concerning the metaphysics of the ancient Hebrews, certainly, who. such [things] they do not pollute by confusion.
Jam Metaphysica vera per illarum Now the true metaphysics ought to. again, linguarum confusionem demum iterum finally break through the confusion of perrumpere debet. Talis Metaphysica those languages. The following adjuvante Arithmetica Cosmo logica B.D. metaphysical, cosmological arithmetic of Bengelii. ruptis repagulis. Domino dante. Bengel. by helping, by destroying doors, rursus emerget. Tum agnoscent homines by the giving Lord, emerges again. First, tremenda Dei. per quae facti & generati men will acknowledge the awe of God. sunt. Ps. 139. tum trement etiam coram through which they are made and created verbo Dei cum laetitia. & superflua. non (Psalm 139). then they will tremble even necessaria. sponte abjicient. quibus in the presence of the word of God with hactenus obfuscata sunt necessaria. happiness, and superflowing, not necessarily, freely they will humble [themselves], by which, to this point, they are necessarilv confused.
Greek substantive [t ô ] or he is using the detlnite article for emphasis. Gadamer refers to Oetinger's numerous translations of other authors in his introduction.
150 PART n COMMENTARY ON THE VIEWS OF FRICKER AND OETINGER
CHAPTER 1
MATHEMATICS APPLIED TO MUSIC THEORY
Between the two of them, Pricker and Oetinger cover numerous topics and issues, from music theory to metaphysics. This commentary will focus on four major aspects of the treatise, covering the views of both Pricker and Oetinger; music theory and mathematics, music perception, epistemology and aesthetics, and the legacy of their ideas. Many of these ideas are clear enough from a reading of the treatise itself; however, several issues require elaboration. This commentary will also serve as a summary of their main views and ideas.
Throughout the commentary, the nomenclature for references to the treatise will be as follows: The “Part” number, designated by a Roman numeral, refers to the four main divisions of the treatise: Part I (Pricker), Part U (Oetinger’s commentary on Part I),
Part in (Pricker), Part IV (Oetinger’s commentary on Part IE)- Section numbers refer to the numbered paragraphs within each part.
The scale which Pricker presents is, essentially, the Just scale, but it contains some slight modifications; namely, he eliminates the Just scale distinction between the
151 minor second (25:24) and the augmented unison (16:15) as well as the augmented fifth
(25:16) and the minor sixth (8:5), opting for the latter, simpler one, in each case.
Furthermore, he uses the Pythagorean minor seventh (16:9) rather than the Just (9:5).
Based on his previous choices, it is not clear why he chooses to use the more complex
Pythagorean ratio in this case. Pricker acknowledges the practical limitations of the Just scale, the need for temperament, and the fact that additional ratios are used in practice.
The following table presents both scales (Pythagorean and Just) along with the modifîed version of Pricker.
Pitch: Pythagorean: (Calculation); Just: (Calculation): Fricker: C 1:1 1:1 1:1 C# 2187:2048 9:8 - 256:243 16:15 16:15 Db 256:243 32:27-9:8 25:24 5:3-8:5 NA* D 9:8 3:2-4:3 9:8 9:8 Eb 32:27 4:3-9:8 6:5 4:3-10:9 6:5 E 81:64 9:8+ 9:8 5:4 4:3-16:15 5:4 F 4 0 2:1-3:2 4:3 4:3 F# 729:512 (9:8)' 45:32 3:2-16:15 4502 Gb NA 64:45 8:5-9:8 64:45 G 3:2 3:2 3:2 G* 6561:4096 (9:8)* 25:16 3:2 + 25:24 NA* Ab 128:81 2:1-81:64 8:5 9:5-9:8 8:5 A 27:16 2:1-32:27 5:3 3:2 + 10:9 5:3 Bb 16:9 2:1-9:8 9:5 2:1-10:9 16:9 * B 243:128 2:1-256:243 15:8 2:1-16:15 15:8 C 2:1 2:1 2:1 Minor WT 10:9 5:4-9:8
Table 2: Ratios for two tuning systems and Pricker’s modifîed compilation. The diamond symbol [♦] indicates a deviation from the Just scale.
‘ Based, in part, on a chart presented in “Interval.” The New Harvard Dictionary of Music. Ed. Don Michael Randel, 1986.
152 Fricker bases his theory on this modifîed scale for psychological and philosophical reasons, as will be seen in the next chapter. Briefly, though, for him the
Just scale represents a psychological and mathematical ideal, more specifically, a schema against which actual music is compared.
Let us consider Flicker's mathematical concepts in more detail. First of all, he categorizes “arithmetic” (as he refers to it) into three types. The first is addition and subtraction, the second is multiplication and division, and the third is the use of exponents. Fricker demonstrates that he can derive all the necessary ratios by use of the first and second types, but is quick to point out that this is an inefficient process. As he demonstrates in Part I, Section 3 (p. 22), once the mind calculates and reduces the ratios through common denominators and the use of exponents, it can simply perceive the ratios as a series of integers. This is one of his explanations for how the mind can quickly make sense of such a complex phenomena as music.
In order to make this process even more convincing psychologically (his ultimate goal), Fricker wishes to establish the exponents as “hard-wired.”^ He accomplishes this by positing a musical basis: 1“ 2® 3^ 5' 7°, which has as its fundamental components the prime numbers 2,3, and S. Through multiplication of these numbers and their various exponential forms, all the necessary ratios can be supplied. The use of these numbers in
music theory was not new; René Descartes, for one, employed them in his Compendium
^ Obviously not Fricker’s term, but a contemporary substitute for “God-given.”
153 of Music (1618). For Descartes, it was only a passing comment. He mentions it once at the end of his calculations for the ratios, and makes nothing more of the matter.^
The next major fîgure to make use of the prime numbers was Leonhard Euler, in his Tentamen novae theoriae musicae [1739].^ Euler’s treatise attempts, somewhat, to be psychological in its treatment, but fails in two ways. First, it does not extend beyond some preliminary comments on the matter in the first chapter. Second, the application of his theory to music is so impractical that to claim it has merit psychologically would be difficult to support.
The gist of his theory is this prescient idea: “all combinations of tones are consonances.’’^ He means this in an almost Schoenbergian manner.^ As he states,
“Several tones occurring simultaneously constitute a composite sound which we call a consonance...we apply the term consonance to all sounds which consist of several simultaneous tones.’’^ Euler approaches the matter as a mathematician, not as a musician.
He recognizes the musical sense of the terms “consonance" and “dissonance,” but blurs the distinction with his mathematical approach. All intervals can be expressed mathematically (as ratios), and can then be further categorized by what amounts to a rating system, termed the “degree of agreeableness.’’ The higher the degree, the more complex the ratio is, and therefore, the more diffîcult it is to perceive.
^ .. .’The whole of variety of high and low in music is derived exclusively from the numbers 2,3, and S. All numbers are composed of these three; steps as well as dissonances can, by division of these three, be reduced to unity.” Descartes, p. 46. * See Smith as well as Robert Bailey, Music and Mathematics: an Interface in the Writings of Leonhard Euler. Diss. State University o f New York at Buffalo, 1980. ’ Smith, p. 12. * Arnold Schoenberg, Stvie and Idea. Berkley: University of Berkeley Press, 1984, p. 261. ’ Smith, p. 101.
154 For instance, from the fundamental note to the third partial, the relationship is 1:3, therefore, this is degree three. All series of numbers are related to unity, one. Four to one (4:1) is actually simpler than 3:1 because 4 is a compound of 2, so its degree is actually two. The formula for this is 1:P, where P is any number, and the degree for any number is calculated by “finding the sum of the prime factors of P and subtracting one less than the number of factors from the sum.”® For example, the degree for 1:72 would be calculated by first expressing 72 as primes: 2x2x2x3x3. The sum of these factors is 12, and the number of factors is S, so the degree is [12-(5-l)] = 8. What happens for a larger integer? The formula is expanded slightly:
1:25,200 is expressed in prime numbers as
l:2^x3^x5^x7' = [(4x2) + (2x3) + (2x5)+7]-[(4 + 2 + 2+ 1)-1] = [exponents multiplied by bases] [addition of exponents]
l:[31]-[9-l] =
1:31-8 = 23 Final: 1:23(degree23)
Then, there is the issue of a series containing multiple integers, such as 45,210,
336,500. The basic formula stands, but there is an additional step due to finding the
Lowest Common Multiple (LCM). Furthermore, Euler clarifies that l:p:p is the same as l:p, since the same number can be easily perceived. However, l:p^ should not be considered the same as l:p:p. In addition, if p, q, r, etc. are not prime numbers, then the calculation is not valid. Before calculating the degree, it is necessary to fînd the LCM, which Euler does as follows:
‘ Smith, p. 81.
155 1cm (45,210,336,500) = 2^ x 3^ x 5^ x 7 = 126,000
At this point, the calculation of the degree is as above;
degree(l:2^x3^x5^x7) = [(4x2) + (2x3) + (3x5) + 7]-[(4 + 2 + 3 + l)-l]
then, [8+6+15+7] - 9 = 27 Final 1:27 (degree27)
Euler becomes increasingly complex from this point on. Although he recognizes, practically, the limits of the musical ratios, his system is applied to large and unwieldy mathematical possibilities. In contrast to Fricker, Euler, in addition to using the prime number seven,^ raises the remaining three prime numbers to much higher powers. He also presents a system of “complete” and “incomplete” consonances, rules for voice- leading (which are abstruse and amusical), and, in the end, a rather unusable system for composition or analysis. In the words of the eighteenth-century mathematician Nicolas
Fuss, the Tentamen “had no great success, as it contained too much geometry for musicians, and too much music for geometers.” Sir James Jeans criticized Euler’s theory more particularly. His comments are apropos, and they are presented in full:
In 1739 the mathematician Euler attempted an explanation on psychological lines, saying that the human mind delights in law and order, and so takes pleasure in discovering it in nature. The smaller the numbers required to express the ratio of two frequencies, the easier it is—such was his argument—to discover this law and order, and so the pleasanter it is to hear the sounds in question. Euler went so far as to propose a defînite quantitative measure of the dissonance of a chord. His plan was to express the frequency ratio of the chord in question by the smallest numbers possible, and then to find the smallest number into which all these could be divided exactly. This last number, he thought, gave a measure of the dissonance of the chord. For example, the frequency ratio of the notes of the common chord C E G c’ is
’ Smith and Bailey both recognize that Euler is hesitant about this in the Tentamen, but reconciles himself to the idea in a later treatise ( 1764). Smith, p. 9
156 4:S:6:8. The measure of dissonance is accordingly 120, since this is the smallest number of which 4,5,6, and 8 are all factors. It is easy to criticize this theory from all sides. In the first place it fails to explain the facts, since it assigns the same measure of dissonance, 120, to the chord of the seventh CEGB (with frequency ratios 8:10:12:15) as to the far less dissonant common chord. Again if we put one note, say E, out of tune by one per cent of its frequency (about a sixth of a semitone) we increase Euler’s measure of dissonance 100-fold; if we now reduce the out-of-tuneness to a tenth of this, we increase the measure of dissonance another tenfold. If one note is only infinitesimally out of tune, the measure of dissonance at once shoots up to infinity, which is a complete reductio ad absurdum. Finally, Euler’s theory fails to explain why we enjoy hearing the common chord, with its 120 units of annoyance, when we could reduce the annoyance to 24 units by dropping E out of the chord, and could eliminate the annoyance altogether by sitting in silence."
Starting from Euler’s use of 2,3, and 5, Fricker then makes a significant departure from Euler’s method and limits the exponents available to each number, closer to
Descartes’ use.'^ This limitation is dictated by what numbers are required to obtain the
Just scale. The ratios presented in Part I, Section 1 (p. 19) can all be obtained by the products of the musical basis. In Part I, Section 8 (p. 28), Fricker demonstrates further, through an additional method, as to why the primes are limited. He multiplies the entire series (1” 2^ 3^ 5* 7°' together to create the product 2880. As just noted, this is finding the LCM for a series of ratio integers. For Fricker, he is demonstrating that his method creates a much lower LCM than Euler’s calculations, and is, therefore, a more psychologically valid system, because of its fewer tones (the “fifteen tones’’ of the scale which Fricker mentions). Herman Henck seems to concur with this view in his recent
“ Smith, p. 13 Herbert Henck states (p. 48) that Fricker had an ongoing “argument” with Euler. Apparently, Fricker wrote three letters to Euler during the period of 1753-1755. He does not state whether Euler responded.
157 article (based on another of Fricker’s manuscript works). But there are however some problems to be explained. This is demonstrated as follows.
Taking 2880 as the LCM for Fricker’s series, Euler’s degree is calculated by
2'*x3^x5‘
[(6x2) + (2x3) + (5)]-[(6 + 2 + l)-l] =
[23] - [8] = 15, therefore
Degree (2880) = 1:15
This number (1:15) matches the total number of tones possible that can be generated within the octave, and matches the total number of ratios possible in the Just scale, as presented by Fricker. Henck adds that fifteen is then reduced to twelve tones by
Fricker to match with the ideal octave. However, this process does not work for the expanded series, which would be calculated as follows:
2‘® X 3^ X 5^ X 7‘ (LCM = the product 65,536 x 243 x 25 x 7 = 2,786,918,400)
[(16x2) + (5x3) + (2x5) + (7)]-[(16 + 5 + 2 + l)-l] =
[64] - [23] = 41, therefore
Degree (2,786,918,400) = 41, not 157.
Actually, Henck (without elaborating the calculation) comments that this series would generate 165 tones, but that this number may be reduced to 157, similarly to the reduction of fifteen tones of the other series to twelve. However, neither number comes close to the previous procedure. Henck explains that the LCM is being reduced, somehow, to obtain these numbers. I assumed that it was by employing Euler’s degree method, because it worked for the Erst, but something else is occurring. It is likely that
Fricker derived the 157 tones similar to his method of obtaining the Efteen tones in
158 Section eight (p. 28). Here, he simply tallies the ratios in order, leaving out any that contain, seven, eleven, and thirteen, exceed one, or are smaller than one half. Therefore,
157 tones could simply be the result of a combination calculation.
Pricker’s use of the prime numbers (preceded by Descartes and Euler) and the use of the number seven, need to be clarified in the context of the senario. Smith argues that
Euler should be included among the list of theorists who extended Zarlino’s senario to include seven. However, this is comparison is not valid. The whole point of the senario is to show a simple elegance in demonstrating the consonances as simple number ratios based on an arithmetic series. Euler’s rather complicated theory, including his use of the number seven, goes far beyond any attempt to express the consonances simply.
Furthermore, the series of primes numbers (including seven) is not an arithmetic series, which is a fundamental requirement in Zarlino’s use of the senario.
Anyone, including Pricker, who recognized the use of the seventh is already surpassing Zarlino’s senario, because Zarlino went to great pains to avoid the seventh.
One must remember that, aesthetically, the senario was only designed to express consonances, even though, practically, other dissonances were used. Obviously, as Euler and Pricker demonstrate, any ratio or interval can be mathematically generated from the prime series; however, this does not mean, according to Zarlino, that it should be, and included in the senario.'^
I might be missing or miscontruing some fundamental point, but I think Lester’s argument in Compositional Theory in the Eighteenth-Century is valid (see Joel Lester, Compositional Theory in the Eighteenth Century. Cambridge, Mass.: Harvard University Press, 1992): we need to better recognize the historical context of earlier theorists’ work. Therefore, I believe that discussions which seek to “expand” Zarlino’s senario, based on later theorists’ work, are ignoring Zarlino’s aesthetic and even metaphyiscal purpose for the senario.
159 The use of 1" and 7° is a matter of philosophical preference. Both of these numbers simply equal one, or “unity,” which is a very old and venerable concept. By bounding his basis with unity, Fricker is appealing to ancient authority and metaphysical beauty. Fricker explains (Part I, Section 7, p. 28) that. T h e rule of limits, evidently, is this: every smaller number, should be raised to the degree, that it surpasses, by size, the product of the elevated factors hrom all the remaining larger numbers.” Therefore, the number two needs to be raised to the sixth power in order to surpass the product of nine
(3f) and five.
In Part I, Sections 10 (p. 31) and 17 (p. 47), Fricker deals with expanding the
series beyond the number five, that is, having seven be a functional part of the prime
series.'^ However, it is not the use of seven that is discussed, really, as much as it is the
use of five raised to the second power. When seven is raised to the power of one, the
boundary must expand, and the previous primes must now be raised to higher powers, in
order to “overcome” the product of the later numbers. Therefore, the new series would
become thus,
r,2 '\3 \ 5f,7',ll°
in order to obtain the upper “unity” boundary, as Fricker explains in Part 1, Section 10 (p.
31).
Fricker finds this series unacceptable from the standpoint of perception. He
states, “1“ , 2'^, 3^, 5 \ 7*, I i“, is not yet able to be taken hold of by the mind.” This
series allows for ratios to arise which would be so complex, that our minds would not be
160 able to resolve them. As Fricker concludes, “without a doubt, therefore, the music of heaven is founded by this basis.” In theory, he is not against this series as a possibility, but merely thinks that human perception has not attained the ability to grasp it, and that we must wait to hear it on the other side of the veil, as it were.
Although he objects to the numerous and impractically small pitch distinctions which would result within the scale, noting that over 157 tones would be possible, he recognizes the use of more complex ratios in the, as he calls it, “Italian style.” One example of this is in Part 1, Section 17 (p. 47), where he demonstrates the resolution of an Augmented 6* chord, making use of the 25:24 half step. However, he also notes that more complex rhythmic values are possible from this series, as he concludes in Part 1,
Section 10 (p. 32): “The Italians in like manner surpass the basis of music which pertains to the tactus.” This statement refers to an example discussed in Part 1, Section 17, where
3 \ or nine, is used within a rhythmic pattern. This raises two issues which need clarification: Pricker’s use of the word “melody” and “tactus” as well as the two separate musical bases: 1" 2^ 3^ 5* 7° and 2“ 3*.
Pricker’s first use of the word melody (Section 11, p. 33) is straight-forward: the use of “successive tones.” More precisely, it refers to any linear motion, melodic and/or rhythmic. This is contrasted with harmony, the “simultaneous” use of tones. He elaborates on these definitions by briefly discussing the importance of counterpoint and references Mattheson’s Vollkommenen Kapellmeister (1739) for authority on the point.
" Although seven becomes a theoretical possibility, it is not used in any calculation in Flicker's treatise; the expansion only makes direct use of five raised to the second power.
161 In Section 12 (p. 34), he adds an interesting element to melody, the mathematical basss of 2°° and 3*. Flicker's opening statement is important:
But now we are concerned with melody itself, which truly constitutes the whole art of music, and results from the composition of successive tones.
First of all, note that he believes melody to be the true “art” of music, an idea adopted from Mattheson. His next statement is equally interesting:
Successions infer duration, and indeed, the measured duration of each and every tone, which, through the notes, dots, and rests of the tones—properly observed in the composition of the songs—completes the entire melody.
Fricker believes that rhythm is the fundamental element of melody; the reminder of Section 12 simply outlines the basics of rhythmic notation. Where he is headed with this idea is to set up the analysis of the opera excerpts presented beginning in Section 14
(p. 36), where he shows a preference for solo singing with thorough-bass accompaniment as well as recitative. This explains his conclusion to Section 12 (p. 35): “Therefore melody is a new music...”
Melody as a “new music” possibly refers to the aria style current in his day, and particularly his take on the Italian style of it. However, Fricker’s statement is curious, because it was really not new. In fact, this concept of melody being the true art of music dates back to Giralamo Mei and the ideas of the Camerata. There must be something more to his term “new” than is first apparent. He expands on the importance of solo singing in the next section.
Thus, I now demonstrate more extensively that all music or song is—as it were—somehow persuasive declamation, as the skillful Mattheson clearly pointed out in the Kem Melodischer Wissenschafft.
162 Here, he is trying to account for the entire effect of all music as based in melody, and that music is an extension or expansion of speech—“persuasive declamation.” This terminology, again, calls to mind Mattheson’s arguments regarding the use of rhetoric in the structure of composition {exordium, narratio, propositio, etc.). Then, he continues with the following:
Since, according to the rules of musicians, all the linked numbers are based on 2°°and 3% this is the true melodic basis of music. But, it is apparent to the senses that melody is a new type of music, which the Germans are accustomed to call dry laughter, which is merely a kind of singing accurately expressed according to the measurements of individual musical notes as if in muted sounds where certainly [there is] no harmony of tones, and for that reason [and to that degree] only melody takes place.
First of all, he explicitly states that two and three are the true basis of music, supplanting the earlier series based on the prime numbers expanded by powers (1,2,3, S,
7). Actually, it is not so much supplanting the previous series, as it is providing the deeper foundation. Fricker had earlier stated that the original series was the basis of harmony:
Come then, let’s hasten to the source of harmony...Therefore, all music depends on the repeated multiplication and division of the three prime numbers 2,3, and 5, flowing in their order from unity. Granted, this is an easy observation, yet nevertheless, it rightfully should be called the basis of all music theory! [Part 1, Section 6, p. 27]
Hence, the Italiantalian style in melody, as it were, follows the true harmonic basis of musicic 2^ 3% 5‘... [Part 1,Section 17, p .48]
In both cases he clearly refers to this series as the harmonic basis, but note that he qualifies his claim about this series being the basis of all music by adding the word theory, which explains and limits its use to the ratios only, in the context of a harmonic analysis. His psychological use and distinction will be made clear in the next chapter.
163 Secondly, he expands somewhat on what he means by “persuasive declamation” by referring to “dry laughter,” that is, recitative, which is made clear by the remaining description. This is, for Fricker, the clearest example of the connection between language and music, since it most closely resembles speech.
In what way are the integers two and three the true basis of music? In the context of his examples, Fricker is discussing primarily rhythm. The numbers two and three are the simplest integers able to express rhythmic relationships. By using rhythm as the fundamental example of the true musical basis, he is furthering his argument (based on
Mattheson) that melody, “persuasive declamation,” is intimately linked to language.'^
This leads to his use of the word tactus. “Tactus” has, in this treatise (or, more properly, this translation) seven meanings: “stroke,” “rhythm,” “measure,” “beat,”
“meter,” “note,” and “taste” (the “sense”). One instance of the word is especially difficult. In Section 17, Fricker refers to a certain “military song” (p. 47) in the
Artaxerxes opera where “5 beats [tactus] make the principal phrase [propositionem] and also the greatest affection.” I have been unable to locate where this takes place in the opera, but he is apparently referring to either an asymmetrical meter (unlikely for this time) or (more likely) an asymmetrical phrase or motive. This example is referenced again in Section 25 (p. 68), where it seems, in the context, to refer more clearly to beats.
However, the mind, in melody, is not at any time able to collect 5 beats adequately enough, as the example in § 17 shows. On the contrary, 3 easily counts twenty times into the very main parts of the triplets, and 2 into uncounted successions—lest I say into infinities—partly in reckoning main parts of the beat, and less into all the main parts or individual parts from the innumerable vibrations of the sounds, partly
" As will be demonstrated, Fricker is taking the connection further. Mattheson means it more simply, in a compositional sense, whereas Fricker expands it to a psychological, even metaphysical, sense.
164 in reckoning the beats themselves, and comparing to their own proportions, for the acquiring a sense or fullness of the whole song.
Immediately after referencing this five “tactus” example from Section 17 (p. 47), he discusses how three provides the division of triplet beats, and two subdivides all other binary patterns. Therefore, since he here refers to the simplest case of ternary versus binary, it would be logical to conclude that in the preceding example the five tacti refer to beats.
In addition to viewing the second and third types of mathematics as more psychologically plausible (efficient). Pricker’s use of the ratio 8:9 = 2^3^ as a God-given measuring rod is unique. As he discusses in Part IQ, Section 6 (p. 102), “...The inference
[of this idea] is of the greatest importance, that the mind actually judges by the third type of mathematics (that is, by the elevation to powers), certainly as much as the musical basis admits it, so that, [it judges] the tone not from such numbers 8 and 9, nor by the products of 2 2 2 and 3 3, but are considered as powers 2^:3*.”
In other words, the mind has an instant recognition of these ratios as represented in exponential form. The bias he holds for the ratio 8:9, or 2^:3", being the measuring rod comes from the fact that this is the first ratio that demonstrates the utility of exponents.
That is, taking the ratios in order, 1:1,1:2,2:3,3:4,5:6,8:9, and putting them into exponential form, 8:9 is the first non-trivial example.
As Fricker explains, raising the number one to any power is ineffective. In addition, raising any number to the power of one is also trivial, because, for example, 2‘ is simply 2 1 or 3* = 3 I. Eight to nine (8:9) is the first ratio that demonstrates the
“advantage” (as Oetinger calls it) of exponents. One could argue that the simpler ratio
165 3:4 would demonstrate this point better, for 3:4 = 3:2f. Fricker rejects this for two reasons: fîrst, only one term of the ratio demonstrates the use of exponents; secondly, this ratio lacks a particular symmetry which Fricker finds attractive, demonstrated in the following figure:
a- 3 -
Figure 39: Inversion of bases and exponents in the ratio 8:9, expressed as 2 : 3
Here, bases become powers and powers become bases. The ratio 8:9 is the first and simplest ratio to have this capacity.
While Flicker's theory is psychologically skillful, intuitive, and even insightful, his method of examining music presents problems. This fact, however, should not diminish the value of his unique insights. The following criticisms are taken from the analyses presented in Part I, Sections 11 19.
The first criticism is the inordinately short length of his musical examples, such as that from the Graun aria of Part I, Section 14 (p. 36), which contains, to start, only three syllables of the melody. The second criticism is that he finds it acceptable to compare nonadjacent integers from a given series of musical ratios. There are several examples of this, but one notable one comes from Part 1, Section 15 (p. 41), where he rewrites a portion of music to fit his analysis. Thirdly, he is guilty of re-composing to suit the point
166 he wishes to make. This occurs in Part I, Section IS, where the rhythm of the musical example is changed slightly to obtain the desired ratios.
167 CHAPTER 2
THEORIES OF MUSIC PERCEPTION
Psychological Thought of the Eiehteenth-Centurv
In attempting to explain the processes of musical perception, Oetinger and Fricker claim to answer some of the major psychological and philosophical issues of their time, namely, the nature of the mind, ideas, perceptions, sensations, dualism versus monism, and empiricism versus rationalism versus materialism. In short, they are considering how one can establish knowledge of reality, that is, an epistemology (which will be dealt with in Chapter 3). To better understand the context of Oetinger's and Pricker’s claims, I will summarize the main psychological ideas of the primary philosophers of the seventeenth and eighteenth centuries. This summary is brief, because many excellent and detailed histories have already been written.
The psychological thought of this period is usually divided into three perspectives: rationalism, empiricism, and materialism. While these modes of thought did develop (roughly) in this order, the historian Daniel N. Robinson refutes the thesis that they developed simply in reaction to one another. Instead he characterizes them as
168 conflicting yet concurrent/ The rationalist perspective desired to establish one’s knowledge of reality on purely rational principles, such as those found in mathematics.
This approach was championed by René Descartes (1596-1650). Descartes began from doubt; he recognized the faultiness of perceptions in regard to external objects. He could, therefore, not trust them; instead, he relied on his most fundamental perception: his internal thoughts. Descartes reasoned that he could not escape the fact that he thinks, therefore, he exists (hence, the well-known expression cogito ergo sum), a conclusion which he refers to as “self-evident.” As the philosopher Stuart Hampshire states, “He could conceive of himself as not having a body, but not as not having a mind.”^
Descartes developed his theory of perception on this premise, that is, that the
mind (the only sure reality) exists separate from the body. This is the doctrine of dualism. Descartes realized that, practically, matter (the body) does exist, and that men
have shared perceptions. These facts are, however, dependent on the proof of God’s
existence, because only from this premise can we trust that we will not be ultimately
deceived about the nature of reality.^ Having separated mind and matter, Descartes
needed to furnish a meeting point for them. He posited that the mind and body meet in
the pineal gland, which was the organ where the soul and body could influence one
another. By means of “animal spirits” (purified blood) coursing through the hollow tubes
* * See Daniel N. Robinson, An Inteliecnia! History of Psychology. 3"* Ed. Madison: The University of Wisconsin Press, 1995, p. 149. ^ See Stewart Hampshire, The Aee of Reason. Boston: Houghton Mifflin, 1956, p. 64. ^ Hampshire, p. 61.
169 f the nerves, the soul pushes upon the muscles and creates actions. Fricker adopted a similar view; he refers to the tubes or ducts of the brain.**
The empiricists were monist in perspective, holding that the mind is an apparent phenomenon, and does not exist separately from the body. This view is associated with the “British empiricists,” most importantly, with Thomas Hobbes (1588-1679), John
Locke (1632-1704), and David Hume (1711-1776). What is fundamentally common to all three is that the ideas of the mind arise from experience, as opposed to being
“innate”—a thought inherent in Descartes’ theory, as well as in the theory of Oetinger and Fricker. The empiricists were also little concerned with the physiology of psychology, i.e., how these phenomenon are based in the body. Hobbes felt that ideas and perceptions arise from the motion of matter—atoms acting on the nervous system; in this, he was prescient of modem cognitive theories. Locke was even less concerned with physiological causes; he was not interested in “motions of our spirits, or alteration of our bodies.”^ Likewise attacking the notion of innate ideas, Locke proposes that we
suppose the mind [at birth] to be, as we say, white paper, void of all characters, without any ideas. How comes it to be furnished?.. .1 answer, in one word, from experience. In that, all our knowledge is founded, and from that it ultimately derives itself.^
Locke discussed sensation and reflection as the two classes of ideas formed in the mind. From these two simple classes, Locke claimed that they are all one needs to explain cognition (my word), no matter how complex. This is accomplished through the association of ideas, building the simple into the more complex. This is an idea shared by
* Part I, Section 10. * Locke, quoted in Morton Hunt, The Storv of Psychology. New York: Doubleday, 1993, p. 77. ‘ Locke, quoted in Hunt, p. 78. Hunt points out that Locke did not use the term tabula rasa, which was
170 earlier philosophers, including Hobbes, but Locke is credited with being the main proponent of associationism.
Hume also dismissed the “incorporeal soul,’’^ and supported the theory of Hobbes and Locke, that the mind consists of perceptions. Hume, like Hobbes and Locke, believed in the associativeness of ideas, but Hume developed the theory further by positing a “uniting principle”; the qualities of resemblance, contiguity, and cause and effect} The significance of this is that Hume viewed association as more than a mere chaining of ideas, but as a regulated mental process. However, he weakened the third quality, cause and effect, by characterizing it as a probability rather than a certitude; that is, we can never be sure about cause and effect, we can only take note that certain events are likely (probable) to follow others.^
Materialism is an intriguing blend of rationalism and empiricism, especially as embodied in the views of Gottfried Wilhelm Leibniz (1646-1716). It is not completely accurate to characterize materialism as this blend, and it is not entirely fair to call Leibniz a materialist, because he did not claim to be one. However, many of his time did consider him to be so and, significant for this discussion, Oetinger also did. Leibniz viewed consciousness on a continuum. He posited that there existed degrees of consciousness, down through animals, all the way to stones. Leibniz turned the dualism problem on its head: rather than thinking of mind as primary and matter as secondary.
actually Aquinas’s translation of a phrase from Aristotle. ’ Hunt, p. 85. * Hunt, p. 85. ’ This is a pragmatic position, similar to that taken by Niels Bohr in the Copenhagen interpretation of Quantum mechanics, which predicts the location and speed of fundamental particles as probabilities. This is a significant departure from the function of Newtonian mechanics.
171 Leibniz conceived of matter itself as conscious, and therefore as the primary thing. The fundamental element of matter, for Leibniz, is the monad. Monads are “conscious entities not extended in space and by definition indivisible."'" When enough of these are massed together, consciousness (what Leibniz calls apperception) arises.
Therefore, Leibniz presents a modified version of dualism: the mind exists as more conscious and the body exists as less conscious. According to historian David J.
Murray, “[The mind and body] run their separate courses but in absolute unison, much as two clocks wound up at the same time will both continue to show the same time afterward.”" This is Leibniz’s concept of “pre-established harmony," which God instills into the monads so that they might work in unison. Also inherent in this view are innate ideas, but Leibniz is specific about two: “whatever is, is" and “it is impossible for a thing to be and not be at the same time."'^ These two principles allow man to have an understanding of fundamental truths, such as those found in mathematics. In this matter of innate ideas, he agrees with Descartes and opposes Locke.
So far, I have primarily presented the rationalist side of materialism. The empiricist side is found in Leibniz’s respect for many of Locke’s ideas, such as associationism. Along with this is Leibniz’s acceptance of learned ideas, based on perceptions, as part of the process for the formation of consciousness. Christian Wolff later expanded this concept.
‘® See David J. Murray, A History of Western Psvcholoev. 2“* Ed. Englewood Cliffs, NJ: Prentice Hall, 1988, p. 127. “ Murray, p. 128. Murray, p. 128.
172 Where materialism differs fmm both rationalism and empiricism is in the fact that it seeks to explain the phenomena of mind and body based on qualities inherent in matter itself. Materialism drew criticism for Leibniz among the more conservative religious thinkers; God is removed from the equation. Most philosophers of the time wanted to view God as active in the workings of the universe and in the lives of men. The pre- established nature of the monads carries the thought that God was the impetus behind the universe, but he no longer interacts.
Christian Wolff (1679-1754) adopted many of Leibniz’s ideas as well as Locke’s.
His views continued a blend of perspectives, which is demonstrated in his distinction between a posteriori (learned) and a priori (innate) ideas. He felt that both categories are valid, although most ideas are acquired a posteriori. Wolffs two books on this subject, entitled Psvchologica Empirica (1732) and Psvcholoeica Rationalis (1734), captured this distinction, and affected the characterization of psychology (as either rational or empirical) into the nineteenth century. Wolf agreed with Leibniz’s solution to the mind-body problem (pre-established harmony), for which he was categorized as a
“materialist” by those that opposed him. Wolffs views were very influential in Germany but censured as “barren” by the Pietist leadership in Halle. As the embodiment of the
Auflkldrung thought, and its threat to conservative religious thought, Wolff was viewed as an enemy by many Pietists, including Oetinger, who ironically, in his earlier years, actually favored Wolffs ideas.
173 The Psychological Theory of Music: Fricker and Oetinger
Oetinger appended Pricker’s music treatise to his larger treatise in order to illustrate his views of the sensus communis, his central topic. First of all, what, exactly, was the sensus communis! Literally, this phrase means “common sense, ” but is not as trivial as the phrase has come to be used. Historically, the sensus communis has most often referred to a wide spread social “conscience," that is, the ability to make sound judgments. In addition, the sensus communis has also referred to perceptual abilities.
Both of these ideas can be traced back to Greek thought, particularly Aristotle, and
Oetinger’s concept included both of these aspects as well. Aristotle’s view also referred to an organ, or ability, which allowed one to perceive a combination of senses at the same time, such as “white” and “sweet.” This coordinating sense, for Aristotle, was something separate from the other senses.
According to the sociologist George Becker, Oetinger conceived of the sensus communis as, “a God-given sensory organ in every human.”*^ Becker further characterizes Oetinger’& view as an “inherent human desire and feeling for truth; [it] grasps in intuitive fashion the totality of every problem or situation.” For Oetinger, the sensus communis allows humans to perceive that which is “the most necessary, the most useful, and the simplest. ” Becker continues his description:
Comprised of a “complex of instincts,” the sensus communis operates at the very border between subjective and objective processes of knowing. It involves that feeling which, [quoting Oetinger] “precedes investigation and piece meal unfolding, and which brings with it certainty and assurance before one disentangles the distinguishing attributes.”
" Becker, ‘The Merton Thesis,” p. 651.
174 In other words, the sensus communis permits one to apprehend the essential truths, or characteristics, of a phenomenon outside of (possibly in spite of) reductive techniques. In the context of the Enlightenment, Oetinger keeps the door open to scientifîc investigation of various phenomena, but clearly feels that there is a higher,
God-given means of both guiding and judging the results of such investigations.
Why is music a good test case for the sensus communisl The simplest explanation is to point out the several times that both Oetinger and Fricker refer to music—a complex phenomenon—as being perceived and understood even by simpletons.
There are three specific instances;
.. .The sequence of those two strokes in § 3, which would be pleasing to ten peasants, since there is hardly one to whom it would be displeasing. (Fricker, Part I, Section 10, p. 31)
However, one should not think that the strained cognition of the philosophers, by constant abstraction, surpasses the cognition of the common people...(Fricker, Part 3, Section 18, p. 112
A continuous reduction of a multiple harmonies into something simple brings it about so that even a peasant immediately perceives something consonant and something dissonant without syllogisms. (Oetinger, Part 4, Section 6, p. 143)'"*
In addition, both writers make specific mention of the sensus communis in music.
Fricker specifically refers to the sensus communis in Part 1, Section 10 (p. 31), and actually identifies it as the “organ” with which the mind judges musical tones:
The mind carries the musical basis in its core, therefore, according to this it judges tones, and by it, the mind itself is limited, as seen in § 7 and 9. And this basis is the organ or instrument of the common musical sense...(Emphasis added)
Mattheson’s arguments about the perception of a “Thuringian peasant” are interesting in this regard. See Lester, p. 172.
175 Oetinger, in his commentary on the fîrst portion of Pricker’s theory, is also quite explicit in his reference to the sensus communis, even comparing its function with the other senses:
[One may] conjecture from music (and also from languages) on the nature of the mind, on the sensus communis, indeed, on the individual sensations.. .Perhaps colors are composed by God from the three primary light powers, as are the three prime forces, then seven, finally twelve, according to the type of musical tones...Thus taste and odor, perhaps, have in themselves the same composition from the three prime forces [the primes: 2,3, S]...Sharp, bitter, and sweet are, perhaps, the three primary tastes, from whence the remaining tastes are composed. Therefore, the mind itself, as it holds the basis of music, even thus holds the basis of colors, of odors, and the basis of taste in itself, not so much in the senses...The mind, perhaps similarly, holds these things in a certain, common, physical sense, as light holds the colors in itself, but in the senses the task is according to a peculiar modification. The common, physical sense is given, through which God provides for men, that without a tedious analysis they are able to Judge according to it, because it agrees with human nature.
Here, Oetinger builds not only on Aristotle’s conception of a coordinating organ, but expands the concept of the prime numbers, or prime “powers” concept to be the foundation of the other senses as well. His tone is admittedly conjectural, but he is clearly referring to the sensus communis.
Let us consider, in more detail, the workings of Pricker’s perceptual ideas. There are four main concepts to his theory: categorical perception, ratio perception (reductive techniques and memory), contrapuntal perception (bass-melody relationship), and symbolic cognition.
Within the field of psychology, categorical perception is defined as the mind’s ability to segment various stimuli into readily perceivable constructs or categories. It can
176 be thought as forcing stimuli to fit into a specific pattern or mold. Visually this is done, for example, with colors. Although numerous shades, such “sky,” “baby,” and “navy” can exist, they are often labeled within the same categorical color, “blue.”
Musically, pitches are fit into categories of pitch classes, depending on the scale used. In India, for example, musicians can distinguish twenty-two different pitch classes within the octave. In Western music, however, there are traditionally only twelve classes
(categories) of tones to work with in the octave. From the standpoint of human perception, the only limit to the number of pitch classes is the “Just Noticeable
Difference (JND).” The JND is the psychological measure of how small a change humans can detect between two sound frequencies. Although the JND differs across the range of human hearing, in general, above 1(XX) Hertz, it represents approximately a .25 percent change in frequency.
Categorical perception has an obvious survival advantage. While crossing the street, it is inefficient and dangerous to perceive, “red, tire, metal, glass, large, fast,” rather than to simply note, “car,” and move out of the way. This process is dependent, however, on experience and the ability to label stimuli correctly.
In the context of Pricker’s theory, categorical perception is the answer to one of his primary issues, which was mentioned earlier: how do people perceive such a complex phenomenon as music, especially uneducated and “simple” people? One version of the answer is presented in the very first section of his treatise. Here, Fricker discusses the
“bending” of certain ratios to fit within the Just scale that he has just presented.
Therefore, although the tritone can exist in two different forms, 45/32 (augmented fourth) and 64/45 (diminished fifth), the mind practically ignores this distinction and simply
177 hears a tritone. Similarly, while the major third is properly calculated as two major seconds (81/64), the mind perceives the Just ratio (S/4 = 80/64) as close enough.
His most direct statement of this process is given in a summary paragraph of
Section 20 (p. 57):
The mind compares the innumerable sounds in music between themselves...although the mind mixes together the tones between themselves—for example and ^^/g»—nevertheless it recognizes either a consonance or dissonance in the tones. It does this from the original numbers of those same ratios, that is, by the prime numbers 2, 3, and 5. Thus, the result is that it distinguishes those tones from themselves, one after another, as specifically as possible. Nor clearly, does it mix together any ratio among itself, but rather, the mind separates those same numbers, even in very large powers or their products, where they are mixed together between themselves, and compares each one to their own class. (Emphasis added)
His basic assumption is a valid psychological idea (although the processes can be questioned), and is a necessary one if he is to account for the musical practices of his day.
In actuality, neither the Pythagorean nor the Just scale were in use, because they would not allow for modulation. Fricker recognizes the need for temperaments, and the use of ratios beyond what Pythagorean numerology would allow, and, therefore, uses the Just scale as a psychological ideal, a categorical construct. He does not need to argue about its impracticality, because he is not claiming its actual use in music. He admits that, “in
[music] theory they use, even now, the smaller whole tone, 9/10” (Section I, p. 20).
In summary. Pricker’s use of categorical perception is rather ingenious. The goal of any theory is to explain the data at hand in the simplest manner. Fricker claims that the mind is processing an enormous and varied amount of stimuli (numerous intervals and mathematical ratios as well as changes in tunings) by reducing everything to a relatively simple construct. As he states, “...All tones of music ought to flow from the
178 basis 2^, 3^, S'. Those ratios obtained from this basis...truly ought to constitute the storehouse of music” (Section 26, p. 75).
This idea of reduction leads to the next main point of Pricker’s theory. In order for the mind to perceive music quickly, the sheer volume of sounds hitting the ear needs to be processed quickly. This technique was demonstrated most clearly in Section 3, where Fricker shows how the mind reduces the ratios of a C7 chord resolving to a F chord, through the common denominator 36, to the series 1,2,3,4, S. The essential idea is that any series of numbers or ratios can be multiplied, divided, increased, decreased, and reduced to the simplest relationship. As Fricker states:
Therefore, what is the activity of the mind in hearing and judging music in one reflexive motion? The multiplication and division of simultaneous and successive tones in a continuous comparison, or the reduction of its fractions to larger and smaller limits—in so far as a distinct perception of individual [ratios] must be drawn out...
Willingly, the ratios of the tones, which had increased to large numbers, arrange themselves a second time, and immediately reduce themselves to the simplest numbers by alteration of one or the other tone... (Section 5, p. 26,26)
Therefore, all music depends on the repeated multiplication and division of the three prime numbers 2,3, and 5, flowing in their order from unity. (Section 6, p. 27)
Again, the basic assumption is sound. In some manner, the mind must be
reducing and forming the stimuli into a percept that one can recognize and label as a
“tone,” or “melody,” or “rhythm,” etc. Current psychology, specifically cognitive
psychology, would take issue with such a simplistic model, but the basic principle Fricker
is driving at has validity. Cognitive psychology has been focusing on neural networks as
the mechanism for explaining the process by which the brain forms a percept of pitch,
179 rhythm, melody, etc. The calculations required to activate weighted nodes, learning, and forward or backward propagation could be viewed as simply more sophisticated models of the essential problem that Fricker clearly recognized.
Fricker advances his model, somewhat, by positing a more basic perceptual mechanism than even the series 2^, 3 \ 5'. As he argues in numerous sections throughout both parts of his treatise, he feels that the interval of the whole-tone (expressed by the ratio 8:9) is the ultimate standard by which tones are measured. His argument, as presented in Part ID, Section 6 (p. 102), is that the mind has an instant recognition of this
“tone not from such numbers 8 and 9, nor by the products of 2 2 2 and 3 3, but are considered as powers 2^3^,” that is, through the highest type of mathematics (the third), i.e., exponents.
Why introduce this? I believe that he is trying to explain how the mind simplifies and speeds up even more the mechanism by which it perceives musical elements. As a
“God-given” construct, it is a further means of explaining a human’s remarkable perceptual abilities, and can be posited to exist in the population at large, thereby connecting to the larger idea of the sensus communis.
Related to this idea is his further reduction of the musical basis 2^, 3^, 5* to merely 2" and 3 \ as discussed in the previous chapter. This argument is not qualitatively different than the former, but is an extension of the desire for more simplicity. As argued above, Fricker is presenting a basically sound argument, and it could be argued that since these extensions of the argument are not essentially different, they are still sound in principle. However, there are two weaknesses to these arguments. First, by characterizing the 8:9 measure as a God-given component, Fricker has removed the 180 argument from falsifiable investigation and relegated it to metaphysics. Second, positing the further-reduced basis of 2°° and 3^ as a psychological idea, based on Mattheson’s theory of Melody, is using Mattheson’s argument in a manner for which it was not designed.
Another component of Pricker’s reductive technique is the role of memory in perception. Pricker’s various musical examples and analyses often rely on taking noncontiguous notes, integers, and ratios as somehow contiguous in the mind. This
practice was briefly criticized from a musical standpoint at the end of Chapter 1, but
needs to be considered in more detail psychologically.
Pricker states the role of memory unequivocally in Part 1, Section 16 (p. 43),
where he marvels “that the mind is able to hold—by means of the memory—some large
series of so many numbers. ” This statement comes on the heels of a demonstration which
regarded nonadjacent notes perceived as if they were adjacent. Again, musically, support
or criticism for this analytical practice is varied. Schenker certainly takes nonadjacenies
as theoretically valid and numerous Set-class analyses employ the same practice, but
what is the psychological validity of these practices? That issue is also debated.
Therefore, looking strictly at the psychological side of Pricker’s demonstrations,
can it be argued that this is a valid practice? Certainly, to a point, it can. Non-linear
processes in cognition are a fundamental part of neural networks. Associative memory
depends on the brain’s ability to link disparate memories and concepts. Abstract
reasoning also is dependent on one’s ability to connect various concepts and memories.
Does this apply to real-time, environmental stimuli, though? It can be argued that
it does. Musically, performance data, specirically that studied in serial-order mechanism
181 studies, demonstrates that performers'^ (and, presumably, listeners) plan several steps ahead, even in a chain of stimuli. When errors are made, researchers refer to them in three basic categories of perseverations, anticipations, and exchanges, a fact which implies that the serial order can be, and often is, broken.
What would be the limit of such linear versus non-linear cognition? With regard to music, psychologist Caroline Palmer has consistently found the phrase boundary to be one such limitation on cognitive memory and planning. Numerous other memory studies have found various time or data limitations as well. In this regard, Schenker's analytic theory has been criticized because it has been demonstrated that the average listener, even musicians, cannot hold extraordinarily long passages of music in their memory, thereby affecting their ability to hear the long-range connections he argued for.'^
Another instance, more closely related to Pricker’s examples, is the work of the psychologist Carolyn Krumhansl. She has argued that listeners discern the key of a tonal work through what she refers to as a “tone proRle" A sufficient amount of tonal, musical stimuli statistically arranges itself into a pattern that highlights the relative importance of scale degrees. The result closely follows the theoretical expectations of the tonic being most prominent and the dominant next so. Music theorists David Butler and Helen
Brown have criticized Krumhansl’s work on the basis that it relies too much on memory, and does not allow for an instantaneous perception of the tonal center. They argue.
" Most of these studies look at piano performance, typing, and speech. “ This criticism could be considered a straw-man argument, since Schenker did not claim his theory to be a cognitive theory, but an aethetic and compositional theory. While he did make claims regarding perception, his arguments cannot be taken in an overtly scientific manner. This is Eugene Narmour’s mistake in Bevond Schenkerism. See Eueene Narmour. Bevond Schenkerism: The Need for Alternatives in Music Analvsis. Chicago: University of Chicago Press, 1977.
182 instead, for the “rare-interval hypothesis,” which claims that rarely occurring intervals, such as the tritone and half-step, are more critical in recognizing the tonal center, without the mind having to wait and perform a note-count.'^
This same criticism could be leveled at Fricker. While certainly the mind does use memory in perception (remembering themes, etc.), it is somewhat musically naïve to argue that just because two, three, or four notes occur in a row, that they all can and should be regarded musically. Fricker provides no justification or limit for why certain relationships would be perceived as such, other than the fact they strike the ear at some point.
The third primary aspect to his theory of perception is the contrapuntal relationship between two or more voices. One could call this a form of “chunking” theory, to borrow the term from Chomskian linguistic theory. The key idea is that the mind is assisted in perceiving complex lines of music, particularly the coloratura solo lines of opera, by the slower, more harmonically fundamental bass (continuo) line. In this way, as used by Fricker, the slower bass line segments the soprano line into smaller rhythmic groupings, which enables the mind to grasp them more easily.
This psychological concept can be related to some of the principles of Gestalt theory, such as “similarity.” Visually, the brain groups stimuli according to items which are perceived as similar. This allows for quick discernment of patterns, such as the‘T ’ in the image below.
" See David Butler and Helen Brown, "Tonal Structure Versus Function: Studies of the Recognition of Harmonic Motion.” Music Perception 2 (1984): 6-24, and Carol Krumhansl.
183 $ $ « f) «1 f) ##0 0 O 0 0 # «« 0 « e «««0 « # « e €> 0 €> e e # e e $
Figure 40; Example of visual Gestalt demonstrating the principle of similarity.
Analogously, the mind will group “like” musical stimuli together. This concept, discussed in Part 1, Sections 14 (p. 36), 23 (p. 62), and 24 (p. 65), exhibits good musical intuition. Since the late Renaissance, the bass line assumed a increasing importance in music theory and practice. Figured bass theory had long recognized the function and importance of the bass line, and with the advent of Rameau’s chord theory in 1722, the bass took on an increased importance, as it helped defîne the “root” of the harmony.
By incorporating the relationship between the bass and upper voices into his theory, Fricker adds an important practical element. Music theorists since Rameau’s day have used the bass as a fundamental part of theoretical training. As the foundation of harmonic progression, it is used to explain how chord progressions should be identified aurally. Rather than simply hearing a series of four individual lines, students are usually taught to hear the bass line fîrst, then add the remaining voices in relation to this.
Furthermore, in contrapuntal exercises, the upper part, which is often more complex, is segmented, just as Fricker does, by listening for the bass line fîrst.
As far as developing a psychological theory is concerned, Fricker has done nothing more than describe a good pedagogical technique. Simply asserting that this is
184 how the mind works does not make it necessarily so. Teachers speak in similar language, but only for the sake of analogy. This does not mean that Fricker’s idea could not be translated into a psychological theory.
Finally, the last component of his theory is that of symbolic cognition, which carries with it the thought of a physiological impact on the mind. This is the most difficult aspect of his theory to delineate, because his discussion of it is not overly explicit. In Part I, Section 23 (p. 62), he hints at an idea of the mind judging tones by a comparison of vibrations, represented by dashed lines. His point in the context is that no matter how disparate the sounds, the mind can still Judge the differences due to the difference of frequency, represented by the length of the line. However, he does state that one should “imagine” this, not that this is literally so.
The next possible suggestion comes in Part 3, Section 3 (p. 22). First of all, in
Section three, he refers to how a “material representation of high and low tones arises in the brain. It is through the vibrations of the different sounds.” He further refers to
“delicate matter” in the brain. How literally should this material representation be taken?
He continues in this same section by stating that an “image, having been made from the impression of two different sounds, cannot help but be a representation of some distance or interval.”
Finally, in Section eleven (p. 33), after describing the physiological cycle or circle of perception, he elaborates that, “the intellect begins the circle by its ratios; the mind, measuring the motion of the heart, advances it by the senses, or rather by perceptions as well as their aHections; finally, the imagination sense ends the circle by its symbolic cognition. These three—the intellect, the mind, and the imagination—establish the
185 length, width, and depth of ideas.” Here, Fricker gives his most direct statement about what he means. A concern of many eighteenth century philosophers was to discover at what point does cognition, the “mind,” connect with the physiological brain. Fricker argues that ideas and perceptions are given form in the imagination “sense,” a physical portion of the brain.
Oetinger expands on this idea, and in some ways takes it further. He often speaks about these musical perceptions in very literal ways. In the first section of his commentary on Fricker’s theory, he refers to the “concords” of musicians as able to be expressed on an “internal clavichord.” Another example is from his commentary on the
Fricker’s psychology theory: “the just distances of the tones, just as in the monochord, are measures in the brain sense through the particular points of some line” (Section 4, p.
137). Oetinger’s most direct statement about this idea is in his commentary on Fricker’s psychological part of the theory (Section 3, p. 128): “A mutation conveys these variations into the brain according to a certain cognition of signs, through which the vibrations of the air make a similar representation of music, which signs could be seen, if someone were able to inspect microscopically into the brain of another attending to the music.” The literalness of his language is striking, almost absurd in today’s perspective.
However, one must remember that he is grappling with a very difficult issue: how, exactly, is the mediation between the mind and body accomplished.
The fundamental idea is that the brain measures impressions left by the stimuli.
Oetinger discusses the cognition of “obtaining distances,” and these distances
(representing ratios) are represented on a “distinct monochord with respect to the signs,” that should exist in the “brain sense.” Again, he mentions that, “in the brain sense [are]
186 the symbolic signs of the distances,” as well as “the just distances of the tones that are given on the monochord.” Fricker discusses in great detail the exact paths and physiology of these measurements in the second part of his theory, particularly section fourteen. In all of these cases, both writers are referring to the mind judging tones through a calculation or comparison of the distances of the tones.
This interpretation of Fricker’s theory is not to say that he was substantially
“ahead of his time.” It is arguing that he was dealing with the problem in a relatively sophisticated manner.
187 CHAPTERS
EPISTEMOLOGY AND AESTHETIC POSITION
Epistemology
In addition to its psychological aspect, Fricker and Oetinger attempt to address the mind-body problem in two ways. First, they discuss the physiological aspect of cognition; second, they explore the concept of the “cycle” or “circle” of life, wherein the physical sensations are, at some point, translated into mental ideas. Both writers admit, though, that this aspect of the theory is more speculative than demonstrated.
For purposes of this discussion, Fricker’s epistemology will not be treated separately from Oetinger’s, for he was admittedly influenced by Oetinger. Indeed,
Fricker acknowledges Oetinger’s influence in writing this psychological portion of his treatise. He writes, “I would have been about to finish this analysis of music, had I not been persuaded by reason of what this author [Oetinger] has written concerning the sensus communis, and been compelled by the psychological theory of this same author in
§ 31. Now too, I ought to examine somewhat the physical or actual vestiges of the description, thus far, of the mind of music, and, in regard to these problems, I ought to express my opinion.” Moreover, Oetinger is considered to be the one influential on later
188 thought. Numerous historians have documented the influence of his phenomenalist ideas on the early Romantic movement. Most notably his philosophy of science influenced
Johann Goethe.
Due to limitations in this study, Hans Gadamer’s introductory essay to the
Inquisitio Sensum Communem will be relied upon for an estimation of Oetinger’s main treatise and ideas. Gadamer’s essay is valuable in that it provides an evaluation of two main points dealt with by Oetinger in the Inquisitio: Oetinger’s overall phenomenalist philosophical position and his treatment of the philosophies of Leibniz and Newton.
Oetinger’s Inquisitio is theologically motivated. His goal is to devise a complete epistemological system based upon the Bible, although he is not averse to using modem scientific ideas to support his theses. Ultimately, though, all theories and philosophies are judged by the standard of the Bible. In order to build this knowledge system,
Oetinger conceives of the sensus communis —the innate, God-given organ which mediates our perceptions and guides our Judgments. This is the phenomenal aspect of his position: one cannot attain true knowledge without a mediating, metaphysical element.
Oetinger’s opposition to Leibniz and support of Newton (a discussion which takes up a major portion of the treatise) touches on some fundamental issues, such as the nature of the soul, the connection of the material and immaterial, and the phenomenalist position. Gadamer finds Oetinger’s opposition to Leibniz remarkable, because Leibniz’s philosophy would seem to be an excellent support for Oetinger’s position. Leibniz’s
189 “monads...offered an ingenious mediation between the traditional teaching of substantial forms and the new mechanical physics”'—a mediation which Oetinger was attempting to accomplish himself.
Both men opposed Cartesian dualism, and Gadamer wonders that Oetinger did not see in Leibniz an ally in this matter. However, Oetinger and Leibniz differ in their means of opposing dualism and in their method of reconciling the material and the immaterial.
Leibniz’s system of monads struck Oetinger as ultimately subversive: “The Leibnizian system insinuates itself with remarkable elegance as truth and as acceptable, yet, nevertheless from the most insufficient premises it concludes too much, clearly esteeming
the holy scriptures less.”^ There are two key reasons for Oetinger’s rejection of Leibniz:
Leibniz’s monads would destroy the influence of the soul on the body, because monads
exist independently and cannot have influence on one another (they are “windowless”)
and Oetinger felt that the monads removed the influence and power of God to activate the
material world.
Instead, Oetinger makes use of Newton’s theory of gravity, specifically the
“action at a distance” concept. Oetinger views gravity as an analogy for the all-
permeating (hence, action at a distance) force of life which drives the “circle of life” and
mediates the connection between the material and the immaterial. God’s life-force
(Geistleiblichkeit) permeates all matter, and the sensus communis is the mediating organ
that allows the body and mind to connect, resulting in perception. Judgment, and
understanding. In Oetinger’s view, God is the active source of the life-force (much like
' Hans Gadamer, Introduction. Inquisitio in Sensum Communem et Rationem. By Friedrich Oetinger. Stuttgart: Friedrich Frommann Verlag, 1964, p. x.
190 Aristotle’s “Prime mover”). In Leibniz’s view, God is no longer playing an active role, because the monads are self-sufficient.
There is an irony in Oetinger’s support for Newton, because Newton rejects metaphysical explanations and recognizes the epistemological boundary between God- given knowledge and empirical knowledge, and the limitations it imposes, that is,
Newton would reject the sensus communis as unnecessarily speculative. Gadamer feels that Oetinger recognizes the “systematic weakness” of his work, and justifies it with the claim that the sensus communis is an innate idea, not a rational idea.
The Mind-Bodv Issue
Let us begin with a summary of Fricker’s and Oetinger’s mind-body theory.
Fricker begins the psychological portion of his theory with an admission that he had not yet considered the mind-body connection, but was prompted to do so by Oetinger’s work.
Fricker’s first point on this issue is that the God-given measure of 8:9 is one mediator between the mind and body, as he states:
...For so great is the power and activity of the mind in perceiving that ratio 8:9, that it penetrates all the way to its periphery and into its brain- sense Through it [the ratio], which is as a wall between the mind and the body, [this] most simple ratio activates and moves, as it were, its body. (Section 8, p. 103)
This quote speaks to one of the main mechanisms which connect the mind and the body, that is, the numerical “powers” which both writers referred to, extending out into the body. Consider, again, another description of the process:
^ Gadamer, Introduction to Inquisitio. p. xi.
191 .. The mind is supposed, in this event, to draw out its activity into the body: on the other hand, in so far as the mind holds itself, and learns, as well as accepts impressions or ideas from outside, rather, to this extent, thus the wheel of life runs; so that external matters [run] through the organs of sense into the brain, through this into the organs of speech, through these [organs] into the heart, through this into the brain a second time, but, [now] as an intellectual [organ]. They leave their vestiges behind, where entirely in later cases as well as even by a reception of ideas, suddenly the mind shows itself more active and follows its method, not [that] of the external. (Section 14, p. 109)
At some point in this cyclical process, the sensations are changed into perceptions. Fricker suggests here, but does not state how exactly, that it is after passing through the organs of speech and the heart, and entering into the brain again that the transformation occurs. The role of the organs of speech holds particular importance for both Fricker and Oetinger. In these organs or faculties, sensations are given mental embodiment. Fricker only elaborates further that it is a topic which has not been
investigated sufficiently, and that it is a God-given process (Section 13). His reason for
suggesting that the organs of speech are necessary relies on an argument from the New
Testament that the Gospel must be heard through language, and therefore, this becomes
the precedent for perception (Section 15-19).
Oetinger elaborates on Fricker’s arguments in his commentary, and his thoughts
are helpful in some ways. He delves somewhat more into the precise moment of
transition, but is still not entirely clear, and admits also that these are speculative issues.
He begins by discussing the nature of the sensations themselves. He discusses various
types of waves and tremors, and then asks, “whether the sensations are mere tremors in
the membranes, or whether something else is added on top of a higher nature, and what
that something of a most delicate, lively, active idea is, which thus modifies, according to
the totality of [the] perception, the tremor in the membranes, either with reference to 192 sight, or to hearing, or to taste. For from the mere undulations and tremors no perception of the whole results” [Part 4, Section 1]. In addition to being a question regarding the nature of the sensations, Oetinger’s question about something “higher” being added on touches, somewhat, on the issue of top-down and bottom-up processes.^
He agrees with Fricker that ideas are immaterial, and that they reside within the
“center” that he has referred to so often throughout his commentary. His comments here begin to deal not only with the influx of sensations on the mind, but with the opposite problem, the influence of ideas on the body. It is really all one and the same problem to him:
The action of the center is with respect to the immaterial. The reaction, however, seems fluid, called animal spirits, or as it pleases Swedenborg, the subordinate mechanisms seek from the meningibus to the innermost winding paths of the brain. Lebnizians themselves conceive no mind without a bodily scheme, for they say that where the sense is given, there the body is given. (Section 2)
This comment touches on two key issues. First of all, the “fluid, animal spirits” is taken directly from Descartes, and suggests that he agrees with Descartes on how the soul or mind influences the body, that is, through these animal spirits. This coincides with the term “delicate material” which both authors make use of. Secondly, he states his disagreement with Leibniz and the use of monads. Since, according to Leibniz, the mind arises from the collection of the monads, and each monad is a soul, so to speak, Oetinger must disagree, because this eliminates, for him, the existence of a soul (mind) separate from and higher than the body.
' Cognitive psychologists speak about bottom-up processes as those which are inherent to all human
193 Oetinger touches on the physiological problem some more while commenting on
Fricker’s third section;
In this paragraph the brain sense is considered as a material instrument, through which the tones themselves are perceived as affecting the harmony under the characteristic materials of the substances...But since not even the swellings of the muscles themselves through the fluid can be explained merely mechanically (for the electricity itself, through which the muscles agree to swell, was not yet dissipated mechanically), then much less will the brain agree with mechanical structure merely sensually. (Section 3)
His point seems to be that there cannot be a simple, mechanical, physiological explanation. Since such an argument will not even explain the motion of muscles due to the animal spirits, how can it be expected to explain the more complex phenomenon of ideas and perceptions? He continues:
For it is not yet certain, what was determined by the mechanical and Geometrical nature in its smallest [parts], or what rules of motion of the simpler corpuscles are just as the rules of motion of the larger [parts]. Not even fire itself can be explained merely mechanically. However, whatever it is in this respect, [that] it is repeating from higher [principles], the mind necessarily feels and recognizes individual pairs of numbers or ratios by a most effective power. Now, however, the cognition of the mind in this time requires symbolic characters, according to which is required, in the brain, some change of more delicate material, where vibrations of the air are regarded as equals of the existing ones with representations of motion—not abstractly, but physically. (Section 3)
The first sentence is an admission that scientists of his day did not understand the physical laws governing the fundamental elements constituting the brain, and therefore, the motions controlling and creating the mind. This is similar to physicist Roger
Penrose’s argument that the understanding of cognition will ultimately be solved only by
thinking; top-down are processes which are learned. 194 the laws of quantum mechanics, since the workings of the brain are ultimately dependent on these principles. The last sentence in the quote is also interesting, because it seems to
mitigate Oetinger’s previous statements about symbolic cognition. He states here that
“cognition in this time requires symbolic characters," as if they constitute an intermediate
step in our understanding. Then, he reiterates the change that takes place in the delicate
material whereby the vibrations are changed into cognitive representations. Finally,
Oetinger presents his most detailed description of the physiological process in section six,
where he places the specific ratios and interval relationships in specific parts of the brain
sense.
In conclusion, the mind-body theory of Oetinger and Fricker attempts to be as
specific as possible, but it is admittedly limited by the knowledge of their day.
Furthermore, due to their philosophical position of dualism, they posit a metaphysical
connection, regardless of their desire to find a physiological connection. The uniqueness
of their position is that they suggest that the ratios themselves act as psychological means
through the physiological mechanism of the sensus communis for musical perception.
Aesthetic Position
The aesthetic position of both Fricker and Oetinger is intimately linked to their
theological and philosophical ideal—simplicity. As already noted, one of their main
desires in developing this musical theory was to account for “common sense” perception
of complex phenomena. As Fricker states, “Consciousness requires, necessarily, the
simplicity of mind," and in the words of Oetinger, “I believe that the most simple and
195 most useful of the familiar of the structures of figures of the science of numbers, into which we might immediately strike by the sensus communis, are not yet open to us.”
What he goes on to argue for is that they are attainable through the Biblical wisdom of
Solomon.
Musical pleasure, according to Fricker, is based on the principle of simplicity, as he writes in Part 1, Section 5, “Where does pleasure arise from? Willingly, the ratios of the tones, which had increased to large numbers, arrange themselves a second time, and immediately reduce themselves to the simplest numbers...” The mind resonates, instinctively, with the beauty found in reducing musical ratios to simple series of numbers. Again, he states in Section 26, “Therefore, if they [the tones] might wish to be understandable to the mind, and be delightful, they will hardly ever be able to be so unless they are very simple.”
Their aesthetic position is in direct contrast to someone such as Euler. In regard to simplicity versus complexity, musicologist Charles Smith comments that Euler “makes clear that music should have much variety and that the highest agreeableness does not lie
in simplicity.”** Furthermore, Euler was plainly against the judgment of those with
simpler tastes or less education. He reveals his bias in the following:
In music, as well as in all other matters, it is most important to follow those whose taste is perfect and whose Judgment of things perceived by the sense is faultless...Thus, for the judgment of matters musical, the required auditors are those endowed with both an acute sense of hearing and precise comprehension, and they also possess such a degree of intellect as to enable them to perceive the order in which the pulses of the air particles strike the ear and firom this to pass to judgment.^
* Smith, p. 14. ^ Smith, p. 67.
196 Oetinger adds some interesting comments to this topic in his summary of
Fricker’s theory in the 1767 article Die Eulerische und Frickerische Philosophie iiber die
Musik. Oetinger remarks that Euler’s theory does not suffîciently define dissonance, and since the mind takes delight in the “pleasingness,” one might conclude that monotony is the most pleasing thing of all. In addition, he regards Euler’s attempt at calculating the measure of beauty a piece holds to be an “insoluble” problem, since “each piece has its own art, and they are heterogeneous.”® This is a remarkably open-minded viewpoint for the time.
Oetinger contrasts this with Fricker’s idea that beauty in music is derived from the contrast of consonance and dissonance. This point is, again, illustrated by contrast with
Euler’s position. In 1731, Euler wrote to Daniel Bernoulli, the renowned mathematician, about his ideas for what would become the Tentamen, which was actually mostly completed by this time:
My main purpose was that I should study music as a part of mathematics and deduce in an orderly manner, from correct principles, everything which can make a fitting together and mingling of tones pleasing. In the whole discussion I have necessarily had a metaphysical basis, wherein the cause is contained why a piece of music can give one pleasure and the basis for it is to be located, and why a thing to us pleasing is to another displeasing.
Bernoulli’s reply correctly identifies the shortcoming of Euler’s approach, and comports with Oetinger’s and Fricker’s criticism of Euler’s theory.
I cannot readily divine wherein that principle should exist, however metaphysical it may be, whereby the reason could be given why one could take pleasure in a piece of music, and why a thing pleasant for us, may for another be unpleasant. One has indeed a general idea of
’ See Breymayer, p. 149.
197 harmony that it is charming if it is well arranged and the consonances are well managed, but, as it is well known, dissonances in music also have their use since by means of them the charm of the immediately following consonances is brought out the better, according to the common saying opposita juxta se posita magis elucescunt [opposites placed together shine brighter]; also in the art of painting, shadows must be relieved by light/
In conclusion, Fricker and Oetinger are both very much men of their time: filled with awe for God, yet keenly aware that the scientific world around requires detailed and falsifiable arguments. The Brevissima represents well their position as dualists who attempt a reconciliation between the mind and body through the example of music perception. Finally, their aesthetic confirms their fundamental viewpoint that all good and true knowledge is ultimately simple and revealed by God.
^ Both quotes from Smith, p. 9.
198 CHAPTER 4
THE LEGACY OF THEHl THOUGHT
Oetinger’s Influence on Romanticism: Schenker and Goethe
Hans Georg Gadamer points out that Oetinger’s influence extended well into the nineteenth century.' In fact, numerous scholars have noted Oetinger’s effect on the early
Romantic movement. It is this effect that I will take up next.
In Erb’s history of Pietism, he articulated a common theme discussed in works on
Pietism, that is, Pietistism’s influence on later German Romantic thought. Erb states with respect to the Pietists of Oetinger’s generation that, “there was much in their pietistic background that fitted well with the new age—first with Enlightenment ideals, then with
Sturm und Drang, and finally with Romanticism. ”^ As we will see, Oetinger’s ideas played a significant part in this interaction.
The attitude and approach that Oetinger adopted for his philosophical inquiry made some of his ideas attractive to certain members of the early Romantic tradition.
Numerous scholars have presented studies which suggested that Pietism, and especially
Oetinger, had a direct influence on early romantic thought, specifically, on Schiller,
' Gadamer, Introduction to Inquisitio. p. v. ^ See Peter C. Erb, Ed. Pietists: Selected Writings. New York: Paulist Press, 1983, p. 24.
199 Goethe, and Hôlderlin/ As Becker has argued, Romantics found Oetinger’s organic language appealing. This harmonized well with their “formulations of a Romantic
Naturphilosophie that stressed the experience, the vitalities, and the impenetrable
mysteries of nature in contrast to what they saw as the barren order of the Newtonian
world.’"* Hayden-Roy offers a cautionary note about how strongly certain scholars have
argued for this influence, but does not dismiss it completely.
It is not my purpose to argue the extent of Oetinger’s influence on Romanticism;
rather to note that Oetinger’s language is very suggestive of the language of Heinrich
Schenker, one of the most influential analytical theorists of the twentieth century. I will
not try to argue a direct influence of Oetinger on Schenker, yet, given the fact that Goethe
did have an influence on Schenker, and others have argued that Oetinger had an influence
on Goethe, it would be judicious to at least consider the Zeitgeist influence of Oetinger on
the concept of organicism.
There are numerous examples of Oetinger’s use of organicism, most of which
come from a later work, Biblisches und Emblematisches Worterbuch (1776). He refers to
Christ as “Zaemach,” an “organic, enlivening principle ” which is a Cabalist concept he
adopted from the Old Testament scripture Zechariah 3:8; 6:12.^ This comports with the
organicism of his hermeneutical approach to scriptural interpretation, as described by
Hayden-Roy:
The words of Scripture yield meaning in the same manner that spirit emanates into bodies: their fundamental structure is a “generative order’' that unfolds as the plant from a seed. The image of the seed
^ Erb, p. 68. * Becker, “The Merton Thesis,” p. 656. ’ Hayden-Roy, p. 59.
200 suggests both semantic plenitude and nexus within plenitude: Oetinger calls the words of Scripture “pregnant," ramifying outward into a vast nexus of sensate meaning. This contrasts with the exegetical approach of those theologians influenced by rationalism, who project onto Scripture a “geometrical order,” as Oetinger calls it, whereby an abstract concept constitutes the hermeneutical center of the Bible. These exegetes, argues Oetinger, “peel off’ (“abschelen”) the pregnant, sensate meaning of Scripture. By reducing the words to abstract concepts they make Scripture “overly distinct” (“iiberdeutlich”), and ultimately construct an arbitrary or idiosyncratic system of doctrine that fails to grasp the whole of truth.^
.. .Spirit is not an abstraction or naked moral truth, but rather an “enveloped” vital principle whose meaning consists in the “unfolding" (“Auswicldung") of the seed-like, dynamically ramifying spirit.’ [Emphasis mine]
Although not applied to music, the similarity of language is striking. In fact,
Auswicklung (unwinding/unfolding, cf. Ausfaltung, unfolding) is Schenker’s precise term to describe the unfolding of harmonic levels.^ Oetinger’s reference to the scriptures being “pregnant” and “ramifying outward” is also akin to Schenker’s numerous discussions about the “procreative” urges of tones^, the egotism of the tone'°, the
“primogeniture” of the fifth overtone", and how
...originating from the life of the tone, this principle penetrates the living organism of musical composition, wherever possible, with the force of an element of Nature.’’ [Emphasis mine]
* Hayden-Roy, p. 60. ^ Hayden-Roy, p. 61. * Heinrich Schenker, Free Composition. Trans, and Ed. Ernst Oster, New York: Longman, 1979, p. 25. ’ Heinrich Schenker, Harmony. Trans, by Elisabeth Mann Borgese. Ed. by Oswald Jonas. Chicago: University of Chicago, 1954, pp.6,24,28-29,84,152. 10 Schenker, , Harmony, p. 115. " Schenker, Harmony, p. 26.
201 The penetration of this principle reminds one, from the previous quote, of the manner in which the “spirit emanates into bodies."'^ Gadamer captures Oetinger’s viewpoint well when he summarizes as follows;
In contrast to the violent anatomization of nature through experiment and calculation, he sees the natural development of the simple into the complex as the universal law of growth of the divine creation, and likewise, of the human spirit/^
Schenker also makes use of the ‘generative’ concept, which is integral to development, as described by Hayden-Roy:
Oetinger recommends abstracting from the emblem...in order to discover the “form,” the generative “spiritual'’ kernel within the “mass,” or sensuous embodiment of the emblem. The believer can see these future things in faith, because the language of faith is emblematic; it signifies visible objects which, although as yet untransformed, nevertheless are connected to the future things through the generative kernel of divine life within them.’^ [Emphasis mine]
Oetinger’s suggestion on how to discover the form—“abstracting from the emblem”—is suggestive of Schenker’s process of abstracting from the “surface” to the “background.”
Terms such as “generative kernel” and “transformation” make explicit how this process works, and their use is similar to that found in Schenker’s explanation of form.
In the music of the early contrapuntal epoch, including even Palestrina, the basic voice-leading events, such as passing tones or neighboring notes, had not yet come to fruition, like flowers in bud. Who would have suspected, at that time, that these phenomena, through the process of diminution, were to become form-generative and would give rise to entire sections and large forms! Although the art of prolongation and
Schenker, Harmony, p.115. " For a further discussion of the procreative metaphor, see Robert Snarrenberg, “Competing Myths: The American Abandonment of Schenker’s Organicism.” Theory. Analysis and Meaning in Music. Ed. Anthony Pople, Cambridge, Mass: Cambridge University Press, 1994, pp. 29-56. " See Hans Gadamer. Truth and Method. New York: The Crossroad Publishing Corporation, 1989, p. 28. Hayden-Roy. p. 64.
202 diminution ultimately expanded and enriched form, it was the force of the rirst passing tone, the first neighboring note, the power of the first structure division which bound form to take on organic unity.
All forms appear in the ultimate foreground; but all of them have their origin in, and derive from, the background. [This is the] ultimate manifestation of that structural coherence which grows out of background, middleground, and foreground.'^
The inherent structure is expanded and developed—transformed'^—through time and the musical levels.
The tendency to propagate the forms of the fundamental structure goes through all voice-leading levels. Hence, such transferred forms appear in greatest abundance in the foreground. Every transferred form has the effect of a self-contained structure ... [Emphasis mine]
This reflects Schenker’s motto semper idem sed non eodem modo, “Always the same, but not in the same manner,”^" in the fact that later levels of a piece are expanded “copies" of the fundamental structure. This concept is elaborated in the following statements:
The total work lives and moves in each diminution, even those of the lowest order. Not even the smallest part exists without the whole.^'
Schenker’s language about the organic connection of various parts of a musical work is even more vivid at times:
Schenker, Free Composition, p. 128. Schenker’s use of the words force and power are reminicent of Oetinger’s use when describing the activity of the prime numbers in music cognition. Schenker, Free Composition, p. 130 "The life of the fundamental line...expands through the middleground, through what I have called the voice-leading and transformation levels...Whatever the manner in which the foreground unfolds, the fundamental structure of the background and the transfomation levels of the middleground guarantee its organic life...The principles of voice-leading, organically anchored, remain the same in background, middleground, and foreground, even when they undergo transformation. ” Schenker, Free Composition, p. 5 ,6 . " Schenker, Free Composition, p. 87. “ Schenker, Free Comiwsition. p. 6. Schenker, Free Composition, p. 98.
203 It should have been evident long ago that the same principle applies both to a musical organism and to the human body: it grows outward from within...The hands, legs, and ears of the human body do not begin to grow after birth; they are present at the time of birth. Similarly, in a composition, a limb which was not somehow bom with the middleground and background cannot grow to be a diminution.^^
As suggested earlier, I am not arguing for a direct link between Oetinger and
Schenker. I am arguing, though, that Oetinger’s influence on Schenker is indirect, through the Romantic movement, and particularly through Goethe. Several studies have
shown the specific influence of Goethe on Schenker,^ and Schenker himself was fond of quoting Goethe. The source of Schenker’s organicism can be found in Goethe’s search
for the “Urpflanze,” which bears a direct correlation to Schenker’s conception of the
“Ursatz.” Goethe was fascinated by the similarity of plant forms within their wide
diversity.
Here where I am confronted with a great variety of plants, my hypothesis that it might be possible to derive all plant forms from one original plant becomes clearer to me and more exciting...All this confirms my botanical fancies, and I am on the way to establishing important new relations and discovering the manner in which Nature, with incomparable power, develops the greatest complexity from the simple.^^
Goethe’s conception of complexity from simplicity matches the earlier quote
concerning Oetinger’s view that scriptures “unfold” like a plant from a seed, that is, that
the specific interpretations [forms] are inherent in the “fundamental,” “generative order.”
“ Schenker, Free Composition, p. 6. “ See Gary W. Don, “Goethe and Schenker,’’ In Theory Only. 10 (1988): 1-14. Allan Keiler, “The Origins o f Schenker’s Thought: How Man is Musical, ” Journal o f Music Theory. 33 (1989): 273-298. William Pastille, “Goethe’s Influence on Schenker’s Muical Ontology. ” Schenker Studies. Ed. Hedi Siegel. Cambridge: Cambridge University Press, 1989. ^ Quoted in Don, p. 2.
204 This is also similar to Pricker's (and Oetinger’s) conception of all the various forms and complexities of music being derived from the most simple and fundamental series of numbers, the musical basis: T , 2^, 3 \ 5% 7°.
Interestingly enough, there is actually one specific example which is common to all three writers, the use of ‘centrifugal’ and ‘centripetal’ force. Fricker and Oetinger discuss the outward (centrifugal) and inward (centripetal) motion—force—of the prime numbers to and from the center of cognition (in this case the sensus communis). This motion is what drives the multiplication and reduction of the numbers which compose the musical ratios in a manner that makes them comprehensible to the mind, resulting in music perception. For Goethe, these forces are behind the development and variation of plant life.“ Gary Don explains:
Metamorphosis is defined as the tendency toward growth, elaboration, and complexity. Spezifikationtrieb is the tendency toward maintaining the specifîc form and nature of the object. It is Spezifikationtrieb that keeps growth from getting out of hand, from changing the plant to a point where its original form is no longer recognizable. Metamorphosis is a centrifugal force and Spezifikationtrieb is a centripetal force. Steigerung is the result of both powers working together.^®
In a similar manner, Fricker and Oetinger limit the mathematical operations of the prime numbers constituting the musical ratios, so that they do not exceed human understanding.
But if multiplication accepts no limits...it will be infinite. However, music of infinite tones would be absurd, on the contrary, multiplication should not progress too far, lest it depart from the rule of simplicity and
^ Schoenberg was also influenced by the organic thought of Goethe, as discussed by Severine Neff, “Schoenberg and Goethe: Organicism and Analysis.” Music Theory and the Exploration of the Past. Eds. Christopher Hatch and David W. Bernstein. Chicago: The University of Chicago Press, 1993.409-433. “ Don, p. 6.
205 intelligence.^
Schenker introduces these forces in his discussion of the fifth-relationship, specifically the over- and under-dominant relationship.
A correct and conscious perception of the development of these fifth- relationships, away from the root tone in both directions, rising centrifugally and falling centripetally, is of paramount importance for the artist.^*
Whether Schenker culled these concepts and terms directly from Goethe or
Oetinger is not paramount to the central issue: Oetinger was clearly expressing concepts which entered into the vocabulary of the Romantics, who, in turn, provided a wealth of aesthetic language employed by both Goethe and Schenker.
Gadamer's Debt to Oetinger. Christensen's Debt to Gadamer
Oetinger was, as is Gadamer, a phenomenologist. This fact is important to keep in mind while considering Gadamer's judgment of Oetinger. Both men sought to rectify the tension between empirical observation and the biases which one brings to those observations. Although Gadamer points out some of the weaknesses of Oetinger's treatise, he feels that Oetinger's basic argument stands: the sensus communis is still necessary for a true understanding of knowledge because calculations cannot guarantee the truth of knowledge.
Gadamer's magnum opus is his Wahrheit und Methode (1960). The point of this work is to clarify the historical use and nature of language in philosophy, and more
^ Flicker, Brevissima. p. 78.
206 specifically, define its role in a hermeneutic sense. Within this context, Gadamer looks to
Oetinger’s formulation as an important corrective in the developing definition of the sensus communis in the eighteenth century. Oetinger was combating Enlightenment methods and philosophy, and argued that the scientific method was not completely objective, and still needed the guiding influence of the sensus communis. Gadamer believes that his viewpoint was essentially correct:
According to Oetinger, the Enlightenment is mistaken if it thinks that it is above this method [the rhetorical method of removing prejudices]. Our [Gadamer’s] investigation will lead us to confirm this view of Oetinger’s. For even though he is attacking.. .the Enlightenment ideal of demonstration—something that is no longer of interest today, or is just starting to be so again—the same thing is true of the modem human sciences and their relationship to “logic.”^^
Thomas Christensen makes use of Gadamer’s hermeneutic method to reconcile the historicist and presentist views of music theory. Gadamer feels that cultural and historical biases should not be shunned, but should be employed in a constant dialectical process of understanding present issues in the context of historical understanding, particularly as it affects our concept of language and meaning. We cannot hope to obtain true objectivity, but should carefully and precisely understand our historical biases.
Christensen applies this perspective to the role of analytical music theories, writing that,
“From a hermeneutic perspective all analytical activity is fully historical.” Therefore, for example, one’s understanding of Rameau’s theories should make full use of the issues relevant to Rameau — in his day. This historical position should not necessarily disuade one from understanding his theories through contemporary eyes, nor from making current
“ Schenker, Harmony, p.43. ^ Gadamer, Truth and Method, p. 27, fit. 42.
207 applications of his theories to music of earlier or later time periods, as long as one is aware of doing so.^*’
Christensen has also brought to light the importance of Pietist thought to eighteenth-century theorists and musicians, namely, to Johann Mattheson and Georg
Sulzer. While, at this time, there is no evidence that Oetinger or Fricker had a direct influence on (or contact with) either Mattheson or Sulzer, the Brevissima and the
Inquisitio could prove to be rich, new sources of material, particularly regarding the concept of Schenkerian “organicism. Moreover, the history of psychology and philosophy could benefit brom the historical position that Oetinger and Fricker represent.
The chain of Oetinger to Gadamer to Christensen is pointed out for the sake of scholarly interest, not to argue a direct influence. It would be a forced point to make too much of the link between Oetinger and Gadamer, let alone the link between Oetinger and
Christensen. However, Christensen’s works on the role of Pietism in the Classical period,^' in conjunction with his use of Gadamer’s rich tradition of hermeneutics and phenomenology, demonstrates how important it is to have a clearer understanding of
” See Thomas Christensen, “Music Theory and its Histories." Music Theory and the Exploration of the Past. Ed. Christopher Hatch and David W. Bernstein. Chicago: The University of Chicago Press, 1993.9- 39. See Thomas Christensen, “Sensus, Ratio, and Pthongos.” Musical Transformation and Musical Intuition: Eleven Essavs in Honor of David Lewin. Eds. Raphael Atlas and Michael Cherlin. Roxbury, Mass.: Ovenbird Press, 1994.1-22 as well as Nancy Kovaleff Baker and Thomas Christensen. Eds. Aesthetics and
208 Pietist thought in this time period. The work of Flicker and Oetinger provides us with an extensive, in-depth, and first-hand source that can illuminate the bridge between Pietist thought and music theory in the eighteenth century.
the Art of Musical Composition in the German Enlightenment: Selected Writings of Johann Georg Sulzer and Heinrich Christoph Kock. New York: Cambridge University Press, 1995.
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214 INDEX arithmetic, 22,23,26,46,59,69,80,98, geometric, 20,98,100,133,136,201 101.102.103.104.105.106.107, Goethe, xiii, 189,200,204,205,206, 108.127.135.136.137.138.142, 212 143.150.153.159 Graun, 37,166 arithmetical, 22,23,26,46,59,69,80, harmony, 18,20,26,27,30,31,33,35, 98.101.102.103.104.105.106.107, 46,47,48,49,53,54,58,62,65,68, 108.127.135.136.137.138.142, 69, 72,80,85, 89,91,100,127,130, 143.150.153.159 131,134,135,136,138,141,142, Bacon, 93,100,127,131,134 149,161,163,172,173,184,194,198 basis, 20,27,28,31,33,34,35,48,58, heaven, 31,138,161 61.75.78, 87,91,94,101,104,138, heavenly, 31,138,161 141,153,157,160,161, 163,164, Hippocrates, 124 165,175,176,179,180,182,197,205 Italian, 36,47,48,65,161,162,163 brain, 99,100,103,105,106,107,108, limit, 116,139,177,182,183,205 109,110, 111, 112,113,116,120, limits, 116,139,177,182,183,205 123,125,126,127,128,130,132, Mattheson, xii, 10,33, 35,64,161,162, 134.135.136.137.138.141.142, 163,164,175,181 170,179,185,186,191,192,193,194 Maupertuis, 94,143,147,150 centrifugal, 67,82,83,84,87,205 melody centripetal, 82,83,84,205 melodia, 44,48,56,68 circle, 67,84,87,93,106,109,112,117, melodiae, 36 120,138,185,188,190 melodiam, 34,37,53,54,75 Descartes, 153,154,159,169,172,193 melodiarum, 33,53 Euler, xii, 16,28,30,32,35,78,136, melodias, 33,127 146,147,154,155,156,157,158, melody, xii, 33,34,35,36,37,41,43, 159,196,197 44,45,46,47,48,53,54,62,65, Ezekiel, 66,94 68,69,73,142,161,162,163,164, Fricker, i, xi, xii, xiii, 1,2,5,6,13,14, 166,176,179 15,16,18,19,23,28,29,30,32,37, mind 41.43.46.48.53.55.56.59.61.78, anima, 20,21,23,31,40,60,61,62, 80,114,120,125,135,151,152,153, 64.65.66.68.69.77.80.82.94, 156,157,158,159,160,161,162, 98,101,103,104,105,109,110, 163,164,165,166,168,170,174, 114,115,116,117,120,124,125, 175,176,177,178,179,180,181, 126,127,133,136,139,141,142 182,183,184,186,187,188,191, animae, 22,26,31,39,60,62,64,65, 192,193,194,195,196,197,205,210 66.69.75.80.82.85.89.93.94,
215 97,98,99,101,103,105,113,114, 160,161,165,190,191,194,201, 122,123,125,126,127,135,136, 203,204,205 137,138,139,141,142,143 root, 26,28,53,59,87, 111, 141,146, animam, 57,66,80,82,100,101,103, 184,206 120.122.123.124.125.126.127, rule, 28,33,47,53,75,77,78, 87,99, 130,139,141 136, 160,205 animi, 30,36,109,112,114 Schenker, xiii, 181,182,199,200,201, animo, 75,106,108 202,203,204,206,207,212,213,214 animum, 37,43,69,106,107 Schoenberg, 205 animus, 21,31,106,107,108,115 sensus communis, 2,13,15,16,94,97, mind, 13,20,21,22,23,26,30,31, 98,114,124,126,142,143, 144,145, 37,38,39,40,43,55,57,60,61, 174,175,176,180,188,189,190, 62,64,65,66,68,69,74,75,77, 191,196,205,206,207 80,82,85,89,93,94,97,98,99, common sense, 144,174,195 100,101,103,104,105,106,107, soul, 21,80,114,117,141,169,171, 108,109,110,112,113,114,116, 189,190,193 117.122.123.124.125.126.127, string, 18,19,20,23,24,31,54,117, 128,130,133,135,136,137,138, 131,133 139,141,142,143,153,156,160, tactus, 23,24,31,34,39,41,48,64,69, 163,164,165,168,169,170,171, 114,161,164 172,173,175,176,177,178,179, beat, 34,39,40,60,65,69,164 180,181,183,185,186,187,188, measure, 34,35,41,43,62,64,85, 190,191,192,193,194,195,196, 101,102,103,104,108,112,124, 197,205,206 127,130,133,134,135,136,137, Newton, 91,92,94,95,130,146,147, 138,139,142,143,145,146,156, 189,190,191 157,164,177,180,191,197 Oetinger, i, xi, xii, xiii, 2 ,3 ,4 ,5 ,6 ,9 , meter, 35,164 13,14,15,16,80,114,115,116,124, note, 11,18,34,35,38,39,40,41,42, 130,142,143,150,151,165,168, 47,49,56,64,155,157,162,163, 171,173,174,175,176,186,188, 164,171,177,183,200,203 189,190,191,192,193,194,195, rhythm, 32,39,48,162,164,167,179 197,199,200,201,202,203,204, strike, 56,143,164,183,196 205,206,207 taste, 47,94,114,116,164,176,193 power, 28,31,37,60,61,62,67,69,77, temperament, 152 80,82,83,84,85,87,89,91,93,99, unity, 27,28,53,54,57,67,68,78,83, 103,106,109,112,117,126,127, 85,87,98,102,106,134,138,154, 133,137,138,139,141,147,149, 155,160,163,179,203 wheel, 67,87,106,109,192
216