Terms & Concepts

1. Abstract algebra {WP}
• Algebraic structures {WP}
  Sets with specific operations acting on their elements
  
• Groups {WP}, Rings {WP}, and Fields {WP}
• Lattice {WP}
• Operations {WP}
• Sets {WP}
  
• See also: Set theory and Group theory
  
2. Acoustics {WP} (Rossing, et al. 2002)
   
• Combination tone {WP}
• Harmonic series {WP}
• Helmholtz, On the Sensations of Tone (1863) {WP} (Helmholtz 1863)
• Musical tone {WP}
• Pitch, Intensity, Duration, and Timbre
• Pure tone {WP}
• Sound
• Waveform modal {WP}
  
• Spectr {WP}
  
• -----
  
• See also: Psychoacoustics {WP}
3. Algebraic topology {WP}
• See also: Abstract algebra (above); Topology (below)
  
• Homeomorphism {WP}
  
• Invariant {WP}
4. Art
   
• Herrad of Landsberg, Hortus deliciarum (12th c.) {WP}
• Philosophy and the seven liberal arts {WP}
• Bible moralisée (c. 1250) {WP}
• God the Geometer {WP}, illumination
• Robert Flood, Temple of Music (1617-1618) {BibliOdyssy}
  
• Johannes Kepler (1671-1730) {BA; WP}
  
• Mysterium Cosmographicum (1596) {WP}
• Platonic solids {WP}, illustration
  
• Athanasius Kircher (1602-1680) {WP}
  
• Musurgia Universalis (1650) {WP}
• Frontispiece {WP}
• William Blake (1757-1827) {BA; WP}
  
• Europe a Prophecy (1794) {WP}
• The Ancient of Days, frontispiece
• Newton (1795), monotype {WP}
  
• Vassily Kandinsky(1866-1944( {MoMA}
  
• Impression III (Konzert) (1911) {MoMA}
• Yellow-Red-Blue (1925) {Centre Pompidou}
  
• M.C. Escher (1898-1972) {BA; WP}
  
• Drawing Hands (1948) {WP}
  
• Ascending Descending (1960) {WP}
• Waterfall (1961) {WP}
• Mobius Strip II (1963) {National Gallery of Art}
• Aaron Koblin (b. 1982), Flight Patterns (2011) {Art Institute of Chicago; TED}
  
5. Analysis {Grove} (Bent & Pople 2001}
   
• Corpus analysis (White 2022)
• Computational music theory {WP}
• Graphic music analysis
• See Straus, The Art of Post-Tonal Analysis (Straus 2022)
• Geometric approaches
  
• Tymoczko, A Geometry of Music (Tymoczko 2011)
• Cohn, Audacious Euphony (Cohn 2012)
• Toussaint, The Geometry of Musical Rhythm (Toussaint 2013)
• Hook, Exploring Musical Spaces (Hook 2023)
6. Ancient Greek philosophers and mathematicians
• Pythagoras {BA}
  
• Socrates {BA}
  
• Plato {BA}
• Allegory of the cave {WP}
• Analogy of the divided line {WP}
  
• Timaeus {WP}
  
• Republic {WP}
  
• Aristotle {BA}
• Euclid {BA}
• Elements {WP}
  
7. Arithmetic {WP}
   
• Modular Arithmetic {WP}
  
8. Boethius, Fundamentals of Music (c. 1490) {WP} (Bauer 1989)
   
9. Category theory {WP; SEP}
   
10. Change ringing {WP}
11. Chaos theory {BA; WP}
    
12. Chromatic circle {WP}
    
13. Circle of fifths {WP}
• Circle-of-fifths transform (McCartin 1998)
  
14. Collections, see Scale Theory
    
15. Color theory
    
• Color music {BA}
  
• Color wheel {BA}
16. Coltrane circle {Open Culture}
17. Combinatorics {WP}
• Binomial theorem {WP}
  
• Pascal's triangle {WP}
  
• Mathematical Objects
  
• Set {MW}; unordered, no duplications
  
• Multiset {MW}; unordered, with duplications
  
• Permutation {MW}; ordered, no duplications
  
• Sequence {MW}; ordered, with duplications
• Necklaces and bracelets {WP; Davies 2009}
  
• Other
• Pigeonhole principle {WP}
• Enumeration {WP}
  
18. Consonance and dissonance {WP}
    
19. Complexity {WP}
    
20. Corpus studies {TCL} (Shanahan, Burgoyne and Ian Quinn 2022)
    
21. Cymatics {WP}
    
22. Data science {WP}
    
23. Diatonic set theory {WP} (Johnson 2008)
• Generated collection {WP}
• Interval cycles {WP}
• Maximal evenness {WP}
• -----
  
• Diatonic dualism {WP}
• Specific and generic intervals {WP}
• Myhill Property
• Deep scale property (jpg) {WP}
• Cardinality equals variety {WP}
• Structure equals multiplicity {WP}
  
24. Dimension {WP}
    
25. Euclid, Elements (c. 300 BCE) {WP} (Fitzpatrick 2008)
• Full text (English): Fitzpatrick 2008
• Manuscript: WP}
  
26. Field {WP}
    
27. Field extensions {WP}
28. Figure-Ground {WP}
    
29. Fourier transform {WP}
    
30. Fractal {WP}
• Fractal dimension {WP}
• Mandelbrot set {WP}
• Sierpinski triangle {WP}
• Self similarity {WP}
31. Function {WP}
• Composition {WP}
  
32. Geometric objects
• Point {WP}
  
• Line {WP}
• Line segment
  
• Circle {WP}
• Polygon {WP}
• List of polygons {WP}
  
• Other
• Double helix {WP}
• Golden ratio {WP}
• Helix {WP}
• Klein bottle {WP}
  
• Line {WP}
• Lissajous curve {WP}
  
• Möbius strip {WP}
• Platonic solids {WP}
• Sphere {WP}
  
• Spiral {WP}
• Tiling, or tessellation {WP}
33. Geometric transformations {WP}
    
• Translation {WP}
  
• Rotation {WP}
• Reflection {WP}
• Glide Reflection {WP}
• -----
• Isometry {WP}
  
• Map {WP}
• Scaling {WP}
  
34. Geometry {WP}
• Euclidean geometry {WP}
• Non-Euclidean geometry {WP}
• Differential geometry {WP}
  
35. Graph theory {WP}
    
• Graph
• Verticies, or nodes
  
• Edges
• Directed vs. undirected
  
• Hamiltonian path {HP}
  
36. Group theory {WP}
    
• Abelian group {WP}
• Cyclic group {WP}
• Dihedral group {WP}
• Group, definition {WP}
• Listing of small groups {WP}
• Symmetry group {WP}
• Wallpaper group {WP}
• -----
  
• Musical groups:
• T_n/T_nI
• PLR group
• Serial-four group (Morris 2007)
37. Harmonic series {WP}
38. Harmony {BA}
39. Illusions {Deutsch}
    
• Musical:
• Shepard tones
• Tritone paradox
  
• Visual:
• Ruben vase {WP}
  
• Nekar cube {WP}
  
• Penrose triangle {WP}
  
• Penrose stairs {WP}
40. Information theory {WP} (Shannon 1948; Pierce 1980)
41. Inversion, see Transformation
42. Isomorphism {WP}
    
43. Klein four-group {WP}
44. Liberal arts {WP}
• Quadrivium (arithmetic, geometry, music, astronomy) {WP}
  
• Trivium (Grammar, Logic, Rhetoric) {WP}
  
45. Line of fifths {JoMM} (Moss et al. 2023)
46. Machine learning {WP}
    
47. Mathematical fields {Mathworld}
    
• Abstract Algebra {WP}
• Group theory {WP}
  
• Algebra {WP}
  
• Arithmetic {WP}
  
• Discrete Mathematics {WP}
• Combinatorics {WP}
  
• Graph theory {WP}
• Number theory {WP}
  
• Set Theory {WP}
  
• Geometry {WP}
• Euclidean geometry {WP}
  
• Non-Euclidean geometry {WP}
  
• Number theory {WP}
  
• Probability {WP} & Statistics {WP}
• Topology {WP}
48. Mathematicians/scientists
    
• List of geometers {WP}
• -----
  
• Pythagoras (c. 570-490 BCE) {BA; WP}
  
• Euclid (fl. 300 BCE) {BA; WP}
  
• Ptolemy (100-170) {BA; WP}
  
• René Descartes (1596-1650) {BA; WP}
• Marin Mersenne (1588-1684) {BA; {WP}
  
• Isaac Newton (1643-1727) {BA; WP}
• Leonhard Euler (1707-1783) {BA; WP}
• Carl Friedrich Gauss (1777-1855) {WP}
• Joseph Fourier (1768-1830) {WP}
  
• Évariste Galois (1811-1832) {BA; WP}
  
• Bernhard Riemann (1826-1866) {BA; WP}
• Georg Cantor (1845-1918) {WP}
  
• Felix Klein (1849-1925) {WP}
• Henri Poincaré (1854-1912) {WP}
  
• Noam Elkies (b. 1966) {WP}
49. Metaphor {WP}
    
50. Microcosm-macrocosm analogy {WP}
51. Music of the spheres {WP}
52. Modernism (Straus 2018)
    
• Conflicting textural layers
• Fractured forms
• Immobilized harmonies
• Radical simplification
  
• Radical complexity
53. Modulation {WP}
54. Music {WP}
    
• Linguistics {WP}
  
• Mathematics {WP}
  
• Philosophy {WP}
• Psychology {WP}
• Science {WP}
  
55. Music theory
    
• Counterpoint {WP}
  
• Neo-Riemannian theory (Cohn 2012)
• Post-tonal theory
• Pitch-class set theory {OMT2}
  
• Twelve-tone theory {OMT2}
  
• Rhythm theory, see below
• Scale theory, see below
  
• Schema theory {WP} (Gjerdingen 2007)
• Geometrical music theory (Hall 2008, Tymoczko 2011, Tymoczko 2013, Toussaint 2013, Tymoczko 2023)
• Transformational theory (Lewin 1987)
• Triadic progressions {OMT2}
56. Musical illusions (Deutsch) {Deutsch}
    
• Pitch circularity {WP}
• Speech-to-song illusion {WP}
  
• Tritone paradox {WP}
  
57. Musicality {WP}
    
58. Negative harmony (Levy 1985)
    
59. Neo-Riemannian theory {WP} (Cohn 2012)
• Tonnetz (Ger., 'tone net') {WP}
60. Nonchord tone, Nonharmonic tone, Embellishing tone {WP}
    
61. Number {WP} (Guedj 1997)
• Figurate numbers {WP}
62. Number theory {WP} (Hardy & Wright 2008/1938)
• Number sets {WP}
  
• Natural numbers () {WP}
• Integers () {WP}
• Rational numbers () {WP}
• Real numbers () {WP}
• Irrational numbers {WP}
• Complex numbers () {WP}
• See also:
• Lists of types of numbers {WP}
• Venn diagram 1 {WP}; Venn diagram 2 {WP}
  
• Modular arithmetic {WP; MW}
  
• mod 12, mod 7, mod 8, mod 16, etc.
  
63. Numerology {WP}
    
64. Operation (mathematics) {WP}
    
65. Pattern
• Pattern in mathematics  (Devlin 1994)
• Pattern in music {JMM} (Conklin 2021)
• Pattern-free music (Rickard 2012)
  
• Repetition in Music (Margulis 2013)
66. Perspective {WP}
    
67. Philosophy {BA; WP}
    
68. Pitch {WP}
• Pitch circularity {WP; Deutsch}
  
• Chroma {WP}
• Helical model of pitch
• Shepard tone {WP}
• Scientific pitch notation {WP}
  
69. Pitch-class set theory (Straus 2016)
    See also: BAIN MUSC 525
    
• Pitch-class set
• Intervals
• opci
• ic
  
• Interval vector
  
• Normal form
• T_n & T_nI
• Prime form
• M_n
  
• Set class
• under T_n
  
• under T_n/T_nI
• under T_nM/T_nMI
  
• Interval cycles
• Maximally-even collections
• Nearly-even collections
• Complement union property
  
70. Polygon
• Polygon diagram
  
• Polygon diagonal {WP}
• List of polygons {WP}
  
71. Platonic solids {WP}
• Tetrahedron
• Cube
• Octahedron
• Dodecahedron
• Icosahedron
  
72. Prime number {WP}
• Coprime integers {WP}
• Euclidean algorithm {WP}
• Factorization {WP}
• Fundamental theorem of arithmetic {WP}
  
• Greatest common divisor {WP}
  
• Ulam spiral {WP}
  
73. Polygon {WP}
• Convex polygon {WP}
  
• Inscribed polygon 
  
• See Concyclic points {WP}
  
• Regular polygon {WP}
  
• Star polygon {WP}
  
• Schläfli symbol {WP}
  
74. Projection {WP}
75. Proportionality {WP}
76. Pythagoreanism {BA}
77. Randomness {WP}
    
• Pseudorandom {WP}
  
• Stochastic music {WP}
  
• Markov model {WP}
  
78. Quadrivium {WP}
    
79. Quadruple hierarachy (Tymoczko 2023)
80. Ratio {WP}
    
81. Real coordinate space (Rⁿ) {WP}
    
82. Reflection {WP}
83. Register {WP}
    
84. Representation {WP}
    
85. Rhythm {BA;  WP}
    The Geometry of Musical Rhythm (GoMR) (Toussaint 2019/2013)
    See also: Cooper & Meter 1960; Lerdahl and Jackendoff 1983; Kramer 1988; Hasty 1997; Arom 2004; Sethares 2007; London 2012; Margulis 2014
    
• Combinatorics {WP; GoMR Ch. 40)
  
• Pigeonhole principle {WP}
  
• Binomial coefficient {WP}
  
• Necklaces and Bracelets (GoMR Ch. 14)
• Cross-rhythm {WP}
• Distance geometry (GoMR Ch. 8)
  
• Euclidean rhythms {WP} (GoMR Ch. 21)
  
• Inter-onset duration
• Adjacent inter-onset intervals (IOIs)
  
• Isochrony {WP} (GoMR CH. 2)
  
• Isomorphisms between rhythms and scales (GoMR Ch. 11)
  
• Meter {WP}
• Hypermeter {OMT2}
  
• Metrical dissonance {OMT2}
• Displacement dissonance
• Grouping dissonance
• Syncopation
  
• Multi-level rhythm
• Ostinato {WP}
• Polyrhythm {WP}
• Syncopation {WP}
  
• Tempo {WP}
• Time-unit box system (TUBS), or box notation {WP}
• Timelines (GoMR Ch. 3)
  
• Clave son {WP} (GoMR Ch. 6)
• Tresillo {WP}
• Bembé
  
• Almost maximally even timelines (GoMR Ch. 20)
  
• Six distinguished timelines (GoMR Ch. 7)
  
• Shiko, Son, Soukous, Rumba, Bossa-Nova, and Gahu
• Rhythmic canons (GoMR Ch. 31)
  
• Visualization of rhythms (GoMR Ch. 35)
  
86. Scale theory (Tymoczko 2008)
    
• Scales, modes, and collections
  
• Scale {WP}
• Diatonic modes {OMT2}
  
• Collections {OMT2}
• Asymmetrical
• Diatonic (DIA)
  
• Pentatonic (PENT)
  
• Acoustic (AC)
  
• Harmonic minor (hm) and harmonic major (HM)
  
• Symmetrical
• Whole tone (WT)
  
• Octatonic (OCT)
  
• Hexatonic (HEX)
  
• Messiaen's Modes of Limited Transposition (MLT) {Morris} (Messiaen 1944)
• Tymoczko, Scale Networks
• Pressing scales (Tymoczko 2004) {Norris.com}
• Scale Finder {MMS}
• Ring, Amazing Amazing Scale Finder (Ring 2009)
• Jazz
• Russell, The Lydian Chromatic Concept of Tonal Organization for Improvisation {WP} (Russell 1953)
• Other
• Slonimsky, Thesaurus of Scales and Melodic Patterns (Slonimsky 1947)
• Persichetti, Twentieth-Century Harmony (Persichetti 1961)
87. Schema theory {WP} (Gjerdingen 2007)
• Galant schemas {OMT2}
• List of galant schemas, selected {OMT2}
  
• Rule of the octave {OMT2}
88. Sequence (mathematics)
• Arithmetic {WP}
• Harmonic {WP}
• Geometric {WP}
• Triangular number {WP}
• Square number {WP}
  
• Fibonacci {WP}
• Prime {WP}
• Other:
  
• Per Nørgård's Infinity Series
  
• -----
  
• On-Line Encyclopedia of Integer Sequences {OEIS}
89. Series (mathematics) {WP}
    
90. Set theory {WP}
    
• Set {WP}
  
• Operation {WP}
  
• Venn diagram {WP}
• Equivalence relation {MW}
  
• See also: Combinatorics
  
91. Sonification {WP}
    
92. Spaces (Hook 2023)
• Musical
• Pitch space
• Pitch-class space
• Contour space
• OPTIC spaces
  
• Mathematical
• Euclidean space {WP}
• Metric space {WP}
• Topological space {WP}
  
93. Symmetry {WP} (du Sautoy 2008)
• See also: Group theory
  
94. Synesthesia {WP}
    
95. Tetractys {WP}
    
96. Tonality {WP}
    
97. Topic theory {TCL} (Danuta 2014)
    
98. Topology {WP} (Andreatta, Amiot and Yust 2025) {See: JoMM 14/2}
    
• See also: Algebraic topology (above)
  
• Manifold {WP}
• Orbifold {WP}
• Topological space {WP}
• Torus {WP; MW}
99. Transformational theory {WP} (Lewin 1987) 
• Transposition: Tn
• Inversion: I, TnI, I_n, and I^x/y
• Multiplication: Mn
• Transformational networks {Roeder 2014}
• Triadic transformations {WP}; e.g.,
  
• P, L & R
• P', L' & R', also called S, N & F
  
• Hexpole (H), or LPL: e.g., C+ -> Ab-
• Chromatic mediants (Kopp 2002; Lehman 2018)
• Cycles and chains
  
• PLR cycle
• PL cycle (HEX)
  
• PR cycle (OCT)
• LR cyele (WH/DIA)
  
• PLR cyele (common tone)
  
• etc.
100. Trigonometry {WP}
     
101. Trivium {WP}
     
102. Tuning theory {WP} (Gann 2019)
     See also: BAIN MUSC 726T
     
• Pythagorean tuning {WP}
• Just intonation {WP}
• Mean-tone temperament {WP}
  
• Equal temperament {WP}
• -----
• Ancient Greek scales (Chalmers 1992)
• Tetrachords
  
• Harmonic series {WP}
  
103. Twelve-tone system {WP} (Morris 2007; Straus 2016)
• The series, or row
• Canonical serial operations
• P, R, I & RI
• Transposition (Tn)
• Other operations: IR, ROT, etc.
104. Visual system {WP}
105. Visualization {WP}
106. Voice leading {WP} (Huron 2016; also: Morris 1998; Tymoczko 2011; Cohn 2012; Straus 2014; Tymoczko 2023)
     
• Efficient voice leading
• Parsimonious voice leading
• Voice leading between scales
• Atonal voice leading
• Voice leading space

Legend
BA - Britannica Academic
MW -Mathworld
TCL - Thomas Cooper Library
WP - Wikipedia

See: MUSC 726G Bibliography

Dictionaries & Encyclopedias

Britannica Academic (BA) –https://academic.eb.com/levels/collegiate

European Mathematical Society, Encyclopedia of Mathematics –  https://encyclopediaofmath.org/wiki/Main_Page

Grove Music Online {Grove} – https://www.oxfordmusiconline.com/grovemusic

Randel, Don, ed. 2003. The Harvard Dictionary of Music, 4th ed. Cambridge, MA: Harvard University Press {Harvard Dictionary}

Weisstein, Eric. ed. 2024. Wolfram MathWorld. Available online at: <https://mathworld.wolfram.com>.