Music and mathematics

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Music theorists often use mathematics to understand musical structure and communicate new ways of hearing music. This has led to musical applications of set theory, abstract algebra, and number theory. Music scholars have also used mathematics to understand musical scales, and some composers have incorporated the Golden ratio and Fibonacci numbers into their work.^[1]

[edit] Connections to set theory

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.

[edit] Connections to 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 in fact possible to describe just intonation in terms of free abelian group.^[2]

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.

[edit] Connections to number theory

Modern interpretation of just intonation is fully based on fundamental theorem of arithmetic.

[edit] 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] However, regarding the listener, the degree to which the application of the golden ratio in music is salient, whether consciously or unconsciously, as well as the overall musical effect of its implementation, if any, is unknown.

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 Béla Bartók's works as being based on two opposing systems: those of the golden ratio and the acoustic scale. In Bartok'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. His use of the ratio gave his music an otherworldly symmetry.

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.

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.

This Binary Universe, an experimental album by Brian Transeau (popularly known as the electronic artist BT), includes a track titled 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.

[edit] Tuning systems

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.

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.

[edit] 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.

[edit] 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 theory unisons and octaves and their multiples are also "perfect" but this terminology is rarely used.)

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

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."

[edit] 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.

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.

[edit] 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.

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.

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)

[edit] 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.

• 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.

[edit] Frequency and pitch

In frequency space, each octave is exactly twice the size of the previous octave.

A given octave may span from 110 Hz to 220 Hz. (span=110 Hz)

The next octave will span from 220 Hz to 440 Hz. (span=220Hz)

The third octave spans from 440 Hz to 880 Hz. (span=440 Hz), ad infinitum.

Each successive octave spans twice the frequency range of the previous octave.

This diagram underscores the exponential nature of octaves, when measured on a linear frequency scale.

[edit] Perception

Human ears interpret all octaves as spanning a range of pitches the same size, even though a sub-bass octave may span 40 Hz and a super-treble octave can span 4000 Hz.

This diagrams presents octaves as they appear to the ear, as equally spaced units.

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
C2	130.81	7.34	7.031	0.0833
C2#	138.59	7.78	7.115	0.0833
D2	146.83	8.24	7.198	0.0833
D2#	155.56	8.73	7.281	0.0833
E2	164.81	9.25	7.365	0.0833
F2	174.61	9.80	7.448	0.0833
F2#	185.00	10.39	7.531	0.0833
G2	196.00	11.00	7.615	0.0833
G2#	207.65	11.65	7.698	0.0833
A3	220.00	12.35	7.781	0.0833

Harmonic Identity	Common Name	Example	Multiple of Fundamental Freq	Ratio (this identity/last octave)
1	Fundamental	A2 - 110Hz	1x	1/1 = 1x
2	Octave	A3 - 220 Hz	2x	2/1 = 2x (also 2/2 = 1x)
3	Perfect Fifth	E3 - 330 Hz	3x	3/2 = 1.5x
4	Octave	A4 - 440 Hz	4x	4/2 = 2x (also 1x)
5	Major Third	C#4 - 550 Hz	5x	5/4 = 1.25x
6	Perfect Fifth	E4 - 660 Hz	6x	6/4 = 1.5x
7	"Perfect Seventh"	G#4 - 770 Hz	7x	7/4 = 1.75x
8	Octave	A5 - 880 Hz	8x	8/4 = 2x (also 1x)

This diagram lists the first 16 harmonic identities, along with their names and frequencies. It is reveals the exponential nature of the octave and the simple fractional nature of the non-octave harmonics.

Harmonic Identity	Common Name	Linear Point Exponential Scale	Linear Point Normalized (linear) Scale
1	fundamental	1/1 = 1x	log2(1.0) = 0.00
2	octave	2/1 = 2x	log2(2.0) = 1.00
3	perfect fifth	3/2 = 1.5x	log2(1.5) = 0.585
4	octave	4/2 = 2x	log2(2.0) = 1.00
5	major third	5/4 = 1.25x	log2(1.25) = 0.322
6	perfect fifth	6/4 = 1.5x	log2(1.5) = 0.585
7	"perfect seventh"	7/4 = 1.75x	log2(1.75) = 0.807
8	octave	8/4 = 2x	log2(2.0) = 1.00

This diagram lists the first 16 harmonic identities, frequencies, and log frequencies.

• The Perfect Fifth is located on the 7th step of 12-TET scale. 7/12 = 0.583... ≈ 0.585....
• The Major Third is located on the 4th step of the 12-TET scale. 4/12 = 0.333... ≈ 0.322....
• The Perfect Fourth (the distance from a Perfect Fifth to it nearest upper octave) is located on the 5th step of the 12-TET. 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 12-TET. 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....

[edit] What other equal tempered scales have harmonic identities 1-8 represented?

^{[original research?]}

The diagram below compares/contrasts several good equal-tempered scales. The frequencies are plotted on a logarithmic scale so that each step is equally spaced. On a linear frequency scale, the steps would exponentially grow in size. It is clear how nearly each scale approximates the exact M3, P5, and P7. (The P7 is seldom used in Western music.) Note: the scale steps are the black bars separating the colored spaces.

[edit] See also

Music portal

Mathematics portal

[edit] References

[edit] External links

• Journal of Mathematics & Music, ed. Thomas Noll and Robert Peck
• Database of all the possible 2048 musical scales in 12 note equal temperament and other alternatives in meantone tunings
• Article by Adriaan Daniël Fokker analysing 31-tet and the harmonic identities
• Website for Musimathics book. Contains a table of content
• Music, Mathematics, Philosophy and tuning
• Google Scholar Search for 'music and mathematics'
• The method for transformation of music into an image through numbers and geometrical proportions
• 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
• LucyTuning, a musical tuning, harmonic, and scalecoding system derived from pi, and the writings of John Harrison

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