Pitch-class Sets Research Articles

Is It Chaotic? An Examination of the Harmonic Materials in Razak Abdul Aziz’s Etude No. 5 for Solo Piano

This paper investigates the adoption of chaos theory into Etude No. 5 by Razak Abdul Aziz. Using pitch-class set to analyse the work, the study unveils several chaotic elements. By providing a unique perspective of investigating musical works from the viewpoint of chaos, this study enriches music scholarship.

Twentieth Century Compositional Techniques Applied to Jazz: Pitch-Class Sets in Jazz Composition and Improvisation

This paper presents a speculative method to coalesce traditional jazz improvisation with techniques and elements of post-tonal composition. Throughout the 20th century, erudite composers came up with several aesthetic and conceptual developments but jazz, until very recently, did not move in the same direction. To maintain a fruitful collaboration between composers and improvisers, if the pallet of compositional techniques available to the jazz composer expands, I would argue that a similar development must occur in the field of improvisation. For composers, this paper proposes creative processes supported by pitch-class set theory and, in the interest of coherence and unity with pre-composed material, it offers a complementary approach to soloists.

Zhao Xiaosheng’s Tai Chi:


 The implementation of China Reform and Opening-up policy in 1978 gave birth to the Chinese new wave music, with an idiosyncratic manner of rebelliousness, individuality and innovation that brought a break-through in Chinese new music of the past. In this paper, we focused on a representative piano solo work Tai Chi (1987) by composer Zhao Xiaosheng who explored the ancient Chinese philosophy I Ching and the Tang Dynasty musical form Tang Daqu using Allen Forte’s pitch class set theory. We analyzed how Zhao employed the Western sonata and Chinese Tang Daqu to reflect the yin-yang dualism, based on the composer’s statement in the interview, researchers’ analyses, and authors’ suggestion. By decoding the pitch-class permutations that correspond to the 64 hexagrams of I Ching, the mathematical logic becomes the main way to combine the I Ching with serial music. Nevertheless, the compatibility between musical expression and commentaries of hexagrams is not reflected in the piano solo Tai Chi. Therefore, we discussed the entanglement of a compositional identity formation based on contradicting elements from that of the traditional Chinese musical culture and Western art music, with questions about the accretion and authenticity in the contemporaneity and musicalization of the ancient I Ching. 

Fingerprints, barcode sequences and quasi-phylogenies-Tools for analysing polyphonic music.

A data flow is presented for visualising the evolution of elementary structures of polyphonic music from early Baroque to late Romantic, using quasi-phylogenies based on fingerprint diagrams and barcode sequence data of 2-tuples of consecutive vertical pitch class sets (pcs). The present methodological study, which sees itself as a proof of concept for a data-driven approach, uses examples of music from the Baroque, the Viennese School and the Romantic era to show that such quasi-phylogenies can be generated from multi-track MIDI (v. 1) files that largely correspond to the eras and the chronology of compositions and composers. The method presented is considered to have the potential to support the analysis of a wide range of musicological questions. In the context of collaborative work on quasi-phylogenies of polyphonic music, a public data archive could be established that provides multi-track MIDI files with contextual data.

Exhaustive chord progressions and their use in music composition

A general approach for the design of mild chord progressions from the n-chords (unordered pitch-class sets of cardinal n) of a p-tonic scale was developed. Four relations between n-chords were considered: equivalence through transformation (transposition or inversion), parsimony (quasi-identity), mildness (equivalence or parsimony) and fuzziness (quasi-equivalence). The results showed that these relations are symmetrical and compatible with any transformation. Therefore, a parsimonious, mild or fuzzy progression of n-chords exists in 48 forms that keep its properties: direct, retrograde, inverted, retrograde inverted and their transpositions. Circular non-redundant exhaustive parsimonious progressions of the n-chords from a p-tonic scale were established for n = 2–5 and p = 2–9. The n-chords and their progressions can be represented in a two-dimensional table of the (n – 1)-chords of the p-tonic scale versus the set classes of its n-chords. Harmony and structural form can thus be deduced from the intrinsic properties of a scale.

FluidHarmony: Defining an equal-tempered and hierarchical harmonic lexicon in the Fourier space

FluidHarmony is an algorithmic method for defining a hierarchical harmonic lexicon in equal temperaments. It utilizes an enharmonic weighted Fourier transform space to represent pitch class set (pcsets) relations. The method ranks pcsets based on user-defined constraints: the importance of interval classes (ICs) and a reference pcset. Evaluation of 5,184 Western musical pieces from the 16th to 20th centuries shows FluidHarmony captures 8% of the corpus's harmony in its top pcsets. This highlights the role of ICs and a reference pcset in regulating harmony in Western tonal music while enabling systematic approaches to define hierarchies and establish metrics beyond 12-TET.

“Atonal” Motifs and the Presentation of the Musical Idea. Approaching a Historically Sensitive Analysis of Arnold Schönberg’s Works between 1909 and 1912

Nothing has conditioned the English-language analytical discourse about Schönberg’s atonal period music more than pitch-class sets. In Remaking the Past, Joseph Straus defines the pitch-class set as “a motive from which many of the identifying characteristics – register, rhythm, order – have been boiled away”. This understanding of atonal motif, which equates it with pitch-class set, remains widely accepted, intimating a type of “common practice” in Schönberg’s atonal music, evidenced by the motivic coherence demonstrated in pitch-class set analyses. This article proposes a different understanding of motif in atonal period works, based on Schönberg’s definition in Fundamentals of Musical Composition and Zusammenhang, Kontrapunkt, Instrumentation, Formenlehre. In these texts he defines the motif as a “rhythmicized phenomenon”, in which “often a contour or shape is significant”. For Schönberg, the motif is the “‘germ’ of the musical idea”. As the article recounts, Schönberg’s writings outline three forms of presentation of the musical idea: Entwicklung (development), Abwicklung (envelopment) and Aneinander-Reihung (juxtaposition). Since either Schönberg or his students referred to each method of presentation in reference to a different stage of the atonal period, an analytical approach that focusses on presentation of the idea not only illuminates something about compositional process, but also assumes that the atonal period was one of great variety and experimentation. The article reveals that pitch-class sets and other analytical hardware can serve as tools of interpretation and criticism, aiding in the periodization and pedagogy of this seminal time in music history.

The missing fundamentals of harmonic theory: Chord roots and their ambiguity in arrangements of jazz standards

Since Lippius and Rameau, chords have roots that are often voiced in the bass, doubled, and used as labels. Psychological experiments and analyses of databases of Western classical music have not produced clear evidence for the psychological reality of chord roots. We analyzed a symbolic database of 100 arrangements of jazz standards (musical instrument digital interface [MIDI] files from midkar.com and thejazzpage.de ). Selection criteria were representativeness and quality.The original songs had been composed in the 1930s and 1950s, and each file had a beat track. Files were converted to chord progressions by identifying tone onsets near beat locations (±10% of beat duration). Chords were classified as triads (major, minor, diminished, suspended) or seventh chords (major–minor, minor, major, half-diminished, diminished, and suspended) plus extra tones. Roots that were theoretically less ambiguous were more often in the bass or (to a lesser extent) doubled. The root of the minor triad was ambiguous, as predicted (conventional root or third). Of the sevenths, the major–minor had the clearest root. The diminished triad was often part of a major–minor seventh chord; the half-diminished seventh, of a dominant ninth. Added notes (“tensions”) tended to minimize dissonance (roughness or inharmonicity). In arrangements of songs from the 1950s, diminished triads and sevenths were less common, and suspended triads more common, relative to the 1930s. Results confirm the psychological reality of chord roots and their specific ambiguities. Results are consistent with Terhardt’s virtual pitch theory and the idea that musical chords emerge gradually from cultural and historic processes. The approach can enrich music theory (including pitch-class set analysis) and jazz pedagogy.

"Tonal" Dodecaphony and Sentential Form: Extracts from Humphrey Searle's Symphony No. 2, Op. 33

In an article written for The Musical Times in 1940, the English composer Humphrey Searle suggested the possibility of there being a "tonal center" in twelve-tone music. This vignette attempts to put analytical flesh on the bones of his argument. Using extracts from his Symphony No. 2, Op. 33, I demonstrate how movement from relative consonance to dissonance and back (effected by means of pitch class–set expansion, shifts of macroharmony, and/or functionally suggestive row-area transpositions) can serve both to generate and to resolve structural tension. This tendency is strengthened by traditional neighbor-note rhetoric (reminiscent of smooth nineteenth-century voice-leading transformations) and careful orchestration, as well as the second extract's use of a Classical formal type: an antecedent + continuation phrase hybrid theme.

Schoenberg’s Atonal Music: Musical Idea, Basic Image, and Specters of Tonal Function Jack Boss

Schoenberg’s Atonal Music: Musical Idea, Basic Image, and Specters of Tonal Function is the ‘prequel’ to Jack Boss’s award-winning monograph Schoenberg’s 12-Tone Music: Symmetry and the Musical Idea (Cambridge, 2014). The book focuses on the composer’s brief but intensely productive middle period, which began in 1908 and was effectively tapering off by 1916, followed by a transition to dodecaphony. Boss’s meticulous and engaging analytical study of selected works from this period, using primarily pitch-class set theory, is certainly a welcome addition to the Schoenberg literature, especially since this enigmatic music is chronically under-researched and ‘perpetually misunderstood’, as the author notes (p. 40). Not only is this atonal—or ‘pantonal’, as Schoenberg preferred—music rather difficult to listen to, but the relative paucity of sketches and other documentary evidence from this period also presents some challenges to scholars. Indeed, Boss’s study serves, at least in part, as something of a corrective to a legacy of scholarship that, at worst, has regarded Schoenberg’s atonal music as essentially unanalysable and has repeated inherited truisms about its idiosyncratic athematicism, and nebulous connections to the composer’s biography and the zeitgeist of early twentieth-century Vienna. But, as Schoenberg’s Atonal Music also clearly demonstrates, there is still considerable ambivalence and ongoing disagreement about this music in the analytic literature.

Milton Babbitt’s E-type All-Combinatorial Hexachord and its Applications to Jazz

Many of Babbitt's ideas have relevance to jazz, to which I can attest from having spent many years exploring that connection, which I illuminate in this article. I had the wonderful experience of studying with Babbitt as an undergraduate at Princeton; he was a huge inspiration to me. In particular I worked extensively on applying integer pitch-class notation (IPC) to jazz, and Babbitt was extremely helpful as I developed my approach. I retrace the unlikely route that led a jazz musician into the heart of Babbitt's world and, in doing so, I hope to show why my jazz development of IPC has taken its own particular turns. Then, building on that foundation, I focus on a particular pc set that Babbitt used and discussed: the E-type all-combinatorial hexachord, also called the augmented scale, and I discuss its applications to jazz. To date, this hexachord is seldom used in jazz, yet it seems tailor-made for jazz composition and improvisation. I discuss a few rare examples where it has already been used in jazz. Then I examine its structure, paying particular attention to those aspects that seem most applicable to jazz. I examine its relationship to pitch class sets (chords and scales) commonly in use in jazz and, using set operations such as intersection, disjunction, and union, I develop a network of related harmonies involving the hexachord. This network serves as a foundation for a harmonic vocabulary for jazz that is novel, yet compatible with existing practice. Finally, I demonstrate how these structures can serve as compositional determinants.

Dig That Lick (DTL): Analyzing Large-Scale Data for Melodic Patterns in Jazz Performances

Dig That Lick (DTL): Analyzing Large-Scale Data for Melodic Patterns in Jazz Performances

Voice Leading Among Pitch-Class Sets: Revisiting Allen Forte’s Genera

The theory of PC-set class genera by Allen Forte was an important contribution to the understanding of similarity relations among PC sets within the tempered system. The growing interaction between the universes of PC-sets and transformational theories has been explored the space between sets of the same or distinct cardinality, by means of voice-leading procedures. This paper intends to demonstrate Forte’s method along with proposals by other authors like Morris, Parks, Straus, Cohn, and Coelho de Souza. Some analysis demonstrates such operations in passages picked from Heitor Villa-Lobos’s works, like the Seventh String Quartet and the First Symphony.

On the use of relational presheaves in transformational music theory

Traditional transformational music theory describes transformations between musical elements as functions between sets and studies their subsequent algebraic properties and their use for music analysis. This is formalized from a categorical point of view by the use of functors where is a category, often a group or a monoid. At the same time, binary relations have also been used in mathematical music theory to describe relations between musical elements, one of the most compelling examples being Douthett's and Steinbach's parsimonious relations on pitch-class sets. Such relations are often used in a geometrical setting, for example through the use of so-called parsimonious graphs to describe how musical elements relate to each other. This article examines a generalization of transformational approaches based on functors , called relational presheaves, which focuses on the algebraic properties of binary relations defined over sets of musical elements. While binary relations include the particular case of functions, they provide additional flexibility as they also describe partial functions and allow the definition of multiple images for a given musical element. Our motivation to expand the toolbox of transformational music theory is illustrated in this paper by practical examples of monoids and categories generated by parsimonious and common-tone cross-type relations. At the same time, we describe the interplay between the algebraic properties of such objects and the geometrical properties of graph-based approaches.

Implementação de um sistema composicional semiaberto a partir da similaridade entre conjuntos de classes de notas

Este trabalho descreve a formalização e aplicação de um sistema composicional, denominado Similare, que articula micro e macroestrutura tomando como base as relações de similaridade entre conjuntos de classes de notas. Após o exame detalhado dos dois referenciais que dão sustentação teórica ao trabalho (sistemas composicionais e similaridade), o sistema foi definido e, a partir dele, duas obras foram planejadas: uma para quarteto de cordas e outra para piano solo. Durante o processo de pesquisa, aplicativos em Python foram elaborados para auxiliar os compositores na manipulação dos conjuntos.

A detailed list and a periodic table of set classes

In this paper, pitch-class sets are analyzed in terms of their intervallic structures and those related by transposition are called a set type. Then, non-inversionally-symmetrical set classes are split into two set types related by inversion. As a higher version of the interval-class vector, I introduce the trichord-type vector, whose elements are the number of times each trichord type is contained in a set type, as well as a trichord-class vector for set classes. By using the interval-class, trichord-class, and trichord-type vectors, a list of set classes and types is developed, including, apart from the usual information, the intervallic structures and the trichord-type vectors. The inclusion of this last characteristic is the most significant difference with respect to previously published lists of set classes. Finally, a compact periodic table containing all set classes is given, showing their main characteristics and relationships at a glance.

Pitch-Class Set Usage and Development in Late-Period Improvisations of John Coltrane

ABSTRACTThe recordings Stellar Regions and Interstellar Space, made by John Coltrane shortly before his death in 1967, are among his least studied and understood works. While these recordings are generally regarded as representative of “free jazz,” three case studies present evidence that Coltrane’s improvisations on these recordings are highly organized, utilizing a structural methodology focused on trichordal pitch-class sets. Musical set theory is used as a primary analytical tool in combination with common jazz harmonic and improvisational practices. Analyses will illustrate the extensive use of Tn-types (0,1,3) in “Iris,” and (0,2,4), and (0,2,5) in “Saturn.” As an exemplar of this methodology, compelling evidence in “Iris” demonstrates the use of Tn-type (0,1,3), as a structural progenitor for both the saxophone improvisation and piano accompaniment for the entire length of the piece. En route, the potential influence of Nicolas Slonimsky’s Thesaurus of Scales and Melodic Patterns (1947) is examined. As an archetype for an improvisational methodology focused on pitch-class sets, the significance of “Iris” resonates throughout Stellar Regions and Interstellar Space and more broadly to Coltrane’s other late-period recordings, challenging the persistent characterization of this music as chaotic and “free.”

SLIDE in a Leading Role: Pitch Structure in Barrie Cabena’s Sonata Festiva

Barrie Cabena’s Sonata Festiva (1971) highlights SLIDE, a chord progression familiar from the neo-Riemannian and twentieth-century Russian music-theoretic literatures. Diverse expressions of SLIDE involve chord pairs, single chords, and melodic lines in textural, tonal, and formal contexts, leading to contemplative, energetic, comedic, and triumphant effects. Aspects of pitch-class set theory help to relate SLIDE to other harmonic phenomena in the work.

Homometry

HOMOMETRY TOM JOHNSON BEFORE THE MATHEMATICS T WAS ALL RIGHT THERE on page 1 of Allen Forte’s seminal 1973 book, The Structure of Atonal Music. Taking an example of the (0,1,4,6) tetrachord as found in Schoenberg’s George Lieder and in Webern’s Six Pieces for Orchestra, Forte wrote: This pitch combination, which is reducible to one form of the allinterval tetrachord, has a very special place in atonal music. It could occur in a tonal composition only under extraordinary conditions, and even then its meaning would be determined by harmonic-contrapuntal constraints. Here, where such constraints are not operative, one is obliged to seek other explanations. Accordingly, in the sections that follow, terminology and notation will be introduced which will facilitate the discussion of certain properties of such combinations. The rst task is to formulate a more general notion to replace that of pitch combination. So the two all-interval tetrachords become the key example of pitchclass sets that go beyond the tonal language. Why? Because they are I 72 Perspectives of New Music homometric. As shown in Example 1, the elements of (0,1,3,7) and (0,1,4,6) have the same six differences, one of each. Twenty pages later, Forte is more speci c: One might conclude that each prime form has a distinct interval vector. That is, however, not the case. For example, the two forms of the all-interval tetrachord, 4-Z15 and 4-Z29, are so constructed that they are not reducible by transposition or by inversion followed by transposition to a single prime form, but they have identical vectors (111111). Thus, he singles out the all-interval tetrachord rather than all the other dissonant chords and clusters of atonal music because it is homometric. He goes on to speak generally of this category, which he calls Z-related pairs: There are 19 such pairs in all. In addition to the one of cardinal number 4, there are three pairs of cardinal number 5 and 15 pairs of cardinal number 6. Always concerned to base his ideas on what he found in actual scores, he goes on to show two Z-related pairs as they occur in three measures of Webern’s Three Short Pieces, Op. 11 No. 1. EXAMPLE 1: ALL-INTERVAL TETRACHORDS Homometry 73 EXAMPLE 2 74 Perspectives of New Music EXAMPLE 3 Homometry 75 Today Forte’s ”Z-related pairs” are generally known as homometric pairs, though the word “homometric” is not yet in the English dictionary. Forte acknowledges in a footnote that these sets were rst described in the literature by David Lewin in 1960, and he might also have gone back to the earlier appearance of this idea in an article by Lindo Patterson in 1944, showing how certain crystal formations are also similar when they have the same sets of differences. Despite the unique and attractive qualities of homometric pairs, very few composers have used them in constructing their harmonies. I have read remarks about how Elliott Carter used the all-interval tetrachords, but really only as details. I am probably the only composer who has produced entire compositions with all-interval tetrachords. The two all-interval tetrachords (0,1,4,6) and (0,1,3,7), with their inversions and transpositions make a list of 48 chords and I wanted to use all 48 in a piano piece called Intervals (2013). The piece consists of short sections that single out each of the six intervals. Example 2 is a diagram of the 48 chords with their tritones singled out, followed by the piano music allowing us to hear the sequence (Example 3). My second piece using only the all-interval tetrachords is a twentyminute opera composed for a production of Grand Magasin in the Festival d’Automne in Paris in 2016. The lyrics, taken from the Essai sur l’entendement of Gottfried Wilhelm Leibnitz, deals with the tiny differences between leaves of the same tree, and the music demonstrates this with two utes and two sopranos. Example 4 shows one section using one of the (0,1,4,6) tetrachords. EXAMPLE 4 76 Perspectives of New Music THE MATHEMATICS The thinking...

Homometry

HOMOMETRY TOM JOHNSON BEFORE THE MATHEMATICS T WAS ALL RIGHT THERE on page 1 of Allen Forte’s seminal 1973 book, The Structure of Atonal Music. Taking an example of the (0,1,4,6) tetrachord as found in Schoenberg’s George Lieder and in Webern’s Six Pieces for Orchestra, Forte wrote: This pitch combination, which is reducible to one form of the allinterval tetrachord, has a very special place in atonal music. It could occur in a tonal composition only under extraordinary conditions, and even then its meaning would be determined by harmonic-contrapuntal constraints. Here, where such constraints are not operative, one is obliged to seek other explanations. Accordingly, in the sections that follow, terminology and notation will be introduced which will facilitate the discussion of certain properties of such combinations. The rst task is to formulate a more general notion to replace that of pitch combination. So the two all-interval tetrachords become the key example of pitchclass sets that go beyond the tonal language. Why? Because they are I 72 Perspectives of New Music homometric. As shown in Example 1, the elements of (0,1,3,7) and (0,1,4,6) have the same six differences, one of each. Twenty pages later, Forte is more speci c: One might conclude that each prime form has a distinct interval vector. That is, however, not the case. For example, the two forms of the all-interval tetrachord, 4-Z15 and 4-Z29, are so constructed that they are not reducible by transposition or by inversion followed by transposition to a single prime form, but they have identical vectors (111111). Thus, he singles out the all-interval tetrachord rather than all the other dissonant chords and clusters of atonal music because it is homometric. He goes on to speak generally of this category, which he calls Z-related pairs: There are 19 such pairs in all. In addition to the one of cardinal number 4, there are three pairs of cardinal number 5 and 15 pairs of cardinal number 6. Always concerned to base his ideas on what he found in actual scores, he goes on to show two Z-related pairs as they occur in three measures of Webern’s Three Short Pieces, Op. 11 No. 1. EXAMPLE 1: ALL-INTERVAL TETRACHORDS Homometry 73 EXAMPLE 2 74 Perspectives of New Music EXAMPLE 3 Homometry 75 Today Forte’s ”Z-related pairs” are generally known as homometric pairs, though the word “homometric” is not yet in the English dictionary. Forte acknowledges in a footnote that these sets were rst described in the literature by David Lewin in 1960, and he might also have gone back to the earlier appearance of this idea in an article by Lindo Patterson in 1944, showing how certain crystal formations are also similar when they have the same sets of differences. Despite the unique and attractive qualities of homometric pairs, very few composers have used them in constructing their harmonies. I have read remarks about how Elliott Carter used the all-interval tetrachords, but really only as details. I am probably the only composer who has produced entire compositions with all-interval tetrachords. The two all-interval tetrachords (0,1,4,6) and (0,1,3,7), with their inversions and transpositions make a list of 48 chords and I wanted to use all 48 in a piano piece called Intervals (2013). The piece consists of short sections that single out each of the six intervals. Example 2 is a diagram of the 48 chords with their tritones singled out, followed by the piano music allowing us to hear the sequence (Example 3). My second piece using only the all-interval tetrachords is a twentyminute opera composed for a production of Grand Magasin in the Festival d’Automne in Paris in 2016. The lyrics, taken from the Essai sur l’entendement of Gottfried Wilhelm Leibnitz, deals with the tiny differences between leaves of the same tree, and the music demonstrates this with two utes and two sopranos. Example 4 shows one section using one of the (0,1,4,6) tetrachords. EXAMPLE 4 76 Perspectives of New Music THE MATHEMATICS The thinking...