Music theory analyzes thepitch, timing, and structure ofmusic. It usesmathematics to studyelements of music such astempo,chord progression,form, andmeter. The attempt to structure and communicate new ways of composing and hearing music has led to musical applications ofset theory,abstract algebra andnumber theory.
While music theory has noaxiomatic foundation in modern mathematics, the basis of musicalsound can be described mathematically (usingacoustics) and exhibits "a remarkable array of number properties".[1]
Though ancient Chinese, Indians, Egyptians and Mesopotamians are known to have studied the mathematical principles of sound,[2] thePythagoreans (in particularPhilolaus andArchytas)[3] of ancient Greece were the first researchers known to have investigated the expression ofmusical scales in terms of numericalratios,[4] particularly the ratios of small integers. Their central doctrine was that "all nature consists ofharmony arising out of numbers".[5]
From the time ofPlato, harmony was considered a fundamental branch ofphysics, now known asmusical acoustics. EarlyIndian andChinese theorists show similar approaches: all sought to show that the mathematical laws ofharmonics andrhythms were fundamental not only to our understanding of the world but to human well-being.[6]Confucius, like Pythagoras, regarded the small numbers 1, 2, 3, and 4 as the source of all perfection.[7]
Without the boundaries of rhythmic structure – a fundamental equal and regular arrangement ofpulserepetition,accent,phrase and duration – music would not be possible.[8] Modern musical use of terms likemeter andmeasure also reflects the historical importance of music, along with astronomy, in the development of counting, arithmetic and the exact measurement of time andperiodicity that is fundamental to physics.[citation needed]
The elements of musical form often build strict proportions or hypermetric structures (powers of the numbers 2 and 3).[9]
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.[10] The common types of form known asbinary andternary ("twofold" and "threefold") once again demonstrate the importance of small integral values to the intelligibility and appeal of music.[11][12]
Amusical scale is a discrete set ofpitches used in making or describing music. The most important scale in the Western tradition is thediatonic 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 theoctave. Theoctave 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 calleddoh orA orSa, as the case may be).
When expressed as a frequency bandwidth an octaveA2–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 orratios between the pitches (known asintervals) 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 written1/1), generally a note which functions as thetonic of the scale. For interval size comparison,cents are often used.
| Common term | Example name | Hz | Multiple of fundamental | Ratio of within octave | Cents within octave |
|---|---|---|---|---|---|
Fundamental | A2 | 110 | 0 | ||
| Octave | A3 | 220 | 1200 | ||
0 | |||||
Perfect fifth | E4 | 330 | 702 | ||
| Octave | A4 | 440 | 1200 | ||
0 | |||||
Major third | C♯5 | 550 | 386 | ||
Perfect fifth | E5 | 660 | 702 | ||
| G5 | 770 | 969 | |||
| Octave | A5 | 880 | 1200 | ||
0 |
There are two main families of tuning systems:equal temperament andjust tuning. Equal temperament scales are built by dividing an octave into intervals which are equal on alogarithmic scale, which results in perfectly evenly divided scales, but with ratios of frequencies which areirrational numbers. Just scales are built by multiplying frequencies byrational numbers, which results in simple ratios between frequencies, but with scale divisions that are uneven.
One major difference between equal temperament tunings and just tunings is differences inacoustical beat when two notes are sounded together, which affects the subjective experience ofconsonance and dissonance. Both of these systems, and the vast majority of music in general, have scales that repeat on the interval of everyoctave, which is defined as frequency ratio of 2:1. In other words, every time the frequency is doubled, the given scale repeats.
Below areOgg Vorbis 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.
5-limit tuning, the most common form ofjust intonation, is a system of tuning using tones that areregular numberharmonics of a singlefundamental frequency. This was one of the scalesJohannes Kepler presented in hisHarmonices Mundi (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',[13] and by theoristJose Wuerschmidt in the 20th century. A form of it is used in the music of northern India.
American composerTerry 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 nochord progression: 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.[14] 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 ofA4 (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 | 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 | diatonictritone | 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 minus an octave, (3:2)2/2 = 9:8.
The just major third, 5:4 and minor third, 6:5, are asyntonic comma, 81:80, apart from their Pythagorean equivalents 81:64 and 32:27 respectively. According toCarlDahlhaus (1990, p. 187), "the dependent third conforms to the Pythagorean, the independent third to the harmonic tuning of intervals."
Westerncommon practice music usually cannot be played in just intonation but requires a systematically tempered scale. The tempering can involve either the irregularities ofwell temperament or be constructed as aregular temperament, either some form ofequal temperament or some other regular meantone, but in all cases will involve the fundamental features ofmeantone temperament. For example, the root of chordii, 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 thesyntonic comma or comma of Didymus, is the key comma of meantone temperament.
Inequal temperament, 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-toneArab tone system), the most common number is 12, which makes up the equal-temperamentchromatic scale. In western music, a division into twelve intervals is commonly assumed unless it is specified otherwise.
For the chromatic scale, the octave is divided into twelve equal parts, each semitone (half-step) is an interval of thetwelfth 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 asmusical 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. The19 equal temperament, first proposed and used byGuillaume 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.Twenty-four equal temperament, with twenty-four equally spaced tones, is widespread in the pedagogy andnotation ofArabic music. However, in theory and practice, the intonation of Arabic music conforms torational ratios, as opposed to theirrational ratios of equally tempered systems.[15]
While any analog to the equally temperedquarter tone is entirely absent from Arabic intonation systems, analogs to a three-quarter tone, orneutral second, frequently occur. These neutral seconds, however, vary slightly in their ratios dependent onmaqam, as well as geography. Indeed, Arabic music historianHabib 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."[15]
53 equal temperament arises from the near equality of 53perfect fifths with 31 octaves, and was noted byJing Fang andNicholas Mercator.
Musical set theory uses the language of mathematicalset theory 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 astransposition andinversion, one can discover deep structures in the music. Operations such as transposition and inversion are calledisometries because they preserve the intervals between tones in a set.
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 anabelian group with 12 elements. It is possible to describejust intonation in terms of afree abelian group.[16][17]
Transformational theory is a branch of music theory developed byDavid 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. 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 aGrassmannian.
Thechromatic scale has a free and transitive action of thecyclic group, with the action being defined viatransposition of notes. So the chromatic scale can be thought of as atorsor for the group.
Some composers have incorporated thegolden ratio andFibonacci numbers into their work.[18][19]
Themathematician andmusicologistGuerino Mazzola has usedcategory theory (topos theory) for a basis of music theory, which includes usingtopology as a basis for a theory ofrhythm andmotives, anddifferential geometry as a basis for a theory ofmusical phrasing,tempo, andintonation.[20]
.jpg)
