View source for Music and mathematics

{{short description|Relationships between music and mathematics}}
[[Image:Spectrogram of violin.png|thumb|upright=1.5|A [[spectrogram]] of a violin waveform, with linear frequency on the vertical axis and time on the horizontal axis. The bright lines show how the spectral components change over time. The intensity colouring is logarithmic (black is −120 dBFS).]]

[[Music theory]] analyzes the [[Pitch (music)|pitch]], timing, and structure of music. It uses mathematics to study [[elements of music]] such as [[tempo]], [[chord progression]], [[Musical form|form]], and [[Meter (music)|meter]]. The attempt to structure and communicate new ways of composing and hearing music has led to musical applications of [[set theory]], [[abstract algebra]] and [[number theory]].

While music theory has no [[axiomatic]] foundation in modern mathematics, the basis of musical [[sound]] can be described mathematically (using [[acoustics]]) and exhibits "a remarkable array of number properties".<ref>[[Reginald Smith Brindle]], ''The New Music'', Oxford University Press, 1987, pp. 42–43</ref>

==History==
Though ancient Chinese, Indians, Egyptians and Mesopotamians are known to have studied the mathematical principles of sound,<ref>Reginald Smith Brindle, ''The New Music'', Oxford University Press, 1987, p. 42</ref> the [[Pythagoreanism|Pythagoreans]] (in particular [[Philolaus]] and [[Archytas]])<ref name="purwins">{{cite book|title=Profiles of Pitch Classes Circularity of Relative Pitch and Key-Experiments, Models, Computational Music Analysis, and Perspectives|pages=22–24|year=2005|url=http://ccrma.stanford.edu/~purwins/purwinsPhD.pdf|last1=Purwins | first1=Hendrik}}</ref> of ancient Greece were the first researchers known to have investigated the expression of [[musical scale]]s in terms of numerical [[ratio]]s,<ref>Plato (trans. Desmond Lee) ''The Republic'', Harmondsworth Penguin 1974, page 340, note.</ref> particularly the ratios of small integers. Their central doctrine was that "all nature consists of [[harmony]] arising out of numbers".<ref>Sir James Jeans, ''Science and Music'', Dover 1968, p. 154.</ref>

From the time of [[Plato]], harmony was considered a fundamental branch of [[physics]], now known as [[musical acoustics]]. Early [[Indian music scale|Indian]] and [[Chinese musicology|Chinese]] theorists show similar approaches: all sought to show that the mathematical laws of [[harmonic]]s and [[rhythm]]s were fundamental not only to our understanding of the world but to human well-being.<ref>Alain Danielou, ''Introduction to the Study of Musical Scales'', Mushiram Manoharlal 1999, Chapter 1 ''passim''.</ref> [[Confucius]], like Pythagoras, regarded the small numbers 1,2,3,4 as the source of all perfection.<ref>Sir James Jeans, ''Science and Music'', Dover 1968, p. 155.</ref>

==Time, rhythm, and meter==

Without the boundaries of rhythmic structure – a fundamental equal and regular arrangement of [[Pulse (music)|pulse]] [[Repetition (music)|repetition]], [[Beat (music)|accent]], [[Phrase (music)|phrase]] and duration – music would not be possible.<ref>Arnold Whittall, in ''The Oxford Companion to Music'', OUP, 2002, Article: ''Rhythm''</ref> Modern musical use of terms like [[Meter (music)|meter]] and [[bar (music)|measure]] also reflects the historical importance of music, along with astronomy, in the development of counting, arithmetic and the exact measurement of time and [[Frequency|periodicity]] that is fundamental to physics.{{Citation needed|date=October 2015}}

The elements of musical form often build strict proportions or hypermetric structures (powers of the numbers 2 and 3).<ref>{{cite web |title=Александр Виноград, Многообразие проявлений музыкального метра (LAP Lambert Academic Publishing, 2013)|url=https://www.morebooks.de/store/gb/book/%D0%9C%D0%BD%D0%BE%D0%B3%D0%BE%D0%BE%D0%B1%D1%80%D0%B0%D0%B7%D0%B8%D0%B5-%D0%BF%D1%80%D0%BE%D1%8F%D0%B2%D0%BB%D0%B5%D0%BD%D0%B8%D0%B9-%D0%BC%D1%83%D0%B7%D1%8B%D0%BA%D0%B0%D0%BB%D1%8C%D0%BD%D0%BE%D0%B3%D0%BE-%D0%BC%D0%B5%D1%82%D1%80%D0%B0/isbn/978-3-659-48005-8}}</ref>

==Musical form==
{{Main|Musical form}}
Musical form is the plan by which a short piece of music is extended. The term "plan" is also used in architecture, to which musical form is often compared. Like the architect, the composer must take into account the function for which the work is intended and the means available, practicing economy and making use of repetition and order.<ref>Imogen Holst, ''The ABC of Music'', Oxford 1963, p. 100</ref> The common types of form known as [[Binary form|binary]] and [[Ternary form|ternary]] ("twofold" and "threefold") once again demonstrate the importance of small integral values to the intelligibility and appeal of music.<ref>{{Cite journal|last1=Dreyfus|first1=Tommy|last2=Eisenberg|first2=Theodore|date=1986|title=On the Aesthetics of Mathematical Thought|url=https://www.jstor.org/stable/40247796|journal=For the Learning of Mathematics|volume=6|issue=1|pages=2–10|jstor=40247796|issn=0228-0671}}</ref><ref>{{Cite journal|last=Crocker|first=Richard L.|date=1963|title=Pythagorean Mathematics and Music|url=https://www.jstor.org/stable/427754|journal=The Journal of Aesthetics and Art Criticism|volume=22|issue=2|pages=189–198|doi=10.2307/427754|jstor=427754|issn=0021-8529}}</ref>

==Frequency and harmony==
[[Image:Chladini.Diagrams.for.Quadratic.Plates.svg|thumb|Chladni figures produced by sound vibrations in fine powder on a square plate. ([[Ernst Chladni]], ''Acoustics'', 1802)]]
A [[musical scale]] is a discrete set of [[pitch (music)|pitch]]es used in making or describing music. The most important scale in the Western tradition is the [[diatonic scale]] but many others have been used and proposed in various historical eras and parts of the world. Each pitch corresponds to a particular frequency, expressed in hertz (Hz), sometimes referred to as cycles per second (c.p.s.). A scale has an interval of repetition, normally the [[octave]]. The [[octave]] of any pitch refers to a frequency exactly twice that of the given pitch.

Succeeding superoctaves are pitches found at frequencies four, eight, sixteen times, and so on, of the fundamental frequency. Pitches at frequencies of half, a quarter, an eighth and so on of the fundamental are called suboctaves. There is no case in musical harmony where, if a given pitch be considered accordant, that its octaves are considered otherwise. Therefore, any note and its octaves will generally be found similarly named in musical systems (e.g. all will be called '''doh''' or '''A''' or '''Sa''', as the case may be).

When expressed as a frequency bandwidth an octave '''A<sub>2</sub>–A<sub>3</sub>''' spans from 110&nbsp;Hz to 220&nbsp;Hz (span=110&nbsp;Hz). The next octave will span from 220&nbsp;Hz to 440&nbsp;Hz (span=220&nbsp;Hz). The third octave spans from 440&nbsp;Hz to 880&nbsp;Hz (span=440&nbsp;Hz) and so on. Each successive octave spans twice the frequency range of the previous octave.

[[Image:4Octaves.and.Frequencies.svg|thumb|upright=2.2|left|The exponential nature of octaves when measured on a linear frequency scale.]]
[[Image:4Octaves.and.Frequencies.Ears.svg|thumb|upright=2.2|left|This diagram presents octaves as they appear in the sense of musical intervals, equally spaced.]]

Because we are often interested in the relations or [[ratio]]s between the pitches (known as [[Interval (music)|intervals]]) rather than the precise pitches themselves in describing a scale, it is usual to refer to all the scale pitches in terms of their ratio from a particular pitch, which is given the value of one (often written '''1/1'''), generally a note which functions as the [[tonic (music)|tonic]] of the scale. For interval size comparison, [[cent (music)|cent]]s are often used.
{{clear}}
[[File:Middle_C,_or_262_hertz,_on_a_virtual_oscilloscope.png|thumb|[[Oscilloscope|Oscillogram]] of [[C (musical note)|middle C]] (262 Hz). (Scale: 1 square is equal to 1 [[millisecond]])]]
[[File:C5_523_Hz_oscillogram.png|thumb|C5, an octave above middle C. The frequency is twice that of middle C (523 Hz).]]
[[File:C3_131_Hz_oscillogram.png|thumb|C3, an octave below middle C. The frequency is half that of middle C (131 Hz).]]

:{|class="wikitable"
!Common<br>term
!Example<br>name
!H<small>z</small>
!Multiple of<br>fundamental
!Ratio of<br>within octave
!Cents<br>within octave
|-
|{{center|Fundamental}}
|style="text-align:center;"|A<sub>2</sub>
|110
|style="text-align:center;"|<math>1x</math>
|style="text-align:center;"|<math>1/1 = 1x</math>
|{{right|0}}
|-
|rowspan=2 style="text-align:center;" |Octave
|rowspan=2 style="text-align:center;" |A<sub>3</sub>
|rowspan=2 |220
|rowspan=2 style="text-align:center;" |<math>2x</math>
|style="text-align:center;"|<math>2/1 = 2x</math>
|{{right|1200}}
|-
|style="text-align:center;"|<math>2/2 = 1x</math>
|{{right|0}}
|-
|{{center|Perfect Fifth}}
|style="text-align:center;"|E<sub>4</sub>
|330
|style="text-align:center;"|<math>3x</math>
|style="text-align:center;"|<math>3/2 = 1.5x</math>
|{{right|702}}
|-
|rowspan=2 style="text-align:center;" |Octave
|rowspan=2 style="text-align:center;" |A<sub>4</sub>
|rowspan=2 |440
|rowspan=2 style="text-align:center;" |<math>4x</math>
|style="text-align:center;"|<math>4/2 = 2x</math>
|{{right|1200}}
|-
|style="text-align:center;"|<math>4/4 = 1x</math>
|{{right|0}}
|-
|{{center|Major Third}}
|style="text-align:center;"|C{{music|#}}<sub>5</sub>
|550
|style="text-align:center;"|<math>5x</math>
|style="text-align:center;"|<math>5/4 = 1.25x</math>
|{{right|386}}
|-
|{{center|Perfect Fifth}}
|style="text-align:center;"|E<sub>5</sub>
|660
|style="text-align:center;"|<math>6x</math>
|style="text-align:center;"|<math>6/4 = 1.5x</math>
|{{right|702}}
|-
|{{center|[[Harmonic seventh]]}}
|style="text-align:center;"|G<sub>5</sub>
|770
|style="text-align:center;"|<math>7x</math>
|style="text-align:center;"|<math>7/4 = 1.75x</math>
|{{right|969}}
|-
|rowspan=2 style="text-align:center;" |Octave
|rowspan=2 style="text-align:center;" |A<sub>5</sub>
|rowspan=2 |880
|rowspan=2 style="text-align:center;" |<math>8x</math>
|style="text-align:center;"|<math>8/4 = 2x</math>
|{{right|1200}}
|-
|style="text-align:center;"|<math>8/8 = 1x</math>
|{{right|0}}
|}

==Tuning systems==
{{Main|Musical tuning|Musical temperament}}

There are two main families of tuning systems: [[equal temperament]] and [[just tuning]]. Equal temperament scales are built by dividing an octave into intervals which are equal on a [[logarithmic scale]], which results in perfectly evenly divided scales, but with ratios of frequencies which are [[irrational number]]s. Just scales are built by multiplying frequencies by [[rational number]]s, which results in simple ratios between frequencies, but with scale divisions that are uneven.

One major difference between equal temperament tunings and just tunings is differences in [[Beat (acoustics)|acoustical beat]] when two notes are sounded together, which affects the subjective experience of [[consonance and dissonance]]. Both of these systems, and the vast majority of music in general, have scales that repeat on the interval of every [[octave]], which is defined as frequency ratio of 2:1. In other words, every time the frequency is doubled, the given scale repeats.

Below are [[Ogg Vorbis]] files demonstrating the difference between just intonation and equal temperament. You might need to play the samples several times before you can detect the difference.
*[[Media:Comparison 550Hz - 554Hz.ogg|Two sine waves played consecutively]] – this sample has half-step at 550&nbsp;Hz (C{{music|#}} in the just intonation scale), followed by a half-step at 554.37&nbsp;Hz (C{{music|#}} in the equal temperament scale).
*[[Media:Major third comparison.ogg|Same two notes, set against an A440 pedal]] – this sample consists of a "[[Dyad (music)|dyad]]". The lower note is a constant A (440&nbsp;Hz in either scale), the upper note is a C{{music|#}} in the equal-tempered scale for the first 1", and a C{{music|#}} in the just intonation scale for the last 1". [[Beat (acoustics)|Phase]] differences make it easier to detect the transition than in the previous sample.

===Just tunings===

[[File:HarmonicIdentities.Names.Frequencies.svg|thumb|upright=2.5|The first 16 harmonics, their names and frequencies, showing the exponential nature of the octave and the simple fractional nature of non-octave harmonics.]]
[[File:Normalized harmonic identities, names, and frequencies.svg|thumb|upright=2.5|The first 16 harmonics, with frequencies and log frequencies.]]

[[5-limit tuning]], the most common form of [[just intonation]], is a system of tuning using tones that are [[regular number]] [[harmonic]]s of a single [[fundamental frequency]]. This was one of the scales [[Johannes Kepler]] presented in his [[Harmonices Mundi]] (1619) in connection with planetary motion. The same scale was given in transposed form by Scottish mathematician and musical theorist, Alexander Malcolm, in 1721 in his 'Treatise of Musick: Speculative, Practical and Historical',<ref>{{cite web|url=https://archive.org/details/treatiseofmusick00malc|title=A treatise of musick, speculative, practical and historical|first1=Alexander|last1=Malcolm|first2=Mr (Joseph)|last2=Mitchell|date=25 May 2018|publisher=Edinburgh : Printed for the author|via=Internet Archive}}</ref> and by theorist [[Jose Wuerschmidt]] in the 20th century. A form of it is used in the music of northern India.

American composer [[Terry Riley]] also made use of the inverted form of it in his "Harp of New Albion". Just intonation gives superior results when there is little or no [[chord progression]]: voices and other instruments gravitate to just intonation whenever possible. However, it gives two different whole tone intervals (9:8 and 10:9) because a fixed tuned instrument, such as a piano, cannot change key.<ref>Jeremy Montagu, in ''The Oxford Companion to Music'', OUP 2002, Article: ''just intonation''.</ref> To calculate the frequency of a note in a scale given in terms of ratios, the frequency ratio is multiplied by the tonic frequency. For instance, with a tonic of [[A440 (pitch standard)|A4]] (A natural above middle C), the frequency is 440&nbsp;[[hertz|Hz]], and a justly tuned fifth above it (E5) is simply 440×(3:2) = 660&nbsp;Hz.

{|class="wikitable"
! Semitone || Ratio || [[Interval (music)|Interval]] || [[Harmonic series (music)|Natural]] || Half Step
|-
! 0
| 1:1 ||  [[unison]] || 480 || 0
|-
! 1
| 16:15 ||  [[semitone]] || 512 || 16:15
|-
! 2
| 9:8 ||  [[major second]] || 540 || 135:128
|-
! 3
| [[sesquiquintum|6:5]] || [[minor third]] || 576 || 16:15
|-
! 4
| [[sesquiquartum|5:4]] || [[major third]] || 600 || 25:24
|-
! 5
| [[sesquitertium|4:3]] || [[perfect fourth]] || 640 || 16:15
|-
! 6
| 45:32 || diatonic [[tritone]] || 675 || 135:128
|-
! 7
| [[Superparticular ratio|3:2]] || [[perfect fifth]] || 720 || 16:15
|-
! 8
| 8:5 || [[minor sixth]] || 768 || 16:15
|-
! 9
| 5:3 || [[major sixth]] || 800 || 25:24
|-
! 10
| 9:5 || [[minor seventh]] || 864 || 27:25
|-
! 11
| 15:8 || [[major seventh]] || 900 || 25:24
|-
! 12
| 2:1 || [[octave]] || 960 || 16:15
|}

[[Pythagorean tuning]] is tuning based only on the perfect consonances, the (perfect) octave, perfect fifth, and perfect fourth. Thus the major third is considered not a third but a ditone, literally "two tones", and is (9:8)<sup>2</sup> = 81:64, rather than the independent and harmonic just 5:4 = 80:64 directly below. A whole tone is a secondary interval, being derived from two perfect fifths minus an octave, (3:2)<sup>2</sup>/2 = 9:8.

The just major third, 5:4 and minor third, 6:5, are a [[syntonic comma]], 81:80, apart from their Pythagorean equivalents 81:64 and 32:27 respectively. According to [[Carl Dahlhaus|Carl]] {{harvtxt|Dahlhaus|1990|p=187}}, "the dependent third conforms to the Pythagorean, the independent third to the harmonic tuning of intervals."

Western [[common practice period|common practice music]] usually cannot be played in just intonation but requires a systematically tempered scale. The tempering can involve either the irregularities of [[well temperament]] or be constructed as a [[regular temperament]], either some form of [[equal temperament]] or some other regular meantone, but in all cases will involve the fundamental features of [[meantone temperament]]. For example, the root of chord '''ii''', if tuned to a fifth above the dominant, would be a major whole tone (9:8) above the tonic. If tuned a just minor third (6:5) below a just subdominant degree of 4:3, however, the interval from the tonic would equal a minor whole tone (10:9). Meantone temperament reduces the difference between 9:8 and 10:9. Their ratio, (9:8)/(10:9) = 81:80, is treated as a unison. The interval 81:80, called the [[syntonic comma]] or comma of Didymus, is the key comma of meantone temperament.

===Equal temperament tunings===
In [[equal temperament]], the octave is divided into equal parts on the logarithmic scale. While it is possible to construct equal temperament scale with any number of notes (for example, the 24-tone [[Arab tone system]]), the most common number is 12, which makes up the equal-temperament [[chromatic scale]]. In western music, a division into twelve intervals is commonly assumed unless it is specified otherwise.

For the chromatic scale, the octave is divided into twelve equal parts, each semitone (half-step) is an interval of the [[twelfth root of two]] so that twelve of these equal half steps add up to exactly an octave. With fretted instruments it is very useful to use equal temperament so that the frets align evenly across the strings. In the European music tradition, equal temperament was used for lute and guitar music far earlier than for other instruments, such as [[musical keyboard]]s. Because of this historical force, twelve-tone equal temperament is now the dominant intonation system in the Western, and much of the non-Western, world.

Equally tempered scales have been used and instruments built using various other numbers of equal intervals. The [[19 equal temperament]], first proposed and used by [[Guillaume Costeley]] in the 16th century, uses 19 equally spaced tones, offering better major thirds and far better minor thirds than normal 12-semitone equal temperament at the cost of a flatter fifth. The overall effect is one of greater consonance. [[Quarter tone|Twenty-four equal temperament]], with twenty-four equally spaced tones, is widespread in the pedagogy and [[Arabic maqam#Notation|notation]] of [[Arabic music]]. However, in theory and practice, the intonation of Arabic music conforms to [[Rational number|rational ratios]], as opposed to the [[Irrational number|irrational ratios]] of equally tempered systems.<ref name="Touma 1996 22–24">{{cite book|last=Touma|first=Habib Hassan|title=The Music of the Arabs|year=1996|publisher=Amadeus Press|location=Portland, OR|isbn=0-931340-88-8|pages=22–24}}</ref>

While any analog to the equally tempered [[quarter tone]] is entirely absent from Arabic intonation systems, analogs to a three-quarter tone, or [[neutral second]], frequently occur. These neutral seconds, however, vary slightly in their ratios dependent on [[Arabic maqam|maqam]], as well as geography. Indeed, Arabic music historian [[Habib Hassan Touma]] has written that "the breadth of deviation of this musical step is a crucial ingredient in the peculiar flavor of Arabian music. To temper the scale by dividing the octave into twenty-four quarter-tones of equal size would be to surrender one of the most characteristic elements of this musical culture."<ref name="Touma 1996 22–24"/>

[[53 equal temperament]] arises  from the near equality of 53 [[perfect fifth]]s with 31 octaves, and was noted by [[Jing Fang]] and [[Nicholas Mercator]].

==Connections to mathematics==
===Set theory===
{{Main|Set theory (music)}}
Musical set theory uses the language of mathematical [[set theory]] in an elementary way to organize musical objects and describe their relationships. To analyze the structure of a piece of (typically atonal) music using musical set theory, one usually starts with a set of tones, which could form motives or chords. By applying simple operations such as [[transposition (music)|transposition]] and [[Melodic inversion|inversion]], one can discover deep structures in the music. Operations such as transposition and inversion are called [[isometries]] because they preserve the intervals between tones in a set.

===Abstract algebra===
{{Main|Abstract algebra}}
Expanding on the methods of musical set theory, some theorists have used abstract algebra to analyze music. For example, the pitch classes in an equally tempered octave form an [[abelian group]] with 12 elements. It is possible to describe [[just intonation]] in terms of a [[free abelian group]].<ref>{{cite web |title=Algebra of Tonal Functions. |url=http://sonantometry.blogspot.com/2007_05_01_archive.html}}</ref><ref>{{cite web|title=Harmonic Limit|url=https://xenharmonic.wikispaces.com/Harmonic+Limit}}</ref>

[[Transformational theory]] is a branch of music theory developed by [[David Lewin]]. The theory allows for great generality because it emphasizes transformations between musical objects, rather than the musical objects themselves.

Theorists have also proposed musical applications of more sophisticated algebraic concepts. The theory of regular temperaments has been extensively developed with a wide range of sophisticated mathematics, for example by associating each regular temperament with a rational point on a [[Grassmannian]].

The [[chromatic scale]] has a free and transitive action of the [[cyclic group]] <math>\mathbb{Z}/12\mathbb{Z}</math>, with the action being defined via [[transposition (music)|transposition]] of notes. So the chromatic scale can be thought of as a [[torsor]] for the group.

===Numbers and series===
Some composers have incorporated the [[golden ratio]] and [[Fibonacci numbers]] into their work.<ref>Reginald Smith Brindle, ''The New Music'', Oxford University Press, 1987, Chapter 6 ''passim''</ref><ref>{{cite web |title=Eric – Math and Music: Harmonious Connections |url=http://www.eric.ed.gov/ERICWebPortal/recordDetail?accno=ED388615}}</ref>

===Category theory===
{{Main|Category theory}}
The [[mathematician]] and [[musicologist]] [[Guerino Mazzola]] has used [[category theory]] ([[topos theory]]) for a basis of music theory, which includes using [[topology]] as a basis for a theory of [[rhythm]] and [[Motif (music)|motives]], and [[differential geometry]] as a basis for a theory of [[musical phrasing]], [[tempo]], and [[Intonation (music)|intonation]].<ref>{{citation|title=The Topos of Music: Geometric Logic of Concepts, Theory, and Performance|last=Mazzola|first=Guerino|year=2018}}</ref>

== Musicians who were or are also notable mathematicians ==
* [[Albert Einstein]] – Accomplished pianist and violinist.
* [[Art Garfunkel]] ([[Simon & Garfunkel]]) – Masters in Mathematics Education, [[Columbia University]]
* [[Brian Cox (physicist)|Brian Cox]] – Professor of particle physics in the School of Physics and Astronomy at the [[University of Manchester]].
* [[Brian May]] ([[Queen (band)|Queen]]) – [[Bachelor of Science|BSc (Hons)]] in Mathematics and Physics, [[Doctor of Philosophy|PhD]] in Astrophysics, both from [[Imperial College London]].
* [[Brian Wecht]] ([[Ninja Sex Party]]) – PhD in particle physics, [[University of California, San Diego]]
* [[Dan Snaith]] – PhD Mathematics, Imperial College London
* [[Delia Derbyshire]] – [[Bachelor of Arts|BA]] in mathematics and music from [[University of Cambridge|Cambridge]].
* [[Donald Knuth]] – Knuth is an organist and a composer. In 2016 he completed a musical piece for organ titled "Fantasia Apocalyptica". It was premièred in Sweden on January 10, 2018
* Ethan Port ([[Savage Republic]]) – PhD Mathematics, [[University of Southern California]]
* Gregg Turner ([[Angry Samoans]]) – PhD Mathematics, [[Claremont Graduate University]]
* [[Jerome Hines]] – Five articles published in ''[[Mathematics Magazine]]'' 1951–1956.
* [[Jonny Buckland]] ([[Coldplay]]) – Studied astronomy and mathematics at [[University College London]].
* [[Kit Armstrong]] – Degree in music and MSc in mathematics.
* [[Manjul Bhargava]] – Plays the [[tabla]], won the [[Fields Medal]] in 2014.
* [[Phil Alvin]] ([[The Blasters]]) – Mathematics, [[University of California, Los Angeles]]
* [[Philip Glass]] – Studied mathematics and philosophy at the [[University of Chicago]].
* [[Robert Schneider]] ([[The Apples in Stereo]]) – PhD Mathematics, [[Emory University]]
* [[Tom Lehrer]] – BA mathematics from [[Harvard University]].
* [[William Herschel]] – Astronomer and played the oboe, violin, harpsichord and organ. He composed 24 symphonies and many concertos, as well as some church music.

==See also==
{{Div col|small=yes}}
*[[Computational musicology]]
*[[Equal temperament]]
*[[Euclidean rhythm]]s (traditional musical rhythms that are generated by [[Euclid's algorithm]])
*[[Harmony search]]
*[[Interval (music)]]
*[[List of music software]]
*[[Mathematics and art]]
*[[Musical tuning]]
*[[Non-Pythagorean scale]]
*[[Piano key frequencies]]
*[[Rhythm]]
*''[[The Glass Bead Game]]''
*[[3rd bridge]] (harmonic resonance based on equal string divisions)
*[[Tonality diamond]]
*[[Tonnetz]]
*[[Utonality and otonality]]
{{Div col end}}
{{Portal inline|Music}}

==References==
{{Reflist}}
*{{wikicite|ref={{harvid|Dahlhaus|1990}}|reference=Dahlhaus, Carl. 1990. ''Wagners Konzeption des musikalischen Dramas''. Deutscher Taschenbuch Verlag. Kassel: Bärenreiter. {{ISBN|9783423045384}}; {{ISBN|9783761845387}}.}}
* [[Ivor Grattan-Guinness]] (1995) "Mozart 18, Beethoven 32: Hidden shadows of integers in classical music", pages 29 to 47 in ''History of Mathematics: States of the Art'', [[Joseph W. Dauben]], Menso Folkerts, [[Eberhard Knobloch]] and [[Hans Wussing]] editors, [[Academic Press]] {{ISBN|0-12-204055-4}}

== Further reading ==
* ''Cool math for hot music - A first introduction to mathematics for music theorists'' by Guerino Mazzola, Maria Mannone, Yan Pang, Springer, 2016, {{ISBN|3319429353}}
* ''Music: A Mathematical Offering '' by  Dave Benson, Cambridge University Press, 2006, {{ISBN|0521619998}}

== External links ==
*[https://arxiv.org/abs/1912.12877 ''Axiomatic Music Theory'' by S.M. Nemati]
*[http://www-personal.umd.umich.edu/~tmfiore/1/musictotal.pdf ''Music and Math'' by Thomas E. Fiore]
*[http://thinkzone.wlonk.com/Music/12Tone.htm Twelve-Tone Musical Scale.]
*[http://sonantometry.blogspot.com Sonantometry or music as math discipline.]
*[https://logosfoundation.org/kursus/music_math.pdf Music: A Mathematical Offering by Dave Benson].
*[https://web.archive.org/web/20070513102544/http://mathdl.maa.org/convergence/1/?pa=content&sa=viewDocument&nodeId=1313&bodyId=1470 Nicolaus Mercator use of Ratio Theory in Music] at [http://mathdl.maa.org/convergence/1/ Convergence]
*[http://sites.google.com/site/abimepublications/home/maths-and-music ''The Glass Bead Game''] Hermann Hesse gave music and mathematics a crucial role in the development of his Glass Bead Game.
*[http://www.aboutscotland.com/harmony/prop.html Harmony and Proportion. Pythagoras, Music and Space].
*[http://web.mit.edu/18.06/www/Essays/linear-algebra-and-music.pdf "Linear Algebra and Music"]
*[http://www.deimos.ca/notefreqs Notefreqs] — A complete table of note frequencies and ratios for midi, piano, guitar, bass, and violin. Includes fret measurements (in cm and inches) for building instruments.
* [https://www.bbc.co.uk/programmes/p003c1b9 Mathematics & Music], BBC Radio 4 discussion with Marcus du Sautoy, Robin Wilson & Ruth Tatlow (''In Our Time'', May 25, 2006)
* [http://orges-leka.de/knn-music/Measuring_note_similarity_with_positive_definite_kernels.pdf Measuring note similarity with positive definite kernels], Measuring note similarity with positive definite kernels

{{Music theory}}
{{Mathematical art}}
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[[Category:Mathematics of music| ]]
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