Diatonicism and Chromaticism in Richard Strauss' Tone

Total Page:16

File Type:pdf, Size:1020Kb

Diatonicism and Chromaticism in Richard Strauss' Tone DIATONICISM AND CHROMATICISM IN RICHARD STRAUSS' TONE-POEM TILL EULENSPIEGELS LUSTIGE STREICHE by PATRICK^MAC NEIL B.A. (Honours in Music), Dalhousie University, 1976 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS IN MUSIC THEORY in THE FACULTY OF GRADUATE STUDIES Department of Music We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA November 1981 @ Patrick Mac Neil, 1981 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of t/rVUuat* The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date Q<tZ&M {j} /(}j>/. DE-6 (2/79) ABSTRACT In the first chapter of this thesis I introduce the diatonic-chromatic continuum. It is an analytical device useful in explaining music which exhibits much diversity. In the case of this tone-poem, it is useful to assimilate the many sections within it, those that are straightforwardly diatonic in a traditional sense, those that are so highly chromatic it is sometimes difficult to determine their main and subsidiary keys, as well as those that lie somewhere between these two extremes. As these three section-types are equally characteristic of the work, the diatonic-chromatic continuum serves as a valuable referential concept. In the second chapter I analyze the above sections noting their main keys, subsidiary keys, and the means by which these keys are established. In the third chapter I return to the concept of the diatonic-chromatic continuum and discuss several sections analyzed in the second chapter. I have collected my musical examples together in the Appendix for easier reference. These examples comprise reductions from the tone-poem, simplifications of these reductions I call sketches, and a major example which is a condensation of the entire work. Finally, my study of Till Eulenspiegels lustige Streiche concentrates on the purely musical and not on the programmatic. TABLE OF CONTENTS Abstract ii Acknowledgement iv Chapter I. Introduction 1 II. Till 11 III. General Remarks and Conclusion 40 Appendix of Musical Examples 52 Bibliography 101 ACKNOWLEDGEMENT I thank Dr. William Benjamin for his advice in the preparation of this thesis, and I thank Richard Strauss for leaving the world so much beautiful music, some of which I have studied to my great delight and musical edification. iv CHAPTER I Introduction The term continuum has been defined as a "continuous thing, quantity, or substance; a continuous series of elements passing into each other." The passage of day into night may be represented as a continuum. Beginning in the late afternoon of a cloudless day, I could measure the intensity of light at regular intervals of time until well into the night. I would obtain a series of readings continuously decreasing in intensity. The more frequently I took the readings the closer they would be to each other, so that eventually the difference between two readings would be either too fine to measure or too slight for the human eye to perceive. I would find no point at 2 which day ends and night begins, but, only and always, points at which intensity-readings are preceded or followed by the next highest or lowest readings. This day-night continuum is a "continuous series" of intensity-readings "passing into each other." The passage from the diatonic to the chromatic may be represented as a continuum. By diatonic I mean a work or portion of a' work which uses as its basis the major and minor scales and the chords which may be formed from the pitches of these scales. By chromatic I mean a work or portion of a work which uses as its basis the major and minor scales one or more - 1 - of the pitches of which have been raised or lowered by a 3 half-step. Chromatic then also refers to the chords formed from the pitches of these altered major and minor scales. As decreasing degrees of light-intensity lead one along the continuum from day into night, so increasing degrees of half- step alteration lead one along the continuum from the diatonic to the chromatic. This .latter: continuum I shall simply call the diatonic-chromatic continuum. This continuum can be seen from the perspective of two musical concepts. One of these concepts is tonality. I define tonality rather broadly as the accentuation of a pitch or group of pitches through musical means.^ But for the sake of my analysis of the tone-poem Till Eulenspiegels lustige Streiche, which henceforth I shall refer to as Till, I define tonality more narrowly. Tonality results from the accentuation of a pitch or of a group of pitches forming a major or minor triad; this accentuation involves approaching this single pitch or at least one pitch of this group of pitches by half- step. In a major scale the passage of the leading-tone to the tonic is an instance of tonality. In a perfect cadence the progression of one major triad to another a perfect fourth above or perfect fifth below is an instance of tonality.^ If a musical composition or a section from it exhibits tonality produced by the pitches of the diatonic scales and the chordal progeny formed from these pitches, this "kind" of tonality can be represented on the diatonic-chromatic continuum at the same point at which a diatonic musical composition or section from it would be represented. Similarly, if a composition - 2 - or section from it exhibits tonality produced by the pitches and chordal progeny of the diatonic scales as well as by the pitches and the chordal progeny which are the result of chromatic alteration of this diatonic material, this "kind" of tonality can be represented further along the continuum at a point where a composition or section from it based on these pitches would be represented. The diatonic-chromatic continuum can also be seen from the perspective of modulation. I define modulation as a transposition of the pitch or group of pitches to be accentuated by musical means. A major scale may be transposed up a perfect fifth. This new scale may be considered a reordering of the pitches of the old scale, the ordering from dominant to dominant replacing that from tonic to tonic. There is also a chromatic alteration as the old subdominant is raised a half-step to become the new leading-tone. This new scale may also be transposed up a perfect fifth. In terms of the original scale with which this modulatory process began, two of its pitches have now been raised a half-step, the subd'ominant and the tonic. If this process continues until the tonic of the new scale is a half-step from the tonic of the original, only two pitches of the original scale will remain unaltered. If the process continues from there, there will always be two pitches of the original scale unaltered although not always the same two pitches.^ The original scale can be represented on the continuum at the "diatonic" end and the transpositions of the original scale progressively making alterations to it can be represented at points along the continuum which are - 3 - progressively closer to the chromatic "end". In a composition containing several sections based on these scales, the passage from one section to another may be described as motion to the left or to the right along the continuum. Also, a scale and its transpositions or two successive sections of a composition may be represented on the continuum by points which are not adjacent. This would be done, for example, when a scale is transposed up or down a half-step or when a section based on such a scale is followed by another section based on the transposition. Instead of there being a progressive alteration of pitch material, many alterations are made at once. This may be described as jumping along the continuum from one point to another more remote from it, or as compressing the continuum so that these two distant points are made adjacent.^ I have taken the time to define the diatonic-chromatic continuum because it is useful for describing Till. In Till there are no sections which are purely diatonic, that is, sections which contain musical material formed strictly from the pitches of the diatonic scales. Instead, each section makes use of one or more of the following: chromatic passing tones, chromatic embellishing chords which decorate diatonic ones, and chromatic chords which serve as dominants for the main key of a section and for subsidiary keys within a section. In some cases these subsidiary keys are chromatically related to the main key of a section. There are also sections within Till which are in a key related chromatically to the keys of sections before and after them, and these sections related chromatically to their neighbours may be diatonic of highly - 4 - chromatic themselves. Thus, in Till there is a mixture of sections which are more or less diatonic or more or less chromatic. I blend the two together on a continuum, various points on which represent various sections of the tone-poem. I feel what I have been saying so far will become clearer when I discuss specific sections from Till during the course of this study.
Recommended publications
  • Computational Methods for Tonality-Based Style Analysis of Classical Music Audio Recordings
    Fakult¨at fur¨ Elektrotechnik und Informationstechnik Computational Methods for Tonality-Based Style Analysis of Classical Music Audio Recordings Christof Weiß geboren am 16.07.1986 in Regensburg Dissertation zur Erlangung des akademischen Grades Doktoringenieur (Dr.-Ing.) Angefertigt im: Fachgebiet Elektronische Medientechnik Institut fur¨ Medientechnik Fakult¨at fur¨ Elektrotechnik und Informationstechnik Gutachter: Prof. Dr.-Ing. Dr. rer. nat. h. c. mult. Karlheinz Brandenburg Prof. Dr. rer. nat. Meinard Muller¨ Prof. Dr. phil. Wolfgang Auhagen Tag der Einreichung: 25.11.2016 Tag der wissenschaftlichen Aussprache: 03.04.2017 urn:nbn:de:gbv:ilm1-2017000293 iii Acknowledgements This thesis could not exist without the help of many people. I am very grateful to everybody who supported me during the work on my PhD. First of all, I want to thank Prof. Karlheinz Brandenburg for supervising my thesis but also, for the opportunity to work within a great team and a nice working enviroment at Fraunhofer IDMT in Ilmenau. I also want to mention my colleagues of the Metadata department for having such a friendly atmosphere including motivating scientific discussions, musical activity, and more. In particular, I want to thank all members of the Semantic Music Technologies group for the nice group climate and for helping with many things in research and beyond. Especially|thank you Alex, Ronny, Christian, Uwe, Estefan´ıa, Patrick, Daniel, Ania, Christian, Anna, Sascha, and Jakob for not only having a prolific working time in Ilmenau but also making friends there. Furthermore, I want to thank several students at TU Ilmenau who worked with me on my topic. Special thanks go to Prof.
    [Show full text]
  • Out of Death and Despair: Elements of Chromaticism in the Songs of Edvard Grieg
    Ryan Weber: Out of Death and Despair: Elements of Chromaticism in the Songs of Edvard Grieg Grieg‘s intriguing statement, ―the realm of harmonies has always been my dream world,‖1 suggests the rather imaginative disposition that he held toward the roles of harmonic function within a diatonic pitch space. A great degree of his chromatic language is manifested in subtle interactions between different pitch spaces along the dualistic lines to which he referred in his personal commentary. Indeed, Grieg‘s distinctive style reaches beyond common-practice tonality in ways other than modality. Although his music uses a variety of common nineteenth- century harmonic devices, his synthesis of modal elements within a diatonic or chromatic framework often leans toward an aesthetic of Impressionism. This multifarious language is most clearly evident in the composer‘s abundant songs that span his entire career. Setting texts by a host of different poets, Grieg develops an approach to chromaticism that reflects the Romantic themes of death and despair. There are two techniques that serve to characterize aspects of his multi-faceted harmonic language. The first, what I designate ―chromatic juxtapositioning,‖ entails the insertion of highly chromatic material within ^ a principally diatonic passage. The second technique involves the use of b7 (bVII) as a particular type of harmonic juxtapositioning, for the flatted-seventh scale-degree serves as a congruent agent among the realms of diatonicism, chromaticism, and modality while also serving as a marker of death and despair. Through these techniques, Grieg creates a style that elevates the works of geographically linked poets by capitalizing on the Romantic/nationalistic themes.
    [Show full text]
  • Kostka, Stefan
    TEN Classical Serialism INTRODUCTION When Schoenberg composed the first twelve-tone piece in the summer of 192 1, I the "Pre- lude" to what would eventually become his Suite, Op. 25 (1923), he carried to a conclusion the developments in chromaticism that had begun many decades earlier. The assault of chromaticism on the tonal system had led to the nonsystem of free atonality, and now Schoenberg had developed a "method [he insisted it was not a "system"] of composing with twelve tones that are related only with one another." Free atonality achieved some of its effect through the use of aggregates, as we have seen, and many atonal composers seemed to have been convinced that atonality could best be achieved through some sort of regular recycling of the twelve pitch class- es. But it was Schoenberg who came up with the idea of arranging the twelve pitch classes into a particular series, or row, th at would remain essentially constant through- out a composition. Various twelve-tone melodies that predate 1921 are often cited as precursors of Schoenberg's tone row, a famous example being the fugue theme from Richard Strauss's Thus Spake Zararhustra (1895). A less famous example, but one closer than Strauss's theme to Schoenberg'S method, is seen in Example IO-\. Notice that Ives holds off the last pitch class, C, for measures until its dramatic entrance in m. 68. Tn the music of Strauss and rves th e twelve-note theme is a curiosity, but in the mu sic of Schoenberg and his fo ll owers the twelve-note row is a basic shape that can be presented in four well-defined ways, thereby assuring a certain unity in the pitch domain of a composition.
    [Show full text]
  • MTO 18.4: Chenette, Hearing Counterpoint Within Chromaticism
    Volume 18, Number 4, December 2012 Copyright © 2012 Society for Music Theory Hearing Counterpoint Within Chromaticism: Analyzing Harmonic Relationships in Lassus’s Prophetiae Sibyllarum Timothy K. Chenette NOTE: The examples for the (text-only) PDF version of this item are available online at: http://www.mtosmt.org/issues/mto.12.18.4/mto.12.18.4.chenette.php KEYWORDS: Orlandus Lassus (Orlando di Lasso), Prophetiae Sibyllarum, chromaticism, counterpoint, Zarlino, Vicentino, diatonicism, diatonic system, mode, analysis ABSTRACT: Existing analyses of the Prophetiae Sibyllarum generally compare the musical surface to an underlying key, mode, or diatonic system. In contrast, this article asserts that we must value surface relationships in and of themselves, and that certain rules of counterpoint can help us to understand these relationships. Analyses of the Prologue and Sibylla Persica demonstrate some of the insights available to a contrapuntal perspective, including the realization that surprising-sounding music does not always correlate with large numbers of written accidentals. An analysis of Sibylla Europaea suggests some of the ways diatonic and surface-relationship modes of hearing may interact, and highlights the features that may draw a listener’s attention to one of these modes of hearing or the other. Received April 2012 Introduction [1.1] Orlandus Lassus’s cycle Prophetiae Sibyllarum (1550s) is simultaneously familiar and unsettling to the modern listener. (1) The Prologue epitomizes its challenges; see Score and Recording 1. (2) It uses apparent triads, but their roots range so widely that it is difficult to hear them as symbols of an incipient tonality. The piece’s three cadences confirm, in textbook style, Mode 8, (3) but its accidentals complicate a modal designation based on the diatonic octave species.
    [Show full text]
  • Chromatic Sequences
    CHAPTER 30 Melodic and Harmonic Symmetry Combine: Chromatic Sequences Distinctions Between Diatonic and Chromatic Sequences Sequences are paradoxical musical processes. At the surface level they provide rapid harmonic rhythm, yet at a deeper level they function to suspend tonal motion and prolong an underlying harmony. Tonal sequences move up and down the diatonic scale using scale degrees as stepping-stones. In this chapter, we will explore the consequences of transferring the sequential motions you learned in Chapter 17 from the asymmetry of the diatonic scale to the symmetrical tonal patterns of the nineteenth century. You will likely notice many similarities of behavior between these new chromatic sequences and the old diatonic ones, but there are differences as well. For instance, the stepping-stones for chromatic sequences are no longer the major and minor scales. Furthermore, the chord qualities of each individual harmony inside a chromatic sequence tend to exhibit more homogeneity. Whereas in the past you may have expected major, minor, and diminished chords to alternate inside a diatonic sequence, in the chromatic realm it is not uncommon for all the chords in a sequence to manifest the same quality. EXAMPLE 30.1 Comparison of Diatonic and Chromatic Forms of the D3 ("Pachelbel") Sequence A. 624 CHAPTER 30 MELODIC AND HARMONIC SYMMETRY COMBINE 625 B. Consider Example 30.1A, which contains the D3 ( -4/ +2)-or "descending 5-6"-sequence. The sequence is strongly goal directed (progressing to ii) and diatonic (its harmonies are diatonic to G major). Chord qualities and distances are not consistent, since they conform to the asymmetry of G major.
    [Show full text]
  • Mystic Chord Harmonic and Light Transformations In
    MYSTIC CHORD HARMONIC AND LIGHT TRANSFORMATIONS IN ALEXANDER SCRIABIN’S PROMETHEUS by TYLER MATTHEW SECOR A THESIS Presented to the School of Music and Dance and the Graduate School of the University of Oregon in partial fulfillment of the requirements for the degree of Master of Arts September 2013 THESIS APPROVAL PAGE Student: Tyler Matthew Secor Title: Mystic Chord Harmonic and Light Transformations in Alexander Scriabin’s Prometheus This thesis has been accepted and approved in partial fulfillment of the requirements for the Master of Arts degree in the School of Music and Dance by: Dr. Jack Boss Chair Dr. Stephen Rodgers Member Dr. Frank Diaz Member and Kimberly Andrews Espy Vice President for Research and Innovation Dean of the Graduate School Original approval signatures are on file with the University of Oregon Graduate School. Degree awarded September 2013 ii © 2013 Tyler Matthew Secor iii THESIS ABSTRACT Tyler Matthew Secor Master of Arts School of Music and Dance September 2013 Title: Mystic Chord Harmonic and Light Transformations in Alexander Scriabin’s Prometheus This thesis seeks to explore the voice leading parsimony, bass motion, and chromatic extensions present in Alexander Scriabin’s Prometheus. Voice leading will be explored using Neo-Riemannian type transformations followed by network diagrams to track the mystic chord movement throughout the symphony. Bass motion and chromatic extensions are explored by expanding the current notion of how the luce voices function in outlining and dictating the harmonic motion. Syneathesia
    [Show full text]
  • Secondary Dominant Chords.Mus
    Secondary Dominants Chromaticism - defined by the use of pitches outside of a diatonic key * nonessential chromaticism describes the use of chromatic non-chord tones * essential chromaticism describes the use of chromatic chord tones creating altered chords Secondary Function Chords - also referred to as applied chords * most common chromatically altered chords * function to tonicize (make sound like tonic) a chord other than tonic * applied to a chord other than tonic and typically function like a dominant or leading-tone chord - secondary function chords can also be used in 2nd inversion as passing and neighbor chords - since only major or minor triads can function as tonic, only major or minor triads may be tonicized - Secondary function chords are labeled with two Roman numerals separated by a slash (/) * the first Roman numeral labels the function of the chord (i.e. V, V7, viiº, or viiº7) * the second Roman numeral labels the chord it is applied to - the tonicized chord * secondary function labels are read as V of __, or viiº of __, etc. Secondary Dominant Chords - most common type of secondardy function chords * always spelled as a major triad or Mm7 chord * used to tonicize a chord whose root is a 5th below (or 4th above) * can create stronger harmonic progressions or emphasize chords other than tonic Spelling Secondary Dominant Chords - there are three steps in spelling a secondary dominant chord * find the root of the chord to be tonicized * determine the pitch a P5 above (or P4 below) * using that pitch as the root, spell a
    [Show full text]
  • Transfer Theory Placement Exam Guide (Pdf)
    2016-17 GRADUATE/ transfer THEORY PLACEMENT EXAM guide! Texas woman’s university ! ! 1 2016-17 GRADUATE/transferTHEORY PLACEMENTEXAMguide This! guide is meant to help graduate and transfer students prepare for the Graduate/ Transfer Theory Placement Exam. This evaluation is meant to ensure that students have competence in basic tonal harmony. There are two parts to the exam: written and aural. Part One: Written Part Two: Aural ‣ Four voice part-writing to a ‣ Melodic dictation of a given figured bass diatonic melody ‣ Harmonic analysis using ‣ Harmonic Dictation of a Roman numerals diatonic progression, ‣ Transpose a notated notating the soprano, bass, passage to a new key and Roman numerals ‣ Harmonization of a simple ‣ Sightsinging of a melody diatonic melody that contains some functional chromaticism ! Students must achieve a 75% on both the aural and written components of the exam. If a passing score is not received on one or both sections of the exam, the student may be !required to take remedial coursework. Recommended review materials include most of the commonly used undergraduate music theory texts such as: Tonal Harmony by Koska, Payne, and Almén, The Musician’s Guide to Theory and Analysis by Clendinning and Marvin, and Harmony in Context by Francoli. The exam is given prior to the beginning of both the Fall and Spring Semesters. Please check the TWU MUSIc website (www.twu.edu/music) ! for the exact date and time. ! For further information, contact: Dr. Paul Thomas Assistant Professor of Music Theory and Composition [email protected] 2 2016-17 ! ! ! ! table of Contents ! ! ! ! ! 04 Part-Writing ! ! ! ! ! 08 melody harmonization ! ! ! ! ! 13 transposition ! ! ! ! ! 17 Analysis ! ! ! ! ! 21 melodic dictation ! ! ! ! ! harmonic dictation ! 24 ! ! ! ! Sightsinging examples ! 28 ! ! ! 31 terms ! ! ! ! ! 32 online resources ! 3 PART-Writing Part-writing !Realize the following figured bass in four voices.
    [Show full text]
  • MUTH M202 Theory
    MUTH M202-003 Theory III General Syllabus Fall, 2018 Instructor: Dr. Marcus St. Julien Office: CM 146 E mail: [email protected] or [email protected] Phone: 504-810-6780 Office Hours: by appointment. Class Meeting Times: Monday/Friday, 11:30 A.M.-12:20 P.M. and Tuesday/Thursday, 11:00 A.M.-12:15 P.M. Course Goals: This course extends the part-writing and analysis techniques of Bach chorale style to include standard devices of chromatic harmony, such as borrowed chords, Neapolitan sixth chords and augmented sixth chords, along with the study of melodic organization, binary forms and sonata form. Musicianship skills are focused on aural perception of materials in major and minor modes, including decorative chromaticism, such as chromatic neighbor tones, and harmonically functional chromaticism, such as secondary dominants, along with rhythmic materials including syncopation at the subdivision level, compound divisions of simple beats and simple divisions of compound beats. Expected Student Learning Outcomes: At the end of the semester, a student must be able to do the following: Demonstrate correct part-writing and analysis of standard devices of chromatic harmony, such as borrowed chords, Neapolitan sixth chords and augmented sixth chords. Demonstrate recognition of standard devices of melodic organization, such as phrase, period, double period [and sentence]. (I still don’t get the concept of “sentence” in music, incidentally!) Demonstrate recognition of various patterns of binary-form organization. Demonstrate the ability to parse pieces in sonata form into their constituent parts, such as exposition, development, recapitulation, first subject, transition and second group. Demonstrate the ability to identify points of modulation and areas of tonicization in binary-form and sonata-form compositions.
    [Show full text]
  • Class Notes for Counterpoint
    Class Notes for Counterpoint Richard R. Randall Department of Music University of Massachusetts, Amherst Fall 2007 i Preface and Acknowledgments This book is designed to provide you with a solid foundation in counterpoint. Our department’s belief is that counterpoint is something that should be part of our every- day music making. It is a way to hear music. It is way to understand music. UMass is unique among music programs in that we teach counterpoint in the first semester of a five semester core curriculum. At other schools, the subject, if taught at all, is often relegated to an elective. I would like to acknowledge the influence of Heinrich Schenker’s Kontrapunkt(1910) and Felix Salzer and Carl Schachter’s Counterpoint in Composition(1969) in preparing these materials. In addition, I would like to thank my counterpoint teacher, Miguel Roig-Francoli. Most importantly, I owe a great deal of thanks to my teaching colleagues Jessica Embry, Adam Kolek, Michael Vitalino, Daniel Huey, and Sara Chung for their hard work, insightful suggestions, and generous help in preparing this text. ii Introduction What is Counterpoint? Lat.: contrapunctus,fromcontra punctum:“against note.” (Fr. contrepoint ; Ger. Kontrapunkt; It. contrappunto) Counterpoint is a broad term for interacting yet independent voices. Since the earliest forms of polyphony, musical textures have been made up of multiple “lines” of music (or “voices”) that combine to form vertical sonorities. Studying counterpoint teaches us how to recognize and understand those lines. Counterpoint is the essence of what we call “voice leading.” The vertical aspect of music is described as “harmonic.” The horizontal aspect of music is described as “melodic,” or “linear” when talking about individual lines and “contrapuntal” when talking about how those melodies interact with each other.
    [Show full text]
  • Theory IV – Study Guide Dr. Amy Dunker Clarke College Dubuque, IA 52001 Classical Serialism Arnold Schoenb
    Theory IV – Study Guide Dr. Amy Dunker Clarke College Dubuque, IA 52001 www.amydunker.com Classical Serialism Arnold Schoenberg composed the first twelve-tone piece in the summer of 1921 (Suite, Op. 25 (completed in 1923). Schoenberg had developed a method of composing with twelve tones that are related only with one another. He saw twelve-tone or serial composition as the natural extension of chromaticism on the tonal system. Anton Webern and Alban Berg: Schoenberg’s two pupils who composed in the twelve- tone method. Tone Row (also called , Row, Set, Basic Set, Series): an arrangement of the twelve pitches of the chromatic scale so that no notes repeat (except immediately after it is heard and trills/tremolos) until all pitches of the row have sounded in order. Dodecaphonic Scale: Twelve tone scale Four Forms of the Tone Row: Prime: The original set (do not confuse this with the terms use in Non-Serial Atonality) Retrograde: The original set in reverse order (i.e. backwards) Inversion: The mirror inversion of the original set Retrograde Inversion: The inversion in reverse order Abbreviations: P=Prime R=Retrograde I=Inversion RI=Retrograde Inversion *In addition, each of the four basic forms has twelve transpositions Order Numbers: numbers assigned to the row which indicate each notes intervallic distance from the first note of the row. The first note of the row is assigned the number zero (0). Twelve-Tone Matrix (“Magic Square”): a method of determining all 48 possible versions of the tone row. To construct a Twelve-Tone Matrix do the following: 1.) Fill in the Prime or Original row across the top ( from left to right) using the row’s order numbers.
    [Show full text]
  • The Consecutive-Semitone Constraint on Scalar Structure: a Link Between Impressionism and Jazz1
    The Consecutive-Semitone Constraint on Scalar Structure: A Link Between Impressionism and Jazz1 Dmitri Tymoczko The diatonic scale, considered as a subset of the twelve chromatic pitch classes, possesses some remarkable mathematical properties. It is, for example, a "deep scale," containing each of the six diatonic intervals a unique number of times; it represents a "maximally even" division of the octave into seven nearly-equal parts; it is capable of participating in a "maximally smooth" cycle of transpositions that differ only by the shift of a single pitch by a single semitone; and it has "Myhill's property," in the sense that every distinct two-note diatonic interval (e.g., a third) comes in exactly two distinct chromatic varieties (e.g., major and minor). Many theorists have used these properties to describe and even explain the role of the diatonic scale in traditional tonal music.2 Tonal music, however, is not exclusively diatonic, and the two nondiatonic minor scales possess none of the properties mentioned above. Thus, to the extent that we emphasize the mathematical uniqueness of the diatonic scale, we must downplay the musical significance of the other scales, for example by treating the melodic and harmonic minor scales merely as modifications of the natural minor. The difficulty is compounded when we consider the music of the late-nineteenth and twentieth centuries, in which composers expanded their musical vocabularies to include new scales (for instance, the whole-tone and the octatonic) which again shared few of the diatonic scale's interesting characteristics. This suggests that many of the features *I would like to thank David Lewin, John Thow, and Robert Wason for their assistance in preparing this article.
    [Show full text]