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Part I. A taxonomic paradigm from ’ “De divisione” applied to the eight modes of music. Part II. “Arioso and Toccata” for euphonium solo, wind ensemble, , and percussion. [Original composition]

Deason, William David, D.M.A.

The Ohio State University, 1992

UMI 300 N. ZeebRd. Ann Arbor, MI 48106

PART I: A TAXONOMIC PARADIGM

FROM BOETHIUS' DE DIVISIONE APPLIED TO

THE EIGHT MODES OF MUSIC

PARTE: ARIOSO AND TOCCATA FOR EUPHONIUM SOLO,

WIND ENSEMBLE, HARP, AND PERCUSSION

A Dissertation

Presented in Partial Fulfillment of the Requirements for

the Degree Doctor of Musical Arts in the

Graduate School of The Ohio State University

by

William David Deason, B. M., M. M.

*****

The Ohio State University

1992

Dissertation Committee

Thomas Wells Approved by

Paul Droste

Burdette Green Adviser Peter King School of Music ACKNOWLEDGEMENTS

I express sincere appreciation to Dr. Burdette Green and Dr. Peter King for their guidance, insight, and suggestions throughout the research on the Document. I extend

sincere appreciation to Dr. Paul Droste for his comments and suggestions during the preparation of the Arioso and Toccata. I also wish to express gratitude to Dr. Thomas

Wells for his flexibility in letting me write a euphonium concerto instead of a piano

concerto. I also wish to say thanks to my mother and sisters, my brother and his family for

not calling me during my dissertation research. They have never called me anyway.

Finally I want to extend a warm hug and a kiss to Mary, Margie, Holly, Kelly, Reggie,

Candy, Tika, and Doc for their patience in waiting for this dissertation to be finished before

putting pressure on me for more cookies. VITA

May 24,1945 Bom, - Sioux Falls, South Dakota

1967 B. M., Florida State University

1968 M. M., Florida State University

1969-1970 Assistant Professor of Music, Lander College, Greenwood, South Carolina

1976-1979 Instructor of Music, Caldwell College, Caldwell, New Jersey

1980-1985 Assistant Professor of Music, CaldweUCollege

1982-1985 Chair, Music Department, Caldwell College

1983-1985 Chair, Fine Arts Division, Caldwell College

1986-1990 Graduate Teaching Associate, School ofMusic, The Ohio State University

1983 Gossamer Rings for Soprano saxophone and Band. premiered by the U.S. Navy Band, Sixth International Saxophone Symposium, Steve Mauk, soloist, Washington, D. C.

1986 Chamber Concerto for Horn and Percussion recorded on Crystal Records by Meir Rimon and the Indiana University Percussion Ensemble

1988 Gossamer Rings awarded third prize at the American College Bandmasters Association Conference, Old Dominion University

1989 Gossamer Rings performed on tour by The Ohio State University Concert Band, Craig Kirchhoff, conductor

iii 1989 Carnival for eight Saxophones and Vibraphone premiered by the Eastman Saxophone Ensemble, Ray Ricker, director

1992 Arioso and Toccata for Euphonium Solo. Wind Ensemble. Harp, and Percussion premiered at the International T.U.B.A. Conference, David Miles, soloist. University of Kentucky

1977-1990 6 ASCAP awards

1978-1984 3 "Meet The Composer" grants

FIELDS OF STUDY

Major Field: Music

Music Composition Thomas Wells History of Theory Burdette Green Schenicerian Theory Gregory Proctor Acoustics David Butler Music CAI Ann Blombach

IV TABLE OF CONTENTS

ACKNOWLEDGEMENTS ii

VITA iii

LIST OF TABLES vi

LIST OF DIAGRAMS vii

LIST OF EXAMPLES viii

CHAPTER PAGE

Introduction 1

I, The Aristotelian Science of Division 11 n. The History of Boethius'De divisione 23 in. Summary of the De divisione 29

IV. A Taxonomic Paradigm Derived from Boethius' De divisione 39

V. The Eight Church Modes 47

VI. The Taxonomic Paradigm of the De divisione applied to the Eight Church Modes 59

Vn. Examples of Division in Five German Music Treatises 78

Conclusion 91

Bibliography 92 LIST OF TABLES

TABLE PAGE

1. A comparision of the quaternary division of numbers by the Pythagoreans and the division of the Universe in the Periphyseon 21

VI LIST OF DIAGRAMS

DIAGRAM PAGE

One. The creation of a definition through the addition of differentiae to a genus as an example of qualitative division 17

Two. The quantitative division of a whole into parts. 17

Three. Five division types of the De divisione. 39

Four. The taxonomic paradigm of the De divisione. 43

Five. A comparision of Byzantine and Western modal organizations. 51

Six. The four authentic modal seats and the four secondary seats with surrounding whole tones and . 53

Seven. llaQ species of fourths and fifths. 54

Eight. The eight modes of music with bracketed species. 56

vu LIST OF EXAMPLES

EXAMPLE PAGE

One. The division of genus into species applied to the Protus, Deuterus, Tritus, and Tetrardus modes 60

Two. The division of genus into species through a difference applied to the Protus maneria. 61

Three. The division of the (subaltern) genus fifth into species through the differentiae of whole tones and . 62

Four. The first and fourth oppositions applied to the Protus modes. 64

Five. A comparison of the Protus authentic and plagal modes with b flat as an accident and as a difference. 65

Six. The division of the whole into parts applied to the eight modes. 66

Seven. The division of the subject into accidents applied to the Protus authentic mode. 69

Eight. The division of the octave species on D into accidents Protus authentic and Tetrardus plagaal. 70

Nine. The subjects Protus authentic and Tritus authentic compared. 71

Ten. The division of accident into subjects applied to the Deuterus authentic mode. 72

Eleven. The division of the gamut by according to the Musica enchiriadis and the De harmonica institutione of Hucbald. 74

Twelve. The division of the gamut according to hexachords. 75

Thirteen. The division of accident into other accidents applied to the tetrachord and hexachord. 76

Vlll INTRODUCTION

Carolingian treatises are practical handbooks rather than theoretical treatises.! Though Boethius’ De musica was the most frequently quoted theoretical source in tenth and eleventh century treatises, there is little in his work which would assist

Medieval singers in the learning of repertoiy. A perusal of the cyclic organization of the De institutione harmonische of Hucbald,- the Dasia notation of the Musica enchiriadis-^ and the mnemonic aids given to help memorize chant in the Epistola de ignoto cantu of

Guido,^ shows that the purpose of Carolingian theorists was primarily utilitarian. Their focus was on practical ways of uniformly and efficiently performing and teaching chant. A rationale based on pure theory could scarcely provide the means to accomplish this goal.

What theoretical structures Carolingian music writers did create can be described as first- order theory in that theoretical relationships were used directly as an aid to establishing uniform performance. A predilection for pure theory is less demonstrable among

Carolingian music writers than among later writers.

Beginning in the eleventh century a more purely theoretical way of looking at music appeared in treatises written by German theorists. This new theoretical way saw music theory as a science that required empirical observation and demonstration. For example.

! Crocker (1972) says that the Frankish musician was singer, teacher, and theorist in that order. See Richard Crocker, "Hermann's Major Sixth," Journal o f the American Musicological Society 25 (1972), p. 29. ^Huchald, De harmonica institutione, Gerbert Scriptores [GS], I , pp. 104-21. Section 1,104a-109a. flection II, 109a - 114b. Section III, 114b to 121a. For an English translation, see Hucbald , Guido , and John on Music : Three Medieval Treatises. Translated by Warrrai Babb. Edited, with Introductions by Claude V. Palisca. (New HavemYale University Press, 1978), pp. 13-44. ^Musica et scholia enchiriadis una cum aliquibus tractatulis adiunctis , Bayerische Akademie der Wissenschaften, Veroffentlichungen der Musichistorischen Konunission 3 (München, 1981). ^The solmisation syllables u t, re , m i, fa , so l , and la from the Ut queant laxis were provided by Guido as an aid in the singing of chant. See the Epistola der ignoto cantu in Gerbert Scriptores , II, pp. 43-50. For an English translation see Guido of Arrezo, Epistola de ignoto cantu , translated by Oliver Strunk inSource Readings in Music History. (New York: W. W. Norton and Co., 1950), pp. 121-5. around the year 1050 Hermannus stated that, in order to be a musician, one must know the

“science of rationally composing songs.”^

Starting with Bemo of Reichenau in the early eleventh century, theorists were preoccupied with organizational methods, the use of a new vocabulary of descriptive terms, and the inner structural relationships of the Medieval gamut. Above all, this concern with the identical modal correspondences which may be found by dividing the gamut into identical hexachords testifies to a more theoretical approach.

This predilection for theorizing has been noted in modem scholarship. In the case of Hermannus, for example, Oesch (1961) criticized his theoretical preoccupation as theorizing for the sake of theorizing.® Fellerer (1956) noted Wilhelm of Hirsau’s persistent use of demonstration for proving theoretical concepts.^

Early Medieval German writers on music were theorists as well as teachers, and they did create a theory of music. They explored the structure of the gamut and modal relationships in such a systematic and logical manner that it is easy to see that a new conceptualization was emerging in early Medieval German music theory.

^Musica Hermanni Contracti, translated by Leonard Ellinwood (Rochester: Eastman School ofMusic, 1936, p. 47: "Oportet autem nos scrire, quod omnis musicae rationis ad hoc spec tat intentio, ut cantilenae rationabiliter componendae..." ®Hans Oesch, Berno and hermann von Reichenau als Musiktheoretiker, Publikationen derS:hweizerischen Musikforschenden Gesellscfiaft , ser. 2, vol. 9 (Bern: Paul Haupt, 1961), as cited in Lawrence Gushee, "Questions o f Genre in Medieval Treatises on Music ," Gattungen der Musik in Einzeldarstellungen : Gedenkscrift Leo Schrade. Edited by Wulf Arlt et al. Bern: Franche Verlag, 1973, p. 413. ^K. G. Fellerer, "Zum Musiktraktat der Wilhelm von Hirasu ", Fs. Wilhelm Fischer, Innsbruck 1956, p. 65. Purpose of this study

This study will demonstrate that a component of Aristotelian logic, the science of division, influenced the ways German theorists of the early Medieval period rationalized the division of plainchant into eight modes or tones. Discussions of the modes in German theory frequently focused on how the Medieval gamut could be divided so that a unified modal system resulted. Division as a science was elevated to a position of extreme utility.

By the eleventh century, German music theorists had gained knowledge about the tools of

Aristotelian division, and that they used these tools in their description of the authentic/plagal pairings which were known later as the “maneriae.”

During the early Medieval period, the Aristotelian science of division was primarily known through Boethius’ translations and monographs. One of his monographs, the De divisions, so concisely outlines Aristotle’s division categories, that it can serve as the basis for the present study. A summary of Boethius’ division types will be presented, and a taxonomic paradigm will be constructed.

Division was not unique to the Greeks. It is reasonable to assume that the process of division would have to be a necessary part in any civilized society. However, Plato and

Aristotle regarded the process of division as a science.^ In doing so, they elevated division to a point where its nature could be examined and demonstrated like any other empirical science.

Thus, treatises by five early Medieval German theorists will be examined for terminological usage and conceptual evidence that they were influenced by Aristotelian logic in the division of the modes into authentic and plagal octave species, and the division of these into species of fourths and fifths. The five treatises are the Prologus in

^Marguerite Deslauriers, "Plato and Aristotle on Division and Definition," Ancient Philosophy, (Pittsburgh: Mathesis Publications), Vol. XX, No. 2, Fall, 1990, p. 216. tonarium of Bemo of Reichenau (ca, 1021-48),^ the Musica of Hermannus Contractus (ca.

1048-54),^® the Musica of Wilhelm of Hirsau (ca. 1 0 6 9 ) ,the De musica of Aribo (1068-

7 8 ),12 and theM iisica of Engelbert of Admont (before 1 3 2 5 ) .Although other German theorists of this period also used species theory, these five writers were specifically chosen for inclusion in this document because they employed a new concept. They rationalized the modes by dividing the gamut according to hexachordsi'i and by further subdividing hexachords into . I will argue that this new approach to the division of the gamut, which Crocker described as a second order theoretical concept,!^ is an illustration of Boethius’ division of accidents into other accidents. Although the Medieval diatonic gamut can be conceptually divided into hexachords or tetrachords, neither the hexachord nor the tetrachord is essential to the existence of the gamut.

However, Boethius’ De divisione is more than merely Aristotelian. Boethius used anthropomorphism to illustrate Aristotle’s view of the opposition of relationship. In this way Boethius grafted a Platonic concept onto Aristotelian division. Anthrpomorphism, which can be traced from the Pythagoreans to Plato, provided the common illustration that the five German theorists used when they discussed the relationship between the authentic and plagal modes.

^Bemo of Reichenau, Prologus in Tonarium , Scriptores ecclesiastici de musica sacra potissmum , edited by (St. Blasien, 1784), [GS], II, pp. 69-93 ^^Musica Hermanni Contracti , op. c it. ^ ^ Willehelmi Hirsaugensis Musica , edited by Dennis Harbinson, Corpus Scriptorum de Musica 23 (Rome: American Institute of Musicology, 1975). ^2joseph Smits van Waesberghe, Aribonis De musica , Corpus Scriptorum de Musica 2 (Rome: American Institute of Musicology, 1951). ^^Engleberti Admontensis De musica , GS, II, pp. 287-369. ^'^Harold Powers, "Mode," The New Grove Dictionary ofMusic and Musicians. , 20 vols., edited by Stanley Sadie. (London: Macmillan Publishers, Ltd., 1980), vol. 12, p. 379. ^^Richard Crocker, "Hermann’s Major Sixth," op. c it., p. 31. The Organon of Aristotle

Aristotelian logic is comprised of several distinct parts, each of which received a measure of attention at various times during the Medieval period. His logical writings, known cumulatively as theOrganon (or instrument),!® include the Categories, De interpretatione, Prior analytics, Posterior analytics, the Topics, and the Sophistici elenchi.

Only the Categories, the De interpretatione, and the Topics are important to the present study.

Prior to the twelfth century only Aristotle’s Categories !"! and the De imerpretatione^^ were known to Medieval scholars. The Categories discusses the divisions of reality as expressed through words that are not combined into sentences. The De interpretatione discusses the linguistic components from which statements are constructed and judgments made.!^ In both treatises Aristotle relies on division to make his tenets clear.

Though division came under the rubric of logic for Aristotle, as an aspect of intellectual thought it was not neatly compartmentalized within any standard academic discipline. From Plato to Boethius and well into the early Medieval period, the science of division was virtually synonymous with dialectic. Plato clearly made this association in the

Sophist, 20 and Boethius rearticulated this association in his Commentary on Cicero’s

!®Sten Ebbesen, "Ancient Scholastic Logic as the Source of Medieval Scholastic Logic," The Cambridge History o f Later Medieval Philosopny , edited by Norman Kretzmann, Anthony Kenny, Jan Pinborg, and Eleonore Stump (Cambridge; Cambridge University Press, 1982), p. 104-5. ^'^Aristotle's Categories and De interpretatione, translated with notes by J. L. Ackrill (Oxford: Clarendon Press, 1963), pp. 3-42. !8 Ibid., pp. 43-68. ^^Aristotle's Categories and De interpretations, op. cit. 20 Plato's Sophist. Translated with Commentary by Seth Benardete (Chicago: University of Chicago Press, 1986), 253 D: "Stranger. Well assert.. .fliat to divide according to genera... is die characteristic of the dialectical science? Theatetus. Yes, well assert it." Topics.'^^ Since both logic and dialectic were parts of the Medieval Trivium, the science of division had a guaranteed position of importance.

Aristotelian Logic in the later Medieval period

The sphere of intellectual activity centering in France and especially Paris during the eleventh and twelfth centuries produced great thinkers ranging from (1079-

1142)22 to Thomas Aquinas (1225-1274) ,23 all of whom are indebted to Aristotle.

During the twelfth and thirteenth centuries, when the Organon was disseminated in

Europe, exciting musical developments were taking place. reached a high degree of development, the precedents of modem tonality were well underway, and musical notation reached an amazing degree of complexity.

Because Aristotle had become the philosophical authority for the later Medieval period, music theorists, like scholars in other disciplines, strove to verify their arguments through the adcqttion of Aristotelian terminology. Among those who were influenced in one way or another by Aristotle, were Johannes de Garlandia,^^ the Anonymous of

^^Boethius In topica Ciceronis Commentarium : "Plato calls dialectic a faculty which is able to separate into many parts that which is one thing just as it often separates a genus through its own different parts down to the final species and often collects things which exist as many into one thing through the reason of their genera. " Boethius In topica Ciceronis Commentariorum , Patrologiae cursus complétas, series latina , accurante J. P. Migne (Paris, 1891), vol. 64, Lib, 1,1045B 1-8: "Plato etiam dialecticam vocat facultatem quae id quod unum est possit in plura partiri, velut soletgenus per proprias differentias usque ad ultimasspecies separari, atque ea quae multa sunt, in unumgeneram ratione colligere." Also see Eleonore Stump, Dialectic and It's Place in the Development o f Medieval Logic (Ithaca: Cornell University Press, 1989), p. 2. 22Abelard's primary work to be influenced by AristoteUan logic was his Dialectica, edited by J. M. de Rijk (Wijsgerigeteksten en studies. I). Of all the pre-Organon logicians, Abelard is conceded to have been the most influenced by Aristotle. See Martin Tweedale, "Abelard and the Culmination of the Old Logic," The Cambridge History o f Later , op. cit., p. 143. 23 Aquinas' writings are full with Aristotelian terminology. The Summa theologiae mentions Aristotle's Metaphysics , the Categories , Physics, De anima , and the Ethics . See Sancti Aquinas Summa Theologiae , Opera omnia, secundum impressionem petri fiaccadori parmae 1852-73 (New York: Musurgia Publishers, 1948), 4 vols. 2'^Erich Reimer, Johaimes de Garlandia : De mensurabili musica , 2 vols., Beihefte zum Archiv fur Musikwissenshaft, X-Xl (Wiesbaden: Steiner, 1972). 1279,“5 Anonymous IV,Jacques de Liege,^^ and Marchetto de Paduar^ Recent studies by Yudkin (1990)-^, Huglo (1990)^°, and Wingell (1990)^^ have focused on the infUuence of Aristotelian logic and philosophy on music theorists during the later Medieval period.

Aristotelian Logic in the Early Medieval Period

The premise of this study is that the Aristotelian science of division influenced early

Medieval music theorists in the same way that his other logical works influenced later theorists. Although less is known about the availability of Aristotelian texts on logic than about other texts, it is possible to show that even in the early Medieval period some scholars had a rudimentary familiarity with Aristotle’s views.

Following models instituted by Alcuin and his pupil Rhabanus Maurus (776-856), cultural centers were founded during the late eighth and early ninth centuries at the abbeys of Saint-Gall, Fulda, Reichenau, and others.^- The few texts on logic available in these new cultural centers included Boethius’ translations of the first two works of the Organon and treatises attributed to Augustine and Alcuin. Though logic was not a central part of the

Anonymous of 1279, De musica mensurata , translated with commentary by Jeremy Yudkin (Chicago: University of Chicago Press, 1991). “^Fritz Reckow, Der Musiktracktat des Anonymous 4 , 2 vols., Beihefte zum Archiv fur Musikwissenschaft , IV-V (Wesibaden, Steiner, 1967). Jacobi Leodiensis Speculum musicae , ed. Roger Bragard, Corpus Scriptorum de musica 3 [CSM] (Rome: American Institute of Musicology). “^Marchetti Padovensis Pomerium , edited by Joseph Vecchi, CSM 6. ^^Jeremy Yudkin, "The Influence of Aristotle on French University Music Texts," Music Theory and Its Sources : Antiquity and the , edited by Andre Barbera (Notre Dame: University of Notre Dame Press, 1990), pp. 173-89. ^^Michel Huglo, "The Study of Ancient Sources ofMusic Theory in the Medieval Universities," Music Theory and Its Sources, ed. Barbera, pp. 150-72. Richard J. Wingell, " Treatises : Speculatio versus institutio," Paradigms in Medieval Thought Applications in Medieval Disciplines, edited by Nancy van Deusen and Alvin E. Ford. Mediaeval Studies, Vol. 3 (Lewiston: The Edwin Lellen Press, 1990), pp. 157-71> Anton Dumitriu,History o f Logic , 4 vols. (Tumbridge Wells, Kent: Abacus Press, 1975, II, p. 15. writings of Augustine and Alcuin, their authority as clerics may have been responsible for the popularity of their treatises during the eighth, ninth, and tenth centuries.^^

On the other hand, a catalogue of holdings at the monastic library of Reichenau as early as the year 822 records copies of the Isagoge of Porphyry, the De interpretatione of

Aristotle,^'^ and a version of Boethius’ translation of Aristotle’s Categories. These logical treatises will be discussed during the course of this document. At the beginning of his

Musica, Hermannus specifically refers to the Isagoge of Porphyry and the Categories of

Aristotle,^^ two important logical texts. It is clear that these texts of the Organon were available to the earliest German music theorists.

The Problem for early Medieval theorists

Despite Aristotle’s influence on theorists during the later Medieval period, the eight mode system, which had been formulated and adopted centuries earlier, was still the classificational norm for polyphonic music.^^ The system was the same one which

Carolingian theorists used in the classification of the chant repertory in their desire to impose an order and unity on the large repertory of monophonic chant. Their goal was to transmit a comprehensive picture of the total chant repertory through representative

^^io)mMaxea\3on, Early Medieval Philosophy (380-1150) (London; Routledge and Kegen Paul, 1983),pp 80-1. The most studied woik on logic from Alcuin onward was the Categoriae decern, which had been attributed during the Middle Ages to Augustine. The Categoriae decern is a Latin epitone on Aristotle's Categories. Alenin's Dialectica shows knowledge, through Boethius' commenatries, of the Isagoge of Porphyry and the De interpretatione of Aristotle. See Marenbon, p. 47. Marenbon, op cit., p. 53. Trans. Ellinwood, Musica Hermanni Contracti , op. cit., p. 19: "For, to make a long story short, just as Grammar is reduce to eight varieties, the Isagoge to five, and the Categories to ten— " Musica Hermanni Contracti, p.l9: "Ut enim ex multis dicamus, sicut grammatica ad VIII isagoge ad V praedicamaita ad X varietates rediguntur...." ^^In this documait the eight modes are regarded as a single system in which they are arranged in four pairs of two, called the authentic and plagal modes, respectively. In each pair the first mode, or the authentic mode, is regarded as superior to the second mode, or the plagal. Yet the two modes in each pair mingle together as one unit. examples, for it would have been an impossible task to unify performance practices without some kind of logical classification model. The problem for early Medieval theorists, then, was to find a taxonomic paradigm that would enable them to classify the chant repertory according to the eight modes.

Organization of this document

The goal of this document is to demonstrate the influence of the Aristotelian science of division on the organization of the eight modes of music by early German theorists.

Chapter I presents the science of division as it is defined by Aristotle. The discussion differentiates between qualitative and quantitative division, and gives examples of each. Because Aristotelinan division features a specialized terminology called the predicables, Aristotle’s definitions follow. Also included are Boethius’ definitions of the predicables whenever they differ or amplify Aristotle’s definitions. An understanding of the predicables is necessary because Boethius used them in the De divisione. Similarly I will use the predicables in the taxonomic paradigm when it is applied to the eight church modes .

Because quaternary division was especially common in Aristotelian division, examples of quaternary division in Aristotle’s writings are included. Chapter I concludes with a comparison of quaternary division examples by the Pythagoreans and by the

Medieval philosopher John Scottus Eriugena. Although the two divisions are separated by about fourteen hundred years, they show such a remarkable similarity that I use them here to show the pervasiveness of quatemay division from the Greek to the early Medieval periods.

Chapter II discusses the background of the De divisione, and includes comments on the preservation and transmission of this treatise. 10

Boethius discusses six categories of division in the De divisione. Five of these categories were used by German theorists in their modal discussions. Chapter III summarizes these five division categories.

Chapter IV presents a taxonomic paradigm derived from the Be divisione, which adds musical nomenclature to the logical terminology.

Chapter V discusses the eight modes of music, focusing on the two primary ways early Medieval theorists rationalized the modes : 1) as a nucleus of tones and semitones around a specific pitch called a “final”, and 2) as octave species constructed of species of fourths and fifths. Species theory was the manner in which early German theorists rationalized the modes. The different ways in which Byzantine and early Western music theorists divided the eight modes using quaternary division is also discussed.

Chapter VI applies the taxonomic paradigm of the Be divisione to the eight modes of music. The purpose of this chapter is to show how the modal system can be rationalized through Boethius’ division types.

Chapter VII dscusses the terminological and conceptual usage of Aristotelian division in the eight mode systems of Bemo, Hermannus, Wilhelm, Aribo, and Englebert.

Quotations from their treatises are included as evidence of the influence of Aristotelian division. This chapter shows the degree to which these five theorists applied the Boethian concept of anthropomorphic opposition in their divisions of the eight modes by means of the division of genus into species through differentiae. CHAPTER I

The Aristotelian Science of Division

Division begins by amassing a collection of things to be classified. The collection is variously called the subject, the germs, or the whole. Once collection is completed, division begins and progresses to the point where further division is impossible. ^

Aristotle’s recognition of the value of division, as well as his observation about the limits of its usefulness, can be found in several logical and biological works.^ The most important extant treatise on division after Aristotle is Boethius’ De divisione. Since the De divisione is based on Aristotelian division, the discussion in this chapter will focus on

Aristotle’s views about division.

Aristotle’s discussions on division feature a specialized set of terms called predicables, and these terms were adopted by Boethius in his logical monographs, including the De divisione. Because the Aristolian science of division cannot be comprehended without an understanding of the predicables, definitions of them are presented before discussing division.

This chapter is concerned with two types of division which are called here qualtitative and quantitative division. Though Aristotle did not specifically name quantitative division, he said in the Categories that genus and species signify substance qualitatively differentiated,^ and his discussions suggest that division is either qualitative or quantitative, depending on the goal. For instance, if one wishes to reach a definition of

^The Works o f Aristotle, ed. Ross. Topica , trans. Pickard-Cambridge, 109b, 16-19. ^Aristotle, Topica , 101b38-102al, 103b27-27,143a34-bl, Prior analytics, 46a31-59, Posterior analytics , 91bl2-15,93b29, 10Gal4-b3,96b25-27a6, and Metaphysics , 1031al2,1038al-4,1045a23-35. On the limitations of division, see the Posterior analytics , 46a31-33. ^Aristotle's Categories and De interpretatione ,tisas. A.ckn\\. p. 5. Categories ,3& 20-21. 12

the subject which is divided, then qualitative division is necessary. If the goal of division is to reach a plurality of parts with no hierarchy, then quantitative division is required.

The number of parts into which either the subject or the genus is to be divided can vary with the individual doing the dividing. Examples of binary, ternary, and quaternary division can be found in Aristotle’s writings. However, if frequency of usage is any indication of usefulness, then quaternary division was the most useful division type for

Aristotle. To illustrate this point examples of quaternary division by Aristotle are included in later discussion.

Although Boethius’ De divisione is a summary of Aristotelian division, Boethius also incorporated an important Platonic device in his treatise: the use of an anthropomorphic illustration to show a division type. Plato derived anthropomorphism from the

Pythagoreans, and Boethius got it from Plato.'^ Medieval scholars acquired Pythagorean anthropomorphism from both Plato and Boethius.

The chapter will conclude with an example of quaternary division by the

Pythagoreans which shows a striking one-to-one correspondence with a division application by one of the most important early Medieval philosophers, John Scottus

Eriugena (d. 877). Both the Pythagoreans and John Scottus accompany their quaternary divisions with anthropomorphic illustrations. After the Pythagorean example is given, a short discussion follows which suggests that the quaternary divisions of the Pythagoreans and John are in reality examples of dual division. The example is given as propeadeutic to the discussion of the dual yet quaternary division of the eight church modes given in

Chapter VI.

^Chadwick, Boethius , op. cit., p. 201. 13

The Predicables

Medieval writers on division inherited from Aristotle and Porphyry a set of six terms which are used in division. These terms, called predicables, were developed by

Aristotle for use in structuring propositions.^ The definitions of the predicables given here are based on those of Aristotle and Boethius.

Kretzmann and Stump (1990) define a predicable as a universal attribute which can be asserted of more than one thing.® They list six predicables: 1) definition, 2) genus, 3) species, 4) difference, 5) accident, and 6) property. With the exception of accident, all of the predicables were used in the discussions of mode by the German theorists that I will examine in chapter VÏI.

Genus and species refer to things or attributes of things. Genus is a comprehensive class which is comprised of species. For Aristotle a genus is “what is predicated in the category of essence of a number of things exhibiting differences in kind.The genus is always of wider denotation than the subclass s p e c i e sIn the De divisione, Boethius defined genus as that which is predicated of more than one thing differing in species in respect of what it (the genus ) is.^ In other words, for Artistotle and Boethius, genus and species share some common attribute or attributes.

Species is a sub-class of . Though a j/jcc/cj usually contains several members which are individualized by some agent or difference, Aristotle said that the species can also be synonymous with individuals, In the De divisione Boethius equated

®M. Bochensk, History o f Formal Logic , p. 51. ® "Peter of Spain: Predicables; Categories," The Cambridge Translations o f Medieval Philosophical Texts. Edited by Norman Kretzmann and Eleonore Stump, p. 80. "^The Works o f Aristotle , ed. Ross. Topica , trans. by Pickaer-Cambridge, 1.5,102a,32-3. «Ibid., 121b, 3-5. De divisione, Patrologiae latina [PL], accurante J.-P. Migne, 880A13-15: "Gennî est quod praedicatur de pluiibus specie differentibus in eo quod est..." ^^The Works o f Aristotle , ed. Ross. Topica , trans. Pickard-Cambridge, op. cit .,154a, 17-18. 14

species with genus, since genus is always the proper whole of the species, but is more general in nature.”ii

A difference is the means by which we present one thing as standing apart from another.i 2 Differences are used to divide the \nXo species. Pi. difference is part of the essence of a subject that distinguishes one species from other species in the same genus. Boethius went a step further and recognized two types of difference : constitutive and divisive. The "constitutive" difference allows a new species to stand in the place of the genus. In this instance, the new then becomes a ge/iws which can be divided into still other species. The new species is called a subaltern genus. The "divisive" difference simply divides the genus.

Accident is an inessential property that may be attributed to a subject Accident is defined by Aristotle as something which may or may not belong to some “self-same thing”, such as in “sitting,” or in possessing the quality of being “ w h i t e . ” Por Boethius, if an attribute is not predicated on the subject’s essence, it is termed an accident.

Property is an accident that has a special relationship to the subject: it must be capable of being predicated convertibly of the subjecL^^ As an example of this, Aristotle says a property of man is that he (man) is capable of learning grammar. If a subject is capable of learning grammar, then he is a man.i^ It can be seen that property is an attribute which is unique to a specific subject.

Ibid., PL 878D13-879A1: ".. .genus semper speciei propriae totum est, ut universalius in natura." ^^Ibid., PL 880A15-B1: "Differentia, quia aliud ab alio distare proponimus." ^^Trans. Pickard-Cambridge, Topica , op. cit., 102b, 6-9. ^'^Boethius, De topicis differentiis, Lib. II, PL 1186C9-12: "Solumvero inesse ad accidens maxime pertinet, nam cum neque utgenus , neque utdiffintio , neque ut proprium inest, sed inest tamen, ut accidens inesse necesse est." ^^Trans. Pickard-Cambridge, Topica, op. c it., 102a, 188ff. l^ibid. 15

A definition is comprised of a genus plus one or more differentia. In the De divisione, Boethius said that division is necessary for a full definition of species, and that a single definition is made of divisions joined t o g e t h e r .

In summary, the predicables can be divided into two classes. On the one hand, genus and species refer to things or attributes of things. On the other, difference, accident, and property are the means by which we assign things or attributes of things to their proper ge/JM5 or species. This process creates a The predicables, then, provide a general rationale for the construction of theoretical descriptions (or definitions

), and this rationale creates the necessity for class-inclusion (as in genus or species ), and class exclusion (as in difference, accident, or property ).

Qualitative and quantitative division

Aristotle’s discussions of division differentiate two categories of division: 1) qualitative division, and 2) quantitative division.

In qualitative division, a hierarchical order of differentiae is successively added^^ to genus being divided.-^ The goal of qualitative division is to reach a of the divided subject which reveals the nature of kinds of things of which the subject is an example.-! When dividing qualitatively, one establishes a series of greater or lesser degrees of differentiae, so that the addition of a difference to the genus “intensifies that

! ^Boethius, De divisione, op. c i t PL 660C13414: "Nam divisionibus junctis una compositurdiffinitio." !®D. F. Henry, "Predicables and Categories," The Cambridge History o f Later Medieval Philosophy , op. cit., p. 132. To use the term "added" in a discussion of division would seem to be a contradiction. Yet Aristotle makes it clear that the addition of differentiae to a genus is the means for isolating the uniqueness of a particular subject which is to be defined. In the Topics he said that "for the fiamer of a definition should first place the object in its genus , and then append its differences." The Works o f Aristotle , edited by S. D. Ross (London: , 1928). Topica , translated by W. A. Pickard-Cambridge, 139a, 28-30. ^^The Works o f Aristotle , ed. Ross. Analytica posteriora , translated by G. R. G. Mure, 96a24-34. ^^Deslauriers, "Plato and Aristotle on Division and Definition ", op. cit., p. 208. 16

character which it has as a given.” — Aristotle said that the genus should fall under the same division as the sp e c ie s^ meaning that both the genus and species are to be divided quantitatively or qualitatively. For Aristotle, the purpose of division is to discover “kinds” of entities which contain the same differentia?-^ In reaching the definition, it is possible to include as many differentiae as one wishes as long as they form a consistent line of division.^

Another purpose of qualitative division is to show logical relations between classes.2^ An example of this is can be seen in the Medieval Porphyrian Tree. The

Porphyrian Tree is an example of qualitative division of the germs Substance. Substance is divided into the species animate and inanimate bodies. Under animate body comes rational and irrational animal. Under rational animal is man, and under man are Socrates, Plato, and particular men.27

When deciding whether to divide qualitatively or not, one should study the class of predicates to see if they admit an increase of degree. If they do, then qualitative division is the most useful division type.28 Diagram one shows how a definition is reached through the successive conjunction ofdifferentiae to a germs creating species. The diagram shows how a species is sought through the addition of a differentia to the highest genus. If a definition of the species is not reached, another differentia must be added to the existing differentiae. A definition is reached when enough differentiae have been added to the highest genus to reach the definition of Sl species.

^“The Works o f Aristotle , ed. Ross. Topica , trans. Pickard-Cambridge, op. cit., 115a, 30-2). 23/Wrf. , 121a, 6-7. 24/W., 143a34-bl 33 The Works o f Aristotle , ed. Ross. Analytica posteriora , trans. Mure, 96b25- 97a6. 3^0. M. Balme, "Aristotle's Use of Division and Differentiae," Philosophical Issues in Aristotle's Biology. (Cambridge: Cambridge University Press, 1987, ) p. 69. 33porphyry thr Phoenician. Isagoge. Translation, Introduction, and Notes by Edward W. Warren (Toronto: The Pontifical Institute of Mediaeval Studies, 1975), p. 35. 33 The Works o f Aristotle , ed. Ross. Topica , trans. Pickard-Cambridge, 115a, 30-32. 17

(higher) çentis — éfffsrentia = species

(higher) (lower) differeniie (a) = species genus — ^ differeniie ( b) + genus

differeniie (a) (higher) (lower) differentia ( b) + genus = species genus- differentia (c) idefinitian)

Diagram one. The creation of a definition through the qualitative addition of differentiae to a senus as an example nn qualitative division.

Quantitative division yields parts of things, not kinds of things, and these parts are related, although not hierarchically. The many parts of a divided whole may be listed or regarded in any order. This process is the reverse of qualitative division in that qualitative division proceeds from many parts to one part {ihedefinition ), while quantitative division proceeds from a singular entity (the whole) to many parts. If the class of predicates of a whole represent quantity rather than quality, that is, if the predicates do not admit an increase of degree, then one must use quantitative division.29 Diagram two illustrates quantitative division of a whole into parts.

part part part

whole

Diagram two. The quantitative division of a whole into parts.

29lbid., 152a, 37-152b,5. 18

Both qualitative and quantitative divisions were used by the Medieval German theorists in their discussions of the modes.

The Stages of division

Division involves three stages. The first stage consists of collecting the things to be classified. Stage two establishs the differences among the things through the addition of differentiae. For Aristotle, these differentiae must be based on naturally occurring conditions.^® Stage three consists of classifying the results of the division according to an external taxonomic paradigm.

After division is completed, representative examples of each of the categories are selected for demonstration. This process is called induction,^! and the greater the number of divisional categories, the more accurate will be the inference concerning a particular category.

Both qualitative and quantitative division were basic to that part of Aristotelian logic known to early Medieval German music theorists. As I will show in Chapter IV, both division types were prominent in the De divisione of Boethius.

Examples of division in Aristotle’s writings

Quaternary division was the division type most frequently applied by Aristotle. A few representative examples of this division type will demonstrate its importance.

As shown in Dumitriu's History o f Logic, Aristotle’s entire corpus of writings can be divided into four areas: 1) logic, 2) rational philosophy, 3) metaphysics, and 4) practical sciences.^2

3®Deslauriers, Plato and Aristotle, op. cit ., p. 212. The Works o f Aristotle , ed. Ross. Topica , trans. Pickard-Cambridge, op. cit., 108b, 10-14. 32Dumitriu,History o f Logic , op. cit.. I, p. 143. 19

In his Topics, Aristotle divided propositions into four types. The elements from which propositions or problems are formed are 1) property, 2) definition, 3) genus, and 4) accident. According to the nature of their predicates, any proposition belongs to one of these four species of propositions.^^

In his Physics, Aristotle outlined four universal conditions which are common to all natural phenomena. The four conditions are 1) movement, 2) place, 3) void, and 4) time.3'^

In his earlier work, the Categories, Aristotle spoke of the four types of “quality,” which include 1) states or conditions, 2) things in virtue of a natural capacity or incapacity,

3) affective qualities and affections, and 4) the shape and figure or external of a thing.^^

Also in the Categories he also spoke of the four types of “things that are.” These include 1) things that are said of a subject and in a subject, 2) things that are in a subject but not of a subaltern subject, 3) things that are both of a subject and in a subject, and 4) things that are neither in a subject nor said of a subject.^^

In book II of the Posterior analytics, Aristotle said that there are four questions which cover the whole sphere of knowledge. These are 1) is the connection of an attribute with a thing a fact, 2) is there a reason for the connection, 3) does a thing exist, and 4) what is the nature of a thing.^?

Trans. Pickard-Cambridge,2b/>/ca , 101b, 23-5. ^'^Aristotle, Th&Physics, with an English translation by Philip H. Wicksteed and Francis Comford, 2 vols. (New York: The Loeb Classical Library, 1929), p. 193. The four conditions are l)movement, 2)place, 3) void, and 4) time. ^^Aristotle's Categories and de interpretatione, translated with notes by J. L. .Ackrill (Oxford: Clarendon Press, 1963), pp. 24-31. 2®Ibid., p. 4. Works o f Aristotle , ed. Ross. Analytica posteriora , translated by G. R. G. Mure. Bk II, 89b, 21-5. 2 0

Aristotle’s famous four primary “causes” for the understanding of particular things are listed in the Metaphysics. The four “causes” are: 1) formal, 2) material, 3) efficient, and 4) final.^® In summary, many examples of quaternary division can be found in Aristotle’s thinking. It is reasonable to infer that, if Aristotle’s defintion of division was adopted by Medieval scholars, then his predilection for quaternary division may also have been equally influential.

The Influence of Greek Quaternary Division on John Scottus Eriugena Quaternary division can be seen in Medieval philosophical treatises, particularly in those by writers trained in logic.^9 Moreover, it is evident that the manner in which some Medieval writers divided a subject was so similar to Greek procedures, that a direct influence is likely. One remarkable example of the commonality between Medieval and Greek quaternary division will illustrate this point. A striking similarity can be seen between the Pythagorean quaternary division of numbers and the quaternary division of the Universe by the important early Medieval philosopher John Scottus Eriugena (d. 877). John, who was associated with , grandson of Charlemagne, conjoined logic with his theological system in a major philosophical work entitled the Periphyseon (c. 870) This text, although primarily a Neo-Platonic work,*^! shows that John possessed a substantial knowledge of Aristotelian

^^Aristotle, T?ie Metaphysics , trans. Tredennick, I. II, 983a, 24 -983b. ^^Marenbon, Early Medieval Philosophy , op . cit., p. 157. ^^Periphyseon is contained in the Patrologiae latina , ed Migne (Paris; Migne, 1844-64), vol. 122. A newer edition is available by I. P. Sheldon-Williams (Dublin: Institute for Advanced Studies, 1969-81). “^^Some of the Neoplatonic elements in the Periphyseon include: l)the trichotomy of the Universe into God, the (Platonic) Forms, and Sensible Matter, 2) a regard for the movement of imiversal nature as consisting of three aspects: pennanance, procession, and return, 3) an echo of Plato's World Soul {PL 122, 476C-7A), and 4) the Soul (human soul) is responsible for creating all that lies below the the intelligible world. See Marenbon, Early Medieval Philosophy, pp. 60-70.. 2 1

logic. The entire content of the Periphyseon is structured around a quaternary division which is similar to the quaternary division of numbers by the Pythagoreans. Shown below is a comparison of the quaternary division of numbers by the Pyth^oreans with an

“anthropomorphic” division of the Universe given in the Periphyseon of John:

Pythagoreans'*- Periphyseon “*3

Monad: That which begets without 1. That which creates and is not being begotten created. Tetrad: That which is begotten and 2. That which is created and does not beget creates. Ogdoad: A three dimensional 3. That which is created and does construct of the sensible not create. world. Hebdomad: That which is neither 4. That which neither creates nor is begotten nor begets. created.

Table one. A comparison of the quaternary division of numbers bv the Pythagoreans and the division of the Universe in the Periphyseon.

Hadot has suggested that this Pythagorean division was known to John through a letter which included a dispute between Candidus and Marius Victorinus over the issue of the Hebdomad, in which God, who is Immutable, is neither created nor creates.'^ John seems to have made some notations in this letter.

If one looks closely at the quaternary division of the Pythagoreans and of John in the Periphyseon, it seems that two of the divisions are unnecessary. These are the

Pythagorean Ogdoad and the Hebdomad, and John’s divisions two and four. All that exists can be justified within the Pythagorean Monad and the Tetrad, and divisions one and three of the Periphyseon. For the Pythagoreans, the Tetrad (the Universe) exists because

^^This view of the Pythagoreans was transmitted through Philo, whose writings indicated that this Pythagorean quaternary division was known to Plotinus and Philolaus, and was hence known in Neoplatonist circles. See The Cambridge History o f Later Greek and Early Medieval Philosophy , ed. Armstrong, p. 522. 43johannis Scoti, De divisione naturae libri quinque , Patrologiae latina , acc. Migne, vol 122 ,441- 1022. ^ P . Hadot, Marius Victorinus: Traites theologiques sur la Trinité (Paris, 1960: Sources chret. 68), pp. 112-24. 2 2

of the existence of the Monad (God). The Monad is the only division that can create

(beget). An identical relation can be seen between divisions one and three of the

Periphyseon. In effect, all existence, whether it is God or the Universe, can be assigned to these two divisions.

In summary, the quaternary division examples of the Pythagoreans and John

Scottus can be seen as actually constituting a dual division type. The Monad of the

Pythagoreans and John’s division number one are “perfect” and “immutable” because they are capable of creating without themselves being created. The Tetrad and John’s number three are “imperfect” and “mutable” because they are created without creating anything themselves.

This point is stressed here because it reflects the Medieval conception of the relationship between the authentic and plagal modes which are discussed in chapters V and

VI of this document. In this conception, the authentic modes are seen as “immutable” and

“perfect”, while the plagal modes are seen as “imperfect” and “mutable.”

C onclusion

Through his definitions and examples, Aristotle demonstrated his valuation of division. Not only his prescriptions of division for others, but his own division of the sciences shows the practical ways he uses division.

The predicables may seem ambiguous to us today, yet Aristotle’s careful definitions of them, as well as the amount of attention he spent explaining division, may help explain why the Medievals regarded division so highly. CHAPTER II

The History of Boethius’ De divisione

Boethius’ De divisione is a technical treatise on the application of classification techniques. Like technical books written for the two other disciplines comprising the early

Medieval trivium, grammar and rhetoric, the De divisione was written for a specific practical purpose. Its purpose is to show how to divide and classify a subject through observation of similarities and differences among the subject's constituent parts, and thereby reach a definition of the subject.

Boethius’ De divisione, a treatise written between 505 and 509,^ is relatively short, if it is compared with some of his better known texts, the De institutione musica, tite De arithmetica, and the Consolation o f Philosophy, as well as some lesser known texts, the

Commentaries on Aristotle’s Categories and the De interpretatione. It would be easy to underestimate the importance of the De divisione. No English translation was available until 1990,2 although an Italian translation and commentary appeared in 1969.^ Scholars interested in the text must still consult the Latin text found in the Migne edition of the

Patrologia latina.^ However, the De divisione, as well as similar early texts on classification, had a long and pervasive influence on Medieval logic and taxonomy. Both this treatise and the De topicis differentiis,^ another monograph of Boethius with which the

^L. M. tie Rijk, "On the Chronology o f Boethius' Works on Logic IVivarium , 2 (1964), p. 47. 2An English translation is available. See Boethius on Division in the Cambridge Translations o f Medieval Philosophical Texts. Volume One; Logic and the Philosophy o f Language. Edited by Norman Kretzmann and Eleonore Stump (Cambridge: Cambridge Univasity Press, 1988), pp. 11-38. 2See the Trattato sulla Divisione , translated by Lorenzo Pozzi (Padua: Liviana, 1969). ^Boethius, De divisione , ed. J.-P. Migne, in cursus complétas , vol. 64,874D-892A; (Tumbolt: Brepols, 1860). ^Boethius, De differentiis topicis, ed. J.-P. Migne, in Patrologia latina , vol. 64,1173B2- 1218C6. An English translation is available. See Boethius's De topicis differentiis . Translated, with Notes and Essays on the Text by Eleonore Stump (Ithaca: Cornell University Press, 1978). 23 24

De divisione was frequently paired,^ were part of the “'logica vetus, ” or “old logic,” the

primary course of study for scholars from the fifth to the twelfth centuries. (The De topicis

differentiis is a detailed study of dialectical and rhetorical topics for the purposes of

argumentation,'^ )

Boethius’ sources for the De divisione

At the beginning of the De divisione, Boethius attested to the importance of his

subject by briefly discussing some of his precedents. He began by referring to a lost

treatise on division by Andronicus of Rhodes,* a Peripatic philosopher of the first century

B.C. Boethius also referred to Porphyry's commentary Plato’s Sophist? also lost. From

these statements, Chadwick has suggested that a major source for the De divisione was probably this Porphyrian commentary, since it was evidently designed to synthesize the

Platonic dialectic of the Sophist with that of Andronicus on division.^® Magee ( 1991) agrees that the Porphyrian commentary was probably a source, even though the text of the

De divisione mentions no specific Platonic dialogues,!^ but questions the work of

Andronicus, due to slim evidence of its e x i s t e n c e .

®Sten Ebbesen, " Ancient Scholastic Logic as the Source o f Medieval Scholastic Logic , " The Cambridge History o f Later Medieval Philosophy , op. cit., p. 122. ^Boethius's De topicis differentiis , trans. Stump, p. 221. * Boethius, De divisione , PL 875D1-4: "Quam magnos studiosis afferat fructus scientia dividendi, quaroque apud peripateticam disciplinam, semper haec fuerit in honore notilia, docet at Andronici, diligentissimi senis, de divisione liber editus..." Andronicus was also the Erst editor of the Aristotelian Organon . See Ebbesen, "Ancient Scholastic Logic," The Cambridge History o f Later Medieval Philosophy , p. 105. ^Ibid., PL 876D1-2. ^®Chadwick, Boethius , op. c it., p. 164. ^ * John Magee, "Boethius' De divisione, text and sources ," Abstracts of American Philological Association Annual Meeting (Chicago, Dec. 18,1991). ^^Ibid. 25

The divisional paradigm used by Boethius can, however, be found in other writers who preceded him, including Albinus, Sextus Empiricus, and Clement of Alexandria.

Reiss (1982) suggested that in the De divisione Boethius intended to “correct” Aristotle's view on the value of division. In the preface to an Italian translation of the De divisione

Pozzi (1969) stated that the Boethian monograph is an original synthesis of the Stoic,

Platonic, and Aristotelian conceptions of division.!^

The fact that the De divisione and the De topicis differentiis were considered original monographs by Boethius may have contributed to their great prestige during the

Medieval period. This is especially the case during the early Medieval period, before the large number of manuscript copies of the De musica and the De arithmetica were circulated.

It was on these latter works that much of Boethius’ later reputation was to rest. Through

Boethius’ logical monographs the Medievals received their basic philosophical concepts, particularly the terminology used in Medieval Logic.

When the entire Aristotelian Organon became available in the early twelfth century in new translations, only the De divisione and the De topicis differentiis were retained in the curriculum at the University of Paris,^’^ but modem scholars agree that Boethius was one

Chadwick, Boethius , op. cit. In Book I, Chapter XII of the De institutione musica , Boethius also mentioned Albinus. In this chaper, Boethius referred to the method of division of voices. He described it as a twofold division, which included: 1) Continuous voice which is interrupted by intervals, and 2) Continuous voice as when speaking or reciting a prose oration. Boethius said that Albinus added a third type which included an intermediate voice, such as in the recitation of heroic poems as in a singing manner. See the De institutione musica , translated by Calvin Bower ^'^Edmund Reiss, Boethius (Boston: Twayne Publishers, 1982), p. 52. ^ ^Lorenzo Pozzi, Trattato sulla divisione . (Padua: Liviana, 1969), p. 9: "Nella dottiina stoica abbiamo una rivalutazione della divisione, rivalutazione che seque quella della dialettica." L. W. Laistner, Thought and Letters in Western Europe A.D. 500 to 900 (London: Methuen and Co. Ltd., 1931), p. 60. 1 ^Osmond Lewry, O.P., "Boethian Logic in the Medieval West ", Boethius, edited by Margaret Gibson (Oxford: Basil Blackwood, 1982), p. 113. See also the Statute o f the Faculté des Arts, 19 march 1255, listed by Michel Huglo, "The Study o f Ancient Sources o f Music Theory in the Medieval Universities ," Music Theory and Its Sources, ed. Barbera, p. 153. 26

of the main philosophical sources for the early Scholastics, and that he must be considered as second only to Augustine as an authority among Christian philosophers.^^

Preservation and transmission of the De divisione

The preservation and transmission of the De divisione is difficult to trace during the centuries immediately after Boethius’ death. A senator and friend of Boethius, Martius

Novatus Renatus, may have been responsible for its earliest preservation.^^ Renatus’ name appears in a series of scribal subscriptions in several early manuscripts of the dialectical treatises. A manuscript of the tenth or eleventh century from Fleury^o contains comments of Theodorus the librarian; this fact indicates that the De divisione had been included. Theodorus claimed to have “corrected” the codex of Renatus.^i Renatus, who was evidently an important figure during Boethius’ life, commissioned Theodorus to copy the dialectical writings at Constantinople.^^

According to Magee, Boethius’ logical monographs were circulated as a corpus probably until the Carolingian period, when individual treatises began to be copied separately.^

In the Bibliotheca bihliothecarum D of 1739, Montfaucon said that he had seen a codex which had been drawn by the scribe Theodorus from an autograph belonging to Flavian, a pupil of Priscian.24 This codex contained a copy of the De divisione. According to

^^"Boethius on Division ", The Cambridge Translations, eds. Kretzmann and Stump, p. 11. l^Chadwick, Boethius , p. 255. 2®0rleans 267; Paris Bibli. Nat. nouv. acq. 1611 (libri 31). fo. 51r. Chadwick, Boethius , p. 256. 22 Reiss, Boethius , op . c it., p. 135. 23Magee, "Boethius' De divisione , text and sources, " o p . cit. 2'^Montfaucon, Bibliotheca bihliothecarum II (Paris, 1739), 1130d, no. 481, as cited by G. Schepss, "Subscriptionen in BoethiushandschriAen ", Baltter fur das bayerische Gymnasialschulwesen 24 (1888), p. 24. 27

Chadwick, the manuscript research of Lorenzo Minio-Paluello has produced two recensions of Boethius’ translation of Aristotle’s Topics?^ A fragment of the second translation was inserted into the middle of the De divisione^ and was preserved in twelve manuscripts. Of these, the earliest one dates from the tenth or eleventh century.^®

Manuscript copies of the De divisione may be found in the Arundel Manuscript

Collections^ at the British Library, as well as in the Harleian Collection,^^ also at the same library. Abelard's short glosses on the De divisione are preserved in the Bibliothèque

Nationale in Paris, and the De divisione was one of the direct sources for Abelard’s

Dialectica?'^ Yet, since the De topicis differentiis existed in about 170 extant manuscripts from the tenth to the eleventh centuries,si it seems reasonable to infer that its companion, the De divisione , must also have been viewed as an important work. Confirmation of this view came from recent research by Magee, who showed that the De divisione was copied at least 195 times, and was widely circulated throughout Europe and Britain.sz

C onclusion

Boethius’ De divisione is a synthesis of division types given by the Greeks. Its structure suggests that it was a handbook designed for pratical application. Since the treatise was known and copied many times during the Medieval period, it must have been considered a useful text in the division and classification of things.

25chadwick, Boethius , op . cit ., p. 138. 26lbid., p. 201. ^^Bibliothecae Aruudelianae , No. 383, fol. 54, and No. 392, fol. 151. '^Bibliothecae Harleianae , No. 3272, f. 5. “^^Biblitheque Nationale , MS. Lat. 13.368, fol. 146r-156r. 20petnis Abaelardus, Dialectica, edited by L. M. De Rijk (Assen: Van Gorcum and Co., 1956), pp. 535- 81. 21l. M. De Rijk, "On the Chronology o f Boethius' Works on Logic IIVivarium , p. 153, note. 1. ^^Magee, "Boethius' De divisione , text and sources ," o p . c it. 28

However, information concerning the location of manuscripts and the availability of the treatise to scholars of the early Medieval period is still lacking. More research needs to be done in this area. CHAPTER III

Summary of the De divisione

Boethius describes six types of division in the De divisione. Only five of these were used by the German theorists discussed in Chapter VII of this study, but since this chapter summarizes the De divisione, all six division types will be included. The taxonomic paradigm which is presented in Chapter IV is based on the five division types that were actually used by the theorists.

The De divisione is not Boethius’ only treatise which features the division types presented here. For example, in Boethius’ Commentaries, three to five division types can be found. However, it is only in his second Commentary on Porphyry’ s Augoge that all six types are found, though they are listed in a different order than in the De divisione.^

General categories of division

In the De divisione, Boethius presents two general categories of division, each of which is subdivided into three types. Each of the six types has specific criteria that identify the type of division to be employed. Of Boethius’ six types of division, four use the predicables, and though the other two do not, they show an obvious indebtedness to

Aristotelian terminology.

^De Rijk, "On the Chronology," pp. 44-5. 29 30

A. Division secundum se.-

1. The first general type of division secundum se is the division of a genus into two or more species ^ by the appending of a difference.^ A difference is part of the essence of a subject that distinguishes a species from all other species in the same genus. The more differentiae that are added to the subject, the more specific the definition of the species becomes. This is the basis of qualitative division in which species are hierarchically related.

2. The second type of general division secundum se is that of a whole divided into its proper parts: ^ “the number of times we separate each out of the whole. One may divide the whole into as many parts as needed. Though this division type appears obvious and simplistic, it is distinct from the division of a genus into species, since the division of a whole into parts is an example of quantitative division. Thus the parts are not hierarchically related. Boethius stresses that the parts are already present in the whole, and that the whole continues to exist after the parts are determined. (Boethius' wording for this division type should be considered carefully, since the division of a whole into parts can be seen as contingent on the “paradox of Increase,” a principle articulated by Henryk after Abelard’s description of the principle in the Dialectica. The paradox states that it is impossible for anything to increase by the addition of parts, because when further parts are adjoined to a

^Boethius, De divisione , PL 878D5-6: "Omnis enim vocis et generis et totius divisio, secundum se divisio nuncupatUT." ^PL 877B8-9: "Est enim divisio generis in species ^PL 879D1; "Nam sicut species ex genere constat et differentia.. ^PL 877B9-10: "Est rursus divisio cum totum in proprias dividitur partes." ^PL 877C12-13; "Totum in partes dividitur quoties in ea ex quibus est compositum unum quodque resolvimus..." Normore points out that, in Medieval philosophy, the application of the principle that numerically distinct things are wholly distinct - that is, they contain no overlap of parts - was a pervasive concept in metaphysics. He cites Abelard, who, in the Theologia Christiana , Bk. HI, pars. 148-53, stated that having no overlap of parts is what characterises numerical distinctness. See Calvin G. Normore, "The Tradition o f Medieval Nominalism ," Studies in Medieval Philosophy, p. 214. ^D. P. Henry, Medieval Logic and Metaphysics (London: Hutchinson University Library, 1972), p. 120. 31

thing, neither that to which the parts are.adjoined, nor the adjoined parts themselves, increase in the sense that they have more parts than they had before.^ I make this point

here in order to show that for Boethius the number of parts contained in a whole is a fixed and inevitable quantity, which can be divided in one and only one way.)

3. The third type of division secundum se is that of an expression (or utterance) that

signifies many things when it is examined and the plurality of its significations is

disclosed.^ This division type is subdivided into two groups, nouns^® and expressions

which consist of nouns and verbs. ^ * Boethius calls the proper signification of a noun or

name a partition of equivocation, while the distribution of an expression is a discrimination

of ambiguity. The division of a whole into parts and the division of an expression are the

two types which do not feature the predicables.

B. Division secundum accidens.

1. The first type in this category is the division of a subject into its accidents

Boethius exemplified this division type by saying that of (the subject) Man, some are black,

some white, and some of an intermediate color.

^Petrus Abaelardus,Dialectica, ed. De Rijk, p. 421,25-37: "Neque enim cum aliquid alicui apponitur, allud quod appositum est, crevit neque illud oui appositum est, cum plures partes quam prius non habeat." ® De divisione, PL 877B10-11: "Est alia cum vox multa significans in signicationes propria recipit sectionem." am following the example of de Rijk, who translates 'nomen' as noun'. See de Rijk, "On the Chronology o f Boethius' Works on Logic I, " p. 46. ^ ^De divisione, PL 878A1-2: "Aut enim unum nomen multa significat, aut oratio jam nomibus verbisque composita. ." ^^PL 877B12-13: ".. .est alia divisio quae secundum accidens fieri dicitur." There are two types of accident: separable and inseparable. In the Aristoteles latinos , these two kinds of accident are given, but they are disthiguished as 'common' and 'particular'. ^^PL 877B13-14: ".. .unus cum subjectum inaccidentia separamus..." PL 878A14-15: ".. .ut cum dicimus omnium homimm alii sunt nigri, alii candidi, alii medii colons." ^^PL 878A14-15: ".. .ut cum dicimus omnium hominun alii sunt nigri, alii candidi, alii medii coloris." 32

2. The second division secundum accidens is the opposite of the first type. This is the division of an accident into subjects. Boethius exemplified this type by saying that, among desirable things, some are located in the soul while others are in the body, since what is sought is an accident of the soul and the body, which are subjects of a goodness located in them.^®

3. The final division of this type is that of an accident into other accidents}'^

Boethius said this type is illustrated by the division of objects into solid and liquid objects.^® He also said that this type of division becomes transformed into one of the other two types of division by accidentP if both accidents are possessed by the same subject.^^

The Oppositions

Near the end of the treatise, Boethius says that the division by accident involves contrarity.2i Thus, the separation of a species from a genus occurs according to one of four Oppositions:^-

1. Oppositions which are antithetical to each other, such as good and evil.^

2. Oppositions which are due to a possession or privation,^^ as when we apply the difference of possession in the division of a g enusP

877B14-15: ".. alius cumaccidens in sunjecta dividimus. " PI, 878B2-4: "Accidentis vero in subjecta sectio evenit, ut es omnium quae expetuntur, alia in anima, alia in corporibus si ta sunt." 878B6-8: " ... et boni quod in anima et in corpore est situm non sunt haec species, sed subjecta." 877B15-16: ".. .tertius cum accidens in accidentia secamus." 878B11-12: " Sed album in dura atque liquida s^aratum est." ^^PL 878B13-14: ".. .sed hujusmodi divisio vicissim semper in alterutra permutatur." 878C11-12: ".. .si utraqus contingerent eidem inesse subjecto..." 890D12-I3 - 891 Al; " Harum autem commune praeceptum est, quidquid ipsorum dividitur in opposita disgregari..." ^^De divisione, PL 882A3-5. : "Secimdum quas igitur harum oppositionum quatuor divisio generis sit, rectissima ratione monstrandum est." ^ P L 881D6: ".. .aut in contraria, ut bonum malo." 2'tpz, 881D7-8: ".. .aut ut habitus et privatio." 883A15-16: "Unde fit utprivationis

3. Oppositions which occur through affirmation or negation.^® Concerning this type Boethius said it is often necessary to devise a species by negation, if a single word does not exist to give (the species) a simple (affirmative) name.^^

4. Oppositions which occur through relationship, as in father, son, master, slave.-®

Although the Oppositions in the De divisione are the same as Aristotle's in the

Categories, Boethius listed them in a different order. However, Boethius did use

Aristotle’s order in his Commentary on Aristotle’s Categories^ Thus, it is possible that maintaining the Aristotelian order had some importance for Boethius in the Commentary.

Of far more importance than the order of listing of the Oppositions is the fact that Boethius chose to illustrate two of the Oppositions with different examples than Aristotle's. Of these two, the example that Boethius used to illustrate the Opposition of relationship - father, son, master, slave - bears no resemblance to Aristotle’s illustration of relationship.

Aristotle exemplified the Opposition of relationship by comparing of a “double and a hair, and “ knowledge with the knowable.”®® The fact that Boethius used an anthropomorphic example to illustrate the Opposition of relationship is significant for music theory because

Medieval theorists viewed the authentic and plagal modes as existing in an anthropormorphic relationship. For example the authentic modes were often called

"teacher" or "master," while the Plagal modes were called "pupil" or "servant."

881D12-882A1: "Tertia oppositio est quae est secundum afRmationem et negationem..." 882B6-9: "Necesse est autem saepe speciem oegatione componere, cum ea quam simplici nomine speciem volumus assignare nullo vocabulo nuncupatur..." 882A2-3. "Quarta secundum relationem, ut pater, Rhus, dominus, servuus." Chapter 10 of the Categories, Aristotle hsted the four Oppositions as follows: 1. relationship 2. contraeity 3. privation and possession 4. affirmation and negation See Aristotle's Categories and De Interpretatione, trans. Ackrill, p. 31. In Chapter 11 of the Categories, Aristotle said that all contraries must either be in the same genus , or in contrary genera , or be themselves genera, ®®Trans. Ackrill, Categories , pp. 31-2. 34

Boethius stressed the importance of applying the Oppositions to the division of genus into species. He said that the division by applying Oppositions as differentiae to genera are num erous,and, in most cases, we equate Oppositions with differentiae.'^- The concern with the division of genus into two species through the difference of Opposition has important consequences for the four-fold division of the eight church modes into authentic and plagal species. This is because the authentic and plagal modes, which are comprised of species of fourths and fifths, are distinct due to the inverted positioning of the fourths and fifths. As I will discuss in Chapter VI, this inverted positioning is an example of Boethius’ first opposition of contraeity.

The Division of a Whole into Parts

Concerning the division of a whole into parts, Boethius says that what we call a whole has many meanings,33 which include qualities of being continuous,^^ non- continuous,35 universal,^^ or consisting of some powers.37 Later he said that a whole is classified as a substance and a form.38 in the process of division, which may be seen as an opposite process from the construction of a definition (discussed below), Boethius said that a genus can be considered to be a whole, while the species can be regarded as a part.39

Differentiae may also be seen as wholes, while a species may be regarded as a part.'*®

31Dc divisione , PL 883C2-3; "In contrariis autem generum multa divisio est." 883C3-4: "Fere enim cuncias differentias in contraria decucimus." 887D5-6; "Quod enim dicimus totum, multipliciter significamus." 3^PL 887D6-7; "Totum namque est quod continuum est..." Chapter 6 of Aristotle's Categories discusses the distinctions between continuous and non-continuous (discrete) quantities. In his discussion of quantity, Aristotle made the distinction quantities whose parts beaar a relative position to each other otiier, and of quantities whose parts do not bear a relative position. See Aristotle, Categories , chapter 6. 33pL 887D8-9: "Dicimus quoque totum quod continuum non est..." 3®PL 887D10-11: "Dicimus quoque totum universale est..." 37pL 888A1-2: "Dicitur quoque totum quod ex quibusdam virtutibus constat..." 888B10-11 : "Fit autem totius divisio in materiam atque formam." 887C13-14: "Sic ergo in divisione genus est totum, species pars." 887C14-15; "Eodem quoque modo 4/j^renriac totum sunt, speczer in quas illae dividuntur partes sunt." 35

Finally Boethius asserted that, in some instances (such as the division of syllables into letters), things are perceived as “parts of the parts."'^'

No matter which of the six types of division one uses, whether: 1) the division of a genus into species, 2) the division of a whole into parts, 3) the division of an expression

(or utterance) into many meanings, 4) a subject intoaccidents, 5) accidents into subjects, or

6) accidents into other accidents, the resulting subdivisions must exhibit some property which properly confirms the appropriateness of the division. Hence the species o f a. genus must possess the property of being in one of the four modes of Opposition. The genus must be seen as having been divided into certain "offsprings of itself, meaning that the division must exhibit the essential nature of the genus, while it must also exhibit a difference. Moreover, the division of a whole is not to be considered as arbitrary, but must be divided into a specific number of parts, since this division is done by quantity

Finally, the parts which constitute the whole should precede their completion, sometimes naturally, and also temporally.'^

Construction of a d efin itio n

After indicating that the demonstration of a definition is discussed in Aristotle’s

Posterior Analytics ,B o e th iu s gives a short but important synopsis of the rules for creating a definition. Boethius restricted his construction of a definition to intermediate

888B3-4: ”.. .alio tamen modo acceptae non partes totius, sed partes partium sunt." 881A4-5: "Ilia vero quae {differentiae ) per se sunt sola ad divisionem generis apta sunt. Haec enim informant et perticiunt uniuscujusque substantiam..." ^^PL 879B3-5: "Generis quoque sectio a totius distributione sejungitur, quod totius divisio secundum quantitatem fit." ‘^P L 879B14-C1: "Partes quae totum jungunt compositi sui perfectionem alias natura tantum, alias ratione quoque temporis an te^ u n t" ^^PL 885D7-11 : "Et illud quidem, an ulla possit diffinitio demonstrari, et quemadmodum per demonstrationem valeat inveniri, et quaecunque de ea subtilius inPostremis Analyticis ab Aristotele tractata sunt praetennittam." In Book II, Ch. 10 of the Posterior Analytics , Aristotle discussed the types of definition . He defined definition as: a) an indemonstrable statement of essential nature, orb) a syllogism of essential nature differing from demonstration in grammatical form, or c) the conclusion of a demonstration giving essential nature. 36

things,'^ since these can be predicated of other genera, species, or individuals.'^’^ It is

significant that Boethius said no definition can be found for either the “higher” g e n e r a ,or the “lower” things, such as individuals, since these lack specifying differentiae^^ His purpose in this section of the De divisione was to demonstrate that the definition of a species consists in the conjoining of a genus with a sufficent number ofdifferentiae to

‘equal’ the s p e c i e s .As an example, he applied his method to the definition of a

“Noun. ”51 This process is to be done through the division of the differentiae of a genus.

As if it were a summary statement, Boethius clarified the relationship among the genus, the speciess, and the definition. He said that a genus is a whole in division, but [the species ]

may be a part in a definition.^^ Boethius went on to say that, in the construction of a definition, the genus and the differentiae are parts, while the species which results from this process can now be considered a whole.53 Boethius was suggesting that the distinctions among genera, species, whole, and parts are not in fact rigid, but that the terms may sometimes be used interchangeably, depending upon whether division of a whole is into quantitative parts or the construction of a definition is being sought.

divisione, PL 886A5-6: "Mediae igitur quae et habeat genera, et de aliis.." 886A6-8: ".. .vel generibus , vel de speciebus, vel de individuis praedicantur sub diffinitionem cadere possunt." 885D-13 - A-2: "Superiores quidem diffinitio nulla complectitur, idcirco quod earum superiora genera inveniri non possunt. " PL 886A2-4: "Porro autem inferiores ut sunt individua, ipsa quoque specificis differentiis carenti." 5®PL 886B8-11; ".. .et postremo toties differentias differentiis distribuimus usque dumomnes junctae generi speciem oscpAidiffinitione describant." Boethius'method of constructing a is similar to Aristotle's discussion on the division of a genus, found in Bk. II, Ch. 13 of the Posterior Analytics. Especially pertinent are the three objects that Aristotle said were to be kept in view during the establishment of a definition by division. These are 1) one should admit only elements in the definable form, 2) these elements must be arranged in the right order, and 3) no elements must be excluded. 51/*L 886B11-12: "Si est nobis diffinire nomen.. 52/»Z, 887B13-14: ".. .quod genus in divisione totum est in diffinitione pars..." 887C16-D2: "In veto eX. genus ei differentiae partes sunt. veto species totum. . ." 37

Criteria for division

A. In the divisional paradigm suggested by Boethius, the following conditions hold for division secundum se. In some cases the conditions must hold for the division.

In other cases the conditions may hold for themselves.

The genus :

1. Must be capable of division into two or more species.

2. Must be divisible into certain “offsprings of itself", whereby the resulting species retain

the nature of the genus.

3. Must be more universal than the species.

4. Must exist prior to the species.

5. Must be divisible by differentiae which are not separable ( thus not using differentiae

which are separable accidents, e.g. sleeping, standing, etc.).

6. Must be divisible according to definite criteria.

7. Must be capable of qualitative division.

8. May be most commonly divided by the first Opposition: contraeity.

The species :

1. Must possess differentiae which vrill distinguish it from other members of the same

genus.

2. Must be the same in substance as the genus.

3. Must be capable of being identified by the name and definition of the genus.

4. May be based on negation only if a simple (affirmative) name does not exist.

5. May be substituted by adifference if no suitable name exists for the species.

6. May resemble a part in a definition. 38

The whole:

1. Must be divisible into parts which are continuous, non-continuous, universal, or

possessing of certain powers.

2. Must be capable of quantitative division.

3. May exhibit the property that the parts themselves are capable of being divided.

4. May resemble the genus before division.

B. The following conditions must hold for division secundum accidens :

The subject:

1. May contain accidents which can be divided by each other if they are possessed by the

same subject.

The accident :

1. Is usually capable of being contrasted with an opposite accident.

C onclusion

This chapter presented the six division types which Boethius gave in the De

divisione. It is important to keep in mind the distinction between qualitative and

quantitative division, since it is in qualitative division where the differentiae establish the

definition of the subject Although the differentiae are usually one of Boethius’ four

Oppositions, the decision as to what constitutes a single differentia is a matter of discretion.

Two people may not agree on what constitutes a specific differentia or on the total number

of differentiae needed for a definition. CHAPTER IV

A Taxonomic Paradigm Derived from Boethius’ De divisione

In this chapter two diagrams are presented. The first diagram shows the relationship among five of Boethius’ six types of division. The second diagram is a taxonomic paradigm which wül be applied to the eight church modes. In the taxonomic paradigm the predicables are combined with modal terminology used in German music treatises.

THE FIVE DIVISION TYPES USED IN MEDIEVAL GERMAN THEORY

Diagram three shows the relationships among five of Boethius’ division types given in the De divisione.

qualitative division

6enijs ( s priori ) Subject I 4.' 1 Ti difference difference difference : occident (constitutive)# (divisive) etc.. I V V species occident occident occident etc I species species p species (Siits?iern\ L part part part \ genus ) t _ J1 _ etc.. — r whole {oposteriori) propertg property

occident quantitative division

Diagram three. Relationships among five division types of the De divisione. 39 40

Division Types in Diagram Three

I. G enus into species..

The division of genus into two or more species through the appending of differentiae to the genus is shown at the left of the diagram. Since the genus exists prior to species, genus is placed above species in the diagram. The differentiae used in the division of genus into species are qualitative, which means that, as differentiae are added to the genus, each differentia brings us closer to identifying the true nature of the genus. ^ In qualitative division, then, successive differentiae are added to the genus until a satisfactory definition of& particular species is reached. Differentiae are added as needed to reach the definition of the species.^ This procedure is illustrated by the arrow pointing to the left at the top of the diagram.

The designation of a difference as either constitutive or divisive is shown on the left side of the diagram.. If a difference is divisive, it distinguishes the various species of a genus from each other. If a difference is constitutive, it becomes or constitutes aspecies.

In some cases a difference can be both constitutive and divisive, which is shown by two small reciprocal arrows.

When a genus is divided into species, ûiespecies may be regarded as subaltern genera, if these species are capable of division into their own species. A subaltern genus is a genus which is lower than the highest genus, but higher than the lowest species. The division of a genus into subalterngenera is shown on the lower left side of the diagram.

1 The Works of Aristotle , ed. Ross. Topica , bans. Pickard-Cambridge, op. c it., 115a, 30-32. ^Division continues until no fiirther division is possible. 41

Since subalterngenera are included under the highest genus, subalterngenera are placed under the genus, as shown at the lower left side of the diagram.

A species exhibits one or more properties, which are attributes unique to that specific species. The division of species by properties is shown on the bottom left side of the diagram. A property may also be an accident, if the property can be removed from a subject without destroying the subject. The far left side of the diagram shows a property which is also an accident. n. Whole into Parts The bottom right side of the diagram shows the division of a whole into parts.

Since the whole is destroyed if any part is destroyed, the existence of the whole is posterior to the parts. Thus the whole is listed below the parts on the diagram, and this division type should be read from the bottom up. As a quantitative division type, it represents the division of a whole into integral parts which are not hierarchically related to each other.

The division of the whole into parts, then, does not reveal the nature or essence of the whole. Since quantitative division is the opposite of qualitative division, it is illustrated by the arrow moving toward the right at the bottom of the diagram.

In certain cases, however, a part may be equivalent to a species. This equivalence is possible when the part itself is capable of being further divided into parts, each of which bears some commonality with the part which was divided.^ The equivalence of a part with a species is shown in the middle of the diagram.

^See Chapter 111, notes 42 and 43. 42

III. Subject into accidents

The division of a subject into accidents is shown on the right side of the diagram.

The accidents are attributes of the subject which may be removed from the subject without destroying i t The arrow pointing to the right signifies that the division of subject into accidents may continue until all theaccidents have been recognized.

IV. A ccident into subjects

The right portion of the diagram also shows the division of an accident into subjects. In this division type, the accident is used to distinguish among subjects which feature that particular accident..

V. A ccident into other accidents

The far right side of the diagram shows the division of an accident into other accidents. In some cases an accident and a difference may be regarded as interchangeabe, just as an accident and a property may also be interchangeable. The interchangeability of an accident with a difference is illustrated in diagram three by small contrary arrows in the upper right side of the diagram.

A TAXONOMIC PARADIGM

Diagram four is similar to diagram three, except that terminology used in discussions of modes in music treatises is added to the logical nomenclature. Diagram four is the actual taxonomic paradigm which will be applied to the eight church modes in chapter V. (accident) hexachord

tetrachord (genus) (accident) g-1 Protus Deuterus Tri tus T etrardus

difference difference n (difference) 1st 2nd 3rd fourth fourth fourth fourth fourth

1st I fourth 4, (accident) (accident) (difference) 4th 4th 3|-j 1st 1 g{ (accident) fourth 2nd fifth fifth fourth fifth fourth fourth I

I (sp ecies) (sp ecies) plagal authentic authentic authentic authentic t Î

Protus Deuterus Tritus Tetrardus (p art) T 1 1 T Whole

Diagram four. The taxonomic paradigm of the De divisione. 44

Discussion of diagram four

The top of the diagram shows the genus Protus divided into the species authentic and plagal modes. The genus Protus, which exists prior to the species, is divided qualitatively through the difference of the contrary positioning of the first species of fourth and first species of fifth.^ This difference can be seen as both a divisive and a constitutive difference, since it both divides the genus Protus, and constitutes the authentic and plagal species. (Obviously this same process can also be applied to the Deuterus, Tritus, and

Tetrardus modes by the differentiae of other species of fourths and fifths.)

The bottom of the diagram shows the Protus, Deuterus, Tritus, and Tetrardus modes as parts of a quantitative whole. In the later Medieval period, the term “maneriae” was used to designate the whole.^ The whole in this diagram is shown to exist posterior to the parts. Just as the designation as Protus, Deuterus, Tritus, and Tetrardus are “parts,” the subdivision of them into constituent authentic and plagal octave species are “parts of the parts.”

The right side of the diagram shows the division of a Subject into accidents through the application of the synemmenon tetrachord.® The appearance of the second species of fourth in the Protus authentic mode, which is shown in the middle of the diagram, illustrates the division of Subject Protus intoaccidents, since two forms of the Protus

^ The tone/semitone construction of the three species of fourth are: first : TST; second: STT, third: ITS. There is a fourth species of fourth which is a duplicate of the first species (TST), but is located in a different position on the gamut. According to Hermannus, this species of fourth has a different constitution and power. See Musica Hermannus Contractus, trans. Ellinwood, p. 27: "Quarta (species ), D, G, in positione prima, in constitutione et postestate quarta..." The four species 'of fifth are: firet:TSTT; second: STTT; third: TTTS; and fourth: TTST. ®See Chapter 2, note 31, for an explanation of the logical origin of the term 'maneriae'. ®The synemmenon tetrachord, which was derived form the Greek Lesser Perfect System and adapted to the Medieval gamut by Hucbald, features a tone/semitone/tone construction on the pitch g. This results in a non diatonic pitch, or b flat 45

authentic mode result. Both of these forms feature the first species of fifth as the bottom fifth, but have two distinct species of fourth at the upper level: the first fourth and the second fourth. Yet both modal forms could be designated Protus authentic.

The far left side of the diagram shows how the division of accident into subjects may be applied to the octave species on a. The usual designation for the octave species on a is as Protus plagal mode. This mode is comprised of a first species fourth (TSS) placed below a first species fifth. However, if the b flat is included, the first species fourth becomes a second species fourth (STT). This results in a second (transposed) Deuterus authentic mode, which means that the Deuterus authentic mode has a second modal seat on a in addition to its normal seat on e. Thus the second fourth divides the octave species on a into two modes; the Protus plagal and a (transposed) Deuterus authentic. This was a process known as transformation (The division of accident into Subjects is also applied to the octave species on g, shown on the far right side of the diagram.) Therefore the second species of fourth, which results from the synemmenon tetrachord, is also illustrative of an accident (the bflatl which is equivalent to a difference (since it creates two modes).

The top portion of the diagram shows the division of accident into other accidents.

I am applying this division type to the division of the hexachord into tetrachords.® The uppermost line on the diagram shows the construction of a hexachord, which is comprised of two tones, a semitone, and two tones. The tetrachord features a tone, a semitone, and a tone. Thus there is a nuclear tetrachord found in the hexachord. In this sense, it is possible to “divide” the hexachord by the tetrachord.

^Guido was among the first writers on music to use transformatio as a descriptive term, which refers to the 'transposition' from one of the usual modal seats found in the finales tetrachord. In chapter 9 of the Micrologus, Guido said: "Utpote si quis vellet antiphonam cuius principium essts in .D., .E., vel in .P., quae sunt alteiius voces, incipeie, mox auditu percipoet quanta diversitatistransformatio heret." ^The creation of a hexachord by the addition of a whole tone at either end of a tetrachord was a unique theoretical construction of the five German theorists included in this study. 46

This division is shown on the second line at the top of the diagram with the letters

T and S. Against the Medieval gamut, the tetrachord and hexachord can both regarded as accidents, since each can be removed from the gamut without destroying it.

C onclusion

Boethius’ De divisione neatly summarizes the Aristotelian science of division. In spite of its brevity it's method is sufficiently comprehensive to apply to many disciplines.

It is perhaps for this reason that the treatise was so valued during the Medieval period.

The remainder of this document wiU show how Boethius’ division types can be applied to the eight church modes. CHAPTER V

The Eight Church Modes

This chapter will discuss the eight church modes from two perspectives: 1) mode as a nucleus of tones and semitones surrounding the final pitches of , and 2) mode as scalar species. Though there were other ways of identifying the mode of a chant, such as the initial tone or tones, medial cadences, range, and modal repercussion (repetition of motivic patterns within a chant), mode as nucleus and mode as species were the two primary means of modal determination in Medieval theory.

There were also two primary Carolingian means of modal determination. The first was to view the concluding pitch of a particular chant ended as the one which determined the mode.i In the other Carolingian conception which I call here mode as nucleus, the final tone is combined with the adjacent tones and semitones which lay on either side of it.

Although the Carolingian Alia musica ^ conjoined the eight modes with the

Boethian seven octave species, the theory of mode as species developed only with

Medieval German theorists. When the eight modes were borrowed by Westrau scholars from the Byzantines, the Carolingian view of mode as nucleus was the prevailing view.

Yet, the later Medievals adopted the German conception of mode as species? Indeed, by the early sixteenth century, the conception of mode as octave species had become

^Dialogues de musica, Source readings in Music History , selected and annotated by Oliver Strunk (New York: W. W. Norton and Co., 1950), I, p. ^Edmund B. Heard, "Mia musica . A cliapter in the History o f Music Theory (PhD. diss.. University of Wisconsin, 1966). ^For example, see Jan Herlinger, The Lucidarium o/Marchetto o f Padua : A Critical Edition, Translation, and Commentary (Chicago: University of Chicago Press, 1977), or Johannes Tinctoris, Uber de natura et proprietate tonorum in Jean Tinctoris, Opera theoretica, CSM, 22 (Rome: American Institute of Musicology, 1975). For an English translation, see Jean Tinctoris, Concerning the Nature and Propriety of Tones, translated by Albert Seay, Colorado College Press Translation, no. 2 (Colorado: Colorado College Music Press, 1967). 47 48

fundamental to modal recognition.^ Although this study is concerned with mode as species, for the sake of completeness a short discussion on mode as nucleus is included.

The chapter concludes with a discussion on the use of anthropomorphic illustrations to show the relationship between the authentic and plagal modes. Though anthropomorphic descriptions are not part of Aristotelian logic, Boethius added this

Platonic element in the De divisione. Anthropomorphism was used frequently by Medieval

theorists as a descriptive device, and I suggest that its use possibly originated with the

Pythagoreans, and was transmitted to Boethius through the Timaeus of Plato.

The “Octoechoi”

The eight church modes comprise part of a system used in the division and classification of monophonic chant during the Medieval period. The earliest Western writers to use the eight modes in the organization of chant were Carolingian. Borrowed by the Carolingians from the Byzantines, the eight mode system appeared in the West during the eighth and ninth centuries. As used by the Byzantines, the modes are known as the

“octoechoi.” The Byzantine limitation of the modes to eight may have had non-musical origins. Werner has stated that the limitation of the modes to eight may have been due to a calendric influence.^

Carolingian scholars were faced with the task of classifying hitherto orally transmitted chant, since it was Charlemagne’s wish to unify performance practice.^

Because notation was only beginning to evolve, the problem of establishing uniformity in

^Dolores Pesce, The Affinities and Medieval Transposition , (Bloomgton: Indiana University Press, 1987), p. 98. ^Eric Werner, "The eight modes of music ()," Hebrew Union College Annual 21, (1948), p. 235. ^Jacques Broussard, The Civilization o f Charlemagne . Translated from the French by Francis Partridge (New York: World University Library, 1968), p. 100. See also F. L. Ganshof, Recherches sur les Capitularies (Paris, 1958). 49

performance was difficult. The hope was that the “octoehoi” used as a model would prove useful in the division and classification of chant. Since the system of eight modes lasted for over 600 years, it can be seen that the Carolingians were successful in their quest for uniformity.

Mode as a nucleus of tones and semitones around a final

The earliest known Western treatise to divide chant according to the eight modes was the Musica disciplinai In this treatise, no discussion of an external construction of mode is given, since no such construction could be done without musical notation.

Recognition of the modes had to be done through memorization of selected representative chants.

Other early Medieval treatises which discuss the division of chant according to the eight modes are those of Hucbald,* the Musica and Scholia enchiriadis treatises,^ the Alia m usicai^ the Dialogus de musica of Pseudo-Odo,^ l and the Commemoratio brevis.

Except for the Alia musica, the primary means of modal designation in these treatises referred to the nucleus of tones and semitones found around four primary “finals” or ending pitches. The four “finals” were identical to the tones of the “finales” tetrachord of the Medieval gamut, and the modal “finals” as a group constitute a tone/semitone/tone

^Aureliani Reomensis Music disciplina , ed Gushee, op. c it. In his Introduction to the Musica disciplina , Gushee adds that he is no longer inclined to grant priority to Aurelian as the first Carolingian theorist. That the Musica disciplina is chronologically the first treatise we possess has little bearing on whether it was the actually the fiist treatise written during this period. See Gushee, pp. 12-3. * Hucbald, De harmonica institutione , Martin Gerbert, ed., Scriptores ecclesiastici de musica sacra potissimum [GS] (St. Blastien, 1784), vol. I, pp. 104-21. For an English translation, Hucbald, Guido, and John on Music , translated by Warren Babb, Music Theory Translation Series 3 (New Haven: Yale University Press, 1978). ^ Hans Schmid, ed., Musica et scholia enchiriadis una cum aliquibus tractatulis adiunctis , Bayerische Akademie der Wissenschaften, Veroffentlichungen dta: Musikhistorischen Kommission 3 (Mucnchen, 1981). Heard, " Alia musica.," op. c it. 1 ^Dialogues de musica , op. c it., I, p. 114. ^^Commemoratio brevis de tonis et psalmis modulandis. Introduction, Critical Edition, and Translation by Terence Bailey (Ottawa: University of Ottawa Press, 1979). 50

pattern which is found on the pitches D, E, F, and G. The four nuclei around these four

“finals” were designated by the Greek nomenclature Protus, Deuterus, Tritus, and

Tetrardus, and the classification of a particular chant was made according to which of these nuclei the chant corresponded. As Crocker observed, modal classification focused upon the identification of the pitches immediately preceding the final in a chant, since this portion would be fresh in the ear when the chant concluded.

In the later Medieval period the nuclei of tones and semitones around the final came to be designated as the “maneriae,” a term derived from Medieval logic. In the ninth century Alia musica, the Greek terms Hypodorian, Hypophrygian, Hypolydian,

Hypomixolydian, Dorian, Phrygian, Lydian, Mixolydian, and Hypermixolydian were used in modal designation.^s in the earliest , the modes were identified by numbering them from I to VIII.^® Modes I and II are the Protus authentic and plagal modes, IH and

IV the Deuterus modes, V and VI the Tritus modes, with VII and VIII the Tetrardus modes.

Chants of the same modal designation may have ranges that center them above or below the primary nucleus of tones and semitones. Thus one chant might have a tessitura lower than the final, while another chant might have a tessitura higher than the final. For this reason, it was necessary to divide each of the four primary modes into two modes.

^^Richard Crocker, "Hermanns's Major Sixth," op cit., p. 30. ^ '^ e term manerias was first used in Logic by Abelard in his Logica ingretientibus , (Geg. Beitrage XXI 3), p. 394. Cited in Novum glossarium mediae Latinitatis , ed. F. Blatt (Hafiiia, 1959), p. 117. The first use of maneriae in a musical treatise was by Bemardus Silvestris in(b&Tractatus de cantu seu correctione antiphonari : "Quatuor enim sunt diversitatis sive maneriae cantuum quibus omnis ipsorum multiplicitas includitur. Hec apud Grecas vocantur protus, deuterus, tritus, tetrardus." Cited inNovum glossarium , ed. F. Blatt (Hafnia, 1959), 117. Also cited in the Lexicon latinitatis medii aevi , ed. A. Blaise (Tumbolt, 1975), p. 559. l^Heard, Alia musica. op. cit., p. 79. ^^Michel Huglo, Les Tonaires , p. 58: "Mais ce qui est plus important pour l'histoire des tonaires est de remarquer quel'Alia musica est, a notre connaissance le premier auteur medieval qui ait tente de lancer un pont entre ce qu'on apellee les modes grecs' antiques et le huit tons ecclesiastiques cette 'relation' établie par 'auteur principal entre les dorien, hypodorien, phrygien, etc. et les tons I, II, II, etc. se retrouvera consignee plus tard dans nombre de ternaires et de traites." 51

yielding the authentic and plagal modes. The Protus, Deuterus, Tritus, and Tetrardus designations, then, each refer to an authentic and plagal pair.

Although Byzantine and Western conceptions of mode featured similar nomenclatures, the manner of dividing the eight modes was different The Byzantines ordered the eight modes into two sets of four, while Western writers divided the eight modes into four pairs of two modes. Diagram five compares the Byzantine and Western manner of ordering the modes with both the modal names and numbers. Byzantine Western Tetrardus êuihenilc 7 Tetrardus pisçal 8 Tritus âiiiPefiiiù 5 authentic 7 Deuterus authentic 3 Tritus piauai 6 Protus authentic 1 authentic 5 Tetrardus plaçai 8 Deuterus plagal 4 Tritus plagal 6 authentic 3 De uterus plagal 4 Protus plagal 2 Protus plagal 2 authentic l

Diagram five. A Comparison of Byzantine and Western modal organizations,

There is always a second location in the diatonic octave where the nucleus of tones and semitones is identical with the nuclei surrounding the finals. This second location is found at the interval of a fifth above or a fourth below the authentic finals or seats, and was later called the affinal position. Theorists recognized that if a chant remained in the higher portion of its modal range it could conclude on the affinal pitch, or the secondary modal seat The identical nuclear similarity of tones and semitones around the final and affinal positions was a critical factor in the pairing of the authentic and plagal modes.

^ ^Harold Powers, "Mode", The New Grove Dictionary o f Music and Musicians , 20 volumes., edited by Stanley Sadie (New York: Macmillan and Co.jn 1980), vol. 12, p. 379. 52

It can be seen that, from the standpoint of intervallic similarity, the authentic/plagal

modal pairs are related. In describing the relationship between the authentics and plagals,

theorists frequently used anthropomorphic nomenclature, such as “teacher and pupil,”^®

“author” or “master,”!^ or, in the case of the authentic mode, the one “endowed with

authority.”-® The term “inferior” was often applied to the plagal modes, which meant that a

plagal mode was lower than its corresponding authentic mode.

Diagram six shows the four authentic modal seats (finals), the plagal secondary

seats (afffinals), and the pattern of tones and semitones which surround each. The primary

seats are encased within the pitch set Ç to a, while the secondary seats are encased within

the set G to e. Each pitch set is comprised of two tones, a semitone, and two tones.-^

^^Musica disciplina , ed. Gushee, op. cit., p. 79:".. .quos quidam latus, quidam autem disciplpulus nuncupant. Quod est evidentius apparent, si volueris segregare a magistro discipuium.. Dialogues de musica , GS, vol. I, p. 258: "M. Quarta D. in quam terminatur modus, qui dicitur authentus protus, id est, auctor vel princeps primus..." 2® Bemonis, Prologus in tonarium , GS, vol. II, p. 68: "Authentos enim Giaeci magistros dicunt, quasi auctoratos, id est, autoritate praeditos, quorum praecellit auctoritas.” See alsoMusica Hermanni Contracti, trans. Ellinwood, p. 32: " But this is done according to their position on the monochord, not according to their dignity, for each one of them is authentic, i.e. author or master, in itself." Musica Hermanni Contracti , p. 32: "Sed hoc secundum monochordi positionem fit, non secundum dignitatem, nam unusquisque eorum per se authenticus est, hoc est auctor sive magister." ^ll am using primary seats to represent the modal finals, and secondary seats for the cofinals. 53

primary seat secondary seat (authentic) (plagal)

Tetrardus ^ » g * ■ " ------1

. . ' » ^ - Tritus ------m--- • ------Z:____ ZZ__L_ ___

------. — . - n — ------^ ------g « # * * ------De uterus ------a ....* ------

. . - - Protus

T T S T T T T S T T

o = modal seat

Diagram six. The four authentic modal seats and the four secondary seats with surrounding whole tones and semitones.

Mode as species

In the early Medieval period, there developed another way of viewing the eight modes. This was to regard the modes as octave species which are in turn comprised of species of fourths and fifths. Prominent among those who regarded mode in this way were early German theorists, includng Bemo, Hermannus, Wilhelm, Aribo, and Englebert.

In species theory, the patterns of whole tones and semitones which surround modal

seats are viewed within an identical interval of fourth or fifth. The or fifth becomes a boundary interval. Although the boundary interval is the same for all nuclei contained within it, the differing pattern of whole tones and semitones contained within the 54

boundary interval means that several scalar varieties of fourths and fifths result. These varieties are called species. Gushee points out that species theory was the hottest theoretical topic during the eleventh century.22

Diagram seven shows the three species of fourth and the four species of fifth, spffcies of fifth . o " ^ ^

Fourth of fourth STTT 5th « - - « ^ JBl Third ITS Third T S T T 4th 5th _o_ -o— ■ nl Second STT Second s t t t 4th 5th ------=— R— e — « ■o -o- First First TST T S T T 4th ' 5th

Diagram seven. The species of fourths and fifths.

In species theory, species of fourths and fifths are combined to form octave species. Seven species of octave may be created by this combination. Only some, however, were considered acceptable. In the Medieval modal view these were octave species that were created from similarly numbered species of fourths and fifths. The

Protus authentic mode (Dorian), for example, is comprised of a first species fifth, or D a, placed below a first species fourth, or ^-D . The Protus plagal mode (Hypodorian) has the same species of fourth and fifth, but their positions are inverted. The Deuterus modes

(Phrygian and Hypophrygian)) feature the second species of fourth, or B-E, and second

^^Gushee, "Questions o f Genreop. cit. 55

species of fifth, or E-b. The Tritus modes (Lydian and Hypolydian) feature the third species of fourth, or C-F, and third species of fifth, or F-c.

The Tetrardus modes (Mixolydian and Hypomixolydian) have two problematic characteristics. The first characteristic of the Tetrardus modes is that they are constructed with a fourth species of fifth, T T S T T as the bottom element, but with a first species of fourth, T S T, as the upper element. The latter is placed on the the pitch G instead of D, as it is in the Protus modes. Hence the Tetrardus modes are constructed of unlike species of fourth and fifth. The second more interesting characteristic is that the Tetrardus plagal mode, which extends ftom D to d, is exactly the same octave species as the Prutus plagal mode. Since Medieval theorists sought a single identity for each octave species, a dual designation of the D-d octave posed a problem for the modal nucleus theory, but could be accomodated by the species theory.

Diagram eight shows the eight mode system divided into four pairs of two modes.

Each pair shows the primary modal designation, such as Dorian, Phyrygian, etc. on the top of each staff. Under each staff appear the related plagal modes, each of which is given the prefix “Hypo.” The brackets show the octave species for each mode. 56

dorian -e- -a-

hypodorian Phrygian

hypophrygian lydian

- e - -f t O * hypol ydian mixolydian

JOL a t. ■ o n .

hypomixol ydian

Diagram eight. The eight modes of music with bracketed octave species.

The anthroporphic view of the authentic/plagal pairs

We now turn to a discussion of the Medieval view of the relationship between the authentic and plagal modes. The fact that anthrpomorphic nomenclature was used in describing the modal pairs has been discussed earlier in this chapto".^ This anthropomoiphic view of modes had a philosophical origin.

This origin is in the Pythagorean concept of duality. The Pythagoreans had established a series of ten ‘oppositions’ as a means of understanding reality.^'^ Foremost among these oppositions was the concept of odd versus even. Odd represented

23See pp. 47-8. 2‘*^Dumitriu, History o f Logic, op cit., vol. I, pp. 74-5. 57

conceptually the Creator, and would hence signify such attributes as “unity,”

“indivisibility,” and “immutability.”^ Even represented conceptually the Creation, and represented “divisibility,” “alterity,” or a “determined” nature.-^

The Pythagoreans also applied dualism to numbers directly. Odd numbers represented “immutability” and “indivisibility,” while even numbers represented

“divisibility” and “alterity.

The Pythagorean conception of “unity” versus “alterity” as applied to odd and even numbers can be seen in Plato’s Timaeus. This cosmological treatise was the most influential work of Plato during the early Medieval period, and it exhibits Pythagorean influence.28 In the Timaeus, Plato said that all Creation displayed alterity as distinct from the Creator, who alone displayed Immutability.^^ Yet for Plato the Creator and the Created existed in an environment of ‘friendship’ or ‘kinship.’^0 The Pythagoreans and Plato described the relationship between odd and even numbers in an anthpomorhic manner.

The philosophical opposition of Unity (or Immutability) and Duality (or Mutability) was known to Boethius, who used it in his De arithmetica.^^ This treatise likely transmitted the anthropomorphic conception of duality to early Medieval thinking.

25 J. A Philip, Pythagoras and Early Pythagoreanism. Phoenix: The Jioumal of the Classical Association of Canada, Vol. Ill (Toronto: The UNiversity of Toronta Press, 1966®, p. 189., Cambridge History o f of Later Greek and Early Medieval Philosophy, op. cit., p. 89. Alexander Polyhistor, The Succession o f Philosophers. ^^Diogenes Laertius, Lives and Doctrines o f the Illustrious Philosophers, VIII, 25. Dumitriu,History of Logic, I, p. 74. J. A. Philipp, Pythagoreans and Early Pythagoreanism (Toronto, 1966).. ^^The Timaeus o f Plato . Ed. R. D. Archer-Hind, op. c it.. 25 Ibid., p. 91. 2®Ibid., p. 105: "And God set soul in the midst thereof and spread her through all its body.. .but for its excellence it was able to be company." 21 Boethius, De arithmetica, Lib. II, ed. Freidlein, p. ?: "Aut enim propriae, immutabilis, ejusdemque substantiae est, quod Deus, vel anima, vel mens est, vel quocunque propriae naturae incorporalitate beatur, aut mutabilis, variabilisque naturae, quod corporibus indubitanter videmus accidere.” 58

C onclusion

Even when the advent of polyphonic music created difficulty in the use of the eight modes as a taxonomic system, the eight mode paradigm remained the normative one among theorists and composers until Glareanus (1547) suggested its replacement by the twelve mode system.^-

Powers (1981) has noted that, even with the obvious superiority for polyphonic music of Glareanus’ new system, Palestrina continued to use the eight mode system when ordering a group of works according to a modal system.33

Medieval theorists divided the modes into authentic and plagal pairs for practical considerations. Yet the description of related authentic and plagal modes as having anthroporphic relationships may also have been regarded from a theological perspective.

This perspective was derived from the Pythagoreans and transmitted through Plato. It was picked up by Boethius and conjoined with Aristotelian division in the De divisione.

32fot an English translation, see Glareanus, Dodecaciiordon . Translation, Transcription, and Commentary by Clement A. Miller. Musicological Studies attd Documents 6 (Rome: American Institute of Musicology, 1958), p. Dodecachordon is available in a facsimile of the 1547 edition, Monuments o f Music and Music Literature in Facsimile 65 (New York: Broude Brothers, 1969). ^^Harold Powers, "Tonal Types atid Modal Categories in Renaissance PolyphonyJournal o f the American Musicological Society 34 (Fall, 1981), p. 467. Powers says that this conservative stance, which took place along with an increased interest in modality during the sixteenth century, shows that the 'evolutionist' view of a historical succession from modality to tonality fails. CHAPTER VI

The taxonomic paradigm of the De divisione applied to the eight church m odes

This chapter will show how Boethius’ division of genus into species, a whole into parts, and subject into accident.s can be applied to the eight church modes. It will also explain how these division tools can rationalize irregular modes of the system. Examples of irregular modes include transposed modes resulting from the process of transformatio, and octave species that are created from dissimilar species of fourths and fifths.^ (Chapter

VU will offer evidence that these division tools were employed by five Medieval German theorists in their discussions on the eight church modes.)

G enus into species

In the De divisione, Boethius said that: ""germ is divided not in meanings, but in offsprings of itself.”^ His implication is that whatever results from the division of the genus retains the “essence” or “nature” of the original genus. The genus “creates” the species over which the germs rules. Example one illustrates the qualitative^ division of the genera Protus, Deuterus, Tritus, and Tetrardus (the maneriae) into authentic and plagal species. The left side of the example shows the number of each mode. The odd numbered modes are listed above the staff in the example, while the even numbered modes are positioned under the staff. The right side of the example shows the authentic modes as

^Guido was among the first writers on music to use transformatio as a descriptive term, referring to a move from one of the original 'seats' of the tetrachord of the finals. In chapter 9 of the Micrologus , Guido said: "Utpote si quis vellet antiphonam cuius principium essts in .D., .E., vel in Jr., quae sunt alterius modi voces, incipere, mox auditu pa^ciperet quanta diversitatis transformatio heret." ^Boethius, De divisione , PL 878D10-11: "Genus v ao non in significationes sed in quasdam a se quodammodo procreationes disjungitur." ^Ibid., PL 879B-7: "Generis veto distributio qualitate perficitur." 59 60

“immutable” or “perfect,” while the plagal modes are “mutable” or “imperfect.”^ In an anthropomorphic description, this concept is an example of Boethius’ division through the opposition of contraeity.^

VII tetrardus authentic V tritus authentic Immutable de uterus authentic spscw s I I I protus authentic I

t t Q I______I II protus plagal Mutable IV deuterus plaqal ^ species VI i^tritus plagal VIII tetrardus plagal

Example one. The division of genus into species applied to the Protus. Deuterus. Tritus. and Tetrardus modes.

At another place in the treatise Boethius specified that the division of a genus into species, “is in the conjunction of a difference.” ® He then further clarified the nature of this division type when he said that the division of a genus into species must be made either through a divisive or constitutive difference? That is, the divisive difference divides the

^The conception of the opposition of "perfection" and "imperfection" was a Pythagorean doctrine which was transmitted to the Medieval West through the Timaeus of Plato. ^See chapter III, page 28 of this document. Boethius actually illustrated the opposition of contraeity as "good" and "evil". ®Ibid., PL 879D3: ".. species vero a genere, differentiae conjunctione." ^See Chapter IV, note 3. It is in the Zk differentiis topicis where Boethius expressly stated that differentiae will be either constitutive or divisive. See Boethius, De differentiis topicis. lib . I, PL 1178 B9-10: "Aut enim constitutiva erit differentia, aut divisiva: sed si constitutiva hierit, quasi generis obtinet locum..." 1178 B13-14: "At si divisiva fuerit, velutspecies consideratur..." 61

genus into species, while the constitutive difference becomes a species^ In some instances

the same difference, can be both divisive and constitutive. In other words, the difference

which actually causes the division of the genus can itself constitute aspecies. Example two

shows the difference which is used to divide the Protus into authentic authentic and plagal

species. This difference is the reversed position of the first species of fourth and fifth.

f o u r t h

[X ------= J a u t h e n t i c 1 V / \ 1 1 ^ h 1 s t ' 1 1 p r o t u s , f i f t h , h ? ------I ^plagal ...... I 1______I 1 s t f o u r t h

Example two. The division of genus into species through a difference applied to the Protus

maneria.

Both the species of fifths and fourths can now be viewed as subaltern genera,

which are genera that are lower in qualitative status than the highest genus, but are higher

than the lowest species. Example three takes the fifth as an illustration and shows the four

species of fifths. Each fifth belongs to a different authentic/plagal modal pair. The

differentiae which divide each species of fifth ftom the others are specific arrangements of

whole tones and semitones.^ Since the fifths were one of the differentiae which divided the

^In the De divisione, Boethius said that, "in general, all differentiae that are of such a sort that there can be no species without them, but in virtue of them [are the ones that] must be used in the division of the genus or in the definition of the species." De divisione , PL 881C8-12: . .et universaliter dicendum est, quae cunquedifferentiae hujusmodi sunt, ut non modo praeter has species esse non possit, sed per eas solas sit, hae vel in divisione generis, vel in speciei diffinitione sumendae sunt." ^See Chapter V, note one. 62

modal pairs from each other, the fifth itself could seen as a divisive difference. Since each species of fifth has its own constitutive difference, specifix patterns can be seen as species

of a new subaltern genus : the Fifth (as a general class), specîes 1st fifth

-> spec-fes (subaltern) 2nd fifth I çeûüs fifth spectes -4» >5> 3rd fifth > T T S T 4-1 spec?es 4th fifth

Example three. The division of the fsubalteml genus fifth into species through the

differentiae of whole tones and semitones.

The division of a genus into species is accomplished by one of Boethius’

oppositions.io Therefore it can be assumed that the oppositions are equivalent to

differentiae. Of the four oppositions, Boethius suggested that the first opposition,

contraeity, is the most prevalent. In the words of Boethius, “ much of the division of

genera occurs in connection with contraries, for we bring almost all differentiae down to

contraries.1 As I indicated earlier in this document, Boethius’ source of the four

^^See Chaper four for a discussion of the four oppositions. Boethius, De divisione , PL 883C2-5: "In contrariis autem multo divisio est. Fere enim cunciasdifferentias in contraria deducimus." l^See Chapter 4, section II. 63

oppositions was Aristotle’s CategoriesA^ However, Boethius ordered the four oppositions differently, and more significantly, he used an analogy different than Aristotle’s to illustrate the fourth opposition. Boethius created an anthropomorphic illustration for this type of opposition, by suggesting that “the fourth opposition occurs through relationship, as father, son, master, slave...

Example four, given below, is similar to example two. However, in this example

Boethius’ concept of opposition is applied to the division of the Protus modes into authentic and plagal species. The example shows that, although the species of fifth and fourth which comprise the two modes are identical, they are in inverted order. Because of this, they show contraeity, which is Boethius’ first opposition. The authentic mode has the fourth placed above the fifth; the plagal has the fifth placed above the fourth. Hence this mutual but reversed positioning of the first fovuth and fifth shows the divisive difference between the two modes. At the same time it can be seen that the fourth and fifth are related, since the fourth contains the same pattern of tones and semitones as the fifth (the fifth contains the fourth within itself). Because the fourth and fifth are related in this way, this division also shows Boethius’ fourth type of Opposition: relationship.^^

^^Aristotle's Categories and De interpretatione, trans. Ackrill, pp. 31-2: " Things are said to be opposed to one another in four ways: as relatives or as contraries or as privation and possession or as affirmation and negation." ^‘^Boethius, De divisione , PL 882A2-3: "Quarta secundum relationem, ut pater, Alius, dominus, servus." In the De topicis differentiis, Boethius gave an even clearer definition of opposition through relationship. He said that "properties of opposites which are related to each other are themselves also related to each other." See Boethius's De topicis differentiis , trans. Stump, p. 56. Boethius, De differentiis topicis , Lib. II, PL 119ID5-6: "... oppositorum ad se relatorum propria, et ipsa ad se referuntur." Aristotle illustrated opposition by relation as in "the double and the half." See Aristotle's Categories and De interpretations, trans. Ackrill, p. 31. ^^See note 17 of Chapter 1 for a discussion of the anthropomorphic nature of relationship in the Periphyseon of John Scottus Eriugena. 64

1st fourth = T S T relationship 1st fifth = T T S T

1st fourth Protus authentic

Protus plagal 1st fifth

1st fourth

Example four. The first and fourth oppositions applied to the Protus modes.

If a difference can be sq>arated from the subject without changing the nature of the subject, this difference can be regarded as a separable a c c id e n t. In the eight mode system, the use of the b flat as an accident could be applied to the Protus authentic and the

Tiitus authentic and plagal modes only. If the flat were used in the Protus plagal and

Tetrardus authentic modes, then the flat would have to be regarded as z difference, because the Medievals after Guido viewed the flat in these modes as changing the nature of the subject Example five compares the Protus authentic with the Protus plagal modes, both of which include the bflat. In the Protus authentic, the flat constitutes an accident, since it does not change the designation of the mode as Protus authentic. In the Protus plagal, the flat constitutes adifference, since it changes the designation from Protus plagal to Deuterus authentic. (This was called transformatio, and will be discussed in the section on division secundum accidens later in this chapter.)

^®This was an attribute of flccirf^nr with which Aristotle, Poiphyry, and Boethius agreed. In the Medieval gamut of music, only the b flat or b natural exhibit this feature, except for the gamut of the Musica enchiriadis, where other chromatic inflections are used. 65

SCCfi^Di Protus authentic -€► □nr ë~ n

tv

Protus plagal (Deuterusauthentic) difference ^ = fi nal

Example five. A comparison of the Protus authentic and plagal modes with b flat as an accident and as a difference.

Whole into parts

In the De divisione, Boethius thoroughly discusses the division of a whole into its constituent parts. He specified four principal ways of division. These include the division of a whole into:

1) continuous parts (such as in a body into its parts)

2) non- continuous parts (such as in the “whole” crowd, the “whole” population,

the “whole” army)!®

3) the whole which is universal (as in “man” or “horse”)

4) the division of a whole that consists of powers (such as in the division of

abstract entities, like the soul, there is the capacity for “understanding,” another

for “sensing,” and another for “living.”^°)

!^Boethius, De divisione , PL 887D6-7: "Totum namque est quod continuum est, ut corpus.. . ” !®Ibid., PL 887D8-10: "Dicimus quoque totum quod continuum non est, ut totum gregem, vel totum populum, vel totum exercitum." !^Ibid., PL 887D10-11: "Dicimus quoque totum quod universale est, ut bominem vel equum." ^Olbid.,PL 888A1-3: "Dicitur quoque totum quod ex quibusdam virtutibus constat, ut animae alia est potentia sapiendi, alia sentiendi alia vegetandi.” 66

It should be kept in mind that, for Boethius, dividing a whole into parts is a quantitative division.-^ The implication is that this type of division, unlike dividing a genus into species, is not hierarchical. For this reason, Boethius may have felt obliged to include a type of division in which all the parts can be regarded as integral (or collective) parts, rather than as divisive (or distributive) parts, that is, a division which would be more closely aligned with the division of g erm into species.

Example six illustrates the division of a whole into non-continuous or discrete parts applied to the eight modes. The parts of the whole are the four pairs of authentic and plagal modes. Since Boethius also said that multiple divisions exist,^- the parts themselves may be divided into ‘parts of the parts’. Hence in this type of division, the individual modes can be regarded as these ‘parts of the parts’. A B Whole Parts Parts of the Parts Tetrardus authentic 1 plagal Tritus authentic 8 mode plagal system Deuterus authentic plagal Protus authentic 4 plagal

Example six. The division of the whole into parts applied to the eight modes.

The Medievals had known Latin grammatical theory from Priscian and Donatus, among others.^ Since early logic and grammar had a common origin, both of these

^^The division of a whole is into integral parts. In the division of genus into species, the species are hoerarchically related. Boethius, De divisione , PL 885B-8: "Fit autem generis ejusdem divisio multipliciter..." ^T he 'parts of speech' for Priscian were name (conqirised of a noun plus an adjective), verb, participle, pronoun, preposition, adverb, inteqection, and conjunction. See D. P. Henry, Commentary on De grammatico : The Historical-Logical Dimensions o f a dialogue o f St. Anselm's (Reidel: Synthèse 67

disciplines dealt with “parts of speech.” The difference was that logic, as inherited from

Aristotle, focused on two of the eight “parts of speech,” the name/verb as a signifier of the subject, and the predicate concepts of the proposition under consideration. In the De divisione, Boethius followed the example of Aristotle by also granting a higher status to names and verbs as “parts of speech.”-'^ What is significant for the present discussion is that, during the Medieval period, there was an attempt to bridge the g ^ between logic and grammar.^ Hence the analogy of using “parts of speech” to illustrate the division of a whole into parts with regard to modal division shows not only the influence of grammar, but of logic as well.

Division secundum accidens

Division secundum accidens operates on three levels: 1) a subject into accidents, 2) accidents into subjects, and 3) accidents into other accidents. In division secundum accidens, the accidents are present in the subject, but they may be removed from the subject without destroying the subject. In this document 1 regard the b fiat, through its availability in the synemmenon tetrachord, as an accident, since the Protus authentic and the Tritus

Historical Library 8), pp. 262-3. Also see D. P. Henry, ’Predicables and CategoriesThe Cambridge History o f Later Medieval Philosophy , p. 133. In the De syllogismo categorico , Boethius contrasts the philosopher's view that names and verbs are primary, with the other six parts being merely supplementary, with that of the grammarians, who simply see eight parts of speech. Boethius, De syllogismo categorico , PL 798C7-14; "Haec enim ex nomine et verbo componi videtur: sed prius utrum nomen et verbum solae partes orationis sint consideremus, an etiam aliae sex, ut grammaticorum opinio fert, an aliquae ex his in verbi et nominis jura vertantur, quod nisi prius constitutum sit, tota propositionum ac deinceps ea ipsa quae ex propositionibus conqwnitur syllogismorum ratio titubabit." The principal grammatical texts studied Insiitutiones grammaticae of Priscian, and the Arr motor and Ars minor of Donatus. During the Carolingian period these were supplemented by texts for school use taken from Bede, Alcuin, and others. The subjects of the Trivium were studied more widely than the Quadrivium during the Carolingian period. Of the Trivium, grammar was the most studied. See Laistner, Thought and Letters in Western Europe , p. 170. ^'^D. P. Henry, " Predicables atid Categories ," The Cambridge History o f Later Medieval Philosophy , p. 133. ^S ten Ebbesen, "Ancient Scholastic Logic as the Source o f Medieval Scholastic Logic ," The Cambridge History o f Later Medieval Philosophy , p. 123. 68

authentic and plagal modes would have the same designation if the b flat were not present

In other words, the b flat sometimes is extrinsic to the subject “diatonic gamut”

Near the end of the De divisione, Boethius says that anything belonging to the division by accident involves contraeity.^® As an illustration of the first division level, or a subject intoaccidents, Boethius uses the example of the subject ‘men’ being divided into black, white, and a intermediate color,because these are accidents for men, not species of men, and Man is their subject, not their genus."^^

The implication of this level of division is that the “nature” or “essence” of the subject remains intact during the division process. The accident or accidents may be removed from Üie subject without the destruction of the subject. Moreover, since the division by accident involves contraeity, the subject cannot possess an accident and its opposite simultaneously.

Subject into accidents

Example seven shows the division of a subject into accidents as applied to the

Protus authentic mode. The subject “Protus authentic” is divided into two vCTsions: 1) with b flat and 2) with b natural. Since the flat and the natural are both accidents, and are therefore extrinsic attributes of the subject, the choice to use one or the other would be made with the knowledge that the designation of the subject as “Protus authentic” remains ubchanged. This is another way of saying that the accident which is present in the subject, may be removed without destroying the subject.

^Boethius, De divisione , PL 890 D10-891A1: "Nunc de his divisionibus dicemus quae per accidens hunt. Harum autemcommune praeceptum est, quidquid ipsorum dividitur in opposite disgregari.. ^^See Chapter 4, note 14 of this document. 2^Ibid., PL 878AI5-878B2: "Haec enim accidentia sunt hominibus, non hominumspecies , et homo his subjectum, non honimgenus est." 69

3C HS- âcciésiîi -HT H»- Subject = Protus _ $ authentic I— 3C $ o Q

Example seven. The division of the subject into accidents applied to the Protus authentic mode.

The eight mode system provides another illustration of the division of a subject into accidents. Since the eight mode paradigm uses seven discrete diatonic pitches, it is necessary to feature two designations for one of the octave species. The octave species which received this dual designation during the Medieval period was the octave species on

D In some treatises of the Medieval period, this octave species was designated as both

Protus authentic and Tetrardus plagal. The distinction between the designations was based on the two different ‘mean’ pitches of the D octave species. The Protus authentic mode has A as it’s “mean” pitch.. The Tetrardus plagal mode, on the other hand, has the pitch

0 , the final, as its “mean” pitch. Both of these modes have, of course, the same combination of first species fifth and fourth as their constituent parts, but the lesser species are in reversed order. Hence in terms of species the Tetrardus plagal mode would appear to be exactly the same as the Protus plagal mode. The difference lies, however, in the “mean” pitch. Example eight shows the Protus authentic and Tetradus plagal modes with their respective “mean” pitches. The dual designation of these two modes represents the division of Subject into accidents. 70

a? £ 1st fourth I------1 Protus ÜCL a authentic 1st fifth £ 4th fifth

Tetrardus E -. 0 " plagal 1st fourth

Example eight. The division of the octave species on D into the accidents Protus authentic and Tetrardus plagal

The division of a subject into accidents is, of course, similar to the division of a genus into species through a difference, when that difference can be viewed as an accident.

Just as the Protus authentic may be divided into two accidents, so may the subject

“Tritus authentic” be divided into two accidents by the synemmenon tetrachord. However, there is an important distinction. If the pitch class b in the Tritus authentic is flatted, then the species of fourth in that mode is changed, and the tritone above the final is eliminated.

This change does not affect the division type, but it may explain the ubiquity of the flat in this mode during the Medieval period. It may also explain why, in the sixteenth century,

Glareanus considered the use of the flat in the Tritus authentic mode as creating a different mode (i.e. transposed Ionian).-^ For Glareanus, the accident would in this instance be regarded as a difference, since the division type would be different. It would constitute the division of genus into species. Example nine, which is a comparison between the Protus authentic and the Tritus authentic modes with the flat present, shows how the flat affects

^^Glareanus, Dodecachordon , ed. Miller, op. cit., pp. 71

the Tritus mode’s species of fourth, and converts it from three whole tones into two whole tones and a semitone.

Tritus authentic Q -, n.

of fourth (-n.) Protus authentic

Example nine. The subjects Protus authentic and Tritus authentic compared.

A ccidents into subjects {transformatio )

If the bflat is used in the octave species on a, or the normal Protus plagal mode, then a second Deuterus authentic mode results. This constitutes a process which was known astransformatio, and the process results in two seats for the Deuterus authentic mode. This process is therefore an illustration of how the accident b flat can be divided into two subjects: the original Deuterus mode on e, and the transposed one on a. Example ten shows a comparison among the Protus plagal, the Deuterus authentic, and the new

(transposed) Deuterus authentic modes. 72

scciâeni = plagal

. Deuterus authentic subject - (transposed) Deuterus authentic ê

Example ten. The division of accident into subjects applied to the Deuterus authentic mode.

The difficulty for music theorists was, of course, that the Deuterus authentic and

Protus authentic modes already had identities in the eight mode system. Hence the result of tramfortnatio led to a dual view of the Deuterus authentic, as weU as the Protus authentic modes, thereby forcing an expansion of the eight mode system. This dual view is similar to the dual view that the Medievals had inherited from the Greeks concerning the relation between odd and even numbers. The application of the attitude toward the authentic versus plagal modes, with the forma: as representing “immutability” or perfection, and the latter as representing “mutability” or imperfection, was then transferred to modal transposition. The new Deuterus authentic on a was “less perfect” than the same mode on its original seat, which, of course, was on e. This may have been among the reasons why Guido considered transformatio as an unacceptable practice.30 His view remained valid until the late Medieval period.

^^Hucbald, Guido, and John on Music : Three Medieval Treatises , translated by Warren Babb, edited by Claude V. Palisca (New Haven: Yale University Press, 1978). Trans. Babb, Micrologus , p. : "Here b flat seems to create a certain confusion and transformation, so that G sounds as Protus^a as Deuterus..." Joseph Smits van Waesberge, Guidonis Aretini Micrologus , Corpus Scriptorum de Musica 73

A ccident into other accidents (Hexachords into Tetrachords)

According to Aristotelian logic, a property and an accident may, under certain conditions, be equivalent predicables, and used interchangably.^^ Such a view is taken in this document with respect to the division of the gamut into tetrachords and hexachords.

The Medieval gamut featured two important attributes which theorists closely associated with the division of the gamut into eight modes: the gamut could be divided into its component tetrachords or hexachords. From Hucbald forward, theorists discussed the gamut and its division into the eight modes with reference to one or the other of these attributes.

The first of these attributes to be discussed was the arrangement of the gamut into tetrachords. The gamut was typically shown to contain four tetrachords, of which the

“primary” tetrachord was the familiar “tetrachord of the finals.”^- The finals of the eight modes constituted this tetrachord, which features a tone/semitone/tone intervallic structure.

This particular intervallic structure was found on two locations within the gamut, at the

“graves” and “excellentes” tetrachords. In the Music enchiriadis the gamut was organized so that all four tetrachords reflected the same structure as the “graves” tetrachord.^^

Example eleven shows a comparison between the division of the gamut according to

4 (Rome: American Institute of Musicology, 1950), pp. 124-5: ".. .ubi quandam confusionem et transformationem videtur facere, ut .G. sonet protum, a. deuterum..." ^^Trans. Pickard-Cambridge, Topica, 102b 21-3: " It is clear on the face of it that there is nothing to prevent an accident from becoming a temporary or a relative property ^^Hans Schmid, ed., Musica et scholia enchiriadis una cum aliquibus tractatulis adiunctis , Bayerische Akademie der Wissenschafren, Veroffentlichungen der Musikhistorischen Kommission 3 (München, 1981), p. 77. ^^Ibid., pp. 89. For a discussion concerning how the vitia (upward or downward semitonal inflections of the middle pitch of each tetrachord) of the Enchiriadis gamut could be used to rationalize "accidentals" without having to employ transformatio , see Charles M. Atkinson, 'From 'Vitium' to 'Tonus acquistus'. On the Evolution o f the Notational Matrix o f Medieval Chant ," Cantus Planus , Budapest, 1990, pp. 181-97. 74

tetrachords in the Musica enchiriadis and that found in the De harmonica institutione of

Hucbald. Nusicd graves finales su peri ores excellentes enchfnsdis TST TST TST TST gamut

Hue bald’s ascendi ng gamut

Example eleven. The division of the pamut bv tetrachord according the the Musica enchiriadis and the De harmonica institutione of Hucbald.^'*

The second division of the gamut was through the hexachord. In the later theoretical literature, the hexachord actually came to be designated by the predicable property?^ The outline of a hexachordal view was first given by Guido, who located it on two positions in the gamut by placing a whole tone on either side of the “graves” and

“superius” tetrachords.^^ The result was to create identical hexachords from Q to a, and from G to e. These became later known as the “natural” and “hard” hexachords, respectively. Both Hermannus Contractus in his Musica?'^ and Wilhelm of Hirsau, in his

^'^Trans. Babb, Hucbald, Guido, and John on Music, p. 33. ^^Pesce reminds us that the actual word 'hexachord' was not used until the sixteenth century. See Dolores Pesce, The Affinities and Medieval Transposition (Bloomington: Indiana University Press, 1987), p. 174. ^^Joseph Smits van Waesberghe, Guidonis Aretini Micrologus , Corpus Scriptorum de Musica 4 (Rome: American Institute of Musicology, 1956), pp. 117-18. ^^For a critical text and English translation, Leonard Bllinwood, Musica Hermanni Contracti (Rochester: Eastman School of Music, 1936.) 75

treatise of the same title,described the “seats of the finals” within the framework of identical hexachords. Wilhelm recognized the potential of a third hexachord within the interval set F to d through the synemmenon tetrachord, which became known as the “soft” hexachord. Hence the eight modes, which had their primary “seats” based in the “natural” hexachord, were described with respect to their position within identical hexachords.

Movement within a specific mode was determined according to one’s position within a particular hexachord by the use of hexachordal syllables. To move from one hexachord to another was called mutation. Example twelve shows the division of the gamut according to identical hexachords.

I I S I T natural o O-

I T T S T T soft natural ITS IT £L-O-

hard i i hard TTSTT TT SIT

Example twelve. The division of the gamut according to hexachords.

We now come to the reason why Boethius’ division of accident into other accidents is applicable in this instance. If the tetrachord and hexachord are both regarded as accidents, then it is possible to show that the hexachord, as an accident, can be "divided" into tetrachords. In the same manner, the tetrachord can be “divided” into the hexachord.

^^Willehelmi Hirsaugensis Musica , edited by Dennis Harbison, Corpus Scriptorum de Musica 23 (Rome: American Institute of Musicology, 1975), p. 59: "Cuicumque horum tetrachordum tonum ex utraque parte addideris, oritur tibi aliqua meta troporum, utpote a Gravibus prima meta est a (gamma) in .E., a Finalibus secunda a .C. in a., a Superioribus tertia a .G. in .e., ab Excellentibus quarta a .c., in aa., quinta per synemmenon nasei poterit ab f . in .d." 76

This division type is possible because the tetrachord is encased within the hexachord. This is precisely the way in which Hermannus Contractus derived his hexachord.^^ Example thirteen shows how the tetrachord is encased within the hexachord.

- e — Q - ± TST T______T

$ ..q . . gl ,

zzc TST

Example thirteen. The division of accident into other accidents applied to the tetrachord and hexachord.

C onclusion

This chapter has attempted to show how five of the six division types outlined in

Boethius’ De divisione could illustrate the division of the eight church modes. Two aspects of this division process should be noted, since they will be significant in the next chapter, in which selected music treatises are examined for evidence of logic. The first aspect is terminological, in which the logical predicables are the semantic tools used to describe the divisional components. The second aspect is conceptual, in that Boethius provided at least one model which could be used to rationalize the way in which the division was

^^Trans. Ellinwood, Musica Hermanni Contracti , p. 57: " Take any tetrachord you wish.. .add a tone at both ends; you then have the limits of the tonal patterns which form the basis of the modes." Musica Hermanni Contracti, p. 57: "Accipe tetrachordum quodciunque voleris.. .addito utrinque tono, habes terminus modorum qui hunt sedes troporum." 77

undertaken. This conceptual model is in Boethius’ use of an anthropomorphic illustration to describe his fourth opposition, that is, the opposition of relationship. The next chapter will show that this illustration was repeated by five German theorists in their efforts to describe the relationship between the authentic and plagal modes. CHAPTER VII

Examples of Division in Five German Music Treatises

The eight church modes were established in the Medieval West during the

Carolingian period. Yet the suggestion that Carolingian music writers were influenced by

Aristotelian logic is not directly supported by theoretical writings. However, when we turn to the early Medieval German writers on music, the influence of logic is apparent. This chapter presents terminological and conceptual evidence that the Aristotelian science of division was used by these writers in their discussions on mode. In support of this tenet, quotations from five German treatises are presented These treatises are the Prologus in tonarium of Bemo of Reichenau, the Musica of Hermannus Contractus, theMusica of

Wilhelm of Hirsau, the De musica of Aribo, and the Musica of Engelbert of Admont.^

Each treatise is examined for evidence of an awareness of the five division types given in the taxonomic paradigm presented in Chapter VI.

Of the five theorists, only two mention directly Aristotelian logic or logical texts which were Peripatetic. These two theorists are Hermannus and Wilhelm. As I mdcated in the Introduction to this document, a listing of the monastic library at Reichenau from the year 822 reveals that several important books on logic were available.^ Both Hermannus and Bemo were in an excellent position to have come into contact with Aristotelian logic.

In the case of Hermannus, confirmation of this contact comes directly from the author. At the beginning of his Musica, he specifically référés to the Isagoge of Porphyry, and the

Categories of Aristotle.^

*The dates of these treatises are given on page 3 of this study. ^See page 4. ^Trans. Ellinwood, Musica Hermanni Contracti, op. cit., p. 19: "For, to make a long story short, just as Grammar is reduced to eight varieties, the Isagoge to five, and the Categories to ten...". Musica 78 79

Although Wilhelm did not mention any logical texts by name, he suggested that an association exists between Peripatetic logic and tetrachordal formation in music. He said

that “ancient and modem authorities of musical art wrote many commendable things, many

useful and extremely necessary, (since) they invented the first paths of this same art, they

also found the clearest source itself, that is, the principal tetrachord, which is a kind of

Peripatetic logic..

G enus into Species through Differentiae

Examples one and two of Chapter VI illustrate the division of the genera Protus,

Deuterus, Tritus, and Tetrardus into authentic and plagal species. Since the authentic and

plagal species have been divided under the highest genus (the Protus, Deuterus “maneria,”

etc.), these species can now be called subaltern genera. The authentic and plagal species

(or subalterngenera ) can be further divided into lesser species. Example three of Chapter

VI showed the division of the subaltern genus “fifth" into four species of fifths according

to the differentiae of varying tone and semitone combinations.

The term species was generally not used in Carolingian music treatises. Yet all of

the German theorists discussed in this chapter used the predicable species in their

discussions of fourths, fifths, and octaves. The consistent use of species among the

Germans testifies to the value which they placed on this logical term.^

Hermanni Contracti, p. 19: "Ut enim ex multis pauca dicamus, sicut grammatica ad VIII isagoge ad V praedicamenta ad X varietetates rediguntur..." ^Willehelmi Hirsaugensis Musica , ed Haibinson, op. ci t., p. 26: "W. Antiqui et modérai musicae arris auctores multa scripsere probabilia, multa utilia et valde necessaria, quin immo primas eiusdem artis invoiere veoas, ipsum quoque linq)idissimum fontem, id est principale tetrachordum quodam Pmpatericae rationis ex parte detexere fossorio..." ^In section five his Prologus , Bano discusses the species of fourths, fifths, and octaves. Bemo Augensis Prologus in tonarium , GS, II, op. cit., p. 67: "Habeat enim diatessaron species tres, diapente quatuor, diapason a VII.” In his Musica JHermannus devotes chapter five to the species of fourth, chapter six to 80

However, it is not the mere use of species that shows the influence of Aristotelian logic. It is in the conceptual view of the German theorists through which specific species of fourths, fifths, and octaves are regarded as members of a larger class of species of

fourth, species of fifth, and species of octave (which we may call subaltern genera ), that

the most striking influence of Aristotelian logic is shown.

By the time of De musica of Engelbert, one can see the terminology of genus into species applied to the modes. In Engelbert’s words; “according to the quality of their genus, (the modes) are called authentic and plagal, that is, the principal and secondary, or authoritative and subjugal.”®

G enus into Species through the First Opposition of Contraeity

The division of genus into species through Boethius’ first opposition of contraeity applied to the modes is a conceptual view which was shared by Bemo and Hermannus.

Their emphasis on symmetrical properties of the eight mode system shows that both theorists valued symmetry in their theory of the modes. Moreover, the symmetrical properties of the modes has little to do with the chant repertory. Instead the concern of

Bemo and Hermannus for symmetry shows that they were interested in the intemal

species of fifth, and chapter seven to the .species of octave. Trans. Ellinwood, Musica Hermanni Contracti , op. cit., p. 27: ".. juinc qualiter ex eo quod principale diximus omnium quadrichordorum, id est diatessaron specierum tam ordo quam proprietas oriatur diligentur inquiramus." P. 29:.. ."videamus quoque de diapente speciebus, an et ipsae de suo hoc est tercio quadiupli passu naturaliter orientes et ordinabiliter procedentes, sicut diateaaeron species in finalibus, ita ipsae disponi detrectent in gravibus." Wilhelm devoted chapter sevai of his treatise to a discussion of the species of fourth, fifth, and octave. Willhelmi Hirsaugensis Musica , ed. Harbinson, op. cit., p. 21: "In quibus chordis species diatessaron, diapenti et diapason naturaliter consistant." Aribo discusses the division of octave species into species of fourths and fifths in chapter eight of the Smits van Waesberghe edition of the De musica. Aribonis De musica, ed. Smits van Waesberghe, op. c it., p. 26: "Prima species diapason constat ex prima diatesseron et diapaite specie, secunda de secundis, tercia de t«ciis, quarta de quartis. Et in his speciebus diatesseron, diapente, diapason naturalem merito miratniir ordinem, quia omnes in eandem desinunt, ex qua incipiunt, et hae sunt sui nominus." ^Engelberti abbatis admontensis De musica, GS, II, p. 343: "A generis et generationis suae dignitate vocantur authenti et plagales, id est principales et secundarii, sive auctorales et subiugales." 8 1

properties of their theoretical system. This concern for intemal structural symmetry of the modes can be seen, then, as a second order of logic ^ and is a unique characteristic of

Medieval German theory.

For Hermannus the Protus and Tetrardus authentic modes are symmetrical in that the Protus authentic mode is characterized by the descent of a tone and the ascent of a first species fifth,* while the Tetrardus authentic is the opposite in that it ascends by a tone, but descends by a fourth species fifth.^ The Protus and Tetrardus authentic modes are described within the range of a sixth, or Q to A. Similarily, the Deuterus mode descends by a and ascends a fourth, while the Tritus mode ascends by a ditone and descends by a fourth. The opposition of these two modes exists within the same sixth, or C to A as do the Protus and Tetrardus authentic modes. It can be seen that this symmetry has its own intemal logic and elegance, and has little to do with the chant repertory. Powers (1980) has also noted the elegence of this symmetrical attribute in the modal theory of Hermannus.^o

Bemo was even more specific in his view of the modes as being opposite, for he simply said that “Protus and Tetrardus are opposites.. ^ and that “the Deuterus and Tritus are entirely opposites.” 12

^See note 14 of the Introduction to this document. *Trans. Ellinwood, Musica Hermanni Contracti , op. cit., p. 57: "Primus modus vocum est qui tono deponitur et prima specie diapente intenditur..." ^ Ibid., p. 58: "Qarrtum modum vocum tono intensum et quarta jpeciV diapente remissum tetrardo aptamus." l®Harold Powers, "Mode’" The New Grove Dictionary o f Music and Musicians , ed. Sadie, op. cit., vol. 12, p. 389. ^^Berno Augiensis Prologus in Tonarium , op cit., p. 78: "Protus.. .et Tetrardus contrarii sunt..." ^^Ibid., p. 78:, "Deuterus et Tritus omnino sunt contrarii..." 82

G enus into Species through the Fourth Opposition of Relationship

The other opposition used by the German theorists was Boethius’ opposition of relationship. As I indicated in Chapter I, Boethius used an anthropormoiphic illustration to show the opposition of relationship, and, in doing so, he conjoined a Platonic concept with

Aristotelian division. Anthropomorphic illustrations were frequently used by Medieval

German theorists to clarify the relationship between the authentic and plagal modes. The use of anthropomorphic illustration is to show that, though the authentic and plagal modes are separate, they are related enough to mingle with each other in an atmosphere of kinship or friendship.!'^

Bemo employed Boethius’ opposition of relationship of authentic and plagal modes when he said, “so...we use Latin names, and call the higher (authentic) ones teachers, and the lower (plagal) ones pupils.”!^

Aribo applies a unique anthropomorphic illustration in his description of the authentic and plagal modes as male and female respectively, and asserts that these modes mingle by sharing a marriage chamber. Thus in Aribo’s words,“the authentics are concordant and discordant with the plagals, as if the four modest brides with their grooms from the same maniage beds came forth and linked two circles of dances, so that those very wedding beds were the centers for their wedding choruses.,.”!®

Wilhelm employed a distinctly anthropomorphic illustration in his description of the relationship among the authentic and plagal modes. His description became increasingly graphic as he discussed each modal pair of the “maneriae.” He said, concerning the

!^See pp. 18-9 of this Document. !^In Chapter V I suggested that the source of this view was Piatos' Tmaeus . Prologus in Tonarium , .op. cit., p. 69: "Sed.. .latinis utamur nominibus, et excelsiores magistros, inferiores vero disciplulos nominemus." ^^Aribonis De musica, ed. Smits van Waesberghe, p. 17-18: "Concordant discordantque autenti cum plagis, quomodo si prooederent de quatuor thalamis totidem nuptae modestae cum suis sponsis, copularentque duos chorearum circulos, ut ipsi thalami matronali choro essent caitra..." 83

relationship between the Protus authentic and plagal; " ...if these two are joined:”^'^

Deuterus:".. .if these two are united:”^^: Tritus: "... if these two are mated:” and

Tetrardus: "if these two are intertwined.W ilh e lm stated that he used anthropomorphic illustrations rather than the musical ones, because he thought that such a process would aid in comprehension of modal relationships. Thus he said, “I have visually and thus more comprehensively recast the important elements of these figures, making them physical more than musical.”2l

Whole into Parts

In the treatises selected for this study, the terminology of a “whole into parts ” as a division type which is quantitatively based is less used than the qualitative division of a genus into species. Yet the terminology of the division of a “whole into parts” can be seen in three of the five treatises. The evidence is strong enough to show that the conception of a whole which is divided into quantitative parts was a concept that was clearly understood during the early Medieval period.

Hermannus and Engelbert applied the division of a whole into parts in their description of the "maneriae,” while Aribo used the description of the whole into parts in his discussion of the species of fourths and fifths.

In section eight of his Musica Hermannus defines mode as “an inflection of many pitches within any one octave, as determined by fixed intervals and fitted into one w h o l e . ” ^ 2 From his wording, Hermannus shows that his conception of a whole exists

^'^Willehelmi Hirsaugensis Musica , ed. Harbinson, op. cit., p. 33: "Qui si iungantur..." Ibid., "Qui si coadunentur..." ^^Ibid., "Qui si copulentur..." 2®Ibid., "Qui si conectantur..." ^^Ibid., p. 34: ".. .manifestus adhuc tam corporis quam chordis visui figurae huius perspicacia subiciatur." 22Trans. Ellinwood, Musica Hermanni Contracti , op. cit., p. 37: "Tropus est inter unumquoque diapason multarum vocum ratis effecta intervallis apta in unum corpus modulatio." 84

only after consideration of the parts. This was an important aspect of Boethius’ division of a whole into parts.

Aribo combined species with the division of a whole into parts and added an ontological hue to this relationship. In chapter eighty seven of his De musica, he said,

“since every complete thing receives its true name from its parts, we ought to consider the same for the fourth and fifth. We know that there are no more than four species of diatessaron and diapente, but since the species of a diapason is made up of the species of its diapente and diatessaron, it cannot be more than the species of its parts.”^ The key phrase in Aribo’s statement is “since every complete thing receives its true name from its parts.”

Aribo is suggesting that since a whole cannot exist without all of its parts, the number of octave species (as a whole) cannot exist without their constituent species of fourths and fifths (as parts). This conception is similar to Boethius’ division of a whole into its proper parts, “the number of times we separate each out of the whole.”^4 Although Boethius’ statement is ambiguous, the implication of both Boethius and Aribo is that, though one may divide a whole into as many parts as one wishes, the whole as an entity cannot exist without all of its parts.

Engelbert’s wording in book four of his Musica shows that he, too, understood

Boethius’ view that the whole succeeds the parts. In chapter ten of book four, where he discusses why there are eight modes, he says that, “ ... in music, lesser consonances create and compound greater ones, just as the parts create their whole... and the diatessaron and the diapente, and again the diapente and the diatessaron exist in accordance with their

^Sm its van Waesberghe, Aribonis De musica , op. cit., p. 35: "Cum enim omne totum de suis partibus nominetur veraciter, idem in tota diapason, et in suis partibus, diatessaron scilicet ac diapente, considerare debemus. Species diatessaron et diapente non plures quam quatuor scimus esse. Sed cum apgcf&r diapason constituantur exspeciebus diapente et diatessaron, non possunt fieri plures, quam partium species." 2‘lSee page 27. 85

species, and make up their species of diapason.. .”25 it can be seen that Engelbert, like

Aribo, combined the division of a genus into species with the division of a whole into parts. By doing so, it is easy for us to see that both Aribo and Engelbert regarded these two division types as being inherently different

Subject into accidents

If a subject features an attribute which can be removed without the destruction of the subject, then that attribute is an accident. An accident is not essential to the subject in the sense that the subject is dependent on the accident for its definition., as would be the case if the attribute were a difference. Although the term accident was not used by the theorists selected for this study, the evidence is clear that they understood the nature of an attribute which is not essential to the definition of a subject.

The following two cases are offered as an example of the division of a subject into accidents : 1) the division of the subject “octave species on D” into the accidents Protus authentic and Tetrardus plagal, and 2) the division of the subject “octave species on D into the accidents Protus authentic with and without the B flat If either of the two modal designations as Protus authentic or Tetrardus plagal were removed, or if the B flat were removed, the subject ‘^oci&vc species on D” remains.

In the first case, the octave species on D, which is comprised of a first species of fifth plus a first species of fourth, was designated in Medieval theory as both a Protus authentic mode and as a Tetrardus plagal mode. The subject “octave species on D” is divided into two accidents, the Protus authentic and Tetrardus plagal modes.

^Engelberti abbatis Admontensis de musica , op. cit., p. 345; "Quod sicut in musica minores consonantiae saciunt, et componunt maiorestangnam partes suum totum. videlicet toni at semitonia, ditonos, et diatessaron,et diapaiti, et iterum diatessaron, et diapenti, et iterum diatessaron et diapenti secundum duasspecies constituunt et comoponuntspecies diapason...." 86

The second case of this division .type is in the regard of two octave species on D the second of which includes the synemmenon tetrachord. The species of fifth would be the same for both octave species, but the species of fourth for the octave containing B natural is a first species of fourth, whereas the synemmenon tetrachord (giving the Bflatl provides a second species of fourth for the second octave species. Yet the designation as Protus authentic applies to both octave species.

Concerning the first case of this division type, Hermannus discussed in section twelve of his Musica the two modes found on D, the Protus auththentic (the Dorian), and the Tetrardus plagal, (the Hypomixolydian). These two modes have exactly the same range.

For Hermannus, the distinction between the two modes could be seen in their respective “mean” pitches.26 These two mean pitches form the point of disjunction for the integral species of fifths and fourths for these two modes.

For the Protus authentic mode, the mean pitch is A, or the cofinal of that mode.

For the Tetrardus plagal mode, the “mean” pitch is G, or the final of the Tetrardus modes.

The distinction between the two modes would be made according to the frequency of one these mean pitches in colons, commas, and conclusions of chants. Frequent use would emphasize the species of fourth, or (D to Gi, or species of fifth, or )D to a). Hermannus felt that repetition of these two “mean” pitches was sufficient for modal identification, for

“these two middle pitches therefore produce, within one octave, the two different modes... in this manner.“27

^^rans. Ellinwood, Musica Hermanni Contracti, op. cit., p. 41: "Cum quatuordecim voces ex quatuor reddantur quadiichordis, necesse est eas secundum pans numeri uatuiam duas mediatates quae et praedictum diezeuxin faciunt habere." 27lbid. : "Hae igitur mediatates inter unum diapason duos diverses de quibus agitur modos hoc ratione conficiunt." 87

Engelbert expained that the distinction between these two modes was in their constituentspecies of fifth. Thus he said that Protus authentic mode is comprised of a first species of fifth and a first species of fourth, while the Tetrardus plagal mode is comprised of a first species of fourth plus a fourth species of fifth.^^

Concerning the second case of the division of subject into accidents, we turn to

Engelbert. In chapter sixteen of book four of the De musica, he referred to the two diapasons that can be formed on D, the second of which is divided through the synemmenon tetrachord. These two diapasons “can pertain to two different types of songs.”29 The synemmenon tetrachord creates two octave species on D, and “allows two forms of ascent and descent as possible in that diapason, that is through b fa or through ^ mi, as one can see.”30

A ccid en ts into subjects

The following is offered as an illustration of Boethius’ division of an accident into subjects. In this example the four species of fifth are regarded as accidents, since it is possible to change any species of fifth into another without altering the designation '^species of fifth.” The accident species of fifth” (first, second, third, and fourth) can be found at three positions in the gamut: 1) on the pitches D, E, F, G; 2) on the pitches A, B, Ç, and D; and 3) on the pitches G, A, B flat, and Ç using the synemmenon tetrachord.

Diagram seven in Chapter V showed the four species of fifth.^^ In the diatonic gamut they are found on the pitches D, E, F, and fî. Each species of fifth also exists at a

“^^Engelberti abbitis Admontensis de musica , op. cit.., p. 348: "Item diapason, quod est a D. in d. deseruit cantui authenti proti, i.e. toni primi, in quantum conqx>nitur ex prima specie diapente, et prima specie diatesseron, et cantui plagalis tetrardi, i. e. tcmi octavi, in quantum componitur ex prima specie diatesseron, et quarta specie diapaite." 25jbid., p. 347: . .in ipsa potest pertinere ad duplicum cantuum." ^^Ibid.:".. .et admittit duplex ascensus et descensus possibilis in illo diapason, videlicet per h.fa, vel per mi, ut patet." 31See page 48. 88

above each of these pitches, or on the pitches A, B, C, and D. This is because the pattern of tones and semitones on these groups of pitches is identical. However, if the pitch B flat is substituted for B natural, a new position on the gamut arises on which a third series of fourths and fifths may be constructed. The B flat is, of course, available through the synemmenon tetrachord.

Hermannus recognized the third positioning of the four species of fifths through the synemmenon tetrachord. Concerning this, he said,"... a mere form of species (of fifth) may be produced if the Greek gamma is added and the synemmenon is inserted, so that there will be a kind of first species, T , A, b, C, D , a second. A, b, Q, D ,.E , a third, b, C,

D , E , F , and a fourth, C, D , E, F, fi.”32

A ccidents into other A ccidents

Example thirteen of chapter VI showed how the tetrachord of the finals, which features a tone/semitone/tone construction, is encased within a hexachord, since a whole tone is added at both ends of the tetrachord. It is possible to say that the tetrachord can be

“divided” into the hexachord. The term “divided” is used in this instance to mean that, since the hexachord contains the tetrachord, the tetrachord can be divided into hexachord with two tones remaining. This is similar to the division of 5 by 4 with 1 as the remainder. If both the hexachord and the tetrachord are regarded as accidents of the gamut, meaning that they can be removed from the gamut without destroying it, I suggest that this can be seen as an example of Boethius’ division of accidents into other accidents. I submit that the application of the division of accident into other accidents is valid here, because the theorists included in this study discussed the tetrachord first, then

^^Trans. Ellinwood., Musica Hermanni Contracti , op. cit .,p. 30: "Ergo, quod constitutio répudiât, forma tantum si gamma Grecum apponatur svnemenon quoque interponatur fieri poterit, ut sit quasi prima C, A. b, C, D. secunda A. b, Ç, 2 , E, tercia b, Ç, 2 , E, E. quarta Ç, 2 , E, E, G-" 89

the hexachord. This division type is valid also because the theorists associated the hexachord with modal recognition.

The division of accident into other accidents as applied to the encasement of the tetrachord within the hexachord is a divisional perspective which all five German theorists shared. This perspective appears to have been unique to them,^^ and is the primary reason why these five writers were chosen for this study.

The best example of this division type can be seen in Hermannus’ Musica. In the section on the modi vocum, he associated the “tetrachord of the finals” with its

“encasement” within a hexachord by adding a whole tone at both ends of this tetrachord.

Hermannus also recognized that this process applies to the identical hexachord found in the affinal position, or at the upper fifth The process of adding whole tones to both ends of these two tetrachords resulted in a hexachord on c, later called the “natural” hexachord, and a hexachortd on g, later called the “hard” hexachord. For Hermannus, this process was the basis of modal construction: “ take any tetrachord you wish...add a tone at both ends. You then have the limits of the tonal patterns which form the basis of the m odes.34 it is evident that Hermannus associated the hexachord with modal recognition, for he later said, “thus it is clear that this recognition of the modes can be perfectly accomplished in any tetrachord by adding a tone at both ends.”^^

In diagram two of chapter IV, I showed how an accident can be equivalent to a property if the property can be taken from the subject without destroying the subject. This, of course, is the definition of an accident.

^^Harold Powers, "Mode," The New Grove Dictionary o f Music and Musicians ed. Sadie, op. cit.. vol. 12, p. 391. ^'^rans. Ellinwood, Musica Hermanni Contracti , op. cit., p. 57. Musica Hermanni Contracti , p. 57" " Accipe tetrachordum quodcumquje voleris.. .addito utrinque tono, babes tmninris modorum qui Sunt sedes troporum." ^^Ibid., p. 60: Ergo banc agnitionem troporum in quodlibet tetracbordo adiecto utrimque tone perfecte fieri manifestum est." 90

In his De musica, Aribo applied the predicable property in his description of the interval of the sixth in which Protus, Deuterus, Tritus, and Tetrardus modal seats are found.^® “The property of the Protus (authentic and plagal) exists so that they are in agreement in their height through the diapente and in their disposition across the tone. The property of the Deuterus (authentic and plagal) exists with a complete diatessaron through its extending and a ditone when it returns. The property of the Tritus (authentic and plagal) exists in such a way that they are concordant during the extension of a ditone and through a diaterssaron during its return. The property of the Tetrardus (authentic and plagal) exists in such a way that they are countered by the Proti, so much as the height of the Proti is equal to the depth of them.”37 in this last statement, where he contrasts the height of the Protus mode with the depth of the tetrardus, Aribo suggests that the Protus authentic mode is opposite to the Tetrardus authentic. The ascending Protus authentic mode ascends by two tones, a semitone, and two tones in its initial species of fifth. The Tetrardus authentic mode descends by two tones, a semitone, and two tones in its initial species of fifth.

In his Musica, Wilhelm also recognized the identity of modal seats as identical tetrachords ‘encased’ within a hexachordal formation, and he expanded this divisional conception to include a whole tone on either side of the synemmenon tetrachord.

However, Wilhelm took this division type a step further when he identified the synemmenon tetrachord as another species of fourth which, if placed within a framework that included a whole tone at either end, would yield an identical hexachord to the two indicated by Hermannus In Wilhelms words, “To whichever of those tetrachords (graves. finales, superiores. and excellentes) you add a tone from either part, the first limit is from

property is an attribute which is capable of predicated convertibly of the subject. ^^Aribonis De musica , ed. Smits van Waesberghe, op. cit., p. 32: "Proprietas est protorum, ut concordent in elevatione per diapente in dispositione per tonum. Proprietas est dueterorum cum integra diatesseron per intensionem, ditono per remissionem. Proprietas est tritorum, ut concordent intensione ditono, diatesseron remissione. Proprietas est tetrachordum, ut protis opponantur, quatenus horum elevatio istorum sit remissio." 91

gamma to E., for the finales, the second limit is from Ç to G, from the superiores. the third limit is from G to e, for the excellentes, the fourth limit is from ç to (Z), the fifth is able to be found through the synemmenon from F to d.^^ As Pesce has pointed out, this was the first recognition of what was later termed the “soft” hexachord.^^

From quotations of five German theorists this chapter has shown that the science of

division as inherited from Aristotle and transmitted by Boethius was a new approach to rationalizing the eight modes. Of the six predicables, genus, species, property, difference, and definition were used by the theorists. Although accident was not used, it is evident that the conception of accident as a property which is not essential to the subject was

understood. Finally, the quotations in this chapter show that Boethius’ five divsion types were, to varying degrees, applied by these five German writers.

^^Willehelmi Hirsaugensis musiica , ed. Harbinson, op. cit., p. 59-60: "Secundum melodiae modum vel institutionem troporum quatuor tetrachorda in monochordo Armantur, id est Gravium A.B.C.D., Finalium D.Ef.G., Superium a.b.c.d., Excellentium d.e.f.g. Cuicumque horum tetrachordorum tonum ex utraque parte addideris, oritur tibi aliqua meta troporum utpote a Gravibuus prima meta est a (gamma) in E., a Finalibus secunda a C., in a., a Superiorbus tertia a G. in e., ab Excellentibus quarta a c. in P., quinta per synemmenon nasci poterit as F. in d." 3^Pesce, The Affinities and Medieval Transposition , op. cit., p. 29. CONCLUSION

Thanks to recent research by Magee,^ it has now been established that Boethius’ De divisione was a widely copied treatise during the Medieval period. In the absence of testimony that any music writer of the early Medieval period had a copy of the De divisione and had actually read it, this study can at best speculate that Aristotelian logic was known to these writers through the Boethian monograph. More research needs to be done on the availability of this treatise to music theorists. Species theory and division reached German theorists of the early Medieval period from some source. I suggest that the De divisione was the likely source.

In summary, the method of classifying of the eight church modes adopted by early

Medieval German music theorists, the terminology used in this classification, and the manner of rationalizing problematic chants, all show such an obvious similarity to the methods proposed by Boethius, that it seems plausible to assume an influence, direct or indirect from the De divisione .

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