Anequal temperament is amusical temperament ortuning system that approximatesjust intervals by dividing anoctave (or other interval) into steps such that the ratio of thefrequencies of any adjacent pair of notes is the same. This system yieldspitch steps perceived as equal in size, due to thelogarithmic changes in pitch frequency.[2]
Inclassical music and Western music in general, the most common tuning system since the 18th century has been12 equal temperament (also known as12 tone equal temperament,12TET or12ET, informally abbreviated as12 equal), which divides the octave into 12 parts, all of which are equal on alogarithmic scale, with a ratio equal to the12th root of 2, ( ≈ 1.05946). That resulting smallest interval,1/12 the width of an octave, is called asemitone or half step. InWestern countries the termequal temperament, without qualification, generally means12TET.
In modern times,12TET is usually tuned relative to astandard pitch of 440 Hz, calledA 440, meaning one note,A, is tuned to 440 hertz and all other notes are defined as some multiple of semitones away from it, either higher or lower in frequency. The standard pitch has not always been 440 Hz; it has varied considerably and generally risen over the past few hundred years.[3]
Other equal temperaments divide the octave differently. For example, some music has been written in19TET and31TET, while theArab tone system uses24TET.
Instead of dividing an octave, an equal temperament can also divide a different interval, like the equal-tempered version of theBohlen–Pierce scale, which divides the just interval of an octave and a fifth (ratio 3:1), called a "tritave" or a "pseudo-octave" in that system, into 13 equal parts.
For tuning systems that divide the octave equally, but are not approximations of just intervals, the termequal division of the octave, orEDO can be used.
Unfrettedstring ensembles, which can adjust the tuning of all notes except foropen strings, and vocal groups, who have no mechanical tuning limitations, sometimes use a tuning much closer tojust intonation for acoustic reasons. Other instruments, such as somewind,keyboard, andfretted instruments, often only approximate equal temperament, where technical limitations prevent exact tunings.[4]Some wind instruments that can easily and spontaneously bend their tone, most notablytrombones, use tuning similar to string ensembles and vocal groups.
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In an equal temperament, the distance between two adjacent steps of the scale is the sameinterval. Because the perceived identity of an interval depends on itsratio, this scale in even steps is ageometric sequence of multiplications. (Anarithmetic sequence of intervals would not sound evenly spaced and would not permittransposition to differentkeys.) Specifically, the smallestinterval in an equal-tempered scale is the ratio:
where the ratior divides the ratiop (typically the octave, which is 2:1) inton equal parts. (SeeTwelve-tone equal temperament below.)
Scales are often measured incents, which divide the octave into 1200 equal intervals (each called a cent). Thislogarithmic scale makes comparison of different tuning systems easier than comparing ratios, and has considerable use inethnomusicology. The basic step in cents for any equal temperament can be found by taking the width ofp above in cents (usually the octave, which is 1200 cents wide), called beloww, and dividing it inton parts:
In musical analysis, material belonging to an equal temperament is often given aninteger notation, meaning a single integer is used to represent each pitch. This simplifies and generalizes discussion of pitch material within the temperament in the same way that taking thelogarithm of a multiplication reduces it to addition. Furthermore, by applying themodular arithmetic where the modulus is the number of divisions of the octave (usually 12), these integers can be reduced topitch classes, which removes the distinction (or acknowledges the similarity) between pitches of the same name, e.g.,c is 0 regardless of octave register. TheMIDI encoding standard uses integer note designations.
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12 tone equal temperament, which divides the octave into 12 intervals of equal size, is the musical system most widely used today, especially in Western music.
The two figures frequently credited with the achievement of exact calculation of equal temperament areZhu Zaiyu (also romanized as Chu-Tsaiyu. Chinese:朱載堉) in 1584 andSimon Stevin in 1585. According to F.A. Kuttner, a critic of giving credit to Zhu,[5] it is known that Zhu "presented a highly precise, simple and ingenious method for arithmetic calculation of equal temperament mono-chords in 1584" and that Stevin "offered a mathematical definition of equal temperament plus a somewhat less precise computation of the corresponding numerical values in 1585 or later."
The developments occurred independently.[6](p200)
Kenneth Robinson credits the invention of equal temperament to Zhu[7][b]and provides textual quotations as evidence.[8] In 1584 Zhu wrote:
Kuttner disagrees and remarks that his claim "cannot be considered correct without major qualifications".[5] Kuttner proposes that neither Zhu nor Stevin achieved equal temperament and that neither should be considered its inventor.[10]
Chinese theorists had previously come up with approximations for12TET, but Zhu was the first person to mathematically solve 12 tone equal temperament,[11] which he described in two books, published in 1580[12] and 1584.[9][13] Needham also gives an extended account.[14]
Zhu obtained his result by dividing the length of string and pipe successively by ≈ 1.059463, and for pipe length by ≈ 1.029302,[15] such that after 12 divisions (an octave), the length was halved.
Zhu created several instruments tuned to his system, including bamboo pipes.[16]
Some of the first Europeans to advocate equal temperament were lutenistsVincenzo Galilei,Giacomo Gorzanis, andFrancesco Spinacino, all of whom wrote music in it.[17][18][19][20]
Simon Stevin was the first to develop 12 TET based on thetwelfth root of two, which he described invan de Spiegheling der singconst (c. 1605), published posthumously in 1884.[21]
Plucked instrument players (lutenists and guitarists) generally favored equal temperament,[22] while others were more divided.[23] In the end, 12-tone equal temperament won out. This allowedenharmonic modulation, new styles of symmetrical tonality andpolytonality,atonal music such as that written with the12-tone technique orserialism, andjazz (at least its piano component) to develop and flourish.
In 12 tone equal temperament, which divides the octave into 12 equal parts, the width of asemitone, i.e. thefrequency ratio of the interval between two adjacent notes, is thetwelfth root of two:
This interval is divided into 100 cents.
To find the frequency,Pn, of a note in 12 TET, the following formula may be used:
In this formulaPn represents the pitch, or frequency (usually inhertz), that is to be calculated.Pa is the frequency of a reference pitch. The indes numbersn anda are the labels assigned to the desired pitch (n) and the reference pitch (a). These two numbers are from a list of consecutive integers assigned to consecutive semitones. For example, A4 (the reference pitch) is the 49th key from the left end of a piano (tuned to440 Hz), and C4 (middle C), and F♯4 are the 40th and 46th keys, respectively. These numbers can be used to find the frequency of C4 and F♯4:
To convert a frequency (in Hz) to its equal 12 TET counterpart, the following formula can be used:
En is the frequency of a pitch in equal temperament, andEa is the frequency of a reference pitch. For example, if we let the reference pitch equal 440 Hz, we can see thatE5 andC♯5 have the following frequencies, respectively:
The intervals of 12 TET closely approximate some intervals injust intonation.[24]The fifths and fourths are almost indistinguishably close to just intervals, while thirds and sixths are further away.
In the following table, the sizes of various just intervals are compared to their equal-tempered counterparts, given as a ratio as well as cents.
| Interval Name | Exact value in 12 TET | Decimal value in 12 TET | Pitch in | Just intonation interval | Cents in just intonation | 12 TET cents tuning error |
|---|---|---|---|---|---|---|
| Unison (C) | 20⁄12 =1 | 1 | 0 | 1/1 =1 | 0 | 0 |
| Minor second (D♭) | 21⁄12 = | 1.059463 | 100 | 16/15 =1.06666... | 111.73 | -11.73 |
| Major second (D) | 22⁄12 = | 1.122462 | 200 | 9/8 =1.125 | 203.91 | -3.91 |
| Minor third (E♭) | 23⁄12 = | 1.189207 | 300 | 6/5 =1.2 | 315.64 | -15.64 |
| Major third (E) | 24⁄12 = | 1.259921 | 400 | 5/4 =1.25 | 386.31 | +13.69 |
| Perfect fourth (F) | 25⁄12 = | 1.33484 | 500 | 4/3 =1.33333... | 498.04 | +1.96 |
| Tritone (G♭) | 26⁄12 = | 1.414214 | 600 | 64/45=1.42222... | 609.78 | -9.78 |
| Perfect fifth (G) | 27⁄12 = | 1.498307 | 700 | 3/2 =1.5 | 701.96 | -1.96 |
| Minor sixth (A♭) | 28⁄12 = | 1.587401 | 800 | 8/5 =1.6 | 813.69 | -13.69 |
| Major sixth (A) | 29⁄12 = | 1.681793 | 900 | 5/3 =1.66666... | 884.36 | +15.64 |
| Minor seventh (B♭) | 210⁄12 = | 1.781797 | 1000 | 16/9 =1.77777... | 996.09 | +3.91 |
| Major seventh (B) | 211⁄12 = | 1.887749 | 1100 | 15/8 =1.875 | 1088.270 | +11.73 |
| Octave (C) | 212⁄12 =2 | 2 | 1200 | 2/1 =2 | 1200.00 | 0 |
Violins, violas, and cellos are tuned in perfect fifths (G D A E for violins andC G D A for violas and cellos), which suggests that their semitone ratio is slightly higher than in conventional 12 tone equal temperament. Because a perfect fifth is in 3:2 relation with its base tone, and this interval comprises seven steps, each tone is in the ratio of to the next (100.28 cents), which provides for a perfect fifth with ratio of 3:2, but a slightly widened octave with aratio of ≈ 517:258 or ≈ 2.00388:1 rather than the usual 2:1, because 12 perfect fifths do not equal seven octaves.[25] During actual play, however, violinists choose pitches by ear, and only the four unstopped pitches of the strings are guaranteed to exhibit this 3:2 ratio.
Five- and seven-tone equal temperament (5TETPlayⓘ and7TETPlayⓘ ), with 240 centPlayⓘ and 171 centPlayⓘ steps, respectively, are fairly common.
5TET and7TET mark the endpoints of thesyntonic temperament's valid tuning range, as shown inFigure 1.
According toKunst (1949), Indonesiangamelans are tuned to5TET, but according toHood (1966) andMcPhee (1966) their tuning varies widely, and according toTenzer (2000) they containstretched octaves. It is now accepted that of the two primary tuning systems in gamelan music,slendro andpelog, only slendro somewhat resembles five-tone equal temperament, while pelog is highly unequal; however, in 1972 Surjodiningrat, Sudarjana and Susanto analyze pelog as equivalent to9TET (133-cent stepsPlayⓘ).[26]
AThai xylophone measured by Morton in 1974 "varied only plus or minus 5 cents" from7TET.[27] According to Morton,
A South American Indian scale from a pre-instrumental culture measured by Boiles in 1969 featured 175 cent seven-tone equal temperament, which stretches the octave slightly, as with instrumental gamelan music.[29]
Chinese music has traditionally used7TET.[c][d]
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Other equal divisions of the octave that have found occasional use include13 EDO,15 EDO,17 EDO, and 55 EDO.
2, 5, 12, 41, 53, 306, 665 and 15601 aredenominators of firstconvergents of log2(3), so 2, 5, 12, 41, 53, 306, 665 and 15601 twelfths (and fifths), being in correspondent equal temperaments equal to an integer number of octaves, are better approximations of 2, 5, 12, 41, 53, 306, 665 and 15601just twelfths/fifths than in any equal temperament with fewer tones.[35][36]
1, 2, 3, 5, 7, 12, 29, 41, 53, 200, ... (sequenceA060528 in theOEIS) is the sequence of divisions of octave that provides better and better approximations of the perfect fifth. Related sequences containing divisions approximating other just intervals are listed in a footnote.[e]
The equal-tempered version of theBohlen–Pierce scale consists of the ratio 3:1 (1902 cents) conventionally aperfect fifth plus anoctave (that is, a perfect twelfth), called in this theory atritave (playⓘ), and split into 13 equal parts. This provides a very close match tojustly tuned ratios consisting only of odd numbers. Each step is 146.3 cents (playⓘ), or.
Wendy Carlos created three unusual equal temperaments after a thorough study of the properties of possible temperaments with step size between 30 and 120 cents. These were calledalpha,beta, andgamma. They can be considered equal divisions of the perfect fifth. Each of them provides a very good approximation of several just intervals.[37] Their step sizes:
Alpha and beta may be heard on the title track of Carlos's 1986 albumBeauty in the Beast.
While traditional equal temperaments—such as 12‑TET, 19‑TET, or 31‑TET—divide the octave into an integral number of equal parts, it is also possible to explore systems that divide the octave into a non-integral (often irrational) number. In such temperaments, the interval between successive pitches is defined by the ratio 2^(1/N), where N is not an integer. This results in irrational step sizes, meaning their multiples never exactly equal an octave.
Such tunings are of interest because, by deliberately sacrificing the octave (i.e., the second harmonic), they can yield a system that offers an improved overall approximation of other intervals in the harmonic series.
For example, in a tuning system based on 18.911‑EDO, the step size is 1200⁄18.911 ≈ 63.45 cents. Approximating the just perfect fifth (with a ratio of 3:2, or about 701.96 cents) requires about 11 steps:
yielding an error of roughly 3 cents.
Similarly, for the just major third (with a ratio of 5:4, or about 386.31 cents), 6 steps are used:
resulting in an error of approximately 5.61 cents.
Thus, although a perfect octave is absent, the consonance of many other intervals in these systems can be significantly higher than in integer-based equal temperaments.
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In this section,semitone andwhole tone may not have their usual 12 EDO meanings, as it discusses how they may be tempered in different ways from their just versions to produce desired relationships. Let the number of steps in a semitone bes, and the number of steps in a tone bet.
There is exactly one family of equal temperaments that fixes the semitone to anyproper fraction of a whole tone, while keeping the notes in the right order (meaning that, for example,C,D,E,F, andF♯ are in ascending order if they preserve their usual relationships toC). That is, fixingq to a proper fraction in the relationshipq t =s also defines a unique family of one equal temperament and its multiples that fulfil this relationship.
For example, wherek is an integer,12kEDO setsq =1/2,19kEDO setsq =1/3, and31kEDO setsq = 2 / 5. The smallest multiples in these families (e.g. 12, 19 and 31 above) has the additional property of having no notes outside thecircle of fifths. (This is not true in general; in 24 EDO, the half-sharps and half-flats are not in the circle of fifths generated starting fromC.) The extreme cases are5kEDO, whereq = 0 and the semitone becomes a unison, and7kEDO, whereq = 1 and the semitone and tone are the same interval.
Once one knows how many steps a semitone and a tone are in this equal temperament, one can find the number of steps it has in the octave. An equal temperament with the above properties (including having no notes outside the circle of fifths) divides the octave into 7t − 2s steps and the perfect fifth into 4t −s steps. If there are notes outside the circle of fifths, one must then multiply these results byn, the number of nonoverlapping circles of fifths required to generate all the notes (e.g., two in 24 EDO, six in 72 EDO). (One must take the small semitone for this purpose: 19 EDO has two semitones, one being 1 / 3 tone and the other being 2 / 3. Similarly, 31 EDO has two semitones, one being 2 / 5 tone and the other being 3 / 5).
The smallest of these families is12kEDO, and in particular, 12 EDO is the smallest equal temperament with the above properties. Additionally, it makes the semitone exactly half a whole tone, the simplest possible relationship. These are some of the reasons 12 EDO has become the most commonly used equal temperament. (Another reason is that 12 EDO is the smallest equal temperament to closely approximate 5 limit harmony, the next-smallest being 19 EDO.)
Each choice of fractionq for the relationship results in exactly one equal temperament family, but the converse is not true: 47 EDO has two different semitones, where one is 1 / 7 tone and the other is 8 / 9, which are not complements of each other like in 19 EDO ( 1 / 3 and 2 / 3). Taking each semitone results in a different choice of perfect fifth.
Equal temperament systems can be thought of in terms of the spacing of three intervals found injust intonation,most of whose chords are harmonically perfectly in tune—a good property not quite achieved between almost all pitches in almost all equal temperaments. Most just chords sound amazingly consonant, and most equal-tempered chords sound at least slightly dissonant. In C major those three intervals are:[38]
Analyzing an equal temperament in terms of how it modifies or adapts these three intervals provides a quick way to evaluate how consonant various chords can possibly be in that temperament, based on how distorted these intervals are.[38][f]
The diatonic tuning in12 tone equal temperament(12TET) can be generalized to any regular diatonic tuning dividing the octave as a sequence of steps T t s T t T s (or somecircular shift or "rotation" of it). To be called aregular diatonic tuning, each of the two semitones ( s ) must be smaller than either of the tones (greater tone, T , andlesser tone, t ). The commaκ is implicit as the size ratio between the greater and lesser tones: Expressed as frequenciesκ = T /t , or ascents κ =T −t .
The notes in a regular diatonic tuning are connected in a "spiral of fifths" that doesnot close (unlike thecircle of fifths in12TET). Starting on the subdominantF (in thekey of C) there are threeperfect fifths in a row—F–C,C–G, andG–D—each a composite of somepermutation of the smaller intervals T T t s . The three in-tune fifths are interrupted by thegrave fifthD–A = T t t s (grave means "flat by acomma"), followed by another perfect fifth,E–B, and another grave fifth,B–F♯, and then restarting in the sharps withF♯–C♯; the same pattern repeats through the sharp notes, then the double-sharps, and so on, indefinitely. But each octave of all-natural or all-sharp or all-double-sharp notes flattens by two commas with every transition from naturals to sharps, or single sharps to double sharps, etc. The pattern is also reverse-symmetric in the flats: Descending byfourths the pattern reciprocally sharpens notes by two commas with every transition from natural notes to flattened notes, or flats to double flats, etc. If left unmodified, the two grave fifths in each block of all-natural notes, or all-sharps, or all-flat notes, are"wolf" intervals: Each of the grave fifths out of tune by adiatonic comma.
Since the comma,κ, expands thelesser tone t = s c , into thegreater tone, T = s c κ , ajust octave T t s T t T s can be broken up into a sequence s c κ s c s s c κ s c s c κ s , (or acircular shift of it) of 7 diatonic semitoness, 5 chromatic semitonesc, and 3 commas κ . Various equal temperaments alter the interval sizes, usually breaking apart the three commas and then redistributing their parts into the seven diatonic semitoness, or into the five chromatic semitonesc, or into boths andc, with some fixed proportion for each type of semitone.
The sequence of intervalss,c, andκ can be repeatedly appended to itself into a greaterspiral of 12 fifths, and made to connect at its far ends by slight adjustments to the size of one or several of the intervals, or left unmodified with occasional less-than-perfect fifths, flat by a comma.
Various equal temperaments can be understood and analyzed as having made adjustments to the sizes of and subdividing the three intervals— T , t , and s , or at finer resolution, their constituents s , c , and κ . An equal temperament can be created by making the sizes of themajor andminor tones (T,t) the same (say, by settingκ = 0, with the others expanded to still fill out the octave), and both semitones (s andc) the same, then 12 equal semitones, two per tone, result. In12TET, the semitone,s, is exactly half the size of the same-size whole tonesT =t.
Some of the intermediate sizes of tones and semitones can also be generated in equal temperament systems, by modifying the sizes of the comma and semitones. One obtains7TET in the limit as the size ofc andκ tend to zero, with the octave kept fixed, and5TET in the limit ass andκ tend to zero;12TET is of course, the case s = c and κ = 0 . For instance:
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