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Mathematics of musical scales

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Title: Mathematics of musical scales  
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Subject: Just intonation, Musical tuning, Harmony, Meantone temperament, Scientific pitch notation, Musical acoustics, Musical temperament, Mathematics and art
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Mathematics of musical scales

Music theorists sometimes use mathematics to understand music, but music has no axiomatic foundation in modern mathematics. Mathematics is "the basis of sound" and sound itself "in its musical aspects... exhibits a remarkable array of number properties", simply because nature itself "is amazingly mathematical".[1] Though ancient Chinese, Egyptians and Mesopotamians are known to have studied the mathematical principles of sound,[2] the Pythagoreans of ancient Greece are the first researchers known to have investigated the expression of musical scales in terms of numerical ratios,[3] particularly the ratios of small integers. Their central doctrine was that "all nature 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 of harmonics 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]

To this day mathematics has more to do with acoustics than with composition, and the use of mathematics in composition is historically limited to the simplest operations of counting and measuring. 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.[7][8]

Time, rhythm and meter

Main article: Meter (music)

Without the boundaries of rhythmic structure – a fundamental equal and regular arrangement of pulse repetitivity, accent, phrase and duration – music would be impossible.[9] In Old English the word "rhyme", derived to "rhythm", became associated and confused with rim – "number"[10] – and modern musical use of terms like meter and measure also reflects 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 form

Main article: 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.[11] The common types of form known as binary and ternary ("twofold" and "threefold") once again demonstrate the importance of small integral values to the intelligibility and appeal of music.

The word "rhyme" was not derived from "rhythm"[contradiction] (see Oxford and Collins dictionaries) but from old English "rime". The spelling of "rime" was later affected by the spelling of "rhythm", although the two are totally different.

Frequency and harmony

A musical scale is a discrete set of pitches 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 A2–A3 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.

Because we are often interested in the relations or ratios between the pitches (known as 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 of the scale. For interval size comparison cents are often used.

Common name Example
of fundamental
within octave
within octave
Fundamental A2,
1/1 = 1x
Octave A3
2/1 = 2x
2/2 = 1x
Perfect Fifth E4
3/2 = 1.5x
Octave A4
4/2 = 2x
4/4 = 1x
Major Third C5
5/4 = 1.25x
Perfect Fifth E5
6/4 = 1.5x
Harmonic seventh G5
7/4 = 1.75x
Octave A5
8/4 = 2x
8/8 = 1x

Tuning systems

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 chord progression: voices and other instruments gravitate to just intonation whenever possible. However, as it gives two different whole tone intervals (9:8 and 10:9) a keyboard instrument so tuned cannot change key.[12] 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 A4 (A natural above middle C), the frequency is 440 Hz, and a justly tuned fifth above it (E5) is simply 440×(3:2) = 660 Hz.

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
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)2 = 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, (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 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.

In equal temperament, 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 keyboards. 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. 24 equal temperament, with 24 equally spaced tones, is widespread in the pedagogy and notation of Arabic music. However, in theory and practice, the intonation of Arabic music conforms to rational ratios, as opposed to the irrational ratios of equally-tempered systems. 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 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."[13]

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 note
Log frequency
log2 f
Log frequency
Distance from
previous note
A2 110.00 N/A 6.781 N/A
A2 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
C3 138.59 7.78 7.115 0.0833
D3 146.83 8.24 7.198 0.0833
D3 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
F3 185.00 10.39 7.531 0.0833
G3 196.00 11.00 7.615 0.0833
G3 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 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).
  • Phase differences make it easier to pick the transition than in the previous sample.

Connections to set theory

Main 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 transposition and 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.

Connections to abstract algebra

Main article: Abstract algebra

Expanding on the methods of musical set theory, some 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.[14]

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, though the result has been controversial.

The chromatic scale has a free and transitive action of the cyclic group \mathbb{Z}/12\mathbb{Z}, with the action being defined via transposition of notes. So the chromatic scale can be thought of as a torsor for the group \mathbb{Z}/12\mathbb{Z}.

The golden ratio and Fibonacci numbers

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 Lendvaï analyzes Béla Bartók's works as being based on two opposing systems: those of the golden ratio and the acoustic scale. In Bartók's Music for Strings, Percussion, and Celesta, the xylophone progression at the beginning of the 3rd movement occurs at the intervals 1:2:3:5:8:5:3:2:1. French composer Erik Satie used the golden ratio in several of his pieces, including Sonneries de la Rose Croix.

The golden ratio is also apparent in the organization of the sections in the music of Debussy's Image, "Reflections in Water", in which the sequence of keys is marked out by the intervals 34, 21, 13, and 8 (a descending Fibonacci sequence), and the main climax sits at the φ position. "Prelude to the Afternoon of a Faun" also reaches a climax point, marked by the entrance of the antique cymbal, at the φ position.

Many of the important musical events in Krzysztof Penderecki's "Threnody for the Victims of Hiroshima" occur at φ positions.

Boards of Canada have discussed using the golden ratio and Fibonacci numbers in their work.[15] Song titles such as "Music is Math", and numerous songs featuring samples of people counting or discussing numbers, also illustrate the influence of mathematics on their music.

Australian composer Scott Sanders, in his piece "Sweets from Dr Phil" uses modular arithmetic to limit the Fibonacci series to a set of digits modulo 2 to modulo 16, performed by 15 instances of the same timbre to reveal a pseudo-melody apparent in the upper notes of the combination of these instruments. A system of directly mapping digits to pitch classes and durations determines the notes each instrument plays.[16][17][18]

The song, "Lateralus", by Tool incorporates the Fibonacci sequence.[19] The theme of the song describes the desire of humans to explore and to expand for more knowledge and a deeper understanding of everything. The lyrics "spiral out", refers to this desire and also to the Fibonacci spiral, which is formed by creating and arranging squares for each number in the sequence's 1,1,2,3,5,8,... pattern, and drawing a curve that connects to two corners of each square. This would, allowed to continue onwards, theoretically create a never-ending and infinitely-expanding spiral. Related to this, the song's main theme features successive time signatures 9/8, 8/8, and 7/8.[20] The number 987 is the sixteenth integer of the Fibonacci sequence.[21]

See also


External links

  • Database of all the possible 2048 musical scales in 12 note equal temperament and other alternatives in meantone tunings
  • by Thomas E. Fiore
  • Twelve-Tone Musical Scale.
  • Sonantometry or music as math discipline.
  • Music: A Mathematical Offering by Dave Benson.
  • Convergence
  • Hermann Hesse gave music and mathematics a crucial role in the development of his Glass Bead Game.
  • Harmony and Proportion. Pythagoras, Music and Space.
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