Music and mathematics 1

Music and mathematics

A spectrogram of a violin waveform, with linear frequency on thevertical axis and time on the horizontal axis. The bright lines show

how the spectral components change over time. The intensitycoloring is logarithmic (black is −120 dBFS).

Music theorists sometimes use mathematics tounderstand music, and although music has no axiomaticfoundation in modern mathematics, mathematics is "thebasis of sound" and sound itself "in its musicalaspects... exhibits a remarkable array of numberproperties", simply because nature itself "is amazinglymathematical".[1] Though ancient Chinese, Egyptiansand Mesopotamians are known to have studied themathematical principles of sound,[2] the Pythagoreansof ancient Greece were the first researchers known tohave investigated the expression of musical scales interms of numerical ratios,[3] particularly the ratios ofsmall integers. Their central doctrine was that "allnature consists of harmony arising out of numbers".[4]

From the time of Plato, harmony was considered a fundamental branch of physics, now known as musical acoustics.Early Indian and Chinese theorists show similar approaches: all sought to show that the mathematical laws ofharmonics and rhythms were fundamental not only to our understanding of the world but to human well-being.[5]

Confucius, like Pythagoras, regarded the small numbers 1,2,3,4 as the source of all perfection.[6]

The attempt to structure and communicate new ways of composing and hearing music has led to musical applicationsof set theory, abstract algebra and number theory. Some composers have incorporated the golden ratio and Fibonaccinumbers into their work.[7]

Time, rhythm and meterMain article: Meter (music)Without the boundaries of rhythmic structure – a fundamental equal and regular arrangement of pulse repetition,accent, phrase and duration – music would not be possible.[8] In Old English the word "rhyme", derived to "rhythm",became associated and confused with rim – "number"[9] – and modern musical use of terms like meter and measurealso reflects the historical importance of music, along with astronomy, in the development of counting, arithmeticand the exact measurement of time and periodicity that is fundamental to physics.

Musical formMain article: Musical formMusical form is the plan by which a short piece of music is extended. The term "plan" is also used in architecture, towhich musical form is often compared. Like the architect, the composer must take into account the function forwhich the work is intended and the means available, practicing economy and making use of repetition and order.[10]

The common types of form known as binary and ternary ("twofold" and "threefold") once again demonstrate theimportance of small integral values to the intelligibility and appeal of music.The word "rhyme" was not derived from "rhythm"Wikipedia:Cleanup (see Oxford and Collins dictionaries) but fromold English "rime". The spelling of "rime" was later affected by the spelling of "rhythm", although the two are totallydifferent.

Music and mathematics 2

Frequency and harmony

Chladni figures produced by sound vibrations in fine powderon a square plate. (Ernst Chladni, Acoustics, 1802)

A musical scale is a discrete set of pitches used in making ordescribing music. The most important scale in the Westerntradition is the diatonic scale but many others have been usedand proposed in various historical eras and parts of the world.Each pitch corresponds to a particular frequency, expressed inhertz (Hz), sometimes referred to as cycles per second (c.p.s.).A scale has an interval of repetition, normally the octave. Theoctave of any pitch refers to a frequency exactly twice that ofthe given pitch. Succeeding superoctaves are pitches found atfrequencies four, eight, sixteen times, and so on, of thefundamental frequency. Pitches at frequencies of half, aquarter, an eighth and so on of the fundamental are calledsuboctaves. There is no case in musical harmony where, if agiven pitch be considered accordant, that its octaves areconsidered otherwise. Therefore any note and its octaves willgenerally be found similarly named in musical systems (e.g.all will be called doh or A or Sa, as the case may be). Whenexpressed as a frequency bandwidth an octave A

2–A

3 spans

from 110 Hz to 220 Hz (span=110 Hz). The next octave willspan from 220 Hz to 440 Hz (span=220 Hz). The third octavespans from 440 Hz to 880 Hz (span=440 Hz) and so on. Each successive octave spans twice the frequency range ofthe previous octave.

Because we are often interested in the relations or ratios between the pitches (known as intervals) rather than theprecise pitches themselves in describing a scale, it is usual to refer to all the scale pitches in terms of their ratio froma particular pitch, which is given the value of one (often written 1/1), generally a note which functions as the tonic ofthe scale. For interval size comparison cents are often used.

The exponential nature of octaves when measured on alinear frequency scale.

This diagram presents octaves as they appear in thesense of musical intervals, equally spaced.

Common name Examplename

Hz

Multipleof fundamental

Ratiowithin octave

Centswithin octave

Fundamental A2,

110

1x 1/1 = 1x 0

Octave A3220

2x 2/1 = 2x 1200

2/2 = 1x 0

Perfect Fifth E4330

3x 3/2 = 1.5x 702

Octave A4440

4x 4/2 = 2x 1200

4/4 = 1x 0

Music and mathematics 3

Major Third C♯5550

5x 5/4 = 1.25x 386

Perfect Fifth E5660

6x 6/4 = 1.5x 702

Harmonic seventh G5770

7x 7/4 = 1.75x 969

Octave A5880

8x 8/4 = 2x 1200

8/8 = 1x 0

Tuning systemsMain articles: Musical tuning and Musical temperament5-limit tuning, the most common form of just intonation, is a system of tuning using tones that are regular numberharmonics of a single fundamental frequency. This was one of the scales Johannes Kepler presented in hisHarmonices Mundi (1619) in connection with planetary motion. The same scale was given in transposed form byScottish mathematician and musical theorist, Alexander Malcolm, in 1721 in his 'Treatise of Musick: Speculative,Practical and Historical',[11] and by theorist Jose Wuerschmidt in the 20th century. A form of it is used in the musicof northern India. American composer Terry Riley also made use of the inverted form of it in his "Harp of NewAlbion". Just intonation gives superior results when there is little or no chord progression: voices and otherinstruments gravitate to just intonation whenever possible. However, as it gives two different whole tone intervals(9:8 and 10:9) because a fixed tuned instrument, such as a piano, cannot change key.[12] To calculate the frequencyof 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 A4 (A natural above middle C), the frequency is 440 Hz, and a justly tuned fifth above it (E5) issimply 440×(3:2) = 660 Hz.

The first 16 harmonics, their names and frequencies,showing the exponential nature of the octave and the

simple fractional nature of non-octave harmonics.The first 16 harmonics, with frequencies and log

frequencies.

Semitone Ratio Interval Natural Half Step

0 1:1 unison 480 0

1 16:15 minor semitone 512 16:15

2 9:8 major second 540 135:128

3 6:5 minor third 576 16:15

4 5:4 major third 600 25:24

5 4:3 perfect fourth 640 16:15

6 45:32 diatonic tritone 675 135:128

7 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

Music and mathematics 4

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 perfectfourth. Thus the major third is considered not a third but a ditone, literally "two tones", and is (9:8)2 = 81:64, ratherthan the independent and harmonic just 5:4 = 80:64 directly below. A whole tone is a secondary interval, beingderived from two perfect fifths, (3:2)2 = 9:8.The just major third, 5:4 and minor third, 6:5, are a syntonic comma, 81:80, apart from their Pythagorean equivalents81:64 and 32:27 respectively. According to Carl Dahlhaus (1990, p. 187), "the dependent third conforms to thePythagorean, the independent third to the harmonic tuning of intervals."Western common practice music usually cannot be played in just intonation but requires a systematically temperedscale. The tempering can involve either the irregularities of well temperament or be constructed as a regulartemperament, either some form of equal temperament or some other regular meantone, but in all cases will involvethe fundamental features of meantone temperament. For example, the root of chord ii, if tuned to a fifth above thedominant, would be a major whole tone (9:8) above the tonic. If tuned a just minor third (6:5) below a justsubdominant degree of 4:3, however, the interval from the tonic would equal a minor whole tone (10:9). Meantonetemperament 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.In equal temperament, the octave is divided into twelve equal parts, each semitone (half-step) is an interval of thetwelfth root of two so that twelve of these equal half steps add up to exactly an octave. With fretted instruments it isvery useful to use equal temperament so that the frets align evenly across the strings. In the European musictradition, equal temperament was used for lute and guitar music far earlier than for other instruments, such asmusical keyboards. Because of this historical force, twelve-tone equal temperament is now the dominant intonationsystem 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 19equal 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 aflatter fifth. The overall effect is one of greater consonance. 24 equal temperament, with 24 equally spaced tones, iswidespread in the pedagogy and notation of Arabic music. However, in theory and practice, the intonation of Arabicmusic conforms to rational ratios, as opposed to the irrational ratios of equally tempered systems. While any analogto the equally tempered quarter tone is entirely absent from Arabic intonation systems, analogs to a three-quartertone, or neutral second, frequently occur. These neutral seconds, however, vary slightly in their ratios dependent onmaqam, as well as geography. Indeed, Arabic music historian Habib Hassan Touma has written that "the breadth ofdeviation of this musical step is a crucial ingredient in the peculiar flavor of Arabian music. To temper the scale bydividing the octave into twenty-four quarter-tones of equal size would be to surrender one of the most characteristicelements of this musical culture."The following graph reveals how accurately various equal-tempered scales approximate three important harmonicidentities: the major third (5th harmonic), the perfect fifth (3rd harmonic), and the "harmonic seventh" (7thharmonic). [Note: the numbers above the bars designate the equal-tempered scale (i.e., "12" designates the 12-toneequal-tempered scale, etc.)]

Music and mathematics 5

Note Frequency (Hz) FrequencyDistance

fromprevious note

Logfrequency

log2 f

Log frequencyDistance fromprevious note

A2 110.00 N/A 6.781 N/A

A♯2 116.54 6.54 6.864 0.0833 (or 1/12)

B2 123.47 6.93 6.948 0.0833

C3 130.81 7.34 7.031 0.0833

C♯3 138.59 7.78 7.115 0.0833

D3 146.83 8.24 7.198 0.0833

D♯3 155.56 8.73 7.281 0.0833

E3 164.81 9.25 7.365 0.0833

F3 174.61 9.80 7.448 0.0833

F♯3 185.00 10.39 7.531 0.0833

G3 196.00 11.00 7.615 0.0833

G♯3 207.65 11.65 7.698 0.0833

A3 220.00 12.35 7.781 0.0833

Below are Ogg Vorbis files demonstrating the difference between just intonation and equal temperament. You mayneed to play the samples several times before you can pick the difference.• Two sine waves played consecutively – this sample has half-step at 550 Hz (C♯ in the just intonation scale),

followed by a half-step at 554.37 Hz (C♯ in the equal temperament scale).• Same two notes, set against an A440 pedal – this sample consists of a "dyad". The lower note is a constant A

(440 Hz in either scale), the upper note is a C♯ in the equal-tempered scale for the first 1", and a C♯ in the justintonation scale for the last 1". Phase differences make it easier to pick the transition than in the previous sample.

Connections to set theoryMain article: Set theory (music)Musical set theory uses the language of mathematical set theory in an elementary way to organize musical objectsand describe their relationships. To analyze the structure of a piece of (typically atonal) music using musical settheory, one usually starts with a set of tones, which could form motives or chords. By applying simple operationssuch as transposition and inversion, one can discover deep structures in the music. Operations such as transpositionand inversion are called isometries because they preserve the intervals between tones in a set.

Music and mathematics 6

Connections to abstract algebraMain article: Abstract algebraExpanding on the methods of musical set theory, some theorists have used abstract algebra to analyze music. Forexample, the pitch classes in an equally tempered octave form an abelian group with 12 elements. It is possible todescribe just intonation in terms of a free abelian group.,Transformational theory is a branch of music theory developed by David Lewin. The theory allows for greatgenerality because it emphasizes transformations between musical objects, rather than the musical objectsthemselves.Theorists have also proposed musical applications of more sophisticated algebraic concepts. Mathematician GuerinoMazzola has applied topos theory to music,Wikipedia:Citation needed though the result has beencontroversial.Wikipedia:Citation needed. The theory of regular temperaments has been extensively developed with awide range of sophisticated mathematics, for example by associating each regular temperament with a rational pointon a Grassmannian.

The chromatic scale has a free and transitive action of the cyclic group , with the action being defined viatransposition of notes. So the chromatic scale can be thought of as a torsor for the group .

Connections to analysisReal and complex analysis have also been made use of, for instance by applying the theory of the Riemann zetafunction to the study of equal divisions of the octave.[13]

References[1] Reginald Smith Brindle, The New Music, Oxford University Press, 1987, pp 42-3[2] Reginald Smith Brindle, The New Music, Oxford University Press, 1987, p 42[3] Plato, (Trans. Desmond Lee) The Republic, Harmondsworth Penguin 1974, page 340, note.[4] Sir James Jeans, Science and Music, Dover 1968, p. 154.[5] Alain Danielou, Introduction to the Study of Musical Scales, Mushiram Manoharlal 1999, Chapter 1 passim.[6] Sir James Jeans, Science and Music, Dover 1968, p. 155.[7] Reginald Smith Brindle, The New Music, Oxford University Press, 1987, Chapter 6 passim[8] Arnold Whittall, in The Oxford Companion to Music, OUP, 2002, Article: Rhythm[9] Chambers' Twentieth Century Dictionary, 1977, p. 1100[10] Imogen Holst, The ABC of Music, Oxford 1963, p.100[11] https:/ / archive. org/ details/ treatiseofmusick00malc[12] Jeremy Montagu, in The Oxford Companion to Music, OUP 2002, Article: just intonation.[13] https:/ / xenharmonic. wikispaces. com/ The+ Riemann+ Zeta+ Function+ and+ Tuning. html

External links• Database of all the possible 2048 musical scales in 12 note equal temperament and other alternatives in meantone

tunings (http:/ / www. harmonics. com/ scales)• Music and Math by Thomas E. Fiore (http:/ / www-personal. umd. umich. edu/ ~tmfiore/ 1/ musictotal. pdf)• Twelve-Tone Musical Scale. (http:/ / thinkzone. wlonk. com/ Music/ 12Tone. htm)• Sonantometry or music as math discipline. (http:/ / sonantometry. blogspot. com)• Music: A Mathematical Offering by Dave Benson (http:/ / www. maths. abdn. ac. uk/ ~bensondj/ html/ music.

pdf).• Nicolaus Mercator use of Ratio Theory in Music (http:/ / mathdl. maa. org/ convergence/ 1/ ?pa=content&

sa=viewDocument& nodeId=1313& bodyId=1470) at Convergence (http:/ / mathdl. maa. org/ convergence/ 1/ )• The Glass Bead Game (http:/ / sites. google. com/ site/ abimepublications/ home/ maths-and-music) Hermann

Hesse gave music and mathematics a crucial role in the development of his Glass Bead Game.

Music and mathematics 7

• Harmony and Proportion. Pythagoras, Music and Space (http:/ / www. aboutscotland. com/ harmony/ prop. html).• "Linear Algebra and Music" (http:/ / web. mit. edu/ 18. 06/ www/ Essays/ linear-algebra-and-music. pdf)

Article Sources and Contributors 8

Article Sources and ContributorsMusic and mathematics  Source: http://en.wikipedia.org/w/index.php?oldid=620738873  Contributors: Aaron Kauppi, Addshore, Algorithme, Alreadytaken4536, Antandrus, BD2412, Barak Sh,Basavarajtalwar, Beefman, Bookuser, Brandoncastro411, Braybaroque, Bruce1ee, CattleGirl, Chris the speller, Commator, Cosprings, DMacks, DerHexer, Dffgd, Dicklyon, Doduz, EditorE,Eflatmajor7th, FMAFan1990, Frank Zamjatin, Gandalf61, Gene Nygaard, Gene Ward Smith, Glenn L, GoingBatty, GreyCat, Hadetor, Hyacinth, J.delanoy, J58f49f, Jarble, Jerry, Joefromrandb,John Cline, Jowa fan, Just plain Bill, KagakuKyouju, Keron Cyst, Kilmer-san, KingsleyIdehen, Klausness, KnowledgeOfSelf, Kylegann, Lalaithion, Lambiam, Lappado, Lewisswinger, Ligia,Loadmaster, MadamIamadam, Madder, Mandarax, Mate Juhasz, Matthias Röder, Mclroy, Mild Bill Hiccup, Mpatel, N57263, Nbarth, Nic bor, Niceguyedc, Nono64, ONEder Boy, Omegatron,P1h3r1e3d13, Phantomsteve, Philip Trueman, Piano non troppo, Picnicpower, Pthag, RationalBlasphemist, Rcsprinter123, Redheylin, RevJPFunk, Rigadoun, RobertG, Robin klein, Ronz,Rracecarr, Salix alba, Sardanaphalus, Sarindam7, Secret Squïrrel, Sk8a h8a, Skew-t, Smhanes, Spellcast, SpikeToronto, StealthFox, Tabletop, Tamfang, Tbhotch, Thehelpfulone, Themfromspace,Thunder852za, Torc2, Tortillovsky, TudorTulok, WadeSimMiser, Waldir, Wavelength, Willow1729, Woodstone, Wyatt915, Yxm011, 191 anonymous edits

Image Sources, Licenses and ContributorsImage:Spectrogram of violin.png  Source: http://en.wikipedia.org/w/index.php?title=File:Spectrogram_of_violin.png  License: GNU Free Documentation License  Contributors: Aquegg,Flugaal, Jacklee, Ma-Lik, Mdd, Mutatis mutandis, Omegatron, Pieter Kuiper, TorschImage:Chladini.Diagrams.for.Quadratic.Plates.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Chladini.Diagrams.for.Quadratic.Plates.svg  License: Public Domain  Contributors:Basavarajtalwar, High Contrast, HyacinthImage:4Octaves.and.Frequencies.svg  Source: http://en.wikipedia.org/w/index.php?title=File:4Octaves.and.Frequencies.svg  License: Creative Commons Attribution-Sharealike 3.0 Contributors: User:BasavarajtalwarImage:4Octaves.and.Frequencies.Ears.svg  Source: http://en.wikipedia.org/w/index.php?title=File:4Octaves.and.Frequencies.Ears.svg  License: Creative Commons Attribution-Sharealike 3.0 Contributors: User:BasavarajtalwarImage:HarmonicIdentities.Names.Frequencies.svg  Source: http://en.wikipedia.org/w/index.php?title=File:HarmonicIdentities.Names.Frequencies.svg  License: Creative CommonsAttribution-Sharealike 3.0  Contributors: User:BasavarajtalwarImage:Normalized.HarmonicIdentities.Names.Frequencies.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Normalized.HarmonicIdentities.Names.Frequencies.svg  License:Creative Commons Attribution-Sharealike 3.0  Contributors: User:Basavarajtalwar

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