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Encyclopedia > Mathematics of musical scales

There are arguments that mathematics can be used to analyse and understand music, and at its core, to compose music itself.[1] Image File history File links This is a lossless scalable vector image. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... // Music is an art form consisting of sound and silence expressed through time. ... Musical composition is: a piece of music the structure of a musical piece the process of creating a new piece of music // A piece of music exists in the form of a written composition in musical notation or as a single acoustic event (a live performance or recorded track). ...

## Connections to group theory GA_googleFillSlot("encyclopedia_square");

Main article: Group theory

For any given octave in equal temperament, the standard musical notes form a commutative group with 12 elements. Group theory is that branch of mathematics concerned with the study of groups. ... This picture illustrates how the hours on a clock form a group under modular addition. ...

It is possible to describe just intonation in terms of free abelian group.[2] In music, just intonation, also called rational intonation, is any musical tuning in which the frequencies of notes are related by ratios of whole numbers. ... In abstract algebra, a free abelian group is an abelian group that has a basis in the sense that every element of the group can be written in one and only one way as a finite linear combination of elements of the basis, with integer coefficients. ...

## Connections to Number theory

Main article: Number theory

Time signatures are usually given in the form $frac{n}{2^m}$ where n and m are positive whole numbers. The most natural and usual signatures are 4/4 and 3/4. Increasing the value of n makes a time signature in some respects more complicated; however, the most complicated time signatures are those for which n is a prime number. For example, 12/8 can be mentally broken down into simply four lots of 3/8, thus rendering it a simple extension of standard time signatures. However, for 13/8, no such reduction is possible, as 13 is prime. For example, the most commonly used time piece in 13/8 is more complicated and unusual than standard time signatures, both to listen to and to perform. Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ... The time signature (also known as meter signature) is a notational device used in Western musical notation to specify how many beats are in each bar and which note value (minim, crotchet, eighth note and so on) constitutes one beat. ... In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors, which are 1 and the prime number itself. ...

Modern interpretation of just intonation is fully based on fundamental theorem of arithmetic. In music, just intonation, also called rational intonation, is any musical tuning in which the frequencies of notes are related by ratios of whole numbers. ... In number theory, the fundamental theorem of arithmetic (or unique factorization theorem) states that every natural number either is itself a prime number, or can be written as a unique product of prime numbers. ...

## The Golden Ratio and Fibonacci Numbers

It is believed that some composers wrote their music using the golden ratio and the Fibonacci numbers to assist them.[3] // Articles with similar titles include Golden mean (philosophy), the felicitous middle between two extremes, and Golden numbers, an indicator of years in astronomy and calendar studies. ... A tiling with squares whose sides are successive Fibonacci numbers in length A Fibonacci spiral, created by drawing arcs connecting the opposite corners of squares in the Fibonacci tiling shown above â€“ see golden spiral. ...

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. James Tenney (August 10, 1934 in Silver City, NM) is an American composer and influential music theorist. ... Glissando (plural: glissandi) is a musical term that refers to either a continuous sliding from one pitch to another (a true glissando), or an incidental scale played while moving from one melodic note to another (an effective glissando). ... Figure 1: Shepard tones forming a Shepard scale, illustrated in a sequencer A Shepard tone, named after Roger Shepard, is a sound consisting of a superposition of sine waves separated by octaves. ...

Ernő Lendvai analyzes Béla Bartók's works as being based on two opposing systems: those of the golden ratio and the acoustic scale.[24] In Bartok's Music for Strings, Percussion, and Celesta, the xylophone progression 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. His use of the ratio gave his music an otherworldly symmetry. Erno Lendvai was one of the first theorists to write on the appearance of the golden section and Fibonacci series and how these are implemented in Bartoks music. ... BÃ©la BartÃ³k in 1927 BÃ©la Viktor JÃ¡nos BartÃ³k (March 25, 1881 â€“ September 26, 1945) was a Hungarian composer, pianist and collector of Eastern European and Middle Eastern folk music. ... Selfportrait of Erik Satie. ...

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, and the main climax sits at the φ position. Claude Debussy, photo by FÃ©lix Nadar, 1908. ...

This Binary Universe, an experimental album by Brian Transeau, includes a track entitled 1.618 in homage to the golden ratio. The track features musical versions of the ratio and the accompanying video displays various animated versions of the golden mean. Brian Wayne Transeau (born October 4, 1971 in Rockville, Maryland) is a trance musician, better known by his stage name, BT. He has been called the Father of Trance for his pioneering in the trance genre [1],[2] and Prince of Dance Music for his multi-instrumentalist skills [3], and...

The math metal band Mudvayne has an atmospheric instrumental track called "Golden Ratio" on its album L.D. 50. Mathematical concepts are also explored in other songs by Mudvayne. A musical scale is a discrete set of pitches used in making or describing music. Typically a scale has an interval of repetition, which is normally the octave. This means that for any pitch in the scale, we have also an equivalent pitch an octave above and an octave below it. While the limits of human hearing are finite, matters are somewhat simplified if we ignore that fact, as is usually done in discussions of theory. 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 all the scale pitches in terms of their ratio from a particular pitch, which is given the value of one (often written 1/1 when discussing just intonation.) This note can be, but is not necessarily, a note which functions as the tonic of the scale. For tunings using irrational numbers (i.e. temperaments) or for interval size comparison cents are often used. This article does not cite its references or sources. ... Mudvayne is an American metal band. ... In music, a scale is a collection of musical notes that provides material for part or all of a musical work. ... Pitch is the perceived fundamental frequency of a sound. ... In music, an octave (sometimes abbreviated 8ve) is the interval between one musical note and another with half or double its frequency. ... For the numerical computation software, see GNU Octave. ... A ratio is a quantity that denotes the proportional amount or magnitude of one quantity relative to another. ... In music theory, the term interval describes the difference in pitch between two notes. ... In music, just intonation, also called rational intonation, is any musical tuning in which the frequencies of notes are related by ratios of whole numbers. ... The tonic is the first note of a musical scale, and in the tonal method of music composition it is extremely important. ... In musical tuning, a temperament is a system of tuning which slightly compromises the pure intervals of just intonation in order to meet other requirements of the system. ... The cent is a logarithmic unit of measure used for musical intervals. ...

The most important scale in the Western tradition is the diatonic scale, but the scales used and proposed in various historical eras and parts of the world have been many and varied. Scales may broadly be classed as scales of just intonation, tempered scales, and practice-based scales. A scale is in just intonation if the ratios between the frequencies for all degrees of the scale are either ratios of small integers, or obtained by a succession of such ratios. It is tempered if it represents an adjustment, or tempering, of just intonation. It is practice-based if it simply reflects musical practice, as for instance various measurements of the tuning of a gamelan might do. In music theory, a diatonic scale (from the Greek diatonikos, to stretch out; also known as the heptatonia prima; set form 7-35) is a seven-note musical scale comprising five whole-tone and two half-tone steps, in which the half tones are maximally separated. ... In music, just intonation, also called rational intonation, is any musical tuning in which the frequencies of notes are related by ratios of whole numbers. ... In musical tuning, a temperament is a system of tuning which slightly compromises the pure intervals of just intonation in order to meet other requirements of the system. ... In music, tuning is the process of producing or preparing to produce a certain pitch in relation to another, usually at the unison but often at some other interval. ... Gamelan - Indonesian Embassy in Canberra A gamelan is a kind of musical ensemble of Indonesian origin typically featuring a variety of instruments such as metallophones, xylophones, drums, and gongs; bamboo flutes, bowed and plucked strings, and vocalists may also be included. ...

## Pythagorean tuning

Main article: Pythagorean tuning

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)²/2 = 9:8. Pythagorean tuning is a system of musical tuning in which the frequency relationships of all intervals are based on the ratio 3:2. ...

## Just intonation

Main article: Just intonation

If we take the ratios constituting a scale in just intonation, there will be a largest prime number to be found among their prime factorizations. This is called the prime limit of the scale. A scale which uses only the primes 2, 3 and 5 is called a 5-limit scale; in such a scale, all tones are regular number harmonics of a single fundamental frequency. Below is a typical example of a 5-limit justly tuned scale, one of the scales Johannes Kepler presents in his Harmonice Mundi or Harmonics of the World of 1619, in connection with planetary motion. The same scale was given in transposed form by Alexander Malcolm in 1721 and theorist Jose Wuerschmidt in the last century and is used in an inverted form 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". Despite this impressive pedigree, it is only one out of large number of somewhat similar scales. In music, just intonation, also called rational intonation, is any musical tuning in which the frequencies of notes are related by ratios of whole numbers. ... In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors, which are 1 and the prime number itself. ... This article is being considered for deletion in accordance with Wikipedias deletion policy. ... In acoustics and telecommunication, the harmonic of a wave is a component frequency of the signal that is an integer multiple of the fundamental frequency. ... The fundamental tone, often referred to simply as the fundamental, is the lowest frequency in a harmonic series. ... Johannes Kepler (December 27, 1571 â€“ November 15, 1630) was a German mathematician, astronomer and astrologer, and a key figure in the 17th century astronomical revolution. ... Harmonice Mundi (1619) is a book by Johannes Kepler. ... Terry Riley â€“ (Portrait by Betty Freeman) Terry Riley (born 24 June 1935) is an American composer associated with the minimalist school. ...

Note Ratio Interval
0 1:1 unison
1 135:128 major chroma
2 9:8 major second
3 6:5 minor third
4 5:4 major third
5 4:3 perfect fourth
6 45:32 diatonic tritone
7 3:2 perfect fifth
8 8:5 minor sixth
9 27:16 Pythagorean major sixth
10 9:5 minor seventh
11 15:8 major seventh
12 2:1 octave

(In theory unisons and octaves and their multiples are also "perfect" but this terminology is rarely used.) In music theory, the term interval describes the difference in pitch between two notes. ... For other uses, see Unison (disambiguation). ... A major second is one of three commonly occuring musical intervals that span two diatonic scale degrees; the others being the minor second, which is one semitone smaller, and the augmented second, which is one semitone larger. ... Sesquiquintum (Latin neuter) (masculine: sesquiquintus) refers to the improper rational fraction It is a superparticular number. ... A minor third is the smaller of two commonly occurring musical intervals that span three diatonic scale degrees. ... Sesquiquartum refers to the improper vulgar fraction The sesquiquartum in musical harmony is the ratio corresponding to the interval ditonus. ... A major third is the larger of two commonly occuring musical intervals that span three diatonic scale degrees. ... Sesquitertium refers to the improper rational fraction It is a superparticular number. ... The perfect fourth or diatessaron, abbreviated P4, is one of two musical intervals that span four diatonic scale degrees; the other being the augmented fourth, which is one semitone larger. ... The augmented fourth between C and F# forms a tritone. ... Sesquialterum (plural: sesquialtera) or sesquialter refers to the improper rational fraction . It is a superparticular number. ... The perfect fifth or diapente is one of three musical intervals that span five diatonic scale degrees; the others being the diminished fifth, which is one semitone smaller, and the augmented fifth, which is one semitone larger. ... A minor sixth is the smaller of two commonly occuring musical intervals that span six diatonic scale degrees. ... The musical interval of a major sixth is the relationship between the first note (the root or tonic) and the sixth note in a Major scale. ... The musical interval of a minor seventh the first note (the root or tonic) and the seventh in a minor scale. ... The musical interval of a Major seventh the first note (the root or tonic) and the seventh, the leading tone, in a major scale. ... In music, an octave (sometimes abbreviated 8ve) is the interval between one musical note and another with half or double its frequency. ...

To calculate the frequency of a note in a scale given in terms of ratios, the frequency ratio is multiplied by the frequency we associate to the unison, which will often be 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 A440 is the 440 Hz tone that serves as the standard for musical pitch. ... MHZ redirects here. ...

440*(3/2) = 660 Hz.

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." The syntonic comma, also known as the comma of Didymus or Ptolemaic comma, is a small interval between two musical notes, equal to the frequency ratio 81:80, or around 21. ... Carl Dahlhaus (June 10, 1928- May 1989), a musicologist from Berlin, has been one of the major contributors to the development of musicology as a scholarly discipline during the post-war era. ...

## Mathematics of musical scales

Western common practice music usually cannot be played in just intonation, even when it is confined to a single key. This is because the supertonic chord, or ii-chord, which is the most important of the minor triads in a major key, serves to bridge between the dominant and subdominant, having a root at once a minor third below the root of the subdominant triad, and hence sharing two of its notes, and a fifth above the root of the dominant triad or dominant seventh chord. The problem becomes still worse when modulation, the key changes so important to common practice music, comes into play. The scale of the Western tradition is by its very nature neither one of just intonation nor one defined only in practice, but is 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. In music the common practice period is a long period in western musical history spanning from before the classical era proper to today, dated, on the outside, as 1600-1900. ... In music or music theory, the supertonic is the second degree of the scale, it is the second note of a diatonic scale. ... In music, the dominant is the fifth degree of the scale. ... In music, the subdominant is the technical name for the fourth tonal degree of the diatonic scale. ... In music, modulation is most commonly the act or process of changing from one key (tonic, or tonal center) to another. ... Well temperament (also circular or circulating temperament) is a type of tempered tuning described in twentieth-century music theory. ... Regular temperament is a system of musical tuning such that each frequency ratio is obtainable as a product of powers of a finite number of generators, or generating frequency ratios. ... An equal temperament is a musical temperament â€” that is, a system of tuning intended to approximate some form of just intonation â€” in which an interval, usually the octave, is divided into a series of equal steps (equal frequency ratios). ... Meantone temperament is a system of musical tuning. ...

Meantone, however, is not the only worthwhile temperament nor is the equal division of the octave into twelve parts the only reasonable way to so divide it. Many other systems of temperament are possible, leading to a variety of harmonic relationships characteristic to them. These characteristics depend on what just intervals, called commas, which differ slightly from the unison become a unison when tempered.

In meantone, for example, the root of a ii-chord regarded as being a fifth above the dominant would be a major whole tone of 9/8 if the fifths were tuned justly, but would be a minor whole tone of 10/9 if it is taken to be a just minor third of 6/5 below a just subdominant degree of 4/3. These are being equated, so meantone temperament is tempering out the difference between 9/8 and 10/9. This means their ratio, (9/8)/(10/9) = 81/80, is tempered to a unison. The interval 81/80, called the syntonic comma or comma of Didymus, is the key comma of meantone temperament, and the fact that it becomes a unison in meantone temperament is a key fact of Western music. The syntonic comma, also known as the comma of Didymus or Ptolemaic comma, is a small interval between two musical notes, equal to the frequency ratio 81:80, or around 21. ...

### Equal temperament

Main article: Equal temperament

In equal temperament, the equal division of the octave into twelve 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 an equal tempering, so that the frets align evenly across the strings. In the European music tradition, equal tempering was used for lute and guitar music far earlier than for other instruments for this reason. An equal temperament is a musical temperament â€” that is, a system of tuning intended to approximate some form of just intonation â€” in which an interval, usually the octave, is divided into a series of equal steps (equal frequency ratios). ... An equal temperament is a musical temperament â€” that is, a system of tuning intended to approximate some form of just intonation â€” in which an interval, usually the octave, is divided into a series of equal steps (equal frequency ratios). ...

Equal tempered scales have been used and instruments built using various other numbers of equal tones. For example, the 19 equal temperament, first proposed and used by Guillaume Costeley in the sixteenth century, uses 19 equally spaced tones, and has better major thirds and far better minor thirds than 12 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 in very widespread use for Arabic music. In music, 19 equal temperament, called 19-tet, 19-edo, or 19-et, is the scale derived by dividing the octave into 19 equally large steps. ... Guillaume Costeley (ca. ... A quarter tone is an interval half as wide (aurally, or logarithmically) as a semitone, which is half a whole tone. ... Arabic music includes several genres and styles of music ranging from Arab classical to Arabic pop music and from secular to sacred music. ...

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 "perfect 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) Image File history File links Size of this preview: 776 Ã— 600 pixelsFull resolution (880 Ã— 680 pixel, file size: 47 KB, MIME type: image/jpeg) Gaul Armstrong (myself) created this file. ...

### Sound samples

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. This page is about the audio compression codec. ...

• 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 "diad". 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.

## References

• Loy, Gareth (2006). Musimathics: The Mathematical Foundations of Music, Vol. 1. The MIT Press. ISBN 0-262-12282-0.

MIT Press Books The MIT Press is a university publisher affiliated with the Massachusetts Institute of Technology (MIT) in Cambridge, Massachusetts. ...

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