sign in sign up
<<
Home , Archicembalo, Augmented fifth, Augmented second, Chromaticism, Equal temperament, Interval class

COMPUTATIONAL ANALYSIS OF QUARTER-TONE COMPOSITIONS BY AND

A thesis submitted to the College of the Arts of Kent State University in partial fulfillment of the requirements for the degree of Master of Arts

by

Andrew M. Blake

May 2020

Table of Contents Page

List of Figures ...... v List of Tables ...... vi CHAPTER I ...... 1 INTRODUCTION ...... 1 The Quarter-Tone System ...... 1 Notation and Intervals ...... 2 Ives’ Some Quarter-Tone Impressions ...... 4 Wyschnegradsky’s Manual of Quarter-Tone ...... 6 Approaches to Analysis of Quarter-Tone ...... 10 The Place of Quarter-Tones in the Series ...... 11 Pitch-Class Set Theory in 24-EDO ...... 14 Computational Analysis ...... 16 Further Review of Related Literature ...... 17 CHAPTER II ...... 24 ANALYSES OF SELECTED EXCERPTS...... 24 Ives’ Three Quarter-Tone Pieces ...... 24 Three Quarter-Tone Pieces - I. Largo ...... 24 Three Quarter-Tone Pieces: III. Chorale ...... 30 Wyschnegradsky’s 24 ...... 34 Prelude 1 in ...... 38 Prelude 4 in ¼-sharp ...... 41 CHAPTER III ...... 45 COMPUTATIONAL TOOLS FOR MICROTONAL ANALYSIS ...... 45 The Limitations of the Humdrum Toolkit ...... 45 Microtonal Hint ...... 46 Microtonal Set-Theory Tools ...... 49

CHAPTER IV ...... 52 COMPUTATIONAL ANALYSES OF SELECTED EXCERPTS ...... 52 Compiling a Small Corpus ...... 52 Analyzing and Interpreting the Output of mhint ...... 53 Ives’ Three Quarter-Tone Pieces ...... 54 Largo ...... 55 Chorale...... 57 Wyschnegradsky’s 24 Preludes: Comparing Preludes I and IV...... 61 CHAPTER V ...... 66 CONCLUSION AND FUTURE WORK ...... 66 Conclusion ...... 66 Future Avenues of Research ...... 67 References ...... 70

v

List of Figures

Figure 1. Spectrum of 24-EDO Accidentals...... 2 Figure 2. Ives' Primary, or "Fundamental Major" Chord...... 5 Figure 3. Ives' Secondary, or "Fundamental Minor" Chord...... 6 Figure 4. Quarter-tone Appoggiaturas (Wyschnegradsky 1932, 12)...... 7 Figure 5. Wyschnegradsky's 13 Quarter-tone Altered Triads (Wyschnegradsky, 1932, 17)...... 7 Figure 6. Quarter-Tone Modulations (Wyschnegradsky 1932, 20)...... 8 Figure 7. Motivic Repetition in Wyschnegradsky's Prelude No. 1...... 10 Figure 8. Wyschnegradsky's "Diatonicized-Chromatic" on C...... 11 Figure 9. Deviation of Interval Approximations in 12-EDO from the Harmonic Series...... 12 Figure 10. Deviation of Interval Approximations in 24-EDO from the Harmonic Series...... 13 Figure 11. Quarter-Tone Passing Motion and primary theme of Ives Largo (A)...... 26 Figure 12. Ives' Primary chord, punctuated by other quartal/quintal sonorities...... 27 Figure 13. Underlying augmented triads in mm. 34-37 of Largo...... 28 Figure 14. "Tonic Prolongation" in mm. 1-4 of Ives' Chorale...... 31 Figure 15. Primary and secondary chords in transition section...... 32 Figure 16. "Parallel diatonic thirds" within an altered ...... 33 Figure 17. Triadic structures and a return to Ives' primary and secondary chords...... 34 Figure 18. 3-step cycles in the C-DC scale...... 35 Figure 19. 3-step cycle "tonic " of Wyschnegradsky's C-DC scale...... 36 Figure 20. "Composing out" the tonic tetrachord in Prelude 1...... 40 Figure 21. The E 1/4-sharp DC scale on which Prelude 4 is based...... 41 Figure 22. Prelude 4, mm. 1-6 with E 1/4-sharp present...... 41 Figure 23. Prelude 4, mm. 1-6 with ostinato omitted...... 42 Figure 24. and syncopation in Prelude 4...... 43 Figure 25. Imitation emphasizing thirds of various sizes in Prelude 4...... 44 Figure 26. Imitation in the coda of Prelude 4...... 44 Figure 27. Calculating a specific interval from generic interval and size in cents...... 47 Figure 28. Wyschnegradsky's Prelude No. 4 (two score)...... 48 Figure 29. Wyschnegradsky's Prelude No. 4 (condensed study score)...... 48 Figure 30. Exponential growth in the number of in N-EDO systems (6-36)...... 50 Figure 31. Exponential growth in the number of N/2-chords in N-EDO (6-36)...... 50 Figure 32. of intervals in Ives' Largo...... 55 Figure 33. Differences in content in sections of Largo...... 57 Figure 34. Intervallic content of Ives' Chorale...... 58 Figure 35. Frequencies of ICs 2.5, 3.5, and 5 in sections of Ives' Chorale...... 60 Figure 36. Primary and secondary chords in the coda of Ives' Chorale...... 61 Figure 37. Intervallic content of Prelude 1...... 62 Figure 38. which skews data heavily toward PCI 5.5...... 63 Figure 39. Accented and Non-Accented Intervallic content of Wyschnegradsky's Prelude 4. ... 64 Figure 40. Harmonic intervals in the first imitative section of Prelude 4...... 65

vi

List of Tables

Table 1. Quarter-Tone Intervals...... 3 Table 2. Sentence-like structures of developing variation in Ives' Largo...... 29 Table 3. Interval Class Table for sc(0, 3, 5.5, 8.5)...... 36 Table 4. Expected prominent harmonic intervals in sections of Ives' Chorale...... 59 1

CHAPTER I

INTRODUCTION

A vast majority of modern Western is understood to be written in a

tuning system called twelve-tone , or 12-ET. In this system, the is

divided into 12 equal parts, referred to in tuning theory as “12 Equal Divisions of the Octave,” or

12-EDO. In the twentieth century, many composers found a renewed interest in exploring other

EDO systems that introduce musical intervals otherwise foreign to the standard 12-EDO. Of

interest to some of these composers was 24-EDO, a system that divides the octave into 24 equal parts rather than 12, or 24 “quarter-tones” rather than 12 . Significant quarter-tone works of the twentieth century include Charles Ives’ Three Quarter-Tone Pieces and Ivan

Wyschnegradsky’s 24 Preludes in Quarter-Tone System, both scored for a pair of tuned a quarter-tone apart. While some music theorists have analyzed these microtonal works from a more traditional standpoint, the introduction of corpus study methodologies offers a means for us to supplement, reinforce, and perhaps reconsider the statements we make about this music.

The Quarter-Tone System

When comparing two EDO systems (such as 12-EDO and 24-EDO), it is important to recognize the arithmetic properties relating the two. For the purposes of interval description, a reasonable metric is the use of cents, or hundredths of a in 12-EDO. When one number of divisions is a of the other, the pitches and intervals of the larger system are a superset of the smaller. In other words, because 24 is divisible by 12, the 24-EDO system retains all of the possible intervals from 12-EDO. For this reason, music which incorporates quarter-tones cannot 2

be disjunct from the intervals of the familiar 12-EDO system. More specifically, any set of three

or more discrete pitches in 24-EDO will always have at least one interval from 12-

EDO. Therefore, the use of 24-EDO should be thought of as an extension of, rather than a

substitute for, our familiar system of Western tuning.

Notation and Intervals

Given that the 12-EDO system is most often limited to five accidentals (natural, flat,

sharp, double-flat, and double-sharp), quarter-tone systems typically add four additional

symbols. A visualization of these accidentals through chromatic quarter-tone motion from C up

to and back is provided in Figure 1 below. From left to right, the new accidentals are the

quarter sharp, three-quarters sharp, quarter flat, and three-quarters flat. Each of these four new pitch classes exist 50 cents (50c) from the 12-EDO pitches on either side.

Figure 1. Spectrum of 24-EDO Accidentals. The quarter-tone system also introduces twelve new non-compound intervals to those in

12-EDO, with each occurring a quarter-step in between pairs of adjacent 12-EDO intervals. These intervals retain their generic size (, second, etc.), but differ from 12-EDO intervals in terms of quality. As there have not been centuries of well-established precedent for naming the qualities of these intervals, the terminology used to name them may differ between various theorists and composers. In order to use one consistent naming system throughout this paper, the non-compound intervals that quarter-tones add to our pitch space are provided in the table below, using the interval qualities described by Ivan Wyschnegradsky in his Manual of

Quarter-Tone Harmony (Wyschnegradsky 1932, 7-8). Alternate names for these intervals from 3 the Xenharmonic Wiki are also provided in order to better illustrate the need for choosing a single naming convention for the purpose of consistency. The complete list of interval names used in this paper can be seen below in Table 1.

Table 1. Quarter-Tone Intervals. Interval Size Interval Name(s) Alternate Interval Names (Cents) (Wyschnegradsky) (Xenharmonic Wiki, 2019) 50 Augmented Unison Up-unison, wide unison Subminor Second (quarter-tone) Downminor second, infra second 150 Neutral Second (three-quarter- Mid second tone) 250 Supermajor Second Upmajor second, ultra second Subminor Downminor third, infra third 350 Mid third 450 Supermajor Third Upmajor third, ultra third Minor Fourth Down-fourth, narrow fourth 550 Major Fourth (Semiaugmented Up-fourth, mid-fourth, wide fourth Fourth) 650 Minor (Semidiminished Down-fifth, mid-fifth, narrow fifth Fifth) 750 Major Fifth Up-fifth, wide fifth, Subminor Sixth Down-sixth, infra sixth 850 Neutral Sixth Mid sixth 950 Supermajor Sixth Up-sixth, ultra sixth Subminor Down-seventh, infra seventh 1050 Neutral Seventh Mid seventh 1150 Supermajor Seventh Ultra seventh, up-seventh, Down-octave, narrow octave

4

Before examining the selected quarter-tone works of Ives and Wyschnegradsky, it is important to examine these composers’ own writings on the subject of quarter-tones. This is especially important when it is considered that the two composers worked independently of each other, with Ives’ work being more experimental (limited only to quarter-tones) while

Wyschnegradsky would later explore other systems such as 72-EDO, as seen in his Arc-en-ciel

Op. 37 (1956), scored for six pianos. For Ives, the essay Some Quarter-Tone Impressions

(written alongside his Three Quarter-Tone Pieces) will be discussed.

Ives’ Some Quarter-Tone Impressions

Charles Ives’ father, George Ives, is acknowledged by the composer as the person who had first introduced him to quarter-tones. George had heard the of a church bell ringing during a thunderstorm, and upon arriving home attempted to recreate the harmony he had perceived using a piano - “I’ve heard a chord I’ve never heard before - it comes over and over but I can’t seem to catch it” (Boatwright 1965, 23). After this experience, George designed a

“quarter-tone machine” - a device composed of many strings that could be tuned to quarter-tone pitches.

In Ives’ writings about quarter-tones, his experiments largely focused on the search for what might be called “fundamental” for 24-EDO - chords which might be considered as stable as major or minor triads, or dominant seventh chords. Having first experimented with a major triad with the third tuned down a quarter-tone, Ives reasoned that such a sonority lacked a sense of balance. By using chords composed of juxtaposed perfect fourths and fifths, however, he arrived at chords with a greater “balance.”

5

In Some Quarter-Tone Impressions, he writes:

Chords of four or more notes, as I hear it, seem to be a more natural basis than triads. A triad [using quarter-tones], it seems to me, leans toward the sound or that the diatonic ear expects after hearing the notes which must form some diatonic interval [the fifth, C-G]. Thus the third note [a tone halfway between E and D sharp] enters as a kind of weak compromise to the sound expected - in other words, a chord out of tune. While if another note is added which will make a quarter-tone interval with either of the two notes [C-G] which make the diatonic interval, we have a balanced chord which, if listened to without prejudice, leans neither way, and which seems to establish an identity of its own (Ives 1925, 111-112).

Charles Ives’ “primary” chord, also referred to as “fundamental major,” consists of two

perfect fifths offset by a neutral third ( interval 3.5). This is the chord Ives proposed

as a possible analogue for “tonic” function from the Western tradition. While Ives spells this

chord in a way that the third is instead written as a larger (C - D ¾-sharp), the

two chord spellings would have no influence on the tuning or perception of this chord on the

quarter-tone piano. This chord notably contains two instances of IC 5 and three instances of IC

3.5 (a detail revisited in the later computational analyses). See Figure 2 below for both Ives’

spelling of the primary chord and an enharmonically equivalent rendering.

Figure 2. Ives' Primary, or "Fundamental Major" Chord. Ives’ is compared by the composer to the dominant from

the Western tradition: it is a sonority less stable and more dissonant than the primary. Referring to this secondary chord also as “fundamental minor,” Ives uses a similar quartal/quintal approach to construction but has five pitches as opposed to the primary chord’s four. The resulting chord is more tightly packed, with the prevalence of the neutral third replaced with the subminor third

(PCI 2.5). In terms of its construction, Ives adds to a quartal (C, , -Flat) a dyad (D ¼-sharp, G ¼-sharp). At the ends of some phrases, especially in the Chorale 6

from his Three Quarter-Tone Pieces, the secondary chord moves to a primary chord much as a

V7 typically moves to I. See Figure 3 below for an example of Ives’ secondary chord built on C,

along with a visualization presenting the chord as two quartal collections.

Figure 3. Ives' Secondary, or "Fundamental Minor" Chord.

Wyschnegradsky’s Manual of Quarter-Tone Harmony

While Ives uses many harmonies that are not based on perfect fourths and fifths in his

Three Quarter-Tone Pieces, his primary and secondary chord would be considered central to his

compositional approach to 24-EDO if only that essay were considered. Wyschnegradsky’s

Manual of Quarter-Tone Harmony, on the other hand, takes a more comprehensive approach to

integrating quarter-tones into our existing musical structures, exploring 24-EDO in a manner less strongly tied to the or fourth. The first part of the Manual deals with quarter-tones

in the already-existing tonal framework, while the second deals with the “free” use of quarter

tones (i.e. in post-tonal music). After providing an explanation of the notations, pitches, and

intervals of 24-EDO, he first provides examples of how “non-harmonic tones” such as passing

tones, neighbor tones, and appoggiaturas, can incorporate quarter-tone

alterations. Wyschnegradsky additionally includes his own recommendations on which quarter-

tone intervals are most ideal for certain cases. For example, he remarks that a quarter-tone

ornament below a pitch is more versatile compositionally than above and is more appropriate in

many cases than a three- (Wyschnegradsky 1932, 12). See Figure 4 below as an

example of quarter-tone appoggiaturas from his Manual. 7

Figure 4. Quarter-tone Appoggiaturas (Wyschnegradsky 1932, 12). Following this “melodic” treatment of quarter-tone pitches, Wyschnegradsky explores the ways in which triads can be altered to incorporate these new intervals. Starting from the triads, he introduces quarter-tone alterations of the fifth and , expanding from two triads to 11 (13, but with two that are transpositions of the others). See Figure 5 below for his generation of these triads. While the quarter-tone flat is often notated as either a horizontally-

mirrored flat sign or a flat with a downward pointing arrow, Wyschnegradsky uses a slightly

different notation (as can be seen in chord 1 of Figure 5). From quarter-tone altered triads,

Wyschnegradsky moves toward similar expansions upon increasingly extended and chromatic harmonies, suggesting quarter-tone alterations for dominant seventh and chords, chords, etc.

Figure 5. Wyschnegradsky's 13 Quarter-tone Altered Triads (Wyschnegradsky, 1932, 17). To conclude the first section, Wyschnegradsky proposes a framework for facilitating

modulations to the 12 new keys introduced by 24-EDO (i.e. C half-sharp Major). This consideration is important because 24-EDO is often conceptualized as two chromatic scales (12-

EDO) a quarter-tone apart. As a result, one could theoretically modulate between two quarter- tone keys using the same established methods from the Western tradition. It is rather the 8

modulation between keys offset by a quarter-tone interval that demands these considerations.

Noting the lack of a tonal relationship between keys on opposing halves of this “double ” spectrum, Wyschnegradsky argues that modulations of diatonic chords must be an abrupt shift between the “semitone” and “quartertone” collections’ chords. In other words, there are no common tones between any two scales if they exist in completely disjunct pitch collections. For this reason, direct modulation, rather than by pivot chord, becomes necessary.

Wyschnegradsky offers a compromise to allow for a functional “pivot chord,” however, noting that a chord with quarter-tone alterations might be used to provide one or more tones common to the new key. See Figure 6 for an example progression.

Figure 6. Quarter-Tone Modulations (Wyschnegradsky 1932, 20). The second section of Manual of Quarter-Tone Harmony considers the use of quarter- tones outside the framework of traditional Western tonal music. This is further divided into two chapters, covering the derivation of new scales and the freely atonal use of quarter-tones respectively. While Wyschnegradsky briefly entertains the idea of non-octave-repeating scales, going so far as to provide scales based on the subminor ninth, neutral sixth, and double octave, the majority of his discussion is concerned with scales of the octave. He defines three categories of scale - “regular,” “semi-regular,” and “irregular.” “Regular” scales are those that divide the octave into some number of equal parts (e.g. Whole-Tone). “Semi-regular” scales employ a hierarchy of divisions in which the octave is first divided into equal parts that are further divided 9

unequally but in a recurring pattern (e.g. Octatonic, Hexatonic). The final category is

“irregular,” containing any scale which is not in the first two groups.

Wyschnegradsky organizes the irregular scales as an extension of a tetrachord-based system. As 24-EDO offers many more pitches within the span of the octave, he provides such transformations as increasing the number of divisions (to create a pentachord, , heptachord, etc.) or altering a tetrachord such that its total intervallic size becomes that of a minor, major, or augmented fourth (as opposed to perfect). Of particular interest among

Wyschnegradsky’s scales is a 13-note scale that he refers to as the “quasi-diatonic.” Constructed from two heptachords spanning the interval of a major fourth and separated by a minor second, he draws significant parallels between this 13-note scale and the diatonic major:

1. Both scales are constructed from two identical, disjunct .

2. When a new major scale is constructed using the upper tetrachord of some initial major

scale as a starting point, the result is a modulation by one “step” in the . In

doing the same with the “quasi-diatonic,” the result is a scale that is one step away on the

“circle of major fourths.”

3. Both scales represent a descending sequence of identical intervals transposed into the

span of an octave. While the major scale can be generated using 7 perfect fourths, the

“quasi-diatonic” can be expressed as 13 minor fourths.

Wyschnegradsky’s “quasi-diatonic” scale is of particular interest for analyses of his 24 Preludes in Quarter-Tone System in Diatonicized , as for this work he utilizes this scale, referring to it as “diatonicized chromaticism.” In other words, a scale similar in step size to the

“chromatic” scale that is generated as a subset of the complete quarter-tone pitch space 10

(“diatonicization”). This scale will be discussed further in the introduction to the analyses of

selected preludes from this set.

Approaches to Analysis of Quarter-Tone Music

Music written in the quarter-tone system introduces pitches and intervals foreign to the

Western tradition. When considered not as a replacement for but rather an extension of the

Western system (12-EDO), it becomes apparent that many existing constructs in

can be readily applied in analyses of this music. Idioms such as “motive” or “pitch center,” for

example, can be extended seamlessly into discussion of music in 24-EDO. The first two

measures of Wyschnegradsky’s “Prelude I” are shown in Figure 7 below. The motive in the first

two beats is repeated exactly, then transposed twice in two more exact repetitions.

Figure 7. Motivic Repetition in Wyschnegradsky's Prelude No. 1. It is also evident from Wyschnegradsky’s organization of his 24 Preludes that there is some pitch

center, or “position,” on which each is based. Figure 8 (shown below) occurs at the top of the

first page of Prelude I, with the text “Position Do.” The accompanying scale in this figure is the

diatonicized-chromatic scale centered on C. While one might be inclined to refer to C here as

the “tonic,” it may be more appropriate to call it a “pitch center” to avoid tonal . 11

Figure 8. Wyschnegradsky's "Diatonicized-Chromatic" Scale on C. While these concepts from Western music (both tonal and post-tonal) do not change significantly

when applied to music in alternate tunings (e.g. 24-EDO, 31-EDO, 72-EDO), other analytical methodologies are best reframed in terms of the tuning in question. How “consonance” and

“dissonance” might be defined in 24-EDO, for example, is a question that brings us back to the harmonic series.

The Place of Quarter-Tones in the Harmonic Series

The history of Western tuning is largely inseparable from the harmonic series. The gradual inclusion of prime-numbered partials above the third () in Zarlino’s

Senario, for example, established the fifth partial (necessary for the 5:4 and 6:5 ). While systems such as 12-EDO provide very close approximations for certain partials of the harmonic series, other partials can be more significantly out of tune. EDO tuning systems’ intervals are typically discussed in terms of cent values, with 1200 cents in a single octave. In terms of cents, 12-EDO provides close approximations for the third (-2c) and fifth partials (+14c). If a higher partial is of a composite (not prime) number, its deviation in cents from any EDO can be expressed as the sum of the deviations of the partials that are its products.

In other words, D (the third partial of G) is the ninth partial above C, or “the third partial of the third partial,” and has a deviation of approximately -4c. In Figure 9, it can be seen that the seventh, , and partials deviate notably from the harmonic series in 12-EDO. 12

Figure 9. Deviation of Interval Approximations in 12-EDO from the Harmonic Series. It is notable in 12-EDO that the third and fifth partials are key components in all

“consonant” intervals. One example of a consonance that is not possible in 12-EDO, the

“concordant seventh,” results from the addition of a pure seventh harmonic to a justly tuned

major triad. The concordant seventh, while impossible on a 12-EDO piano due to the significant deviation from the harmonic series of the seventh partial (-31c), occurs in a cappella vocal styles

such as the barbershop tradition, where becomes more feasible. While it is not

necessarily notated in the music performed by barbershop groups, the use of the concordant

seventh in a 4:5:6:7 is ubiquitous in the style (Gann 2019, 116).

From the above diagram of the harmonic series’ deviation in just intonation from 12-

EDO, it becomes apparent that the next three prime partials (7, 11, 13) are not as closely approximated. In 24-EDO, however, each of the octave is an interval of 50 cents, rather than 100. By utilizing certain intervals from 24-EDO, it becomes possible to approximate sonorities built on the higher partials of the harmonic series. In Figure 10, it can be observed that 13

the pure eleventh and thirteenth become less distant from their closest approximations

in 24-EDO. This is especially the case for the eleventh, which is approximately 49 cents flatter than its 12-EDO approximation but within one cent of its counterpart in 24-EDO.

Figure 10. Deviation of Interval Approximations in 24-EDO from the Harmonic Series. As a result of the more accurate approximations in 24-EDO for the eleventh and

thirteenth harmonics when compared to 12-EDO, more approximations of pure intervals become

available. In terms of intervals in which one pitch class is the fundamental, four additional

sonorities derived from the harmonic series become available, referred to here by the names

Wyschnegradsky used. The eleventh partial adds the major fourth (11:8) and its inversion, the minor fifth (16:11). The thirteenth adds the neutral third (16:13) and its inversion, the neutral sixth (13:8). Additionally, the seventh harmonic is approximated with slightly more accuracy, but deviating in a different direction from its 12-EDO approximation (19 cents flat rather than 31

cents sharp). Wyschnegradsky remarks in his Manual of Quarter-Tone Harmony that sixth-tones

(intervals of approximately 33 cents occurring in 36- and 72-EDO) offer approximations of the

seventh and thirteenth partials that are even closer to pure intonation (Wyschnegradsky 1932, 14

33). It is of interest that the intervals produced by the eleventh and thirteenth harmonics are new

species of thirds, fourths, fifths, and sixths, notably a superset of the generic intervals typically

considered “consonant” in Western music. While these intervals may not sound consonant to a listener unfamiliar with them, their structural significance in 24-EDO is somewhat of an

extension of that of the intervals generated from the third and fifth harmonics observed today in

12-EDO.

Pitch-Class Set Theory in 24-EDO

Unlike the Western tonal tradition (12-EDO today), many of the intervals and harmonies possible in 24-EDO do not have names ubiquitously associated with them. In other words, the term “perfect fifth” is readily understood by trained musicians. Such an established theoretical tradition does not yet exist for microtonal systems such as 24-EDO, although to discuss this

music it is important to use consistent nomenclature. Regarding interval names, the solution is

simple enough: choose one system (e.g. Wyschnegradsky’s Manual) to use consistently. This is

feasible because there are a reasonably finite number of intervals, even with the inclusion of

quarter-tones. The classification of harmonies, or more broadly, tone combinations, however, quickly becomes a Herculean task if one assumes that each tone combination should have a name as clearly associated with it such as “major triad.” Even in 12-EDO, this is not the case.

Allen Forte’s pitch-class set theory, while designed for use with post-tonal music, allows us to

circumvent this issue. In exchange for the abstraction of information associated with voicing and

inversion of a pitch collection (i.e. “ major triad with open voicing”), this

framework allows for the concise summarization of a pitch collection’s intervallic content, such

as sc(037). 15

The problem with using Forte’s set theory-based approach as it is for music in 24-EDO, however, is that it is a system designed “from the ground up” with 12-EDO in mind. Rather than

only designing a 24-EDO analogue of the pitch-class set theory model, I chose to first derive a generic, or “flexible,” system that could be adapted for any n-EDO system. I first developed a model of the standard set theory system in the Python 3 programming language. Rather than using constant numbers in this process (e.g. an IC Vector is comprised of 6 numbers), however, I expressed the parameters of the system as some function of 12 (the system’s cardinality). This approach culminated in a model I refer to as the “flexible system,” as it can be used to compute set classes in any EDO. More details on the software implementation of this model can be found in Chapter III.

Generally, set class theory in 24-EDO differs from the 12-EDO model in a few significant ways:

1. There are 24 pitch classes, rather than 12.

a. As a result, a set can consist of any number of pitch classes between 0 and

23. While one could number them 0 through 23, I use a system to keep

the notation more similar to what is familiar.

i. To add clarity to the naming of collections, I use commas and spaces as

delimiters between pitch classes. (i.e. sc(0, 3.5, 7)). This is similar to the

notation of a microtonal collection used by Dmitri Tymozcko (Tymozcko

2011, 86).

2. There are 11 interval classes, rather than 6. As with the pitch classes, these include

decimal numbers such as IC 1.5. 16

. Due to the number of interval classes in 24-EDO, it might be helpful to express interval

class vectors as a table or list, or even in prose for greater clarity. For example, “sc(0, 0.5, 1) has

2 instances of IC 0.5 and 1 instance of IC 1” vs. <2,1,0,0,0,0,0,0,0,0,0>

3. There are 24 possible transpositions and inversions of any pitch collection, rather than 12.

. There are also twice as many potential by inversion and transposition.

Using this set theory model in 24-EDO, it becomes possible to discuss pitch collections in more meaningful ways. The following sample questions concerning the relationship between two sets become answerable for any pair of pitch collections s1 and s2:

1. If s1 and s2 are of the same cardinality, are they Z-related?

2. Is s1 a subset or superset of s2?

3. What subsets or pitch classes do s1 and s2 have in common?

4. Are s1 and s2 complementary?

While the “flexible system” model also allows for systems that include only specified pitch classes of an EDO, this is a feature primarily of use if working in a system where the octave is not divided by homogeneous intervals. For example, set class space in just intonation might be modeled as a reasonably small subset of 1200-EDO, such that a justly tuned major triad is of sc(0, 3.86, 7.02). In such a case, one would only consider the existence of a specific pitch class when it occurs in music or is necessary for a part of the model to be constructed. In a 5-limit just intonation model (allowing only partials that are multiples of 1, 2, 3, and 5), for example, intervals such as 6:7 would not be necessary for inclusion.

Computational Analysis

In addition to more traditional approaches to analysis, this paper will apply computer-

aided analysis techniques to the chosen repertoire by Ives and Wyschnegradsky. These two 17

works will be used as exampled for model studies in analysis and visualization. While toolkits

such as Humdrum do not have built-in support for analysis of , I have

developed some basic utilities to perform the necessary tasks. Two primary tools are designed

specifically for use with microtonal materials. The first is the set class calculator implementing

the “flexible system” I have previously described, allowing the user to calculate set classes,

apply transpositions/inversions, and generate a list of subsets from a starting pitch

collection. The second utility is my own clone of Huron’s “hint” from the Humdrum toolkit,

named “mhint” as an abbreviation of “microtonal-hint.” The mhint script is designed to replicate

the functionality of hint, but is extended to work with data in the **pitch format in order to

utilize its capability to represent microtonal deviations of +-99c, as well as user-defined tuning

systems (for which I have implemented 24-EDO). More technical specifications on both my set

class calculator and mhint can be found in Chapter III, and the model studies in Chapter IV.

Further Review of Related Literature

While Ives and Wyschnegradsky were composers of the Twentieth century, musical

structures considered “microtonal” today can be seen in treatises from the Ancient Greeks. The tetrachord was a foundational interval of Ancient Greek music theory, consisting of a perfect fourth (4:3 harmonic ) and two additional pitches further dividing this interval. The notes dividing the tetrachord were movable, however, and various positionings were classified as different “genera” by (Mathiesen, 2001). The three basic categories of genera were the diatonic, chromatic, and , the latter of these consisting of a major third on top and two “dieses” (an interval comparable in size to a semitone) at the bottom. These ideas on tuning theory, while less prevalent in the earlier European treatises on music, would become a subject of interest for music theorists from the Sixteenth Century to the present day. 18

Nicola Vicentino (1511-c1576), an Italian theorist and composer, proposed analogues of the eight diatonic modes constructed from chromatic and enharmonic genera (Kaufmann &

Kendrick, 2001). In ascending order, the intervals of the diatonic genus (the tetrachord from which the modes are constructed) are two major seconds and a semitone. The chromatic genus consists of two semitones and a minor third, and the enharmonic two dieses (quarter-tones) and a major third. Vicentino designed the , a with two keyboards, to play music which uses these additional genera. Vicentino’s interest in using the as a melodic interval differs from other theorists of the Renaissance and Baroque periods, who instead viewed these “enharmonic keyboards” as a way to mitigate the problem of enharmonic equivalent pitches in quarter-comma meantone tuning (Meeùs, 2001). In quarter-comma meantone, the

“black keys” of the piano could only be tuned to one pitch (e.g. a key tuned to B-flat would not be appropriate to use for A#). A simple enharmonic keyboard would then have 17 keys, dividing each of the black keys of a 12-note keyboard into two. While this does design introduces the quarter-tone to keyboards, it occurs as a difference between selected pairs of notes which would be considered “enharmonically equivalent” in 12-EDO. It is not until the equal-tempered idiom of 12-EDO is extended to 24 that the more familiar 50c quarter-tone of today emerges.

The compositional idiom of scoring quarter-tone music for two pianos emerged largely in the twentieth century. While many composers of this time period sought sound worlds outside of the Western tonal system, Charles Ives and Ivan Wyschnegradsky were two composers who found these new sounds in an extension of Western equal temperament: 24-EDO. In two of the earlier articles on quarter-tone music, we can see how this subject was viewed in the early twentieth century. Artur Holde’s “Is There a Future for Quarter-Tone Music?” discusses the practical limitations of composition with microtones. Most notable among these is the issue of 19

human performers accurately replicating these pitches - a roadblock which can be sidestepped by

the use of mechanical instruments such as the “quarter-tone piano duet” approach taken by Ives and Wyschnegradsky in their scores (Holde, 1938). Albert Welleck and Theodore Baker’s

“Quarter-Tones and Progress” also discusses the challenges of creating music in systems of quarter- and sixth tones. Aside from limitations of the human ear to automatically adjust microtones to more “correct” sounding pitches, it is noted that quarter-tones become impractical in the upper register of the violin due to how close the performer’s fingers would need to be in order to play these intervals (Welleck & Baker, 1926). During this time period, composers such as Alois Hába and George Rimsky-Korsakov (grandson of Nikolai Rimsky-Korsakov) were also

using quarter-tones, and were a part of what is referred to as George Korsakov’s “Circle of

quarter-tone music” (Ader 2009, 32-33).

As Ives and Wyschnegradsky had explored quarter-tone composition independently of

each other, it is important first to investigate how they framed this musical space, as well as

some writings by more recent scholars. Charles Ives’ approach, while incorporating the use of

quarter-tones, does not distance itself as much from 12-EDO as Wyschnegradsky in his partial

reliance on the perfect fifth as an important structural interval. In the preface to Three Quarter-

Tone Pieces, Ives describes the third movement (Chorale), which uses his two harmonies extensively: “It attempts to work along the pure quarter-tone harmonic lines outlined in the second section of this paper, and is based principally on a primary and secondary chord” (Ives,

1925). Ivan Wyschnegradsky, on the other hand, approached quarter-tone composition as an extension of the Western Classical tradition. In his Manual of Quarter-Tone Harmony,

Wyschnegradsky first considers the use of quarter-tone inflections for non-harmonic tones and ornamentations, and gradually expands toward quarter-tone altered triads and free composition 20

with quarter-tones (Wyschnegradsky, 1932). In the preface to his 24 Preludes in the Quarter-

Tone System, Wyschnegradsky describes the construction of an asymmetrical 13-tone scale,

which he considers similar in function to the similarly asymmetrical diatonic scales in 12-

EDO. This scale is constructed from two heptachords (7-note segments) offset by a whole-tone,

in the same way Western diatonic scales are constructed from two tetrachords. Wyschnegradsky

describes this approach with the term “13-tone diatonicized chromaticism” (Wyschnegradsky,

1979).

While Ives’ Three Quarter-Tone Pieces was among the later compositions of his life, some have postulated that his experiences with quarter-tones originate from when he was much younger. In his article, Ives’ Quarter-Tone Impressions, Howard Boatwright remarks that Ives was originally introduced to quarter-tones by his father, George Ives. There are also references to quarter-tones in the Epilogue from Ives’ Essays Before a Sonata. Boatwright’s article also gives some insight to an interdependence between and harmony in Ives’ approach to composition in quarter-tone space: “It seemed to him that ‘a pure quarter-tone melody needs a pure quarter-tone harmony not only to back it up but to help generate it’” (Boatwright, 1965). In terms of other analyses of these quarter-tone works, perhaps the most in-depth analysis today is

Myles Skinner’s dissertation, in which he analyzes quarter-tone works of four composers

(including both Ives’ Three Quarter-Tone Pieces and Wyschnegradsky’s 24 Preludes). Skinner draws an interesting parallel between Wyschnegradsky’s preludes and J.S. Bach’s Well-

Tempered Clavier, for example - each of the 24 preludes is built on a scale centered on one of the

24 quarter-tone pitches (Skinner 2007, 142-143). Skinner also analyzes music by Alois Hába and Easley Blackwood in this dissertation, also exploring the translation of Neo-Riemannian ideas into quarter-tone systems. 21

Concerning computational methods of music analysis, the model studies in this paper

serve to demonstrate the extensibility of computational techniques to microtonal works. By

demonstrating the use of these analytical tools on selected excerpts, future corpus studies of

microtonal repertoire become feasible. A corpus study is an analysis of a large body of musical

works – a process typically assisted by computational tools today. However, the philosophical

idea of corpus research has been traced as far back as 1927 by David Temperley and Leigh

VanHandel: “Perhaps the earliest true example is the work of Jeppesen (1927), who compiled

counts of various contrapuntal features in the music of Palestrina” (Temperley & VanHandel

2013, 1). The Humdrum toolkit was developed by David Huron in the 1980s and has been used

by theorists and musicologists to conduct studies on large collections of music (Huron,

2002). Parameters such as pitch and intervallic content are highly applicable in computational

music analysis.

Key-finding algorithms, for example, have been developed in the Humdrum toolkit as a machine learning-based approach: provide a set of pieces with known keys to “train” an algorithm, then apply the algorithm to a “test set” of pieces (also with known keys) to evaluate the accuracy of the algorithm. Many of these models have used pitch class data, which is compared to the expected distribution of each of the twelve pitch classes relative to a given key

(Albrecht & Shanahan, 2013). In other words, a piece composed by Bach in the key of C Major might be expected to have more occurrences of C and G than of Db and F#. A recent study extends the concept of tonic identification to other scales, such as the harmonic minor and harmonic major scales. Interestingly, the study also includes some predictions of “tonic” for microtonal scales generated without reference to any cultural or musical tradition (Milne et. al.,

2015). The authors of this study remark, however, that while they had generated predicted tonics 22

for these scales, the data does not yet exist for this data to be evaluated critically. Algorithms for

harmonic analysis of Western tonal music have also been explored by David Temperley, in the context of designing a computer program to replicate the harmonic analyses of musicians and listeners (Temperley 1997, 31-34). While this research is rooted in 12-EDO music, it is possible that such an algorithm could be extended for use with music utilizing quarter-tones as more of an expansion of our tonal system than a point of departure.

Rhythm and meter offer another dimension to computational analyses and have been the focus of a study by Huron, examining onset synchrony in Bach’s two-part inventions (Huron,

1993). This study speaks to the power of the corpus study approach, as it answers questions about voice-leading and the perceptibility of independent polyphonic lines centuries after these compositional idioms had been codified. Both these rhythmic and pitch parameters offer a means to compare and contrast not only differing works, but also smaller sections within one work. For example, one of the demonstrations of computational analysis of the microtonal pieces in this paper uses interval class content to visualize the differences between sections. It would also be possible to use selected parameters to measure on a continuous spectrum how starkly a piece changes from one measure to the next, which may serve as a suitable heuristic for computationally detecting shifts in musical form. A similar methodology used by Matthias

Mauch, Robert MacCallum, Mark Levy, and Armand Leroi for the analysis of changes in styles over time may be feasible for this purpose. This study extracted features from popular songs from 1960 to 2010, measuring their similarities to each other in a search for punctuating “breaks” in the otherwise continuous data. These breaks were found to correlate

with specific “revolutionary” periods in the development of popular music. As an example of

how this approach might be used to analyze an individual work, examining the pitch class 23 content of a Classical sonata exposition from each measure to the next would likely result in such a “break” in the data when chromatic pitches are introduced for modulation, possibly indicating a new subsection of the exposition.

24

CHAPTER II

ANALYSES OF SELECTED EXCERPTS

Ives’ Three Quarter-Tone Pieces

In Ives’ Three Quarter-Tone Pieces, the balance between the two halves of quarter-tone

space described in Some Quarter-Tone Impressions (the 12-tone aggregate, along with a

transposition of itself up a quarter-tone) is musically realized. It is important to note that Ives’

harmonic language in these pieces is not exclusively that of his own primary and secondary

quarter-tone harmonies - at times, he uses in a “tonal” manner a diatonic collection in which a

part of the scale is transposed up by a quarter-tone. At other times, he uses what might be described as “,” in the sense that two familiar triadic structures are superimposed at an interval that includes a quarter-tone offset.

The editor of Three Quarter-Tone Pieces remarks that while Ives suggests one piano should be tuned a quarter-tone sharp, the typical practice of most piano tuners is to instead tune one a quarter-tone flat. While all three pieces are scored for two pianos, Ives had originally conceived of the first and third (Largo and Chorale) as compositions for a single “quarter-tone” piano, likely consisting of two keyboards (perhaps in the manner of an organ console) played by a single performer. The thicker texture of the second movement (Allegro), in contrast to the scoring of the other movements, is indicative of a work intended for two performers rather than one.

Three Quarter-Tone Pieces - I. Largo

The Largo from Ives’ Three Quarter-Tone Pieces might be described as an exploration of different sound worlds available to an individual pianist with a quarter-tone piano. A significant limitation in composing for this medium is that because the intended “quarter-tone” piano would 25

have two stacked keyboards, that each hand is effectively restricted to playing on one “side” of

the available pitch spectrum. For the majority of the Largo, only two of the four available staves

include notated pitch at a time (further reflecting the idea of one instrument with a single

performer). In terms of structure, the piece can be interpreted as a ternary form in which each of

the three sections consists of two contrasting sonic ideas. The similarities between the sections

of this ternary labeled as A (mm. 1-17) and A’ (mm. 43-65) occur in both their thematic

materials and fundamental harmonic language. Despite any preference Ives might have harbored

for his own “primary” and “secondary” quarter-tone harmonies, the majority of this movement

makes use of other sonorities available in 24-EDO.

The first section of Largo (mm. 1-17) is based not on the quarter-tone harmonies derived by Ives, rather emphasizing and expanding upon a dichotomy between the pitch collections available in the two 12-tone aggregates comprising this quarter-tone space. The primary sonority

established in mm. 1-6 is that of superimposed augmented triads. In mm. 1-4, a primary

harmony consisting of the triad F+ (raised by a quarter-tone) over G+. In m. 5, both of these triads are shifted upwards under a transposition that could be labeled as T7 based on Ives’ chosen

spellings of the two triads. Ives uses measure 6 as a means of consolidating these triads into the

same 12-tone subset of 24-EDO, under T0.5. On the keyboard raised by a quarter-tone, the triads

D+ and E+ might be said to “converge” into Eb+ in m. 7. A melody first heard in mm. 4-5 is

expanded through a process of developing variation in mm. 7-9, 10-12, 12-15, and 15-17 (the last of which transitions into the B section of the overall form). Throughout these iterations of the original thematic material, quarter-tone passing and neighbor tones are gradually introduced, followed by transpositions of the thematic material by quarter-tone intervals. See Figure 11 below for a highlighted example of the primary thematic material (mm. 9-11). Note the dashed 26 lines between the two treble staves, used by Ives to highlight the first instance of quarter-tone passing motion.

Figure 11. Quarter-Tone Passing Motion and primary theme of Ives Largo (A). The relationship of mm. 17-18 is of particular interest, as the sustained chord in m. 17 is a G

Major triad, moving to an open fifth on C-G in m. 18. This might be comparable to a V-I relationship - a construct ubiquitous in the Western tonal tradition. Ives’ use of an open fifth is also interesting, as it creates ambiguity as to whether the “” will be major or minor.

After the initial downbeat of the B section, however, it quickly becomes apparent that the

“tonic harmony” prepared by the G major triad is an alien sonority to a listener who expects either a major or minor triad. Rather, Ives establishes C as a tonal center through its use as a while consistently returning to his “primary” tetrachord: C, E¼-flat, G, and B¼-flat.

This sonority occurs in measures 18, 20-22, and 24-25. The sonorities that separate each instance of Ives’ primary tetrachord (e.g. C, D ¼-sharp, G, A ¼-sharp) fall outside Ives’ two proposed fundamental chords for quarter-tone music. However, each of these sonorities from mm. 18-25 are composed of the same fundamental intervallic structures: on each “side” of the 27

quarter-tone keyboard, any sounding pitch forms either a quartal or quintal harmony with another. In cases where three pitches sound, a results from a spatially abstract juxtaposition of two of these perfect intervals: [C, G, D] and [C, G, F]. In this way, no sounding

pitch is left to stand alone, rather all pitches in this selection are reinforced by a quartal or quintal

harmony. In Figure 12 below, the harmonies in mm. 18-23 are labeled as either P (Ives’

“primary'' tetrachord) or Q (a contrasting quintal/quartal harmony).

Figure 12. Ives' Primary chord, punctuated by other quartal/quintal sonorities. In mm. 26-28, Ives uses a melodic sequence of ascending quarter tones to smoothly move

from D to a melody starting on E ¼-sharp (m. 29). The section from mm. 29-41 is best

described as an example of developing variation. Two short motivic cells form the initial

material of this section (m. 29): [E, F#, C#, D#] (raised by a quarter-tone) and [C, A, A#,

C]. Initially, the keyboard which is raised by a quarter-tone plays only one pitch at a time

against harmonies from the other “side” of 24-EDO. It is of interest that the first sonority of

each measure in mm. 29-33 contains harmonies Wyschnegradsky might classify as “augmented

triad with a quarter-tonally raised third”. In mm. 34-41, Ives uses a different small motive and

continues the process of developing variation. Here, the lowest-sounding three pitches at all 28

times (when considered across both keyboards) make up an , and at times two

augmented triads sound concurrently between the two sides of the quarter-tone pitch space. See

Figure 13 below for mm. 34-37 of Largo, where these underlying augmented triads are

highlighted.

Figure 13. Underlying augmented triads in mm. 34-37 of Largo. Ives’ emphasis on the augmented triad here serves to unify this section of the music with the first, ultimately leading to a “dual ” in m. 42 where a rapid ascent on one keyboard is answered by one on the other. Mm. 43-49 might be considered analogous to mm. 1-6, based on its emphasis on a juxtaposition of augmented triads on the two contrasting 12-EDO subsets of

24-EDO. Measure 50 to the end of the piece is analogous to mm. 7-19, although it exhibits several key differences. Rather than repeating the initial 3-note melodic motive at the same pitch level, the motive occurs three times at consistently lower levels of transposition. This basic idea is stated (melody starting on B ¼-flat) and repeated exactly at the transposition of one quarter- tone lower (starting on B-flat). The third statement of the idea, which provides further development, then occurs on A¼-sharp (transposition by yet another descending quarter-tone).

The Largo ends with a “V - I” gesture, with a G major triad moving to C major (mm. 63-65). A 29

B¼-sharp continues to sound through this final , preventing the movement from

“escaping” the quarter-tone pitch space in which it exists.

Overall, the Largo from Ives’ Three Quarter-Tone Pieces demonstrates a consistent

pattern in Ives’ use of developing variation: a basic idea (typically 1-2 measures in length) is stated, and is repeated either exactly, or with slight variation. A third statement (which

sometimes resembles the basic idea) then develops in a manner that moves the music further

from the original idea. This structure is comparable to the idiom of the “sentence,” with the

exception of the tonal connotations inherent in using this label. Table 2 below provides

prominent examples of this sentence-like structure throughout Ives’ Largo, categorized as a two-

part presentation followed by continuation material.

Table 2. Sentence-like structures of developing variation in Ives' Largo. Phrase Presentation Continuation Mm. 6-17 Mm. 6-9 + mm. 10-12 Mm. 13-17 Mm. 18-28 Mm. 18-21, mm. 22-25 Mm. 26-28 Mm. 29-33 M. 29 + m. 30 Mm. 31-33 Mm. 34-42 Mm. 34-35 + mm. 36 + Mm. 38-42 37 Mm. 38-42 (nested within mm. 34- M. 38 + m. 39 Mm. 40-42 42) Mm. 43-49 M. 43 + m. 44 Mm. 45-49 Mm. 50-60 Mm. 50-51 + mm. 52-53 Mm. 54-57 + mm. 58- 60 30

Three Quarter-Tone Pieces: III. Chorale

The Chorale from Ives’ Three Quarter-Tone Pieces is perhaps the simplest of the three

movements, from a technical and analytical perspective. Unlike the Largo, the majority of this

movement utilizes Ives’ primary and secondary quarter-tone sonorities. Ives’ more consistent use of these chords allows for what might be called “tonal implications” - when Ives moves between two different chords in this system, gestures similar in behavior to non-harmonic tones from Western tonal music (e.g. passing tones, suspensions) can be observed. In the context of this approach, non-harmonic would be defined as “existing outside of either of Ives’ two fundamental quarter-tone chords.” See Figure 14 below for the first four measures of the chorale. In this section, Ives first establishes the primary chord built on C (P-C), using two short intermediate progressions in what results as a “tonic prolongation” of sorts. 31

Figure 14. "Tonic Prolongation" in mm. 1-4 of Ives' Chorale. In the transitional material between the first and second sections (mm. 10-15), Ives uses progressions consisting almost exclusively of his primary and secondary chords, with the exception of some unaccented passing tones in the highest voice. In this passage, it can be observed that Ives treats the secondary chord as a harmony with greater tension, perhaps analogous to the dominant seventh chord of traditional tonality. To extend this idea of implied tonality further, Ives might have intended phrases that end on the primary and secondary chords to function in the same way as the idiomatic authentic and half cadence. See Figure 15 for this transition section, in which the primary and secondary chords are labeled P and S. 32

Figure 15. Primary and secondary chords in transition section. The fermata over a secondary chord in m.13 might be considered as a “half cadence,”

with the final primary chord of the excerpt as an “authentic cadence.” If this labeling convention

is considered, it is of interest that chord roots of the two are G and C respectively.

Additionally, the two progressions from secondary to primary chords include bass (not

necessarily root) motion similar to V  I. There are also notable contrasts to Western tonality in

this section, namely in the - parallelism, especially parallel fourths and fifths, are abundant in this writing. For this reason, the chords may be argued as discrete units of perception similar to the “clang” postulated by in Meta+Hodos, rather than as 4 or

5 independent voices (Tenney 1964, 33). Perception of this as a structural unit is also consistent with Dibben’s experiments that extended the hierarchical ideas of Lerdahl and Jackendoff into atonal works (Dibben, 1999).

Three of the four sections in Ives’ Chorale make significant use of the primary and secondary chords, while the second (mm. 16-27) makes use of a contrasting pitch collection that might be described as a “distorted” diatonic collection. This section is of interest in its use of a 33

quarter-tone sonority Ives had rejected as a fundamental structure for music in this system, consisting of two stacked neutral thirds (Ives 1925, 112). The scale Ives uses is comparable to a

G major scale with the C, D, E, and F# all raised by a quarter-tone. Based on the structure of this scale, no triad from 12-EDO is possible, rather seven possible triadic combinations similar to

Wyschnegradsky’s quarter-tonally altered triads (Wyschnegradsky 1932, 17). The diatonic

treatment of this collection is perhaps most visible in mm. 21-22, where Ives uses a sequence of

what could be called “diatonic thirds” melodically. See the excerpt from m. 22 in Figure 16 as

an example of this.

Figure 16. "Parallel diatonic thirds" within an altered G Major scale. In mm. 26-27, closing material for this “diatonic-like” section is heard, ending on a triad that can

be described as “G Major with the fifth raised a quarter-tone.” This chord built on G is used to

transition back into the sound world built on Ives’ primary and secondary chords. See Figure 17

below for this excerpt. 34

Figure 17. Triadic structures and a return to Ives' primary and secondary chords. The third section of Chorale (mm. 30-40) presents thematic material similar to the first

section, only using many chords that fall into neither of the pitch spaces of the first two sections.

Major and minor thirds from 12-EDO are more common in this section, while perfect fifths are

much less frequently used. Throughout this section, C can be perceived as an almost ubiquitous

pedal point throughout, providing grounding for the more dissonant harmonies Ives uses. From

measure 48 to the end, Ives reaffirms his primary and secondary chords, most notably with a long sequence of secondary chords moving in parallel motion. According to Ives, this section presents a theme that first moves in quarter-tones, then in semitones, and ends with motion in whole steps. A brief coda (mm. 58-59) returns to Ives’ primary chord, further indicating that the composer intended this to be the more “resolved” of the two in terms of sound.

Wyschnegradsky’s 24 Preludes

Of Wyschnegradsky’s 24 Preludes, it is of note that he wrote one for each of the 24 possible pitch centers in 24-EDO. Each is primarily composed using the 13-tone “diatonicized- chromatic” scale Wyschnegradsky presents in his preface for the work, although the pitch center 35

sometimes shifts for brief moments. This scale is also referred to by Wyschnegradsky as the

“quasi-diatonic” in Manual of Quarter-Tone Harmony (Wyschnegradsky 1932, 27-28). Myles

Skinner abbreviated this as the “DC” scale in his analysis of the 24 Preludes, and from this I would propose the use of scale names such as “C-DC,” or “E ¼-Sharp-DC” when discussing them. Skinner further proposes that Wyschnegradsky uses pitch collections derived from

“cycles” of scale steps. More specifically, Skinner examines the use of 3-step cycles of the DC scale. The visualization of these 3-step cycles is based on a figure from Skinner’s dissertation, re-created here using the C-DC scale as opposed to the original example which uses G-DC

(Skinner 2007, 163). The upper staff presents the C-DC scale with the first three cycles illustrated with dashed slurs. The lower staff moves through the complete cycle, such that C is the first and last pitch.

Figure 18. 3-step cycles in the C-DC scale. The use of such cycles to generate the “tonic tetrachord” which Skinner identified in

Wyschnegradsky’s preludes results in DC scale degrees 1, 4, 7, and 10. For example, the tonic tetrachord of the C-DC scale (shown below) consists of the pitches C, E-flat, F ¼-sharp, and A

¼-flat. Here, the solid slurs indicate minor thirds, and the dashed slurs indicate major fourths 36

Figure 19. 3-step cycle "tonic tetrachord" of Wyschnegradsky's C-DC scale. If the tetrachord derived above were to be expressed as a set using my Flexible System

model for 24-EDO, it would be classified as sc(0, 3, 5.5, 8.5). This set can be generated by

superimposition of two minor thirds or major fourths: {{0, 3}, {5.5, 8.5}} or {{0, 5.5}, {3, 8.5}}

respectively. It has one axis of inversional - in the prime form [0, 3, 5.5, 8.5], this set

is symmetrical under I8.5, with the pitch class pairs [0, 8.5] and [3, 5.5] mapping onto each

other. As interval class vectors become less intuitive in 24-EDO, the intervallic content of this set is outlined in Table 3 below.

Table 3. Interval Class Table for sc(0, 3, 5.5, 8.5). Interval Class Number of Occurrences Interval Names 2.5 1 Supermajor Second, Subminor Third 3 2 Minor Third 3.5 1 Neutral Third 5.5 2 Major Fourth

37

When considering the use of this collection as a “tonic” tetrachord, it exhibits several

interesting properties in relation to the harmonic series. This set (and Wyschnegradsky’s DC

scale) does not include the pitch a perfect fifth above the root - in fact, it does not contain any perfect fourths or fifths at all. This tetrachord differs significantly at first glance from those chords which comprise the building blocks of Western tonal structures: the triad (major or minor) and the seventh chord (e.g. dominant seventh). The set is very even, however, and could be compared to the more familiar 12-EDO collection sc(0369), or “fully chord”: sc(0, 3, 5.5, 8.5) results from selecting an [0, 3] dyad from sc(0369) and transposing it down a quarter-tone (T-0.5 or T11.5). Due to the symmetries of these collections, however, the

same set class (although a different pitch collection) results from transposing the selected dyad

up by a quarter-tone.

If one were to attempt to fit this tetrachord into one harmonic series using only the first

six partials as a measuring stick (e.g. Zarlino’s Senario), the resulting sonority could not be

deemed satisfactory. The only “consonant” interval against the tonic is a minor third, with the

major fourth and neutral sixth existing as either dissonances or consonances respectively which

are far enough “out of tune” that they become almost unrecognizable. Many microtonal systems

of pitch organization (such as 24-EDO), however, consider harmonics much higher than the sixth

as new potential “consonances.” Viewed through the lens of the higher partials of the harmonic

series (specifically the 11th and 13th), the intervals of this tetrachord come into a much clearer

focus: the intervals sounding against the lowest pitch of this tetrachord (0) are a minor third (3),

major fourth (5.5), and neutral sixth (8.5). The minor third is already established as a consonant

6:5 ratio, but the major fourth and neutral sixth find their place among the consonances of the

upper partials in the 11:8 and 13:8 respectively. 38

Prelude 1 in C

The first prelude in Wyschnegradsky’s set of 24 utilizes the C-DC scale (perhaps an homage to Bach’s Well-Tempered Clavier). In terms of form, this prelude can be analyzed as a two-part form with rounding. The terms “binary” and “ternary,” while describing two- and three-part forms, are avoided in these analyses due to the tonal connotations implied by their

use. The first section (mm. 1-14) consists of a theme (mm. 1-7) that is next repeated with a

slight alteration (mm. 8-14). The second section (mm. 15-26) begins with material that contrasts

from the first, ending the prelude with a return of the primary material (mm. 21-26).

The primary motive in the first section of this prelude consists of two juxtaposed minor

thirds, with each “filled in” with half-step passing tones (e.g. C and E-flat connected through C#

and D). This is followed by a downward half-step line, which connects A ¼-flat in its descent to

F ¼-sharp. The four pitches that lie on either end of these two chromatic clusters (C, E-flat, F ¼-

sharp, A ¼-flat) constitute what Skinner identifies as Wyschnegradsky’s “tonic” tetrachord. An

additional pitch, one quarter-tone below the highest note of the motive, results in what

Wyschnegradsky refers to as an “expressive” unison (Wyschnegradsky 1932, 29).

If the A section is bisected into mm. 1-7 and mm. 8-14, there are significant parallels in how Wyschnegradsky develops this motive. The motive is first stated and repeated in m. 1 then

3 transposed to start on C# and F /2-sharp in m. 2. In mm. 3-4, this pattern restarts with the

motive on C, but the final iteration is instead transposed to start on G ¼-sharp. While it is

interesting to note that the transposition in m. 2 is replaced by its inverted equivalent in m. 4

(transposition by PCIs 5.5 and 6.5 respectively), it is equally significant that the highest

statement of the motive in these measures moves up by a half-step, mirroring the earlier transpositions from C to C#. 39

This material in mm. 1-4 is followed by a B ¼-flat in mm. 5-6 that sustains as additional

pitches are arpeggiated below it. This chord consists (in descending order) of B ¼-flat, F, B ¼-

sharp, F ¼-sharp, C, G ¼-flat, and A ¼-flat. This hexachord can be classified under the set class

sc(0, 0.5, 1.5, 3.5, 5.5, 6.5). It contains no more than two of any interval class, and includes

every IC but 2.5 and 4.5. For this reason, this chord is less readily understood as a set class as it

is as a more “diatonic” in the C-DC Scale: each pitch in this descending sequence

is six “steps” of the C-DC scale lower than the last, with the exception of the A ¼-flat on the

bottom. In this way, it can be thought of as largely a “diatonic sequence” in the context of the C-

DC scale.

When the A section repeats, the final motive statement in m. 11 starts on A ¼-flat (an

additional half-step up from the G ¼-sharp from the first iteration of this thematic

material). Mm. 12-13 are a similar statement of the descending arpeggio from mm. 5-6, this time

starting on C#. The arpeggio consists of C#, G ¼-sharp, D, A ¼-flat, E-flat, A ¼-sharp, B ¼-

sharp, and F ¼-sharp. The only interval in this arpeggio which does not fall strictly into a

“descending 6-step diatonic cycle) of the C-DC scale is the between A ¼-sharp

and B ¼-sharp.

The A-section can also be framed in terms of a process of “composing out” the initial

tetrachord: C, E-flat, F ¼-sharp, A ¼-flat (Straus 1987). In mm. 6 and 13, two pitches are

accented through rhythmic repetition - C and E-flat. Additionally, the lowest pitches of the arpeggiated chords (resulting in part from the single “break” in the otherwise homogeneous 6- step cycle) are A ¼-flat and F ¼-sharp respectively. It also may be of interest that these pitch transformations arise from opposing transpositions of IC 3. The rhythmically accented pitch is

transposed up, while the bass pitch is transposed down. See Figure 20 below for an example of 40

how this process of “composing out” occurs. It is also of note that the bass progression from

measures 1 to 13 moves from C to F ¼-sharp (the major fourth). As Wyschnegradsky chose to order the preludes along the cycle of major fourths, this gesture is comparable to the idiomatic

first section of a tonally open binary form (which in major keys lends itself to a cadence on the

relative dominant).

Figure 20. "Composing out" the tonic tetrachord in Prelude 1. The B section begins with a brief digression from the primary material of the A section in

mm. 14-18. The most significant difference of this section is the prominence of leaps of the

seventh (as opposed to the primacy of half-step motion in the A section). Wyschnegradsky alternates here between leaps (the inversion of the minor second, which was ubiquitous in the previous material) and emphasis of the pitches C and then D through agogic accent and rhythmic repetition. The major-seventh motive is then used to descend into the register of E-flat 2, with mm. 18-20 outlining a return of the “tonic tetrachord,” this time with

fewer passing tones included. Mm. 21-26 is a return of the thematic material from the beginning. Here, the only transposition of the motive that occurs is by ascending octave. At the peak of this ascending line, a pentachord is repeated rhythmically, with the piece ending on C. 41

This pentachord can be analyzed as four stacked major fourths and is explored further in Chapter

IV.

Prelude 4 in E ¼-sharp

Figure 21. The E 1/4-sharp DC scale on which Prelude 4 is based. Wyschnegradsky’s fourth Prelude is based on the E ¼-sharp-DC Scale, as shown in

Figure 21 above. This prelude is similar to the first in how it can be formally subdivided.

Unlike the first, however, it exhibits a greater degree of metrical dissonance as a result of the frequent meter changes. There is also a “tonic” ostinato underscoring all sections of this movement: a repeated E ¼-sharp. This ostinato can be clearly seen in the repeated pitch from mm. 1-6 in Figure 22 below, which is best compared to my own reduction of this excerpt that omits the ostinato for greater clarity of the melodic material in Figure 23.

Figure 22. Prelude 4, mm. 1-6 with E 1/4-sharp ostinato present. 42

Figure 23. Prelude 4, mm. 1-6 with ostinato omitted. The A of this prelude, mm. 1-8, is based on these short melodic figures that are underscored by the E ¼-sharp ostinato. This ostinato continues through the second section (mm.

9-20), which presents contrasting musical materials in terms of rhythm, pitch, and . Rhythmically, the second section adds rhythmic interest through polyrhythm and syncopation. In Figure 24 below, the are highlighted. An example of the syncopation in this section can be observed in the lower staff of m. 10. This section also incorporates a greater variety of meter changes, including multiple measures in complex meters such as 13/8 and 11/8. 43

Figure 24. Polyrhythm and syncopation in Prelude 4. The B section of Prelude no. 4 is also interesting for its melodic and contrapuntal materials, especially in Wyschnegradsky’s use of quasi-imitative procedures. To better illustrate the salient (non-ostinato) features of this prelude, a reduction with the ostinato omitted is used in

the further music examples of this prelude. The melodic material introduced in mm. 9-20 is based on a descending scale that incorporates both descent by semi- and quarter-tone intervals. This material begins in m. 9, and an imitation of it begins on the last of m. 12.

While at first glance the “comes” may appear to consist of the pitch material of the “dux,” further examination of the melodic intervals in these lines reveals a different phenomenon - the comes is nearly identical to the dux in both contour and melodic interval. In other words the comes is almost an exact transposition of the dux such that it starts on E instead of E ¼-sharp. As a consequence of the rhythmic distance between these two entrances, a counterpoint consisting almost exclusively of varying species of thirds occurs. This descending chromatic line of various thirds continues until an eventual arrival at F ¼-sharp (dux) against A ¼-sharp (comes), allowing the lower pitch to smoothly descend to E ¼-sharp for the return of the A-section material. The beginning of this imitation can be seen in the below excerpt of the reduced score 44

(Figure 25), which has generic interval labels to highlight the centricity of the third as a

contrapuntal interval.

Figure 25. Imitation emphasizing thirds of various sizes in Prelude 4. The thematic material of the A section returns in mm. 21-26, and a brief coda based on the

descending scalar motive first heard in mm. 3-4 is prepared by a brief silence in m. 27. In mm.

28-32, this descending motive is repeated in a five-layer canon, with each iteration moving the theme down an additional octave. The first three measures of this coda are shown in Figure 26 below, with each new entrance of this theme labeled with a corresponding number (1-5).

Figure 26. Imitation in the coda of Prelude 4.

45

CHAPTER III

COMPUTATIONAL TOOLS FOR MICROTONAL ANALYSIS

The Limitations of the Humdrum Toolkit

Humdrum is a collection of Unix scripts that allow for computer-aided analysis of music. Some of the basic tools include “hint,” which calculates harmonic intervals, and “mint,” which generates a list of melodic intervals in a piece of music. Music is first encoded into a text format such as **kern (the standard format) or **pitch. Humdrum additionally includes a **pc format for pitch class data, and has some tools for manipulation of pitch-class sets

(**pcset). While it is possible to encode music directly in these formats, there is also a script that allows for conversion of musicXML files into **kern data, which can then be transformed into data formats such as **pitch, **pc, etc. Humdrum tends to process files rather quickly, as many of the tools are built using text manipulation programs such as “awk.” For example, hint processes an input file line-by-line, calculating the harmonic intervals between pitches that are encoded as sounding concurrently. Command line flags such as “c,” which instructs the script to print all compound intervals (e.g. M9 or P11) as their non-compound equivalents (M2 or P4),

can be used to adjust the format of the output data. This allows users experienced with the Unix

command line to quickly extract musical data in a way that best fits the needs of their research.

While the Humdrum toolkit is powerful for quickly analyzing large bodies of music

(corpora), using the full range of tools at one’s disposal quickly becomes less than ideal for

microtonal music. Humdrum was designed primarily for music written in 12-EDO, and the

source code of many Humdrum tools is reflective of this design choice. For example, hint does

not include any consideration of microtonal intervals. Despite this, the **pitch format was

designed with a built-in detuning option: by notating a pitch as C4+2 or C4-50, one can represent 46

detunings of middle C at +2c and -50c (down ¼-tone) deviations from 12-EDO. Whereas it

might be more sustainable in the long term by designing a **kern-like encoding with support for

microtonal pitches, it would be possible to use the existing standard for microtonality for

analyses that do not rely on the richer set of symbols possible in **kern.

Microtonal Hint

While Humdrum’s tools are not fully compatible with microtonal music, the design of

alternate Humdrum-like tools is very much possible. By studying the structure of Humdrum’s

music encoding, as well as how the input data and command line flags map to different outputs, I

managed to design a clone of hint (mhint) in the Python 3.5 programming language. After

verifying this clone produced an identical output to that of Humdrum for identical **pitch input

data, I began to adjust my code to allow for microtonally altered **pitch information. Rather

than designing a system that only uses microtones in 24-EDO, my goal was to design a script

that could use any tuning system that can be completely expressed in terms of “cent deviations

from 12-EDO.”

Another important consideration for designing a script to identify microtonal intervals is

the variety of naming conventions seen not only for 24-EDO, but also other EDOs and systems

more analogous to just intonation. The identification of the generic interval (interval based on

pitch spellings) remains unchanged in mhint, it is rather interval quality that requires a

calculation more finely-grained than that used in hint. This script calculates the quality of an

interval by calculating two cent-values for pairs of pitch objects, to which microtonal deviations

are applied. The generic interval and the cent difference of these two pitches are used in a nested dictionary to identify the interval quality. In computer science, a dictionary is a data structure

that stores data as key-value pairs. The generic interval is paired with a dictionary of possible 47

cent-values and corresponding interval qualities. As an example of this “lookup” process,

consider the interval of C4 and G4 in Figure 27 below. The interval between C4 and G4, for

example, is a generic fifth. This value is used to retrieve a dictionary, which maps interval size

in cents to corresponding qualities for fifths. This interval is 700c, resulting in a final specific

interval of “perfect fifth.”

Figure 27. Calculating a specific interval from generic interval and size in cents. While mhint was originally designed for use with quarter-tones, I implemented a flexible system for definition of tuning systems to better “future-proof” the program for ease of use with other tuning systems and interval quality naming conventions. There is a function that saves a

“tuning system” file, allowing for the definition of any number of possible interval qualities.

While mhint is a tool with greater microtonal flexibility than hint, there are some important considerations and drawbacks of using mhint. While mhint can be run from the Unix command line, it is implemented in Python 3.5 instead of awk. As a result of the extra calculations required for mhint to calculate microtonal intervals, mhint processes music with less time efficiency than hint. A more significant limitation of mhint, however, is its current lack of compatibility with Unix command line “pipes.” A pipe in Unix programming is a way to link the output of one command to the input of another. For example, the command “pitch Largo.krn

| hint > Largo.hint” converts the Largo.krn file into **pitch data, processes this data using hint, and saves the result in the file Largo.hint. This command is not possible using mhint currently, as it cannot yet accept input data using the Unix pipe. As a result of these constraints, mhint as a

standalone script is better suited for extracting data from smaller corpora or individual 48

works/movements. If used as part of a script, however, mhint is just as capable of performing

these tasks as hint.

An important consideration when using mhint to analyze music scored for two quarter- tone pianos is the ambiguity of a composer’s intended generic intervals. For example, consider

Figures 28 and 29. Figure 28 is the two-piano score of Wyschnegradsky’s Prelude No. 4 (piano

II tuned down ¼-tone), while Figure 29 is the edition that uses only one system - a perceptible

“quarter-tone piano.”

Figure 28. Wyschnegradsky's Prelude No. 4 (two piano score).

Figure 29. Wyschnegradsky's Prelude No. 4 (condensed study score). While there is no question that Wyschnegradsky intended the first pitch of this work to be

E ¼-sharp (as he indicates “position Mi ¼-sharp” in the prelude subtitle), the melodic interval

between this tonic pitch and the B in this measure becomes unclear. While the sounding interval

is the same in both editions, the interval is presented generically as a fourth in Figure 28 and as a 49

fifth in Figure 29. To remove this ambiguity entirely, I use a sed script to convert mhint data to

pitch-class interval (PCI) data. An example of this is the augmented fourth or diminished fifth

from 12-EDO: regardless of how this interval is spelled, it has the sound of PCI 6.

Microtonal Set-Theory Tools

While not as applicable in the analyses from this paper, the concept of pitch class set

theory can be readily applied in microtonal space using the “flexible system” model outlined in

the introduction of this thesis. To more effectively model the relationships between pitch class

sets in microtal systems, I developed a model in the Python 3.5 programming language. In

addition to identifying the set class of a pitch collection in quarter-tone music, this calculator can work with any microtonal system that can be expressed as cent deviations from equal-tempered

(12-EDO) pitch classes. Additionally, it can generate a “Forte list” for other EDO systems - comprehensive lists of all possible set classes in a selected system. The algorithm chosen for this, however, sacrifices processing speed to ensure generation of a comprehensive list.

The list generator begins with the empty set and sc(0), which exist in any EDO system. The calculator then determines the set class for every possible dyad, trichord, etc., adding it to a list whenever a previously un-catalogued set is found. To save processing time for larger sets, the algorithm iterates through a previously-generated list of set classes of complementary size. For example, rather than testing every possible 8-pitch collection in 12-

EDO, it is possible to iterate over the list of tetrachords and calculate the complement for each

(as there is a one-to-one mapping between a pitch class set and its complement). Even with this reduction of processing time, the time-complexity of this algorithm is currently prohibitive to systems of a larger size than 24-EDO. In programming, time-complexity refers to the relationship between some parameter of a task and the amount of time required to complete 50 it. This algorithm runs in what is called “exponential” time: as the size of N increases while dealing with N-EDO systems, the computation time required increases exponentially. See

Figures 30 and 31 for examples of this exponential growth. Both figures consider EDO systems from 6 to 36 (36 chosen as the next highest 12-based EDO after 24), with the first showing the exponential increase in the number of possible hexachords in these systems. The second shows what might be described as the “peak” set size for computation - for example, hexachords in 12-

EDO or 12-pitch collections in 24-EDO.

Figure 30. Exponential growth in the number of hexachords in N-EDO systems (6-36).

Figure 31. Exponential growth in the number of N/2-chords in N-EDO (6-36). 51

As can be seen in the above figures, EDO systems beyond 24-EDO quickly become less

feasible for the creation of comprehensive set class lists. An alternative approach for higher- order systems I added to this calculator instead only considers certain pitch classes when examining combinations. For example, the calculator can generate a list of set classes that occur within Wyschnegradsky’s 13-tone DC-scale (a subset of 24-EDO) much more quickly than a list of all 24-EDO sets classes. These lists have uses that fall out of the scope of this paper, but may nonetheless be of use to composers interested in finding ways to “modulate” between two different EDO systems, perhaps by use of some smaller subset of pitches shared between the two.

52

CHAPTER IV

COMPUTATIONAL ANALYSES OF SELECTED EXCERPTS

With the aid of tools designed to work with microtonal music, these quarter-tone works

by Ives and Wyschnegradsky can be examined computationally. Using this small of a sample

(which is not large enough to be considered a “corpus”), it is not possible to draw large-scale conclusions about the two composers’ handling of 24-EDO. However, tools such as mhint still make it possible to compare the harmonic interval content between contrasting excerpts of music. Rather than using computational methods to explain the structure of the music in this paper, the mhint script will be used to extract data that will be discussed in the context of my analyses of the four selections (whereas a true corpus study would formulate testable hypotheses). Another analytical technique possible with data extraction tools such as hint/mhint is that of data visualization through two-dimensional graphs. Most often, these graphs model individual musical parameters over time, and are inspired by the visualizations used by James

Tenney in Meta+Hodos (Tenney 1964, 33). Due to the more “chordal” texture throughout Ives’

Three Quarter-Tone Pieces, the changes in harmonic interval prevalence over time might be useful for visualizing the differences between formal sections.

Compiling a Small Corpus

In order to process these microtonal pieces using mhint, it is first important to encode them in a format that the script can interpret. Rather than typing the music manually, I used the two-piano editions of these works as a starting point. While it would be more ideal to enter the music using quarter-tone accidentals, I found Dorico’s support for MusicXML files to be lacking in detail: any microtonal pitch classes are treated as pitches in 12-EDO when Dorico exports

MusicXML data. The Humdrum script musicxml2hum allows for the conversion of this data 53 into a **krn file with four staves (two for each piano). After converting the **krn data into

**pitch, I split the file into two halves, one with each piano, which allows for the application of the necessary “detuning” through use of a sed (Unix stream editor tool) script. At this time, the two halves are re-merged into a file that can be processed by mhint.

Analyzing and Interpreting the Output of mhint

For the purposes of these analyses, only intervals between simultaneously-articulated pitches (referred to in this paper as “accented” intervals) will be considered. All selections will be processed with the command-line arguments “a,” “c,” and “u.” The “a” flag specifies that all harmonic intervals at a specific point in time should be calculated, not just those between vertically adjacent pitches. The “c” flag converts all compound intervals into their non- compound equivalents, allowing for the intervals to be tallied in a way that assumes octave equivalence. The “u” flag removes all perfect unison and perfect octave intervals from the output.

After generating the **mhint (microtonal harmonic interval) data, the output was further processed to remove barlines and unnecessary interpretation records, with each interval in the output given its own line in a text file. The sortcount script was then used to create a tally of how many times each harmonic interval occurs in the selected excerpt or section of music. In order to avoid skewing of this interval data by differences in excerpt lengths, the of intervals is converted into the of the total intervallic content that each interval comprises. For example, a short movement may have 10 accented perfect fifths out of 200 total intervals, whereas a longer movement may have 20 out of 1,000. If interval frequencies were not considered in terms of a proportion, a misguided argument that the longer movement emphasizes 54 the perfect fifth more than the shorter. In terms of proportion, however, this interval would be

5% of the short movement but only 2% of the longer.

Ives’ Three Quarter-Tone Pieces

Both the Largo and Chorale of Ives’ Three Quarter-Tone Pieces incorporate multiple approaches to composition in 24-EDO. While it is possible to examine the complete harmonic interval content of these movements, Humdrum implements a tool that allows us to isolate specific sections of the music - “yank.” By using yank, it is possible to extract harmonic interval data from different sections of the music for comparison. While less fine-grained than Tenney’s graphs of musical parameters (which might operate on a per-beat or per-note level), key features of the music can be graphed against each other to illustrate the differences between sections. For example, sections based on Ives’ primary and secondary chords would likely have more instances of perfect fifths than sections based primarily on augmented triads. It is possible to use either pitch-class interval (PCI) or interval class (IC) data for these visualizations, considering two factors: the “resolution” of the data (using 23 categories for PCI vs. 12 for IC), and the visual clarity of the chart.

When examining the harmonic interval content of only one movement or section, only one set of data points is needed (23 for PCI data, or 12 for IC). In these cases, a bar chart of PCI frequencies is a more ideal visualization. When considering how this intervallic data changes over time, however, this “higher resolution” data can easily transform into visual for a reader. For example, a chart of all 23 PCIs in 7 distinct section of Ives’ Largo would consist of

23 * 7 (161) points of data. In order to better draw comparisons between sections (e.g. to show how certain prominent intervals change in frequency), it becomes preferable to not only use IC 55

data, but to focus on only the most relevant 2 or 3 ICs. This is a significant tradeoff, as the

resolution of the data is drastically reduced to better highlight the more salient features.

Largo

For Ives’ Largo, the primary visualization goal is to demonstrate the differences between sections which are based on the augmented triad sonority and sections which are not. As can be

seen in Figure 32 below, the two most frequent intervals found in Largo are PCIs 4 and 8. After

PCI 7, the next three most common PCIs are 2.5, 6.5, and 10.5 - intervals that would constitute

an augmented triad if occurring simultaneously against the same pitch (as seen in the opening of

the movement).

Figure 32. Frequencies of intervals in Ives' Largo. More specifically, 23.3% of the harmonic intervals in the Largo are major thirds, and 11.4% are

minor sixths (enharmonic augmented fifths). This is consistent with the expected use of

augmented triads, as an augmented triad in closed-position voicing would consist of two major

thirds and one . Conversely, perfect fifths are an interval that would not occur

frequently in a section based on augmented triads. For this reason, the rudimentary visualization 56 of formal differences will plot the frequencies of two interval classes against each other: ICs 4 and 5.

In the analysis of Ives’ Largo, one section (mm. 18-28) is based much more on Ives’ primary chord, while the remainder of the movement utilizes the augmented triad frequently. Using the frequency of interval classes 4 and 5, it is possible to produce a visualization that allows us to examine data for consistency or inconsistency with our analytical observations, such as:

1. Mm. 1-6 uses superimposed augmented triads as the primary sonority.

2. Mm. 43-49 is almost exclusively made up of augmented triads.

3. Mm. 7-16, 29-42 and 50-65 use augmented triads frequently, but are not as strictly

limited to the use of these sonorities. We might therefore expect other intervals to be

more prevalent here than in the sections named above.

4. Mm. 18-28 are based on Ives’ primary chord.

By graphing the frequencies of these two interval classes across these discrete sections of the movement, it is possible to visualize information that would be expected to correlate with these observations (e.g. that mm. 18-28 will see a sharp increase in IC 5 and decrease in IC

4). Figure 33 below is a graph that shows how the frequencies of IC 4 (solid line) and IC 5

(dashed line) differ between sections of this movement. 57

Figure 33. Differences in Interval Class content in sections of Largo. While the above figure was originally modeled as a line chart to avoid using colors other

than black and white, the use of a scatter diagram better shows the calculated data points. As

expected from the observations listed above, Mm. 18-28 is the only section that prominently uses

IC 5 more frequently than IC 4, and is also the only section based on Ives’ more quintal “primary

chord.” The decisions regarding the segmentation of the form into discrete sections for the

creation of this model were made based on my previous analytical observations regarding how

the piece changes over time. It would also be possible to write a script to generate this

information on a per-measure basis, but this in turn carries a risk of adding too much visual

“noise” in the data. This would be a more standard means of measuring the change in these two

parameters over time, and might be of use in studies that attempt to discern formal divisions in music based on such data.

Chorale

The Chorale from Ives’ Three Quarter-Tone Pieces is based much more heavily on his

primary and secondary chords, which are more quintal and quartal respectively. In terms of 58

intervallic content, it is not enough to use IC 5 as a “litmus test” for frequent usage of a specific

chord, as this interval is common to both sonorities. A different discriminating interval is

necessary to estimate the prevalence of one or both of these chords in sections of this music: the

primary chord contains three instances of IC 3.5 while the secondary contains four instances of

IC 2.5. These intervals are the neutral third and subminor third respectively, although it is of

interest that Ives’ own spellings of these chords are both as species of second. For the Chorale,

therefore, there are three interval classes to measure and compare: IC 5 to indicate the possible

use of either of these two sonorities, and ICs 3.5 and 2.5 to better discriminate between the two.

To create this visualization, Humdrum’s yank script will once again be used to isolate sections of music. As can be seen in Figure 34 below, the three most common accented harmonic intervals in Ives’ Chorale are the neutral third, perfect fifth, and supermajor second

(subminor third).

Figure 34. Intervallic content of Ives' Chorale. 59

In addition to the three most frequent intervals being of interest in the context of Ives’ primary and secondary chords, the next three most frequent intervals in this chart are the inversions of these. In other words, more than of all accented harmonic intervals (with

1,632 in total) are of the three interval classes of interest in this visualization study. Table 4 below defines the formal units of music that will be examined for these three interval classes, along with a description based on my analysis of the work of what sonorities are primarily used in each.

Table 4. Expected prominent harmonic intervals in sections of Ives' Chorale. Measures Main Sonority from Analysis Expected Frequent ICs 1-9 Primary Chord 3.5, 5 10-15 Primary and Secondary Chords 2.5, 3, 5 16-27 “Distorted” Diatonicism Various species of 3rd/5th 27-29 Primary Chord 3.5, 5 30-40 Neither (many 12-EDO 3rds/7ths) Unclear 40-48 Primary and Secondary Chords 2.5, 3.5, 5 49-57 Secondary Chord 2.5, 5 58-59 Primary Chord 3.5, 5

60

Figure 35 below plots the frequencies of these three interval classes in each of these

sections. The solid line represents IC 5, the dashed line IC 3.5, and the dotted line IC 2.5.

Figure 35. Frequencies of ICs 2.5, 3.5, and 5 in sections of Ives' Chorale. As can be seen in the above diagram, two sections that do not utilize either the primary or secondary chord (as described in the previous table) are marked with a stark decrease in the frequency of IC 5, which is a significant component of both harmonies. In the other sections of this work, the lines representing ICs 2.5 and 3.5 can be used to estimate the usage of the primary and secondary chords respectively. This is perhaps most clearly seen in mm. 27-29 and 58-59, where a large gap between the frequencies of ICs 3.5 and 2.5 indicates that the secondary chord might occur somewhere in these short sections, although they remain predominantly composed of the primary chord. As an example of this, consider mm. 58-59 as seen in figure 36 below. All chords in these two measures are Ives’ primary chord, excepting the first, which instead is a secondary chord built on F. This secondary chord is better understood as the last chord of the previous material, which is only included in these two measures as a result of segmenting the 61

movement by measure numbers (as opposed to a more “precise” segmentation by measure and

beat numbers).

Figure 36. Primary and secondary chords in the coda of Ives' Chorale. As can be seen, these elisions between musical sections present a challenge when using musical parameters such as harmonic intervallic content as a heuristic to discern both small- and

large-scale formal boundaries. For this reason, it is critical that any computational analyses of

small datasets (such as the movements chosen for this paper) refrain from making broad

conclusions, rather using the computational tools to better visualize and support analytical

conclusions.

Wyschnegradsky’s 24 Preludes: Comparing Preludes I and IV.

Many of the Wyschnegradsky preludes are very sparse harmonically when compared to

the Ives pieces. For example, Prelude No. 1 only has 197 simultaneously-articulated harmonic intervals, while Ives’ Chorale has 1632. Figure 37 below shows the distribution of harmonic intervals in Prelude No. 1. 62

Figure 37. Intervallic content of Prelude 1. As can be seen in the above figure, the most frequently-used interval by Wyschnegradsky in this prelude is PCI 5.5 - the major fourth interval that corresponds closely to the 11:8 ratio in the harmonic series. The second-most frequent interval is PCI 11, and the third is PCI 6.5. It could be argued that both of these intervals’ use is also indicative of Wyschnegradsky’s frequent use of the major fourth in this prelude: PCI 6.5 (the minor fifth) is its inversion, and the major seventh is the resultant interval of superimposing two minor fourths. This is a dangerous conclusion to draw, however, as there are other factors influencing these intervals. In mm. 24-

26, a pentachord is articulated not one, but seven times in a row. This pentachord is shown in

Figure 38 below, and in abstract pitch space it consists of four stacked major fourths: G ¼-sharp,

D, A ¼-flat, E-flat, and A ¼-sharp. 63

Figure 38. Pentachord which skews data heavily toward PCI 5.5. The other notable number in this chart is that PCI 0.5 makes up 4% of the total accented

harmonic interval content of this prelude. Wyschnegradsky describes the superimposition of one

or more pitches, each a quarter-tone apart, as a functional “expressive unison,” making a similar

case for the use of either expanded or contracted by a quarter-tone. While an expressive unison may consist of two or more pitches, its function as a single, blurred pitch can be a confound when analyzing this music.

Wyschnegradsky’s Prelude No. 4 is much different texturally from Prelude No.

1. Whereas the harmonic interval content of Prelude No. 1 consists of a handful of expressive and a few vertical chords as shown in Figure 39, Prelude No. 4’s harmonic intervals arise in only two sections of the selection: the two-voice descending chromatic imitation (mm. 9-

20 and mm. 27-32). Rather than examining the “accented intervals” in this passage, I instead used the “ditto” humdrum command to examine intervals between pitches that are notated as

sounding simultaneously (as opposed to starting at the same time-point). Figure 39 below shows the resulting distribution of harmonic pitch-class intervals for Prelude No. 4. In the imitative section, I removed the E ¼-sharp ostinato to bring greater focus to the imitative foreground melody. 64

Figure 39. Accented and Non-Accented Intervallic content of Wyschnegradsky's Prelude 4. A few features of this figure stand out notably - there is a significant cluster of intervals

centered on the neutral third (PCI 3.5). Expanding this to the intervals within a quarter-tone of the neutral third, more than half of the prelude’s intervallic content is accounted for. Two of the eleven interval classes (1.5 and 6) are absent entirely in this diagram. While this data is already indicative of Wyschnegradsky’s possible compositional process in this prelude, it is possible to segment this data to produce a “higher-resolution” picture of what is harmonically in the two harmonically interesting sections of this prelude. By using yank, a specific range of measures in this work can be isolated, allowing us to examine mm. 9-20 with greater clarity.

Isolating mm. 9-20, it can be confirmed that the significant cluster of intervals close to

PCI 3.5 originates from this section. This data is consistent with my previous analysis of this section as a two-voice counterpoint using almost exclusively species of thirds. If one considers

PCIs 2.5 and 4.5 as species of thirds, this generic sounding interval accounts for 97% of the 65

harmonies between the two moving voices in mm. 9-20. See Figure 40 below for a chart of these

results.

Figure 40. Harmonic intervals in the first imitative section of Prelude 4. It is also possible to explain differences in the two preludes’ data by considering the textures of these two preludes. While both preludes are largely melodic in nature, Prelude No. 1 uses multiple chords consisting of “stacked” intervals from the C-DC scale. As a result of this,

Wyschnegradsky’s frequent use of PCI 5.5 (often in 6-step cycles) results in an apparent emphasis of PCI 11. In Prelude No. 4, however, Wyschnegradsky does not emphasize any particular chord or means of chord construction. Rather, the harmonies in this prelude arise as a product of counterpoint between two or more parts. As the cluster of intervals around PCI 3.5 originates in a section with a two-voice texture, superimposed thirds are not a compositional possibility. 66

CHAPTER V

CONCLUSION AND FUTURE WORK

Conclusion

Music written in microtonal systems such as 24-EDO is much less established in the

Western tradition than that of 12-EDO. This results in not only a vacuum of any pre-existing microtonal style, but also of analytical approaches for its analysis. At the same time, many of the rudimentary features of 12-EDO, such as pitch intervals, collections/set classes, and even scales, can be reconsidered in such a way to better make sense of this quarter-tone music. The other necessary consideration for Western microtonal music is a lack of any “primary” system or model for these compositions. While composers would collaborate and discuss ideas with each other (e.g. Charles Ives and , or George Rimsky-Korsakov and Ivan

Wyschnegradsky), the composers of this music largely carved out their own musical niche. This can clearly be seen in the quarter-tone works of Ives and Wyschnegradsky, as the former was more hesitant to distance himself from the perfect fifth – a foundational interval in 12-EDO music.

Through analysis of these works as one might approach the analysis of other post-tonal works (with considerations as necessitated by the tuning system employed), quarter-tone structures such as Ives’ primary and secondary chords or Wyschnegradsky’s diatonicized- chromatic scale come into greater focus. As these structures are identified, it becomes possible to compare composers’ use of the numerous compositional possibilities afforded by 24-EDO.

Through both these new constructs and less tuning-reliant musical idioms such as rhythm and motive, it becomes possible to discern musical form in these works. 67

While the analyses performed in Chapter II of this thesis do not require the use of

computational methodologies, tools I have developed for use with microtonal music have proven

useful in my study of these works. In Ives’ Largo and Chorale from Three Quarter-Tone Pieces,

for example, striking differences in the composer’s use of harmonic intervals can be observed

using my “mhint” script. These differences become even more apparent when comparing Ives’

harmonies to those of Wyschnegradsky, as the latter tends to prioritize the minor third and major

fifth in his music. The selected excerpts also show some of the dangers in relying on a

computer-based analysis tool such as mhint – there will always be some pieces or excerpts of music that may confound an analytical tool. For example, if one considered only the excerpt from Ives’ Chorale that uses neutral triads, mhint would reveal a high proportion of perfect fifths

and neutral thirds. If the analyst does not consult the score during this analysis, those measures

could be classified incorrectly.

Future Avenues of Research

Perhaps the most open-ended direction for research following this thesis would be the

development of additional computer-based tools for microtonal analysis. While hint and its

clone (mhint) are powerful scripts, they implement only one task, “identify harmonic intervals.”

The Humdrum toolkit includes many other powerful scripts, such as “mint,” for identification of

melodic intervals, and “diss,” which calculates sensory dissonance (a task more related to music cognition and psychology). Other tools, such as “timebase,” which normalizes rhythms to a consistent duration throughout a piece, or “yank,” which isolates some part or excerpt of data, are already suitable for use with microtonal music (as evidenced by the use of yank in Chapter

IV). 68

To this end, the development of corpora of microtonal works (both for quarter-tone piano and other instrumentations/EDOs) is another direction in which the research presented in this thesis can further expand. By compiling a corpus consisting of all three of Ives’ quarter-tone

pieces and all 24 Wyschnegradsky Preludes, future studies to more definitively examine the

differences between these collections would become feasible. Easley Blackwood’s Twelve

Microtonal Etudes for Media, with one etude in each EDO from 13 to 24,

would be another work of interest, as the flexibility of both mhint and the microtonal pitch-class

set calculator would allow for both to be examined in a similar manner to those of Ives and

Wyschnegradsky. These microtonal corpora, once compiled, could be published for studies by

other interested scholars, with the source code of mhint and the set class calculator made freely

available on a public repository such as GitHub (as Humdrum is published).

These systems, while designed for use with systems built on equal divisions of the

octave, may be extensible to other systems of organizing pitch such as ’s “extended

just intonation.” This would most likely be achieved with a **krn- or **pitch-like representation

that allows for additional accidentals. While this would increase the complexity of the notation,

it would also allow for a greater exactness in the definition of intervals, making the use of

frequency ratios viable (e.g. an interval from C to E could be identified as either a “major third”

or simply as “5:4”). In designing such a system, corpus-based approaches to analyzing music by

composers such as Ben Johnston and also becomes possible.

Finally, the expansion of constructs such as pitch-class set theory into microtonal systems

may be possible with other non-computational theoretical ideas, especially those separable from

Western tonality. A Neo-Riemannian way of thinking about how one microtonal pitch collection

can transform into another may also be feasible in these systems of organizing pitch. Other 69

properties from set theory, such as intersection, may have significant implications for composers

interested in “modulation” between different tuning systems. Just as the fully-diminished

seventh chord functioned as a “magic door” through which composers of the Nineteenth century

could modulate to many key areas, pitch collections that exist in multiple tuning systems

simultaneously could serve a similar open-ended “pivot” function for composers interested in

such techniques.

The quarter-tone system, explored to a great extent in the twentieth century, offers a wide

gamut of new musical possibilities. In the quarter-tone works of Charles Ives and Ivan

Wyschnegradsky, one can see only the approaches taken by two composers who utilized this

palette of new sonic colors. While the field of music theory has had centuries to develop

canonical ideas for common practice (tonal) music, post-tonal music did not become prominent

until the twentieth century. Even in the scope of twentieth century and contemporary music,

many composers have not explored the possibilities allowed by 24-EDO, let alone the countless

additional tuning systems available today. By expanding the computational subset of the music

theory field into the realm of this quarter-tone music, it becomes possible for both composers and theorists to attain a greater understanding of how this music works. 70

References

Ader, Lidia. "Microtonal Storm and Stress: Georgy Rimsky-Korsakov and Quarter-Tone Music in 1920s Soviet Russia." Tempo 63, no. 250 (2009): 27-44.

Albrecht, Joshua and Daniel Shanahan. “The Use of Large Corpora to Train a New Type of Key- Finding Algorithm: An Improved Treatment of the Minor Mode.” (2013).

Boatwright, Howard. "Ives' Quarter-Tone Impressions." Perspectives of New Music 3, no. 2 (1965): 22-31. doi:10.2307/832500.

Dibben, Nicola. "The Perception of Structural Stability in Atonal Music: The Influence of Salience, Stability, Horizontal Motion, Pitch Commonality, and Dissonance." Music Perception: An Interdisciplinary Journal 16, no. 3 (1999): 265-94

Gann, Kyle. The Arithmetic of Listening: Tuning Theory and History for the Impractical Musician. Urbana, IL: University of Illinois Press, 2019.

Holde, Artur. "Is There a Future for Quarter-Tone Music?" The Musical Quarterly 24, no. 4 (1938): 528-33.

Huron, David. "Interval-Class Content in Equally Tempered Pitch-Class Sets: Common Scales Exhibit Optimum Tonal Consonance." Music Perception: An Interdisciplinary Journal 11, no. 3 (1994): 289-305.

———. "Music Information Processing Using the Humdrum Toolkit: Concepts, Examples, and Lessons." Computer Music Journal 26, no. 2 (2002): 11-26.

———. "Note-Onset Asynchrony in J. S. Bach's Two-Part Inventions." Music Perception: An Interdisciplinary Journal 10, no. 4 (1993): 435-43. doi:10.2307/40285582. JSTOR.

Ives, Charles, “Some Quarter Tone Impressions.” 1925. In Essays Before a Sonata, The Majority, and Other Writings by Charles Ives, ed. Howard Boatwright. New York: W.W. Norton, 1970, 105-119.

———. Three Quarter-Tone Pieces. 1925. New York: C.F. Peters, 1968.

Kaufmann, Henry W., and Robert L. Kendrick. “Vicentino, Nicola.” Grove Music Online. 2001.

Mathiesen, Thomas J. “Genus.” Grove Music Online. 2001.

Mauch, Matthias, Robert M. Maccallum, Mark Levy, and Armand M. Leroi. “The Evolution of Popular Music: USA 1960–2010.” Royal Society Open Science 2, no. 5 (2015): 150081. https://doi.org/10.1098/rsos.150081.

Meeùs, Nicolas. "Enharmonic keyboard." Grove Music Online. 2001. 71

Milne, Andrew J., Robin Laney, and David B. Sharp. "A Spectral Pitch Class Model of the Probe Tone Data and Scalic Tonality." Music Perception: An Interdisciplinary Journal 32, no. 4 (2015): 364-93.

Skinner, Myles Leigh. "Toward a Quarter -Tone Syntax: Analyses of Selected Works by Blackwood, Hába, Ives, and Wyschnegradsky." Order No. 3244260, State University of New York at Buffalo, 2007.

Straus, Joseph N. "The Problem of Prolongation in Post-Tonal Music." Journal of Music Theory 31, no. 1 (1987): 1-21.

Temperley, David. "An Algorithm for Harmonic Analysis." Music Perception: An Interdisciplinary Journal 15, no. 1 (1997): 31-68.

Temperley, David, and Leigh VanHandel. "Introduction to the Special Issues on Corpus Methods." Music Perception: An Interdisciplinary Journal 31, no. 1 (2013): 1-3.

Tenney, James. “Meta+Hodos.” 1964. In From Scratch: Writings in Music Theory, ed. Larry Polansky, Lauren Pratt, Rabert Wannamaker, Michael Winter. Urbana, IL: University of Illinois Press, 2015, 13-97.

Tymoczko, Dmitri. A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice. New York: Oxford University Press, 2011.

Wellek, Albert, and Theodore Baker. "Quarter-Tones and Progress." The Musical Quarterly 12, no. 2 (1926): 231-37.

Wyschnegradsky, Ivan. Manual of Quarter-Tone Harmony. 1932. ed. Noah Kaplan. Brooklyn, NY: Underwolf, 2017.

———. 24 Preludes in Quarter-tone System in 13-tone Diatonicized Chromaticism. Leipzig: M.P. Belaieff, 1979.

“24Edo.” Xenharmonic Wiki. 2019. https://en.xen.wiki/w/24edo.