Dictionary## Mathematics of musical scales |

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^{[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 of harmonics and 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. Some composers have incorporated the Golden ratio and Fibonacci numbers into their work.
## Time, rhythm and metreMain article: Meter (music)
Without the boundaries of rhythmic structure - a fundamental equal and regular arrangement of accent, phrase and duration - music would be impossible.^{[9]} In Old English the word "rhyme", derived from "rhythm", became associated and confused with rim - "number"^{[10]} - and modern musical use of terms like metre and measure also reflect the historical importance of music, along with astronomy, in the development of counting, arithmetic and 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, 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, practising economy and making use of repetition and order. ## Frequency and harmonyA Because we are often interested in the relations or ratios between the pitches (known as -
Harmonic IdentityCommon Name Example Multiple of Fundamental FreqRatio (this identity/last octave)Linear Point Normalized (linear) Scale1 Fundamental A _{2}- 110Hz1 *x*1/1 = 1 *x*log _{2}(1.0) = 0.002 Octave A _{3}- 220 Hz2 *x*2/1 = 2 *x*(also 2/2 = 1*x*)log _{2}(2.0) = 1.003 Perfect Fifth E _{3}- 330 Hz3 *x*3/2 = 1.5 *x*log _{2}(1.5) = 0.5854 Octave A _{4}- 440 Hz4 *x*4/2 = 2 *x*(also 1*x*)log _{2}(2.0) = 1.005 Major Third C# _{4}- 550 Hz5 *x*5/4 = 1.25 *x*log _{2}(1.25) = 0.3226 Perfect Fifth E _{4}- 660 Hz6 *x*6/4 = 1.5 *x*log _{2}(1.5) = 0.5857 Harmonic seventh G _{4}- 770 Hz7 *x*7/4 = 1.75 *x*log _{2}(1.75) = 0.8078 Octave A _{5}- 880 Hz8 *x*8/4 = 2 *x*(also 1*x*)log _{2}(2.0) = 1.00
- The
**perfect fifth**is located on the 7th step of the chromatic scale. 7/12 = 0.583... ≈ 0.585.... - The major third is located on the 4th step of the chromatic scale. 4/12 = 0.333... ≈ 0.322....
- The
**perfect fourth**(the distance from a perfect fifth to its nearest upper octave) is located on the 5th step of the chromatic scale. 5/12 = 0.416... ≈ 1 (the octave) - 0.585... (the perfect fifth) = 0.414.... - The minor third (the distance from a major third to its nearest upper perfect fifth) is located on the 3rd step of the chromatic scale. 3/12 = 0.25 ≈ 0.585 (the perfect fifth) - 0.322 (the major third) = 0.263....
- No note on the 12-tet represents the 7th harmonic identity, because no integer divided by 12 will yield a number like 0.807....
## Tuning systemsMain articles: Musical tuning and Musical temperament
5-limit tuning, the most common form of Just intonation, is a system of tuning using tones that are regular number harmonics of a single fundamental frequency. This was one of the scales Johannes Kepler presented in his Harmonice Mundi (1619) in connection with planetary motion. The same scale was given in transposed form by Alexander Malcolm in 1721 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
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 81:64 = (9:8)², rather than the independent and harmonic just 5:4, directly below. A whole tone is a secondary interval, being derived 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 equivalents 81:64 and 32:27 respectively. According to Carl Dahlhaus (1990, p. 187), "the dependent third conforms to the Pythagorean, the independent third to the harmonic tuning of intervals." Western In 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 The following graph reveals how accurately various equal tempered scales approximate three important harmonic identities: the major third (5th harmonic), the perfect fifth (3rd harmonic), and the "harmonic seventh" (7th harmonic). [Note: the numbers above the bars designate the equal tempered scale (I.e., "12" designates the 12-tone equal tempered scale, etc.)] -
Note Frequency (Hz) Frequency Distance from previous noteLog frequency log_{2}*f*Log frequency Distance from previous noteA _{2}110.00 N/A 6.781 N/A A _{2}#116.54 6.54 6.864 0.0833 (or 1/12) B _{2}123.47 6.93 6.948 0.0833 C _{2}130.81 7.34 7.031 0.0833 C _{2}#138.59 7.78 7.115 0.0833 D _{2}146.83 8.24 7.198 0.0833 D _{2}#155.56 8.73 7.281 0.0833 E _{2}164.81 9.25 7.365 0.0833 F _{2}174.61 9.80 7.448 0.0833 F _{2}#185.00 10.39 7.531 0.0833 G _{2}196.00 11.00 7.615 0.0833 G _{2}#207.65 11.65 7.698 0.0833 A _{3}220.00 12.35 7.781 0.0833
Below are Ogg Vorbis files demonstrating the difference between just intonation and equal temperament. You may need 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 just intonation 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 some of the concepts from mathematical set theory 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 ## Connections to abstract algebraMain article: Abstract algebra
Expanding on the methods of musical set theory, many theorists have used abstract algebra to analyze music. For example, the notes in an equal temperament octave form an abelian group with 12 elements. It is possible to describe 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 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. Mathematician Guerino Mazzola has applied topos theory to music The chromatic scale has a free and transitive action of , with the action being defined via ## Connections to number theoryMain article: Number theory
Modern interpretation of just intonation is fully based on fundamental theorem of arithmetic. ## The golden ratio and Fibonacci numbersIt is believed that some composers wrote their music using the golden ratio and the Fibonacci numbers to assist them. James Tenney reconceived his piece "For Ann (Rising)", which consists of up to twelve computer-generated tones that glissando upwards (see Shepard tone), as having each tone start so each is the golden ratio (in between an equal tempered minor and major sixth) below the previous tone, so that the combination tones produced by all consecutive tones are a lower or higher pitch already, or soon to be, produced. Ernő Lendvai analyzes The golden ratio is also apparent in the organization of the sections in the music of Tool use a large number of numerical references to the Fibonacci sequence and the Golden ratio in the song "Lateralus" from the album of the same name. The delivery of the words in the song are timed so that the syllables follow the Fibonacci sequence as well (1,1,2,3,5,8,5,3,2,1,1).
## See also**Equal temperament****Interval (music)**- Musical tuning
- Piano key frequencies
- 3rd Bridge (harmonic resonance based on equal string divisions)
## References**^**Reginald Smith Brindle,*The New Music*, Oxford University Press, 1987, pp 42-3**^**Reginald Smith Brindle,*The New Music*, Oxford University Press, 1987, p 42**^**Plato, (Trans. Desmond Lee)*The Republic*, Harmondsworth Penguin 1974, page 340, note.**^**Sir James Jeans,*Science and Music*, Dover 1968, p. 154.**^**Alain Danielou,*Introduction to the Study of Musical Scales*, Mushiram Manoharlal 1999, Chapter 1*passim*.**^**Sir James Jeans,*Science and Music*, Dover 1968, p. 155.**^**Reginald Smith Brindle,*The New Music*, Oxford University Press, 1987, Chapter 6*passim***^**"Eric - Math and Music: Harmonious Connections". http://www.eric.ed.gov/ERICWebPortal/recordDetail?accno=ED388615.**^**Arnold Whittall, in*The Oxford Companion to Music*, OUP, 2002, Article:*Rhythm***^***Chambers' Twentieth Century Dictionary*, 1977, p. 1100**^**Imogen Holst,*The ABC of Music*, Oxford 1963, p.100**^**Jeremy Montagu, in*The Oxford Companion to Music*, OUP 2002, Article:*just intonation*.**^**"Algebra of Tonal Functions.". http://sonantometry.blogspot.com/2007_05_01_archive.html.**^**Fibonacci Numbers and The Golden Section in Art, Architecture and Music
## External links- Database of all the possible 2048 musical scales in 12 note equal temperament and other alternatives in meantone tunings
*Music and Math*by Thomas E. Fiore- Twelve-Tone Musical Scale.
- Sonantometry or music as math discipline.
- Music: A Mathematical Offering by Dave Benson.
- Nicolaus Mercator use of Ratio Theory in Music at Convergence
- Finding the natural and pentatonic scales through discrete numbers
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