Mathematical Music Theory

Algebraic, Geometric, Combinatorial, Topological and Applied Approaches to Understanding Musical Phenomena

Mariana Montiel, Robert W Peck

• Publisher: WSPC
• Year: 2018
• ISBN:
  9789813235328
• Topic: Technology & Engineering
• Subtopic: Acoustical Engineering
• Index: Technology & Engineering

Questions about variation, similarity, enumeration, and classification of musical structures have long intrigued both musicians and mathematicians. Mathematical models can be found from theoretical analysis to actual composition or sound production. Increasingly in the last few decades, musical scholarship has incorporated modern mathematical content. One example is the application of methods from Algebraic Combinatorics, or Topology and Graph Theory, to the classification of different musical objects. However, these applications of mathematics in the understanding of music have also led to interesting open problems in mathematics itself.

The reach and depth of the contributions on mathematical music theory presented in this volume is significant. Each contribution is in a section within these subjects: (i) Algebraic and Combinatorial Approaches; (ii) Geometric, Topological, and Graph-Theoretical Approaches; and (iii) Distance and Similarity Measures in Music.

remove Contents:

• Section I:
  • From Musical Chords to Twin Primes (Jack Douthett, David Clampitt and Norman Carey)
  • Hypercubes and the Generalized Cohn Cycle (Jack Douthett, Peter Steinbach and Richard Hermann)
  • Associahedra, Combinatorial Block Designs and Related Structures (Franck Jedrzejewski)
  • Rhythmic and Melodic L-canons (Jeremy Kastine)
  • The Fibonacci Sequence as Metric Suspension in Luigi Nono's II Canto Sospeso (Jon Kochavi)
  • One Note Samba: Navigating Notes and Their Meanings Within Modes and Exo-modes (Thomas Noll)
  • Difference Sets and All-Directed-Interval Chords (Robert W Peck)
  • Harmonious Opposition (Richard Plotkin)
• Section II:
  • Orbifold Path Models for Voice Leading: Dealing with Doubling (James R Hughes)
  • Reflections on the Geometry of Chords (Thomas A Ivey)
  • Theoretical Physics and Category Theory as Tools for Analysis of Musical Performance and Composition (Maria Mannone)
  • Intuitive Musical Homotopy (Aditya Sivakumar and Dmitri Tymoczko)
  • Geometric Generalizations of the Tonnetz and Their Relation to Fourier Phases Spaces (Jason Yust)
  • Deterministic Geometries: A Technique for the Systematic Generation of Musical Elements in Composition (Brent A Milam)
• Section III:
  • Flamenco Music and Its Computational Study (Francisco Gómez)
  • Examining Fixed and Relative Similarity Metrics Through Jazz Melodies (David J Baker and Daniel Shanahan)
  • In Search of Arcs of Prototypicality (Daniel Shanahan)

Readership: Students and researchers in Mathematical Music Theory.Mathematics and Music;Algebra;Geometry;Topology;Graph Theory;Combinatorics;Distance and Similarity Measures;Discrete Fourier Transform0 Key Features:

• It includes the most prominent authors in the field
• It gathers a gamut of the most recent work in the field, which is something very difficult to find in one volume
• It will appeal to mathematicians, music theorists, and computer scientists. Within mathematics, it offers a variety of areas and techniques related to musical phenomena that cannot be found together in other volumes

Section I

Chapter 1

From Musical Chords to Twin Primes

Jack Douthett*, David Clampitt^† and Norman Carey^‡

* University of New Mexico, Albuquerque, NM, USA
^† The Ohio State University, Columbus, OH, USA
^‡ CUNY Graduate Center, New York, NY, USA

1.1.Introduction

A freshman music student is often taught by rote that each of the diatonic intervals (seconds through sevenths) come in two flavors: seconds and thirds in major and minor, fourths in perfect and augmented, fifths in diminished and perfect, and so on. The young musician learns to identify these intervals aurally and reproduce them in song. While this pairing of intervals for each scale degree might seem curious, within it lurks some non-trivial mathematics. Moreover, the musician might notice that within the diatonic scale there are three different triads (major, minor, and diminished) and every triad has three different notes, and that seventh chords come in four types and there are four notes in every seventh chord. But in general that is as far as it goes. And to try to attach meaning to the fact that the cardinalities of the diatonic scale (white keys) and its pentatonic complement (black keys) are twin primes would seem like weird numerology. But as it turns out, contemporary mathematical music theory draws these observations together. This chapter will trace these connections, collating some new results with some from the existing literature in music theory, mathematics, and physics.

1.2.Myhill’s Property and Well-Formed Scales

Musical scales are defined to be finite sets of pitches ordered according to pitch height (increasing fundamental frequencies), and periodic at some interval (usually the octave, associated with the frequency ratio 2:1). We understand scales to be equivalent under musical transposition, i.e., translation of all its points by a constant. In music theory there are two ways of conceptualizing scales, which we may call continuous background and discrete background, or continuous and discrete for short (both are discrete sets). A continuous scale is an ordered set of real-valued points on a mod 1 circle. If we order these points from smallest to largest, then a scale is defined as this ordered set or a cyclic rotation thereof, to within transposition. In Fig. 1.1(a), the scale is D = {0, log₂(9/8), log₂(3/2)}, where the base 2 logarithms reflect the fact that there is a natural equivalence relation in music based on the octave. The discrete model is similar except that the circumference is divided into c integer parts labeled 0 through c – 1 (equivalent to rational points 0, 1/c, . . . , (c – 1)/c, but the clock face diagram is conventional in music theory). Reflecting their application in music theory, these numbers are called pitch classes (pcs). In Fig. 1.1(b), c = 12, and the scale is D = {0, 2, 7} (according to music theory convention, the mod 12 pitch class 0 represents the note C, so D contains notes C, D, and G).

The scales in Fig. 1.1 have a property not generally found in an arbitrary scale. We call the intervals between adjacent points on the circle step intervals, and we measure generic intervals according to the number of step intervals they span. Thus step intervals have span 1, intervals that skip one intervening point have span 2, and so forth. Following Clough and Myerson [8, 9], and Clough and Douthett [7] we define the spectrum of a generic interval to be the set of specific interval sizes corresponding to the generic interval of a given span, and we designate this set by enclosing the span k in angle brackets. Note in Fig. 1.1(a) that the step intervals come in two sizes (the first spectrum is the doubleton

1

= {log₂(9/8), log₂(4/3)}), as is the case for the spectrum of span 2 (

2

= log₂(3/2), log₂(16/9)}). This is also true of Fig. 1.1(b), where

1

= {2, 5} and

2

= {7, 10}, the interval sizes understood to be modulo 12. If we let d be the cardinality of the discrete scale with background chromatic of cardinality c (1 ≤ d < c), then when each spectrum

k

, 1 ≤ k ≤ d – 1, is a doubleton, we say the scale has Myhill’s Property (MP), as initially defined in [8, 9]. This is the property observed in the introduction that every nonzero diatonic in...