Music Theory and Mathematics

Music Theory and Mathematics: Chords, Collections, and Transformations

JACK DOUTHETT
MARTHA M. HYDE
CHARLES J. SMITH
Copyright Date: 2008
Edition: NED - New edition
Published by: Boydell and Brewer,
Pages: 278
https://www.jstor.org/stable/10.7722/j.ctt14brswb
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  • Book Info
    Music Theory and Mathematics
    Book Description:

    The essays in Music Theory and Mathematics: Chords, Collections, and Transformations define the state of mathematically oriented music theory at the beginning of the twenty-first century. The volume includes essays in diatonic set theory, transformation theory, and neo-Riemannian theory -- the newest and most exciting fields in music theory today. The essays constitute a close-knit body of work -- a family in the sense of tracing their descent from a few key breakthroughs by John Clough, David Lewin, and Richard Cohn in the 1980s and 1990s. They are integrated by the ongoing dialogue they conduct with one another. The editors are Jack Douthett, a mathematician and music theorist who collaborated extensively with Clough; Martha M. Hyde, a distinguished scholar of twentieth-century music; and Charles J. Smith, a specialist in tonal theory. The contributors are all prominent scholars, teaching at institutions such as Harvard, Yale, Indiana University, and the University at Buffalo. Six of them (Clampitt, Clough, Cohn, Douthett, Hook, and Smith) have received the Society for Music Theory's prestigious Publication Award, and one (Hyde) has received the ASCAP Deems Taylor Award. The collection includes the last paper written by Clough before his death, as well as the last paper written by David Lewin, an important music theorist also recently deceased. Contributors: David Clampitt, John Clough, Richard Cohn, Jack Douthett, Nora Engebretsen, Julian Hook, Martha Hyde, Timothy Johnson, Jon Kochavi, David Lewin, Charles J. Smith, and Stephen Soderberg.

    eISBN: 978-1-58046-747-6
    Subjects: Music, Mathematics

Table of Contents

  1. Front Matter
    (pp. i-iv)
  2. Table of Contents
    (pp. v-vi)
  3. Preface
    (pp. vii-x)
    Charles J. Smith
  4. Introduction
    (pp. 1-8)
    Norman Carey, Jack Douthett and Martha M. Hyde

    The essays in this collection celebrate the work of the late John Clough, a revolutionary musical thinker and a pioneer in enquiry into the nature of diatonic systems. Clough, who held the Slee Chair of Music Theory at University at Buffalo (SUNY) for many years, brought to music theory new perspectives in four roughly chronological phases.

    The first phase of Clough’s work, in the late 1970s, focused on the definition of diatonic sets and an enquiry into interval cycles and sequences; he became interested in extending Allen Forte’s atonal methodology to the diatonic system. Milton Babbitt and Carlton Gamer, among...

  5. Chapter One “Cardinality Equals Variety for Chords” in Well-Formed Scales, with a Note on the Twin Primes Conjecture
    (pp. 9-22)
    David Clampitt

    Researchers have, in the past several decades, used formal approaches to diatonic theory in an attempt to show why the features of certain pitch collections have had such appeal for composers.¹ The results relate either to what musicians have discovered they can do with a given collection—through moves, routines, or processes within the collection, or through manipulation of the collection itself—or to how a given collection functions cognitively, based upon measures of symmetry versus asymmetry, simplicity versus complexity, or information versus redundancy.²

    Investigations of the first type use transformational theory and analysis. For example, harmonic triads, and the...

  6. Chapter Two Flip-Flop Circles and Their Groups
    (pp. 23-48)
    John Clough

    In neo-Riemannian theory, an essential construct is the cycle (or circle) of triads, alternately major and minor. Figure 2.1 shows four such cycles, called “hexatonic systems” by Cohn, which partition the set class (sc) of consonant triads into four subclasses of six triads each.¹ The four systems are transpositions of each other, and they embody relations traceable to Riemann:Parallel(P) (equivalent to Riemann’sQuintwechsel) andLeittonwechsel(L) apply alternately as we make our way around any of the circles.² The meanings of these terms, based on double common-tone retention, are evident from the context.

    Cohn also describes threeoctatonic...

  7. Chapter Three Pitch-Time Analogies and Transformations in Bartók’s Sonata for Two Pianos and Percussion
    (pp. 49-71)
    Richard Cohn

    This paper describes an unusually strong relationship between pitch and rhythm in the first movement of Béla Bartók’s Sonata for Two Pianos and Percussion, composed in 1937. The movement contains four distinct themes, three of which are used in dialogue with the classical sonata tradition. The fourth theme is a nine-note motto from the movement’sLentoopening which reappears as an up-tempo ostinato in the development section. Following in the Beethoven tradition,¹ the four themes are strongly individuated in both their tonal characteristics and their rhythmic profiles. Were we to represent the themes on a harmonic map, and, independently, on...

  8. Chapter Four Filtered Point-Symmetry and Dynamical Voice-Leading
    (pp. 72-106)
    Jack Douthett

    Until now, diatonic systems and neo-Riemannian transformations have generally been considered separately, and group and graph theoretic approaches have dominated the neo-Riemannian and transformational theory literature.¹ In this paper, an alternative approach will be explored; techniques similar to those used in the study ofdynamical systemsin science will be adopted to study neo-Riemannian theory and its connection to diatonic theory.

    Dynamical systems are probably best known today for the fractals they sometimes generate (e.g., Koch’s snowflake, the Dragon curve, Mandelbrot’s set, the Julia set, etc.), but fractals are only part of this field of study. As Strogatz puts it...

  9. Chapter Five The “Over-Determined” Triad as a Source of Discord: Nascent Groups and the Emergent Chromatic Tonality in Nineteenth-Century German Harmonic Theory
    (pp. 107-136)
    Nora Engebretsen

    Neo-Riemannian theory’s demonstration of the susceptibility to group-theoretic interpretation of elements of Hugo Riemann’s theories has sparked a reappraisal of nineteenth-century harmonic theory, focusing on its nascent grouptheoretic content. This is evident in the resurgence of interest in Riemann’sSchritt/Wechselsystem (which can be interpreted as a formulation of a group isomorphic to the neo-Riemannian LPR group and familiar Tn/TnI group), and in Richard Cohn’s explorations of connections between neo-Riemannian theory and Carl Friedrich Weitzmann’s 1853 monograph on the augmented triad.¹ In his essay “Introduction to Neo-Riemannian Theory: A Survey and a Historical Perspective,” Cohn presents a broader view,...

  10. Chapter Six Signature Transformations
    (pp. 137-160)
    Julian Hook

    Among the many distinctive features of the SchubertValse sentimentalein A major reproduced in figure 6.1, the various adventures of the four-eighth-note pickup figure (first appearing as C#–E–D–C# in measure 2) merit special attention, as does the sudden appearance of the key of C# major in the second strain.¹ We consider the latter first. This chromatic mediant C# major could be explained in traditional terms as a product of mixture, a sort of parallel substitution for the diatonic mediant, C# minor. This explanation could be modeled using neo-Riemannian transformations as shown in the simple network of...

  11. Chapter Seven Some Pedagogical Implications of Diatonic and Neo-Riemannian Theory
    (pp. 161-173)
    Timothy A. Johnson

    John Clough’s contributions have tremendous potential for the teaching of music theory at all levels. He used a rigorous style more inviting to professional music theorists than to students, but his work can be relevant and interesting even to beginners. Beginning students need some help, however—not because the work of Clough and his collaborators lacks clarity, but because the students lack the necessary background.

    This chapter introduces several of Clough’s ideas to beginning students in music theory. Some of my prior work suggests methods and materials for introducing certain aspects of diatonic theory to beginning music students.¹ My textbook...

  12. Chapter Eight A Parsimony Metric for Diatonic Sequences
    (pp. 174-196)
    Jonathan Kochavi

    Sequences use familiar harmonies but combine them with a grammar quite separate from functional harmony. Let us assume that we are working in diatonic pitch-class space, and suppose each sequential unit is to be made of up exactly two chords. Building on preliminary work from 1996, John Clough explored the application and adaptation of neo-Riemannian theory to such a context.¹ Using the nomenclature of Hook’s uniform triadic transformations, Clough proposed that the binary opposition of major and minor triads that underlies so much neo-Riemannian theory could be extended to chord types found in such sequences.² Instead of dichotomizing harmonic triads...

  13. Chapter Nine Transformational Considerations in Schoenberg’s Opus 23, Number 3
    (pp. 197-221)
    David Lewin

    Schoenberg’s Opus 23 , no. 3 has a clear subject, with a clear prime transpositional level: Bb₄–D₄–E₄–B₃–C#₄. The subject, with its thematic registral contour, opens the piece in the right hand. It recurs, transposed up 7 semitones, to open the next section of the piece (m. 6, right hand), and, an octave below its prime transpositional level, in the left hand of measure 6. It recurs, at its prime transpositional level, as a cantus firmus that opens the next section of the piece (mm. 9–10,ruhig). It recurs an octave below its opening level, below...

  14. Chapter Ten Transformational Etudes: Basic Principles and Applications of Interval String Theory
    (pp. 222-244)
    Stephen Soderberg

    The following study is based on a general definition of astringof any objects such as that of Eric Weisstein: “A string of lengthnon an alphabetlofmobjects is an arrangement ofnnot necessarily distinct symbols froml. There aremnsuch distinct strings.”¹ We will begin by defining the alphabet as a set of intervals between pitches in order to introduce six string transformations. We will then begin to redefine the alphabet in a series of studies taking us from a traditional pitch-centric viewpoint toward basic digital manipulations of electro-acoustic source material.

    A...

  15. Works Cited
    (pp. 245-252)
  16. List of Contributors
    (pp. 253-256)
  17. Index
    (pp. 257-262)
  18. Back Matter
    (pp. 263-269)