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1724:, 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
1387:: 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. 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
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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
1682:
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) = 81:64, rather than the independent and harmonic just 5:4 = 80:64 directly below. A whole tone is
216:
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
300:
spans from 110 Hz to 220 Hz (span=110 Hz). The next octave will span from 220 Hz to 440 Hz (span=220 Hz). The third octave spans from 440 Hz to 880 Hz (span=440 Hz) and so on. Each successive octave spans twice the frequency range of the previous octave.
1811:
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
273:
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
1853:
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
1905:
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
261:
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
1372:(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', and by theorist
1771:
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.
38:
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
328:) 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
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3336:
1953:
274:
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
2812:
1880:
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
1740:, 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
3117:
4127:
1276:
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.
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2843:— 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.
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2192:- 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
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1760:. 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.
1334:
The first 16 harmonics, their names and frequencies, showing the exponential nature of the octave and the simple fractional nature of non-octave harmonics.
195:
also reflects the historical importance of music, along with astronomy, in the development of counting, arithmetic and the exact measurement of time and
1902:. The theory allows for great generality because it emphasizes transformations between musical objects, rather than the musical objects themselves.
1704:
usually cannot be played in just intonation but requires a systematically tempered scale. The tempering can involve either the irregularities of
2884:
2174:- Astronomer and played the oboe, violin, harpsichord and organ. He composed 24 symphonies and many concertos, as well as some church music.
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1383:
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
225:("twofold" and "threefold") once again demonstrate the importance of small integral values to the intelligibility and appeal of music.
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4219:
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Profiles of Pitch
Classes Circularity of Relative Pitch and Key-Experiments, Models, Computational Music Analysis, and Perspectives
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105:
Though ancient
Chinese, Indians, Egyptians and Mesopotamians are known to have studied the mathematical principles of sound, the
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1298:
2856:
1697:, p. 187), "the dependent third conforms to the Pythagorean, the independent third to the harmonic tuning of intervals."
4782:
4142:
4112:
3552:
2768:
2755:
1269:, which is defined as frequency ratio of 2:1. In other words, every time the frequency is doubled, the given scale repeats.
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For the chromatic scale, the octave is divided into twelve equal parts, each semitone (half-step) is an interval of the
202:
The elements of musical form often build strict proportions or hypermetric structures (powers of the numbers 2 and 3).
4709:
4265:
70:. The attempt to structure and communicate new ways of composing and hearing music has led to musical applications of
1265:. Both of these systems, and the vast majority of music in general, have scales that repeat on the interval of every
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Equally tempered scales have been used and instruments built using various other numbers of equal intervals. The
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1862:, one can discover deep structures in the music. Operations such as transposition and inversion are called
1748:. In western music, a division into twelve intervals is commonly assumed unless it is specified otherwise.
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2441:"Александр Виноград, Многообразие проявлений музыкального метра (LAP Lambert Academic Publishing, 2013)"
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125:, particularly the ratios of small integers. Their central doctrine was that "all nature consists of
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2711:(1995) "Mozart 18, Beethoven 32: Hidden shadows of integers in classical music", pages 29 to 47 in
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Hermann Hesse gave music and mathematics a crucial role in the development of his Glass Bead Game.
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1254:, which results in simple ratios between frequencies, but with scale divisions that are uneven.
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1803:, frequently occur. These neutral seconds, however, vary slightly in their ratios dependent on
1242:. Equal temperament scales are built by dividing an octave into intervals which are equal on a
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879:
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This diagram presents octaves as they appear in the sense of musical intervals, equally spaced.
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Without the boundaries of rhythmic structure – a fundamental equal and regular arrangement of
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1690:, 81:80, apart from their Pythagorean equivalents 81:64 and 32:27 respectively. According to
1246:, which results in perfectly evenly divided scales, but with ratios of frequencies which are
257:
used in making or describing music. The most important scale in the
Western tradition is the
17:
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a secondary interval, being derived from two perfect fifths minus an octave, (3:2)/2 = 9:8.
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117:) of ancient Greece were the first researchers known to have investigated the expression of
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or some other regular meantone, but in all cases will involve the fundamental features of
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One major difference between equal temperament tunings and just tunings is differences in
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is entirely absent from Arabic intonation systems, analogs to a three-quarter tone, or
1725:
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1346:
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184:
164:, like Pythagoras, regarded the small numbers 1,2,3,4 as the source of all perfection.
106:
1305:". The lower note is a constant A (440 Hz in either scale), the upper note is a C
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30:
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were fundamental not only to our understanding of the world but to human well-being.
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theorists show similar approaches: all sought to show that the mathematical laws of
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Cool math for hot music - A first introduction to mathematics for music theorists
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3179:
2849:, BBC Radio 4 discussion with Marcus du Sautoy, Robin Wilson & Ruth Tatlow (
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1508:
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1380:
1321:
differences make it easier to detect the transition than in the previous sample.
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when two notes are sounded together, which affects the subjective experience of
1239:
355:
237:
Chladni figures produced by sound vibrations in fine powder on a square plate. (
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35:
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C5, an octave above middle C. The frequency is twice that of middle C (523 Hz).
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1784:. However, in theory and practice, the intonation of Arabic music conforms to
1273:
374:
C3, an octave below middle C. The frequency is half that of middle C (131 Hz).
71:
2531:
2484:
187:
and duration – music would not be possible. Modern musical use of terms like
3784:
3371:
3351:
3020:
2846:
2065:
1823:
1395:, and a justly tuned fifth above it (E5) is simply 440×(3:2) = 660 Hz.
308:
The exponential nature of octaves when measured on a linear frequency scale.
196:
161:
110:
94:
86:
1776:, with twenty-four equally spaced tones, is widespread in the pedagogy and
1289:
in the just intonation scale), followed by a half-step at 554.37 Hz (C
4204:
1376:
in the 20th century. A form of it is used in the music of northern India.
270:
of any pitch refers to a frequency exactly twice that of the given pitch.
217:
and making use of repetition and order. The common types of form known as
4538:
4457:
4384:
3597:
3199:
3097:
3025:
2111:
2013:
1447:
1357:
153:
114:
2673:
The Topos of Music: Geometric Logic of
Concepts, Theory, and Performance
2492:
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4323:
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3238:
3152:
3030:
2816:
2557:. Edinburgh : Printed for the author – via Internet Archive.
2281:
1549:
137:
126:
2975:
2539:
2507:
1744:), the most common number is 12, which makes up the equal-temperament
3204:
2258:
2017:
1960:
1665:
1428:
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267:
263:
157:
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369:
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90:
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29:
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4096:
Viewpoints: Mathematical
Perspective and Fractal Geometry in Art
3211:
3194:
4208:
2979:
2866:
2862:
1728:
or comma of
Didymus, is the key comma of meantone temperament.
50:, timing, and structure of music. It uses mathematics to study
2750:
by
Guerino Mazzola, Maria Mannone, Yan Pang, Springer, 2016,
2555:"A treatise of musick, speculative, practical and historical"
1342:
The first 16 harmonics, with frequencies and log frequencies.
1866:
because they preserve the intervals between tones in a set.
2859:, Measuring note similarity with positive definite kernels
97:) and exhibits "a remarkable array of number properties".
2553:
Malcolm, Alexander; Mitchell, Mr (Joseph) (25 May 2018).
2062:) – Masters in Mathematics Education, Columbia University
2857:
Measuring note similarity with positive definite kernels
1959:
of notes. So the chromatic scale can be thought of as a
1807:, as well as geography. Indeed, Arabic music historian
89:
foundation in modern mathematics, the basis of musical
2148:) – Mathematics, University of California, Los Angeles
1686:
The just major third, 5:4 and minor third, 6:5, are a
1391:(A natural above middle C), the frequency is 440
1250:. Just scales are built by multiplying frequencies by
2688:. Deutscher Taschenbuch Verlag. Kassel: Bärenreiter.
2044:
Musicians who were or are also notable mathematicians
1926:
1849:
Musical set theory uses the language of mathematical
1311:
in the equal-tempered scale for the first 1", and a C
1176:
1131:
1107:
1050:
1026:
965:
941:
882:
858:
794:
749:
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668:
644:
585:
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516:
459:
435:
2273:(harmonic resonance based on equal string divisions)
2012:) for a basis of music theory, which includes using
320:
Because we are often interested in the relations or
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3125:
3013:
2830:
Harmony and
Proportion. Pythagoras, Music and Space
2763:by Dave Benson, Cambridge University Press, 2006,
2583:
2581:
2216:(traditional musical rhythms that are generated by
3993:The Drawing of Geometric Patterns in Saracenic Art
1947:
1199:
1154:
1116:
1073:
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988:
950:
905:
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817:
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444:
289:When expressed as a frequency bandwidth an octave
4128:Goudreau Museum of Mathematics in Art and Science
136:, harmony was considered a fundamental branch of
2658:"Eric – Math and Music: Harmonious Connections"
2592:. Portland, OR: Amadeus Press. pp. 22–24.
1234:There are two main families of tuning systems:
2813:Nicolaus Mercator use of Ratio Theory in Music
1317:in the just intonation scale for the last 1".
4220:
4123:European Society for Mathematics and the Arts
3297:Mathematica: A World of Numbers... and Beyond
2991:
2878:
2807:Music: A Mathematical Offering by Dave Benson
2371:, Harmondsworth Penguin 1974, page 340, note.
1884:with 12 elements. It is possible to describe
1353:, is a system of tuning using tones that are
1283:– this sample has half-step at 550 Hz (C
8:
3277:List of works designed with the golden ratio
2467:Dreyfus, Tommy; Eisenberg, Theodore (1986).
2154:- Studied mathematics and philosophy at the
336:of the scale. For interval size comparison,
2686:Wagners Konzeption des musikalischen Dramas
2644:, Oxford University Press, 1987, Chapter 6
2512:The Journal of Aesthetics and Art Criticism
2469:"On the Aesthetics of Mathematical Thought"
2395:Introduction to the Study of Musical Scales
1948:{\displaystyle \mathbb {Z} /12\mathbb {Z} }
332:), generally a note which functions as the
27:Relationships between music and mathematics
4227:
4213:
4205:
3762:
3386:
2998:
2984:
2976:
2885:
2871:
2863:
2325:, Oxford University Press, 1987, pp. 42–43
2090:– PhD Mathematics, Imperial College London
3337:Cathedral of Saint Mary of the Assumption
2802:Sonantometry or music as math discipline.
2713:History of Mathematics: States of the Art
2124:- Degree in music and MSc in mathematics.
2114:) - Studied astronomy and mathematics at
1941:
1940:
1932:
1928:
1927:
1925:
1898:is a branch of music theory developed by
1795:While any analog to the equally tempered
1299:Same two notes, set against an A440 pedal
1180:
1175:
1135:
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1106:
1054:
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354:(262 Hz). (Scale: 1 square is equal to 1
1917:has a free and transitive action of the
1694:
1397:
379:
3883:Vier Bücher von Menschlicher Proportion
2311:
93:can be described mathematically (using
2397:, Mushiram Manoharlal 1999, Chapter 1
2338:, Oxford University Press, 1987, p. 42
2052:- Accomplished pianist and violinist.
1971:Some composers have incorporated the
1818:arises from the near equality of 53
7:
4789:
1955:, with the action being defined via
4801:
4068:Journal of Mathematics and the Arts
2508:"Pythagorean Mathematics and Music"
1281:Two sine waves played consecutively
4082:Making Mathematics with Needlework
1822:with 31 octaves, and was noted by
25:
3908:I quattro libri dell'architettura
1720:. For example, the root of chord
4800:
4788:
4777:
4776:
4764:
4187:
4186:
3432:Self-portrait in a Convex Mirror
3116:
2294:
1295:in the equal temperament scale).
199:that is fundamental to physics.
4685:Computational complexity theory
2761:Music: A Mathematical Offering
2473:For the Learning of Mathematics
4143:National Museum of Mathematics
3895:Regole generali d'architettura
2100:in mathematics and music from
324:between the pitches (known as
1:
2569:The Oxford Companion to Music
2425:The Oxford Companion to Music
2180:- Five articles published in
1792:of equally tempered systems.
1774:Twenty-four equal temperament
1767:, first proposed and used by
1364:. This was one of the scales
1301:– this sample consists of a "
18:Normalizing the musical scale
3668:Garden of Cosmic Speculation
2614:"Algebra of Tonal Functions"
2588:Touma, Habib Hassan (1996).
2506:Crocker, Richard L. (1963).
2076:in Mathematics and Physics,
2367:Plato (trans. Desmond Lee)
2080:in Astrophysics, both from
2028:as a basis for a theory of
2016:as a basis for a theory of
4855:
4735:Films about mathematicians
3185:Islamic geometric patterns
2836:"Linear Algebra and Music"
2797:Twelve-Tone Musical Scale.
1986:
1873:
1842:
1834:Connections to mathematics
1349:, the most common form of
1223:
209:
85:While music theory has no
4758:
4304:Philosophy of mathematics
4244:
4182:
4006:A Mathematician's Apology
3114:
2964:
2900:
2671:Mazzola, Guerino (2018),
2348:Purwins, Hendrik (2005).
2116:University College London
1732:Equal temperament tunings
1263:consonance and dissonance
1101:
1098:
1092:
1089:
1074:{\displaystyle 7/4=1.75x}
906:{\displaystyle 5/4=1.25x}
719:
716:
710:
707:
510:
507:
501:
498:
129:arising out of numbers".
4740:Recreational mathematics
4118:The Bridges Organization
2640:Reginald Smith Brindle,
2334:Reginald Smith Brindle,
2204:Computational musicology
989:{\displaystyle 6/4=1.5x}
692:{\displaystyle 3/2=1.5x}
4839:Mathematics and culture
4625:Mathematical statistics
4615:Mathematical psychology
4585:Engineering mathematics
4519:Algebraic number theory
3980:The Grammar of Ornament
3932:Nature's Harmonic Unity
3842:De prospectiva pingendi
2967:List of music theorists
2847:Mathematics & Music
2287:Utonality and otonality
2082:Imperial College London
1896:Transformational theory
1708:or be constructed as a
286:, as the case may be).
168:Time, rhythm, and meter
4771:Mathematics portal
4620:Mathematical sociology
4600:Mathematical economics
4595:Mathematical chemistry
4524:Analytic number theory
4405:Differential equations
4133:Institute For Figuring
4045:The 'Life' of a Carpet
3870:A Treatise on Painting
2782:Axiomatic Music Theory
2684:Dahlhaus, Carl. 1990.
2590:The Music of the Arabs
2427:, OUP, 2002, Article:
2319:Reginald Smith Brindle
2234:List of music software
2164:- BA mathematics from
1949:
1712:, either some form of
1343:
1335:
1201:
1200:{\displaystyle 8/8=1x}
1156:
1155:{\displaystyle 8/4=2x}
1118:
1075:
1037:
990:
952:
907:
869:
819:
818:{\displaystyle 4/4=1x}
774:
773:{\displaystyle 4/2=2x}
736:
693:
655:
610:
609:{\displaystyle 2/2=1x}
565:
564:{\displaystyle 2/1=2x}
527:
484:
483:{\displaystyle 1/1=1x}
446:
375:
367:
359:
317:
309:
246:
121:in terms of numerical
40:
4750:Mathematics education
4680:Theory of computation
4400:Hypercomplex analysis
4014:George David Birkhoff
3988:Ernest Hanbury Hankin
3856:De divina proportione
3836:Piero della Francesca
3815:Leon Battista Alberti
3402:Piero della Francesca
3041:Hyperboloid structure
2709:Ivor Grattan-Guinness
2571:, OUP 2002, Article:
2457:, Oxford 1963, p. 100
2414:, Dover 1968, p. 155.
2384:, Dover 1968, p. 154.
2254:Piano key frequencies
2249:Non-Pythagorean scale
2156:University of Chicago
2060:Simon & Garfunkel
2026:differential geometry
1950:
1756:instruments, such as
1702:common practice music
1362:fundamental frequency
1341:
1333:
1202:
1157:
1119:
1076:
1038:
991:
953:
908:
870:
820:
775:
737:
694:
656:
611:
566:
528:
485:
447:
373:
365:
346:
315:
307:
253:is a discrete set of
236:
229:Frequency and harmony
33:
4829:Mathematics of music
4730:Informal mathematics
4610:Mathematical physics
4605:Mathematical finance
4590:Mathematical biology
4529:Diophantine geometry
3939:Frederik Macody Lund
3810:Filippo Brunelleschi
3691:Hamid Naderi Yeganeh
3553:La condition humaine
2423:Arnold Whittall, in
2183:Mathematics Magazine
1924:
1816:53 equal temperament
1788:, as opposed to the
1765:19 equal temperament
1718:meantone temperament
1174:
1129:
1105:
1048:
1024:
963:
939:
880:
856:
792:
747:
723:
666:
642:
583:
538:
514:
457:
433:
4834:Mathematics and art
4745:Mathematics and art
4655:Operations research
4410:Functional analysis
4164:Mathematical beauty
4089:Rhythm of Structure
4032:Gödel, Escher, Bach
3828:De re aedificatoria
3459:The Ancient of Days
3078:Projective geometry
3007:Mathematics and art
2823:The Glass Bead Game
2567:Jeremy Montagu, in
2265:The Glass Bead Game
2239:Mathematics and art
1753:twelfth root of two
1710:regular temperament
1230:Musical temperament
4690:Numerical analysis
4299:Mathematical logic
4294:Information theory
4169:Patterns in nature
4026:Douglas Hofstadter
3652:Desmond Paul Henry
3642:Bathsheba Grossman
3574:The Swallow's Tail
3495:Giorgio de Chirico
3367:Sydney Opera House
3222:Croatian interlace
2792:by Thomas E. Fiore
2719:, Menso Folkerts,
2218:Euclid's algorithm
2166:Harvard University
1967:Numbers and series
1945:
1890:free abelian group
1845:Set theory (music)
1809:Habib Hassan Touma
1769:Guillaume Costeley
1680:Pythagorean tuning
1379:American composer
1344:
1336:
1248:irrational numbers
1197:
1152:
1117:{\displaystyle 8x}
1114:
1071:
1036:{\displaystyle 7x}
1033:
986:
951:{\displaystyle 6x}
948:
903:
868:{\displaystyle 5x}
865:
815:
770:
735:{\displaystyle 4x}
732:
689:
654:{\displaystyle 3x}
651:
606:
561:
526:{\displaystyle 2x}
523:
480:
445:{\displaystyle 1x}
442:
376:
368:
360:
318:
310:
247:
41:
4816:
4815:
4415:Harmonic analysis
4202:
4201:
4055:
4054:
4019:Aesthetic Measure
3890:Sebastiano Serlio
3864:Leonardo da Vinci
3754:
3753:
3746:Margaret Wertheim
3407:Leonardo da Vinci
2973:
2972:
2721:Eberhard Knobloch
2412:Science and Music
2410:Sir James Jeans,
2382:Science and Music
2380:Sir James Jeans,
2357:. pp. 22–24.
2214:Euclidean rhythms
2209:Equal temperament
1979:into their work.
1977:Fibonacci numbers
1828:Nicholas Mercator
1790:irrational ratios
1758:musical keyboards
1738:equal temperament
1714:equal temperament
1677:
1676:
1385:chord progression
1368:presented in his
1244:logarithmic scale
1236:equal temperament
1215:
1214:
396:
142:musical acoustics
132:From the time of
60:chord progression
52:elements of music
16:(Redirected from
4846:
4804:
4803:
4792:
4791:
4780:
4779:
4769:
4768:
4700:Computer algebra
4675:Computer science
4395:Complex analysis
4229:
4222:
4215:
4206:
4190:
4189:
4040:Nikos Salingaros
3763:
3731:Hiroshi Sugimoto
3681:Robert Longhurst
3627:Helaman Ferguson
3582:Crockett Johnson
3511:Circle Limit III
3480:Danseuse au café
3387:
3357:Pyramid of Khufu
3120:
3000:
2993:
2986:
2977:
2887:
2880:
2873:
2864:
2717:Joseph W. Dauben
2705:
2676:
2675:
2668:
2662:
2661:
2654:
2648:
2638:
2632:
2631:
2628:"Harmonic Limit"
2624:
2618:
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2610:
2604:
2603:
2585:
2576:
2565:
2559:
2558:
2550:
2544:
2543:
2503:
2497:
2496:
2464:
2458:
2455:The ABC of Music
2451:
2445:
2444:
2437:
2431:
2421:
2415:
2408:
2402:
2393:Alain Danielou,
2391:
2385:
2378:
2372:
2365:
2359:
2358:
2356:
2345:
2339:
2332:
2326:
2316:
2301:Music portal
2299:
2298:
2297:
2277:Tonality diamond
2229:Interval (music)
2172:William Herschel
2094:Delia Derbyshire
2030:musical phrasing
1954:
1952:
1951:
1946:
1944:
1936:
1931:
1876:Abstract algebra
1870:Abstract algebra
1742:Arab tone system
1706:well temperament
1398:
1374:Jose Wuerschmidt
1370:Harmonices Mundi
1316:
1315:
1310:
1309:
1294:
1293:
1288:
1287:
1252:rational numbers
1206:
1204:
1203:
1198:
1184:
1161:
1159:
1158:
1153:
1139:
1123:
1121:
1120:
1115:
1080:
1078:
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1072:
1058:
1042:
1040:
1039:
1034:
1007:Harmonic seventh
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987:
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843:
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530:
529:
524:
489:
487:
486:
481:
467:
451:
449:
448:
443:
394:
380:
340:are often used.
76:abstract algebra
21:
4854:
4853:
4849:
4848:
4847:
4845:
4844:
4843:
4819:
4818:
4817:
4812:
4763:
4754:
4704:
4661:
4640:Systems science
4571:
4567:Homotopy theory
4533:
4500:
4452:
4424:
4371:
4318:
4289:Category theory
4275:
4240:
4233:
4203:
4198:
4178:
4174:Sacred geometry
4147:
4113:Ars Mathematica
4101:
4051:
3962:
3915:
3902:Andrea Palladio
3798:
3791:De architectura
3750:
3706:Antoine Pevsner
3686:Jeanette McLeod
3637:Susan Goldstine
3586:
3445:
3439:
3376:
3362:Sagrada Família
3323:
3265:
3133:Algorithmic art
3121:
3112:
3108:Wallpaper group
3046:Minimal surface
3009:
3004:
2974:
2969:
2960:
2927:
2896:
2891:
2853:, May 25, 2006)
2777:
2744:
2742:Further reading
2683:
2680:
2679:
2670:
2669:
2665:
2656:
2655:
2651:
2639:
2635:
2626:
2625:
2621:
2612:
2611:
2607:
2600:
2587:
2586:
2579:
2573:just intonation
2566:
2562:
2552:
2551:
2547:
2505:
2504:
2500:
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2448:
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2347:
2346:
2342:
2333:
2329:
2317:
2313:
2308:
2295:
2293:
2291:
2199:
2128:Manjul Bhargava
2050:Albert Einstein
2046:
2006:category theory
2002:Guerino Mazzola
1991:
1989:Category theory
1985:
1983:Category theory
1969:
1963:for the group.
1922:
1921:
1915:chromatic scale
1886:just intonation
1878:
1872:
1847:
1841:
1836:
1786:rational ratios
1746:chromatic scale
1734:
1366:Johannes Kepler
1351:just intonation
1328:
1313:
1312:
1307:
1306:
1291:
1290:
1285:
1284:
1259:acoustical beat
1232:
1224:Main articles:
1222:
1211:
1172:
1171:
1166:
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1103:
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1085:
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1021:
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1000:
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703:
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431:
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384:
298:
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231:
214:
208:
170:
140:, now known as
109:(in particular
103:
28:
23:
22:
15:
12:
11:
5:
4852:
4850:
4842:
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4831:
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4811:
4810:
4798:
4786:
4774:
4759:
4756:
4755:
4753:
4752:
4747:
4742:
4737:
4732:
4727:
4726:
4725:
4718:Mathematicians
4714:
4712:
4710:Related topics
4706:
4705:
4703:
4702:
4697:
4692:
4687:
4682:
4677:
4671:
4669:
4663:
4662:
4660:
4659:
4658:
4657:
4652:
4647:
4645:Control theory
4637:
4632:
4627:
4622:
4617:
4612:
4607:
4602:
4597:
4592:
4587:
4581:
4579:
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4468:
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4454:
4453:
4451:
4450:
4445:
4440:
4434:
4432:
4426:
4425:
4423:
4422:
4420:Measure theory
4417:
4412:
4407:
4402:
4397:
4392:
4387:
4381:
4379:
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4372:
4370:
4369:
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3995:
3985:
3984:
3983:
3970:
3968:
3964:
3963:
3961:
3960:
3959:
3958:
3956:The Greek Vase
3948:
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3936:
3935:
3934:
3923:
3921:
3917:
3916:
3914:
3913:
3912:
3911:
3899:
3898:
3897:
3887:
3886:
3885:
3878:Albrecht Dürer
3875:
3874:
3873:
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3860:
3859:
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3752:
3751:
3749:
3748:
3743:
3741:Roman Verostko
3738:
3733:
3728:
3723:
3718:
3716:Alba Rojo Cama
3713:
3708:
3703:
3698:
3693:
3688:
3683:
3678:
3673:
3672:
3671:
3662:Charles Jencks
3659:
3654:
3649:
3647:George W. Hart
3644:
3639:
3634:
3629:
3624:
3619:
3614:
3609:
3600:
3594:
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3588:
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3557:
3556:
3544:
3543:
3542:
3535:
3528:
3521:
3514:
3502:
3497:
3492:
3491:
3490:
3483:
3474:Jean Metzinger
3471:
3470:
3469:
3462:
3449:
3447:
3441:
3440:
3438:
3437:
3436:
3435:
3423:
3421:Albrecht Dürer
3418:
3417:
3416:
3404:
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3393:
3391:
3384:
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3197:
3192:
3182:
3177:
3172:
3167:
3162:
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3160:
3155:
3150:
3140:
3138:Anamorphic art
3135:
3129:
3127:
3123:
3122:
3115:
3113:
3111:
3110:
3105:
3100:
3095:
3094:
3093:
3088:
3080:
3075:
3070:
3069:
3068:
3066:Camera obscura
3063:
3053:
3048:
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3033:
3028:
3023:
3017:
3015:
3011:
3010:
3005:
3003:
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2860:
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2844:
2838:
2833:
2827:
2819:
2810:
2804:
2799:
2794:
2790:Music and Math
2786:
2784:by S.M. Nemati
2776:
2775:External links
2773:
2772:
2771:
2758:
2743:
2740:
2739:
2738:
2729:Academic Press
2706:
2678:
2677:
2663:
2649:
2633:
2619:
2605:
2598:
2577:
2560:
2545:
2524:10.2307/427754
2518:(2): 189–198.
2498:
2459:
2453:Imogen Holst,
2446:
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2403:
2386:
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2307:
2304:
2290:
2289:
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2279:
2274:
2268:
2261:
2256:
2251:
2246:
2244:Musical tuning
2241:
2236:
2231:
2226:
2224:Harmony search
2221:
2211:
2206:
2200:
2198:
2195:
2194:
2193:
2187:
2175:
2169:
2159:
2149:
2139:
2125:
2119:
2108:Jonny Buckland
2105:
2091:
2085:
2063:
2053:
2045:
2042:
1987:Main article:
1984:
1981:
1968:
1965:
1943:
1939:
1935:
1930:
1888:in terms of a
1874:Main article:
1871:
1868:
1843:Main article:
1840:
1837:
1835:
1832:
1820:perfect fifths
1801:neutral second
1733:
1730:
1726:syntonic comma
1695:Dahlhaus (1990
1688:syntonic comma
1675:
1674:
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1576:
1573:
1568:
1563:
1559:
1558:
1555:
1552:
1546:
1543:
1539:
1538:
1535:
1532:
1530:perfect fourth
1527:
1522:
1518:
1517:
1514:
1511:
1506:
1501:
1497:
1496:
1493:
1490:
1485:
1480:
1476:
1475:
1472:
1469:
1464:
1461:
1457:
1456:
1453:
1450:
1445:
1442:
1438:
1437:
1434:
1431:
1426:
1423:
1419:
1418:
1415:
1410:
1405:
1402:
1355:regular number
1347:5-limit tuning
1327:
1324:
1323:
1322:
1296:
1226:Musical tuning
1221:
1220:Tuning systems
1218:
1217:
1216:
1213:
1212:
1209:
1207:
1196:
1193:
1190:
1187:
1183:
1179:
1168:
1167:
1164:
1162:
1151:
1148:
1145:
1142:
1138:
1134:
1124:
1113:
1110:
1100:
1097:
1094:
1091:
1087:
1086:
1083:
1081:
1070:
1067:
1064:
1061:
1057:
1053:
1043:
1032:
1029:
1019:
1016:
1013:
1010:
1005:
1002:
1001:
998:
996:
985:
982:
979:
976:
972:
968:
958:
947:
944:
934:
931:
928:
925:
922:
919:
918:
915:
913:
902:
899:
896:
893:
889:
885:
875:
864:
861:
851:
848:
845:
837:
834:
831:
830:
827:
825:
814:
811:
808:
805:
801:
797:
786:
785:
782:
780:
769:
766:
763:
760:
756:
752:
742:
731:
728:
718:
715:
712:
709:
705:
704:
701:
699:
688:
685:
682:
679:
675:
671:
661:
650:
647:
637:
634:
631:
628:
625:
622:
621:
618:
616:
605:
602:
599:
596:
592:
588:
577:
576:
573:
571:
560:
557:
554:
551:
547:
543:
533:
522:
519:
509:
506:
503:
500:
496:
495:
492:
490:
479:
476:
473:
470:
466:
462:
452:
441:
438:
428:
425:
422:
419:
416:
413:
412:
411:within octave
407:
406:within octave
402:
397:
391:
386:
296:
292:
259:diatonic scale
230:
227:
210:Main article:
207:
204:
169:
166:
119:musical scales
102:
99:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4851:
4840:
4837:
4835:
4832:
4830:
4827:
4826:
4824:
4809:
4808:
4799:
4797:
4796:
4787:
4785:
4784:
4775:
4773:
4772:
4767:
4761:
4760:
4757:
4751:
4748:
4746:
4743:
4741:
4738:
4736:
4733:
4731:
4728:
4724:
4721:
4720:
4719:
4716:
4715:
4713:
4711:
4707:
4701:
4698:
4696:
4693:
4691:
4688:
4686:
4683:
4681:
4678:
4676:
4673:
4672:
4670:
4668:
4667:Computational
4664:
4656:
4653:
4651:
4648:
4646:
4643:
4642:
4641:
4638:
4636:
4633:
4631:
4628:
4626:
4623:
4621:
4618:
4616:
4613:
4611:
4608:
4606:
4603:
4601:
4598:
4596:
4593:
4591:
4588:
4586:
4583:
4582:
4580:
4578:
4574:
4568:
4565:
4563:
4560:
4558:
4555:
4553:
4550:
4548:
4545:
4544:
4542:
4540:
4536:
4530:
4527:
4525:
4522:
4520:
4517:
4515:
4512:
4511:
4509:
4507:
4506:Number theory
4503:
4497:
4494:
4492:
4489:
4487:
4484:
4482:
4479:
4477:
4474:
4472:
4469:
4467:
4464:
4463:
4461:
4459:
4455:
4449:
4446:
4444:
4441:
4439:
4438:Combinatorics
4436:
4435:
4433:
4431:
4427:
4421:
4418:
4416:
4413:
4411:
4408:
4406:
4403:
4401:
4398:
4396:
4393:
4391:
4390:Real analysis
4388:
4386:
4383:
4382:
4380:
4378:
4374:
4368:
4365:
4363:
4360:
4358:
4355:
4353:
4350:
4348:
4345:
4343:
4340:
4338:
4335:
4333:
4330:
4329:
4327:
4325:
4321:
4315:
4312:
4310:
4307:
4305:
4302:
4300:
4297:
4295:
4292:
4290:
4287:
4286:
4284:
4282:
4278:
4272:
4269:
4267:
4264:
4260:
4257:
4255:
4252:
4251:
4250:
4247:
4246:
4243:
4238:
4230:
4225:
4223:
4218:
4216:
4211:
4210:
4207:
4195:
4194:
4185:
4184:
4181:
4175:
4172:
4170:
4167:
4165:
4162:
4160:
4159:Droste effect
4157:
4156:
4154:
4150:
4144:
4141:
4139:
4138:Mathemalchemy
4136:
4134:
4131:
4129:
4126:
4124:
4121:
4119:
4116:
4114:
4111:
4110:
4108:
4106:Organizations
4104:
4098:
4097:
4093:
4091:
4090:
4086:
4084:
4083:
4079:
4077:
4076:
4075:Lumen Naturae
4072:
4070:
4069:
4065:
4064:
4062:
4058:
4046:
4043:
4042:
4041:
4038:
4034:
4033:
4029:
4028:
4027:
4024:
4020:
4017:
4016:
4015:
4012:
4008:
4007:
4003:
4002:
4001:
3998:
3994:
3991:
3990:
3989:
3986:
3982:
3981:
3977:
3976:
3975:
3972:
3971:
3969:
3965:
3957:
3954:
3953:
3952:
3949:
3945:
3942:
3941:
3940:
3937:
3933:
3930:
3929:
3928:
3927:Samuel Colman
3925:
3924:
3922:
3918:
3910:
3909:
3905:
3904:
3903:
3900:
3896:
3893:
3892:
3891:
3888:
3884:
3881:
3880:
3879:
3876:
3872:
3871:
3867:
3866:
3865:
3862:
3858:
3857:
3853:
3852:
3851:
3848:
3844:
3843:
3839:
3838:
3837:
3834:
3830:
3829:
3825:
3823:
3822:
3818:
3817:
3816:
3813:
3811:
3808:
3807:
3805:
3801:
3793:
3792:
3788:
3787:
3786:
3783:
3779:
3776:
3775:
3774:
3771:
3770:
3768:
3764:
3761:
3757:
3747:
3744:
3742:
3739:
3737:
3736:Daina Taimiņa
3734:
3732:
3729:
3727:
3724:
3722:
3721:Reza Sarhangi
3719:
3717:
3714:
3712:
3709:
3707:
3704:
3702:
3699:
3697:
3694:
3692:
3689:
3687:
3684:
3682:
3679:
3677:
3674:
3670:
3669:
3665:
3664:
3663:
3660:
3658:
3655:
3653:
3650:
3648:
3645:
3643:
3640:
3638:
3635:
3633:
3632:Peter Forakis
3630:
3628:
3625:
3623:
3620:
3618:
3615:
3613:
3610:
3608:
3604:
3601:
3599:
3596:
3595:
3593:
3589:
3583:
3580:
3576:
3575:
3571:
3569:
3568:
3564:
3563:
3562:
3561:Salvador Dalí
3559:
3555:
3554:
3550:
3549:
3548:
3547:René Magritte
3545:
3541:
3540:
3536:
3534:
3533:
3529:
3527:
3526:
3522:
3520:
3519:
3518:Print Gallery
3515:
3513:
3512:
3508:
3507:
3506:
3503:
3501:
3498:
3496:
3493:
3489:
3488:
3487:L'Oiseau bleu
3484:
3482:
3481:
3477:
3476:
3475:
3472:
3468:
3467:
3463:
3461:
3460:
3456:
3455:
3454:
3453:William Blake
3451:
3450:
3448:
3442:
3434:
3433:
3429:
3428:
3427:
3424:
3422:
3419:
3415:
3414:
3413:Vitruvian Man
3410:
3409:
3408:
3405:
3403:
3400:
3398:
3397:Paolo Uccello
3395:
3394:
3392:
3388:
3385:
3383:
3379:
3373:
3370:
3368:
3365:
3363:
3360:
3358:
3355:
3353:
3350:
3348:
3345:
3343:
3340:
3338:
3335:
3334:
3332:
3330:
3326:
3320:
3319:
3318:Pi in the Sky
3315:
3313:
3312:
3308:
3306:
3305:
3301:
3299:
3298:
3294:
3292:
3291:
3290:Mathemalchemy
3287:
3285:
3284:
3280:
3278:
3275:
3274:
3272:
3268:
3262:
3259:
3257:
3254:
3252:
3249:
3245:
3242:
3241:
3240:
3237:
3235:
3232:
3228:
3225:
3223:
3220:
3218:
3215:
3214:
3213:
3210:
3206:
3203:
3201:
3198:
3196:
3193:
3191:
3188:
3187:
3186:
3183:
3181:
3178:
3176:
3173:
3171:
3168:
3166:
3163:
3159:
3158:Vastu shastra
3156:
3154:
3151:
3149:
3148:Geodesic dome
3146:
3145:
3144:
3141:
3139:
3136:
3134:
3131:
3130:
3128:
3124:
3119:
3109:
3106:
3104:
3101:
3099:
3096:
3092:
3089:
3087:
3084:
3083:
3081:
3079:
3076:
3074:
3073:Plastic ratio
3071:
3067:
3064:
3062:
3061:Camera lucida
3059:
3058:
3057:
3054:
3052:
3049:
3047:
3044:
3042:
3039:
3037:
3034:
3032:
3029:
3027:
3024:
3022:
3019:
3018:
3016:
3012:
3008:
3001:
2996:
2994:
2989:
2987:
2982:
2981:
2978:
2968:
2963:
2957:
2954:
2952:
2949:
2947:
2944:
2942:
2939:
2937:
2934:
2932:
2929:
2926:
2923:
2921:
2918:
2916:
2913:
2911:
2908:
2906:
2903:
2902:
2899:
2895:
2888:
2883:
2881:
2876:
2874:
2869:
2868:
2865:
2858:
2855:
2852:
2848:
2845:
2842:
2839:
2837:
2834:
2831:
2828:
2825:
2824:
2820:
2818:
2814:
2811:
2808:
2805:
2803:
2800:
2798:
2795:
2793:
2791:
2787:
2785:
2783:
2779:
2778:
2774:
2770:
2766:
2762:
2759:
2757:
2753:
2749:
2746:
2745:
2741:
2737:
2736:0-12-204055-4
2733:
2730:
2726:
2722:
2718:
2714:
2710:
2707:
2703:
2702:9783761845387
2699:
2695:
2694:9783423045384
2691:
2687:
2682:
2681:
2674:
2667:
2664:
2659:
2653:
2650:
2647:
2643:
2642:The New Music
2637:
2634:
2629:
2623:
2620:
2615:
2609:
2606:
2601:
2599:0-931340-88-8
2595:
2591:
2584:
2582:
2578:
2574:
2570:
2564:
2561:
2556:
2549:
2546:
2541:
2537:
2533:
2529:
2525:
2521:
2517:
2513:
2509:
2502:
2499:
2494:
2490:
2486:
2482:
2478:
2474:
2470:
2463:
2460:
2456:
2450:
2447:
2442:
2436:
2433:
2430:
2426:
2420:
2417:
2413:
2407:
2404:
2400:
2396:
2390:
2387:
2383:
2377:
2374:
2370:
2364:
2361:
2353:
2352:
2344:
2341:
2337:
2336:The New Music
2331:
2328:
2324:
2323:The New Music
2320:
2315:
2312:
2305:
2303:
2302:
2288:
2285:
2283:
2280:
2278:
2275:
2272:
2269:
2267:
2266:
2262:
2260:
2257:
2255:
2252:
2250:
2247:
2245:
2242:
2240:
2237:
2235:
2232:
2230:
2227:
2225:
2222:
2219:
2215:
2212:
2210:
2207:
2205:
2202:
2201:
2196:
2191:
2188:
2185:
2184:
2179:
2176:
2173:
2170:
2167:
2163:
2160:
2157:
2153:
2150:
2147:
2143:
2140:
2137:
2133:
2129:
2126:
2123:
2122:Kit Armstrong
2120:
2117:
2113:
2109:
2106:
2103:
2099:
2095:
2092:
2089:
2086:
2083:
2079:
2075:
2071:
2067:
2064:
2061:
2057:
2056:Art Garfunkel
2054:
2051:
2048:
2047:
2043:
2041:
2039:
2035:
2031:
2027:
2023:
2019:
2015:
2011:
2007:
2003:
2000:
1996:
1995:mathematician
1990:
1982:
1980:
1978:
1974:
1966:
1964:
1962:
1958:
1957:transposition
1937:
1933:
1920:
1916:
1911:
1909:
1903:
1901:
1897:
1893:
1891:
1887:
1883:
1882:abelian group
1877:
1869:
1867:
1865:
1861:
1857:
1856:transposition
1852:
1846:
1838:
1833:
1831:
1829:
1825:
1821:
1817:
1813:
1810:
1806:
1802:
1798:
1793:
1791:
1787:
1783:
1779:
1775:
1770:
1766:
1761:
1759:
1754:
1749:
1747:
1743:
1739:
1731:
1729:
1727:
1723:
1719:
1715:
1711:
1707:
1703:
1698:
1696:
1693:
1689:
1684:
1681:
1672:
1669:
1667:
1664:
1661:
1658:
1657:
1653:
1650:
1648:
1647:major seventh
1645:
1642:
1639:
1638:
1634:
1631:
1629:
1628:minor seventh
1626:
1623:
1620:
1619:
1615:
1612:
1610:
1607:
1604:
1601:
1600:
1596:
1593:
1591:
1588:
1585:
1582:
1581:
1577:
1574:
1572:
1571:perfect fifth
1569:
1567:
1564:
1561:
1560:
1556:
1553:
1551:
1547:
1544:
1541:
1540:
1536:
1533:
1531:
1528:
1526:
1523:
1520:
1519:
1515:
1512:
1510:
1507:
1505:
1502:
1499:
1498:
1494:
1491:
1489:
1486:
1484:
1481:
1478:
1477:
1473:
1470:
1468:
1465:
1462:
1459:
1458:
1454:
1451:
1449:
1446:
1443:
1440:
1439:
1435:
1432:
1430:
1427:
1424:
1421:
1420:
1416:
1414:
1411:
1409:
1406:
1403:
1400:
1399:
1396:
1394:
1390:
1386:
1382:
1377:
1375:
1371:
1367:
1363:
1359:
1356:
1352:
1348:
1340:
1332:
1325:
1320:
1304:
1300:
1297:
1282:
1279:
1278:
1277:
1275:
1270:
1268:
1264:
1260:
1255:
1253:
1249:
1245:
1241:
1237:
1231:
1227:
1219:
1208:
1194:
1191:
1188:
1185:
1181:
1177:
1170:
1169:
1163:
1149:
1146:
1143:
1140:
1136:
1132:
1125:
1111:
1108:
1088:
1082:
1068:
1065:
1062:
1059:
1055:
1051:
1044:
1030:
1027:
1020:
1017:
1011:
1008:
1004:
1003:
997:
983:
980:
977:
974:
970:
966:
959:
945:
942:
935:
932:
926:
923:Perfect Fifth
921:
920:
914:
900:
897:
894:
891:
887:
883:
876:
862:
859:
852:
849:
838:
833:
832:
826:
812:
809:
806:
803:
799:
795:
788:
787:
781:
767:
764:
761:
758:
754:
750:
743:
729:
726:
706:
700:
686:
683:
680:
677:
673:
669:
662:
648:
645:
638:
635:
629:
626:Perfect Fifth
624:
623:
617:
603:
600:
597:
594:
590:
586:
579:
578:
572:
558:
555:
552:
549:
545:
541:
534:
520:
517:
497:
491:
477:
474:
471:
468:
464:
460:
453:
439:
436:
429:
426:
420:
415:
414:
408:
403:
398:
392:
387:
382:
381:
378:
377:
372:
364:
357:
353:
349:
345:
341:
339:
335:
331:
327:
323:
314:
306:
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251:musical scale
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46:analyzes the
45:
37:
32:
19:
4805:
4793:
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4762:
4695:Optimization
4557:Differential
4481:Differential
4448:Order theory
4443:Graph theory
4347:Group theory
4191:
4094:
4087:
4080:
4073:
4066:
4060:Publications
4044:
4030:
4018:
4004:
3992:
3978:
3955:
3951:Jay Hambidge
3944:Ad Quadratum
3943:
3931:
3906:
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3882:
3868:
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3850:Luca Pacioli
3840:
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3789:
3777:
3701:Hinke Osinga
3696:István Orosz
3666:
3657:Anthony Hill
3612:Scott Draves
3607:Erik Demaine
3591:Contemporary
3572:
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3537:
3530:
3523:
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3509:
3505:M. C. Escher
3485:
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3426:Parmigianino
3411:
3342:Hagia Sophia
3316:
3309:
3302:
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3288:
3281:
3233:
3165:Computer art
3143:Architecture
3103:Tessellation
3086:Architecture
3036:Golden ratio
2930:
2894:Music theory
2850:
2822:
2789:
2781:
2760:
2747:
2725:Hans Wussing
2712:
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2190:Donald Knuth
2181:
2178:Jerome Hines
2152:Philip Glass
2146:The Blasters
2136:Fields Medal
2130:- Plays the
2010:topos theory
1999:musicologist
1992:
1973:golden ratio
1970:
1919:cyclic group
1912:
1908:Grassmannian
1904:
1894:
1879:
1848:
1814:
1797:quarter tone
1794:
1782:Arabic music
1762:
1750:
1735:
1721:
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1685:
1678:
1467:major second
1378:
1360:of a single
1345:
1326:Just tunings
1271:
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1233:
401:fundamental
329:
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248:
242:
215:
212:Musical form
206:Musical form
201:
171:
131:
107:Pythagoreans
104:
84:
44:Music theory
42:
4807:WikiProject
4650:Game theory
4630:Probability
4367:Homological
4357:Multilinear
4337:Commutative
4314:Type theory
4281:Foundations
4237:mathematics
4000:G. H. Hardy
3803:Renaissance
3773:Polykleitos
3711:Tony Robbin
3622:John Ernest
3617:Jan Dibbets
3567:Crucifixion
3390:Renaissance
3244:Mathematics
3217:Celtic knot
3180:Fractal art
3082:Proportion
3056:Perspective
2931:Mathematics
2920:Composition
2851:In Our Time
2817:Convergence
2479:(1): 2–10.
1900:David Lewin
1609:major sixth
1590:minor sixth
1509:major third
1488:minor third
1381:Terry Riley
1240:just tuning
835:Major Third
417:Fundamental
399:Multiple of
356:millisecond
348:Oscillogram
197:periodicity
36:spectrogram
4823:Categories
4635:Statistics
4514:Arithmetic
4476:Arithmetic
4342:Elementary
4309:Set theory
3974:Owen Jones
3821:De pictura
3726:Oliver Sin
3676:Andy Lomas
3525:Relativity
3256:String art
3170:Fiber arts
3051:Paraboloid
2951:Set theory
2946:Psychology
2941:Philosophy
2936:Musicology
2925:Definition
2905:Aesthetics
2769:0521619998
2756:3319429353
2306:References
2271:3rd bridge
2162:Tom Lehrer
2142:Phil Alvin
2134:, won the
2088:Dan Snaith
2074:BSc (Hons)
2038:intonation
1864:isometries
1851:set theory
1839:Set theory
1812:culture."
1417:Half Step
1274:Ogg Vorbis
1272:Below are
177:repetition
72:set theory
4562:Geometric
4552:Algebraic
4491:Euclidean
4466:Algebraic
4362:Universal
3785:Vitruvius
3759:Theorists
3539:Waterfall
3444:19th–20th
3372:Taj Mahal
3352:Parthenon
3329:Buildings
3283:Continuum
3251:Sculpture
3227:Interlace
3021:Algorithm
2841:Notefreqs
2727:editors,
2532:0021-8529
2485:0228-0671
2102:Cambridge
2066:Brian May
2004:has used
1860:inversion
1824:Jing Fang
1548:diatonic
1358:harmonics
326:intervals
243:Acoustics
162:Confucius
154:harmonics
111:Philolaus
95:acoustics
87:axiomatic
4783:Category
4539:Topology
4486:Discrete
4471:Analytic
4458:Geometry
4430:Discrete
4385:Calculus
4377:Analysis
4332:Abstract
4271:Glossary
4254:Timeline
4193:Category
3920:Romantic
3598:Max Bill
3532:Reptiles
3347:Pantheon
3304:Octacube
3270:Artworks
3212:Knotting
3200:Muqarnas
3098:Symmetry
3026:Catenary
3014:Concepts
2910:Analysis
2493:40247796
2197:See also
2138:in 2014.
2112:Coldplay
2014:topology
1778:notation
1700:Western
1557:135:128
1474:135:128
1448:semitone
1408:Interval
1401:Semitone
1314:♯
1308:♯
1292:♯
1286:♯
842:♯
404:Ratio of
352:middle C
144:. Early
115:Archytas
54:such as
4795:Commons
4577:Applied
4547:General
4324:Algebra
4249:History
4152:Related
3766:Ancient
3500:Man Ray
3446:Century
3382:Artists
3239:Origami
3153:Pyramid
3031:Fractal
2915:Aspects
2282:Tonnetz
2186:1951–6.
2022:motives
1550:tritone
1413:Natural
1090:Octave
708:Octave
499:Octave
388:Example
255:pitches
245:, 1802)
223:ternary
193:measure
158:rhythms
150:Chinese
138:physics
127:harmony
101:History
4496:Finite
4352:Linear
4259:Future
4235:Major
3967:Modern
3603:Martin
3466:Newton
3261:Tiling
3205:Zellij
3175:4D art
2956:Tuning
2767:
2754:
2734:
2700:
2692:
2646:passim
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2540:427754
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2429:Rhythm
2399:passim
2259:Rhythm
2036:, and
2024:, and
2018:rhythm
1961:torsor
1673:16:15
1666:octave
1654:25:24
1635:27:25
1616:25:24
1597:16:15
1578:16:15
1537:16:15
1516:25:24
1495:16:15
1455:16:15
1429:unison
1267:octave
383:Common
322:ratios
268:octave
266:. The
264:octave
219:binary
185:phrase
181:accent
146:Indian
123:ratios
66:, and
39:dBFS).
4723:lists
4266:Lists
4239:areas
3778:Canon
3234:Music
3190:Girih
3126:Forms
3091:Human
2536:JSTOR
2489:JSTOR
2355:(PDF)
2132:tabla
2070:Queen
2034:tempo
1805:maqam
1545:45:32
1444:16:15
1404:Ratio
1319:Phase
409:Cents
390:name
385:term
338:cents
334:tonic
189:meter
174:pulse
134:Plato
91:sound
68:meter
56:tempo
48:pitch
3605:and
3195:Jali
2765:ISBN
2752:ISBN
2732:ISBN
2723:and
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2690:ISBN
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2020:and
1997:and
1993:The
1975:and
1913:The
1858:and
1826:and
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1066:1.75
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850:550
783:1200
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191:and
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