Английская Википедия:Equal temperament

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12 tone equal temperament chromatic scale on Шаблон:Sc, one full octave ascending, notated only with sharps. Шаблон:Audio

An equal temperament is a musical temperament or tuning system that approximates just intervals by dividing an octave (or other interval) into steps such that the ratio of the frequencies of any adjacent pair of notes is the same. This system yields pitch steps perceived as equal in size, due to the logarithmic changes in pitch frequency.[1]

In classical music and Western music in general, the most common tuning system since the 18th century has been 12 equal temperament (also known as 12 tone equal temperament, 12 Шаблон:Sc or 12 Шаблон:Sc, informally abbreviated as 12 equal), which divides the octave into 12 parts, all of which are equal on a logarithmic scale, with a ratio equal to the 12th root of 2, ( Шаблон:Radic ≈ 1.05946 ). That resulting smallest interval, Шаблон:Sfrac the width of an octave, is called a semitone or half step. In Western countries the term equal temperament, without qualification, generally means 12 Шаблон:Sc.

In modern times, 12 Шаблон:Sc is usually tuned relative to a standard pitch of 440 Hz, called A 440, meaning one note, [[A (musical note)|Шаблон:Sc]], 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.[2]

Other equal temperaments divide the octave differently. For example, some music has been written in [[19 equal temperament|19 Шаблон:Sc]] and [[31 equal temperament|31 Шаблон:Sc]], while the Arab tone system uses 24 Шаблон:Sc.

Instead of dividing an octave, an equal temperament can also divide a different interval, like the equal-tempered version of the Bohlen–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 term equal division of the octave, or Шаблон:Sc can be used.

Unfretted string ensembles, which can adjust the tuning of all notes except for open strings, and vocal groups, who have no mechanical tuning limitations, sometimes use a tuning much closer to just intonation for acoustic reasons. Other instruments, such as some wind, keyboard, and fretted instruments, often only approximate equal temperament, where technical limitations prevent exact tunings.[3] Some wind instruments that can easily and spontaneously bend their tone, most notably trombones, use tuning similar to string ensembles and vocal groups.

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General properties

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In an equal temperament, the distance between two adjacent steps of the scale is the same interval. Because the perceived identity of an interval depends on its ratio, this scale in even steps is a geometric sequence of multiplications. (An arithmetic sequence of intervals would not sound evenly spaced and would not permit transposition to different keys.) Specifically, the smallest interval in an equal-tempered scale is the ratio:

<math>\ r^n = p\ </math>
<math>\ r = \sqrt[n]{p\ }\ </math>

where the ratio Шаблон:Mvar divides the ratio Шаблон:Mvar (typically the octave, which is 2:1) into Шаблон:Mvar equal parts. (See Twelve-tone equal temperament below.)

Scales are often measured in cents, which divide the octave into 1200 equal intervals (each called a cent). This logarithmic scale makes comparison of different tuning systems easier than comparing ratios, and has considerable use in ethnomusicology. The basic step in cents for any equal temperament can be found by taking the width of Шаблон:Mvar above in cents (usually the octave, which is 1200 cents wide), called below Шаблон:Mvar, and dividing it into Шаблон:Mvar parts:

<math>\ c = \frac{\ w\ }{ n }\ </math>

In musical analysis, material belonging to an equal temperament is often given an integer 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 the logarithm of a multiplication reduces it to addition. Furthermore, by applying the modular arithmetic where the modulus is the number of divisions of the octave (usually 12), these integers can be reduced to pitch classes, which removes the distinction (or acknowledges the similarity) between pitches of the same name, e.g., Шаблон:Mvar is 0 regardless of octave register. The MIDI encoding standard uses integer note designations.

General formulas for the equal-tempered interval

Шаблон:Missing information

Twelve-tone equal temperament

Шаблон:Main 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.

History

The two figures frequently credited with the achievement of exact calculation of equal temperament are Zhu Zaiyu (also romanized as Chu-Tsaiyu. Chinese: Шаблон:Lang) in 1584 and Simon Stevin in 1585. According to F.A. Kuttner, a critic of giving credit to Zhu,[4] 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.[5]Шаблон:Rp

Kenneth Robinson credits the invention of equal temperament to Zhu[6] and provides textual quotations as evidence.[7] In 1584 Zhu wrote:

I have founded a new system. I establish one foot as the number from which the others are to be extracted, and using proportions I extract them. Altogether one has to find the exact figures for the pitch-pipers in twelve operations.[8][7]

Kuttner disagrees and remarks that his claim "cannot be considered correct without major qualifications".[4] Kuttner proposes that neither Zhu nor Stevin achieved equal temperament and that neither should be considered its inventor.[9]

China

Файл:乐律全书全-1154.jpg
Zhu Zaiyu's equal temperament pitch pipes

Chinese theorists had previously come up with approximations for 12 Шаблон:Sc, but Zhu was the first person to mathematically solve 12 tone equal temperament,[10] which he described in two books, published in 1580[11] and 1584.[8][12] Needham also gives an extended account.[13]

Zhu obtained his result by dividing the length of string and pipe successively by Шаблон:Math , and for pipe length by Шаблон:Math ,[14] such that after 12 divisions (an octave), the length was halved.

Zhu created several instruments tuned to his system, including bamboo pipes.[15]

Europe

Some of the first Europeans to advocate equal temperament were lutenists Vincenzo Galilei, Giacomo Gorzanis, and Francesco Spinacino, all of whom wrote music in it.[16][17][18][19]

Simon Stevin was the first to develop 12 Шаблон:Sc based on the twelfth root of two, which he described in van de Spiegheling der singconst (Шаблон:Circa), published posthumously in 1884.[20]

Plucked instrument players (lutenists and guitarists) generally favored equal temperament,[21] while others were more divided.[22] In the end, 12-tone equal temperament won out. This allowed enharmonic modulation, new styles of symmetrical tonality and polytonality, atonal music such as that written with the 12-tone technique or serialism, and jazz (at least its piano component) to develop and flourish.

Mathematics

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One octave of 12 Шаблон:Sc on a monochord

In 12 tone equal temperament, which divides the octave into 12 equal parts, the width of a semitone, i.e. the frequency ratio of the interval between two adjacent notes, is the twelfth root of two:

<math> \sqrt[12]{2\ } = 2^{\tfrac{1}{12}} \approx 1.059463 </math>

This interval is divided into 100 cents.

Calculating absolute frequencies

Шаблон:See also To find the frequency, Шаблон:Math, of a note in 12 Шаблон:Sc, the following formula may be used:

<math>\ P_n = P_a\ \cdot\ \Bigl(\ \sqrt[12]{2\ }\ \Bigr)^{ n-a }\ </math>

In this formula Шаблон:Math represents the pitch, or frequency (usually in hertz), you are trying to find. Шаблон:Math is the frequency of a reference pitch. The indes numbers Шаблон:Mvar and Шаблон:Mvar are the labels assigned to the desired pitch (Шаблон:Mvar) and the reference pitch (Шаблон:Mvar). These two numbers are from a list of consecutive integers assigned to consecutive semitones. For example, Шаблон:ScШаблон:Sub (the reference pitch) is the 49th key from the left end of a piano (tuned to 440 Hz), and Шаблон:ScШаблон:Sub ([[middle C|middle Шаблон:Sc]]), and Шаблон:ScШаблон:MusicШаблон:Sub are the 40th and 46th keys, respectively. These numbers can be used to find the frequency of Шаблон:ScШаблон:Sub and Шаблон:ScШаблон:MusicШаблон:Sub:

<math>P_{40} = 440\ \mathsf{Hz}\ \cdot\ \Bigl( \sqrt[12]{2}\ \Bigr)^{(40-49)} \approx 261.626\ \mathsf{Hz}\ </math>
<math>P_{46} = 440\ \mathsf{Hz}\ \cdot\ \Bigl( \sqrt[12]{2}\ \Bigr)^{(46-49)} \approx 369.994\ \mathsf{Hz}\ </math>

Converting frequencies to their equal temperament counterparts

To convert a frequency (in Hz) to its equal 12 Шаблон:Sc counterpart, the following formula can be used:

<math>\ E_n = E_a\ \cdot\ 2^{\ x}\ \quad </math> where in general <math> \quad\ x\ \equiv\ \frac{ 1 }{\ 12\ }\ \operatorname{round}\!\Biggl( 12\log_{2} \left(\frac{\ n\ }{ a }\right) \Biggr) ~.</math>

Шаблон:Math is the frequency of a pitch in equal temperament, and Шаблон:Math is the frequency of a reference pitch. For example, if we let the reference pitch equal 440 Hz, we can see that Шаблон:ScШаблон:Sub and Шаблон:ScШаблон:MusicШаблон:Sub have the following frequencies, respectively:

<math>E_{660} = 440\ \mathsf{Hz}\ \cdot\ 2^{\left(\frac{ 7 }{\ 12\ }\right)}\ \approx\ 659.255\ \mathsf{Hz}\ \quad </math> where in this case <math> \quad x = \frac{ 1 }{\ 12\ }\ \operatorname{round}\!\Biggl(\ 12 \log_{2}\left(\frac{\ 660\ }{ 440 }\right)\ \Biggr) = \frac{ 7 }{\ 12\ } ~.</math>
<math>E_{550} = 440\ \mathsf{Hz}\ \cdot\ 2^{\left(\frac{ 1 }{\ 3\ }\right)}\ \approx\ 554.365\ \mathsf{Hz}\ \quad </math> where in this case <math> \quad x = \frac{ 1 }{\ 12\ }\ \operatorname{round}\!\Biggl( 12 \log_{2}\left(\frac{\ 550\ }{ 440 }\right)\Biggr) = \frac{ 4 }{\ 12\ } = \frac{ 1 }{\ 3\ } ~.</math>

Comparison with just intonation

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Comparison of intervals in 12-TET with just intonation

The intervals of 12 Шаблон:Sc closely approximate some intervals in just intonation.[23] 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 Шаблон:Sc Decimal value in 12 Шаблон:Sc Cents Just intonation interval Cents in just intonation Difference
Unison ([[C (musical note)|Шаблон:Sc]]) Шаблон:BigШаблон:Sup = Шаблон:Big 1 0 Шаблон:Sfrac = Шаблон:Big 0 0
Minor second ([[D♭ (musical note)|Шаблон:ScШаблон:Music]]) Шаблон:BigШаблон:Sup = Шаблон:Radic 1.059463 100 Шаблон:Sfrac = Шаблон:Big 111.73 -11.73
Major second ([[D (musical note)|Шаблон:Sc]]) Шаблон:BigШаблон:Sup = Шаблон:Radic 1.122462 200 Шаблон:Sfrac = Шаблон:Big 203.91 -3.91
Minor third ([[E♭ (musical note)|Шаблон:ScШаблон:Music]]) Шаблон:BigШаблон:Sup = Шаблон:Radic 1.189207 300 Шаблон:Sfrac = Шаблон:Big 315.64 -15.64
Major third ([[E (musical note)|Шаблон:Sc]]) Шаблон:BigШаблон:Sup = Шаблон:Radic 1.259921 400 Шаблон:Sfrac = Шаблон:Big 386.31 +13.69
Perfect fourth ([[F (musical note)|Шаблон:Sc]]) Шаблон:BigШаблон:Sup = Шаблон:Radic 1.33484 500 Шаблон:Sfrac = Шаблон:Big 498.04 +1.96
Tritone ([[G♭ (musical note)|Шаблон:ScШаблон:Music]]) Шаблон:BigШаблон:Sup = Шаблон:Sqrt 1.414214 600 Шаблон:Sfrac= Шаблон:Big 609.78 -9.78
Perfect fifth ([[G (musical note)|Шаблон:Sc]]) Шаблон:BigШаблон:Sup = Шаблон:Radic 1.498307 700 Шаблон:Sfrac = Шаблон:Big 701.96 -1.96
Minor sixth ([[A♭ (musical note)|Шаблон:ScШаблон:Music]]) Шаблон:BigШаблон:Sup = Шаблон:Radic 1.587401 800 Шаблон:Sfrac = Шаблон:Big 813.69 -13.69
Major sixth ([[A (musical note)|Шаблон:Sc]]) Шаблон:BigШаблон:Sup = Шаблон:Radic 1.681793 900 Шаблон:Sfrac = Шаблон:Big 884.36 +15.64
Minor seventh ([[B♭ (musical note)|Шаблон:ScШаблон:Music]]) Шаблон:BigШаблон:Sup = Шаблон:Radic 1.781797 1000 Шаблон:Sfrac = Шаблон:Big 996.09 +3.91
Major seventh ([[B (musical note)|Шаблон:Sc]]) Шаблон:BigШаблон:Sup = Шаблон:Radic 1.887749 1100 Шаблон:Sfrac = Шаблон:Big 1088.270 +11.73
Octave ([[C (musical note)|Шаблон:Sc]]) Шаблон:BigШаблон:Sup = Шаблон:Big Шаблон:Big 1200 Шаблон:Sfrac = Шаблон:Big 1200.00 0

Seven-tone equal division of the fifth

Violins, violas, and cellos are tuned in perfect fifths (Шаблон:Sc for violins and Шаблон:Sc 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 Шаблон:Radic to the next (100.28 cents), which provides for a perfect fifth with ratio of 3:2, but a slightly widened octave with a ratio of ≈ 517:258 or ≈ 2.00388:1 rather than the usual 2:1, because 12 perfect fifths do not equal seven octaves.[24] 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.

Other equal temperaments

Шаблон:See also

Five-, seven-, and nine-tone temperaments in ethnomusicology

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Approximation of 7 Шаблон:Sc

Five- and seven-tone equal temperament (5 Шаблон:Sc Шаблон:Audio and {{7 Шаблон:Sc}}Шаблон:Audio ), with 240 cent Шаблон:Audio and 171 cent Шаблон:Audio steps, respectively, are fairly common.

5 Шаблон:Sc and 7 Шаблон:Sc mark the endpoints of the syntonic temperament's valid tuning range, as shown in Figure 1.

  • In 5 Шаблон:Sc, the tempered perfect fifth is 720 cents wide (at the top of the tuning continuum), and marks the endpoint on the tuning continuum at which the width of the minor second shrinks to a width of 0 cents.
  • In 7 Шаблон:Sc, the tempered perfect fifth is 686 cents wide (at the bottom of the tuning continuum), and marks the endpoint on the tuning continuum, at which the minor second expands to be as wide as the major second (at 171 cents each).

5 tone and 9 tone equal temperament

According to Kunst (1949), Indonesian gamelans are tuned to 5 Шаблон:Sc, but according to Hood (1966) and McPhee (1966) their tuning varies widely, and according to Tenzer (2000) they contain stretched octaves. It is now accepted that of the two primary tuning systems in gamelan music, slendro and pelog, only slendro somewhat resembles five-tone equal temperament, while pelog is highly unequal; however, in 1972 Surjodiningrat, Sudarjana and Susanto analyze pelog as equivalent to 9-TET (133-cent steps Шаблон:Audio).[25]

7-tone equal temperament

A Thai xylophone measured by Morton in 1974 "varied only plus or minus 5 cents" from 7 Шаблон:Sc.[26] According to Morton,

"Thai instruments of fixed pitch are tuned to an equidistant system of seven pitches per octave ... As in Western traditional music, however, all pitches of the tuning system are not used in one mode (often referred to as 'scale'); in the Thai system five of the seven are used in principal pitches in any mode, thus establishing a pattern of nonequidistant intervals for the mode."[27] Шаблон:Audio

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.[28]

Chinese music has traditionally used 7 Шаблон:Sc.Шаблон:EfnШаблон:Efn

Various equal temperaments

Шаблон:More citations needed section

Файл:16-tet scale on C.png
Easley Blackwood's notation system for 16 equal temperament: Intervals are notated similarly to those they approximate and there are fewer enharmonic equivalents.[29] Шаблон:Audio
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Comparison of equal temperaments from 9 to 25[30]Шаблон:Efn
19 EDO
Many instruments have been built using 19 EDO tuning. Equivalent to Шаблон:Sfrac comma meantone, it has a slightly flatter perfect fifth (at 695 cents), but its minor third and major sixth are less than one-fifth of a cent away from just, with the lowest EDO that produces a better minor third and major sixth than 19 EDO being 232 EDO. Its perfect fourth (at 505 cents), is seven cents sharper than just intonation's and five cents sharper than 12 EDO's.
23 EDO
23 EDO is the largest EDO that fails to approximate the 3rd, 5th, 7th, and 11th harmonics (3:2, 5:4, 7:4, 11:8) within 20 cents. But it does approximate ratios between them (including the justly-tuned 6/5 minor third) very well, making it attractive to microtonalists seeking unusual harmonic territory.
24 EDO
24 EDO, the quarter-tone scale, is particularly popular, as it represents a convenient access point for composers conditioned on standard Western 12 EDO pitch and notation practices who are also interested in microtonality. Because 24 EDO contains all the pitches of 12 EDO, musicians employ the additional colors without losing any tactics available in 12 tone harmony. That 24 is a multiple of 12 also makes 24 EDO easy to achieve instrumentally by employing two traditional 12 EDO instruments tuned a quarter-tone apart, such as two pianos, which also allows each performer (or one performer playing a different piano with each hand) to read familiar 12 tone notation. Various composers, including Charles Ives, experimented with music for quarter-tone pianos. 24 EDO also approximates the 11th and 13th harmonics very well, unlike 12 EDO.
26 EDO
26 is the lowest number of equal divisions of the octave that almost purely tunes the 7th harmonic (7:4). Although it is a meantone temperament, it is a very flat one, with four of its perfect fifths producing a neutral third rather than a major third. 26 EDO has two minor thirds and two minor sixths and could be an alternate temperament for barbershop harmony.
27 EDO
27 is the lowest number of equal divisions of the octave that uniquely represents all intervals involving the first eight harmonics. It tempers out the septimal comma but not the syntonic comma.
29 EDO
29 is the lowest number of equal divisions of the octave whose perfect fifth is closer to just than in 12 EDO, in which the fifth is 1.5 cents sharp instead of 2 cents flat. Its classic major third is roughly as inaccurate as 12 EDO, but is tuned 14 cents flat rather than 14 cents sharp. It also tunes the 7th, 11th, and 13th harmonics flat, by roughly the same amount, allowing 29 EDO to match intervals such as 7:5, 11:7, and 13:11 very accurately. Cutting all 29 intervals in half produces 58 EDO, which allows for lower errors for some just tones.
31 EDO
31 EDO was advocated by Christiaan Huygens and Adriaan Fokker and represents a standardization of quarter-comma meantone. 31 EDO does not have as accurate a perfect fifth as 12 EDO (like 19 EDO), but its major thirds and minor sixths are less than 1 cent away from just. It also provides good matches for harmonics up to 11, of which the seventh harmonic is particularly accurate.
34 EDO
34 EDO gives slightly lower total combined errors of approximation to 3:2, 5:4, 6:5, and their inversions than 31 EDO does, despite having a slightly less accurate fit for 5:4. 34 EDO does not accurately approximate the seventh harmonic or ratios involving 7, and is not meantone since its fifth is sharp instead of flat. It enables the 600 cent tritone, since 34 is an even number.
41 EDO
41 is the second-lowest number of equal divisions of the octave with a better perfect fifth than 12 EDO. Its classic major third is more accurate than 12 EDO and 29 EDO, at six cents flat. It is not meantone, so it distinguishes 10:9 and 9:8, along with the classic and Pythagorean major thirds, unlike 31 EDO. It is more accurate in the 13 limit than 31 EDO.
46 EDO
46 EDO provides major thirds and perfect fifths that are both slightly sharp of just, and many say that this gives major triads a characteristic bright sound. The harmonics up to 11 are within 5 cents of accuracy, with 10:9 and 9:5 a fifth of a cent away from pure. As it is not a meantone system, it distinguishes 10:9 and 9:8.
53 EDO
53 EDO has only had occasional use, but is better at approximating the traditional just consonances than 12, 19 or 31 EDO. Its extremely accurate perfect fifths make it equivalent to an extended Pythagorean tuning, and it is sometimes used in Turkish music theory. It does not, however, fit the technical requirements of meantone temperaments, which put good thirds within easy reach, via the cycle of fifths. In 53 EDO, the very consonant thirds are instead reached by using a Pythagorean diminished fourth (C-FШаблон:Music), as it is an example of schismatic temperament, like 41 EDO.
58 EDO
58 equal temperament is a duplication of 29 EDO, which it contains as an embedded temperament. Like 29 EDO it can match intervals such as 7:4, 7:5, 11:7, and 13:11 very accurately, as well as better approximating just thirds and sixths.
72 EDO
72 EDO approximates many just intonation intervals well, providing near-just equivalents to the 3rd, 5th, 7th, and 11th harmonics. 72 EDO has been taught, written and performed in practice by Joe Maneri and his students (whose atonal inclinations typically avoid any reference to just intonation whatsoever). It can be considered an extension of 12 EDO because 72 is a multiple of 12. 72 EDO does not accurately approximate the 13th harmonic or most simple ratios involving 13. It contains six copies of 12 EDO starting on different pitches, three copies of 24 EDO, and two copies of 36 EDO, which are themselves multiples of 12.
96 EDO
96 EDO approximates all intervals within 6.25 cents, which is barely distinguishable. As an eightfold multiple of 12, it can be used fully like the common 12 EDO. It has been advocated by several composers, especially Julián Carrillo.[31]

Other equal divisions of the octave that have found occasional use include 15 EDO, 17 EDO, and 22 EDO.

2, 5, 12, 41, 53, 306, 665 and 15601 are denominators of first convergents of logШаблон:Sub(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 15601 just twelfths/fifths than in any equal temperament with fewer tones.[32][33]

1, 2, 3, 5, 7, 12, 29, 41, 53, 200, ... Шаблон:OEIS 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.Шаблон:Efn

Equal temperaments of non-octave intervals

The equal-tempered version of the Bohlen–Pierce scale consists of the ratio 3:1 (1902 cents) conventionally a perfect fifth plus an octave (that is, a perfect twelfth), called in this theory a tritave (Шаблон:Audio), and split into 13 equal parts. This provides a very close match to justly tuned ratios consisting only of odd numbers. Each step is 146.3 cents (Шаблон:Audio), or Шаблон:Radic.

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 called alpha, beta, and gamma. They can be considered equal divisions of the perfect fifth. Each of them provides a very good approximation of several just intervals.[34] Their step sizes:

Alpha and beta may be heard on the title track of Carlos's 1986 album Beauty in the Beast.

Proportions between semitone and whole tone

Шаблон:More citations needed section In this section, semitone and whole 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 be Шаблон:Mvar, and the number of steps in a tone be Шаблон:Mvar.

There is exactly one family of equal temperaments that fixes the semitone to any proper fraction of a whole tone, while keeping the notes in the right order (meaning that, for example, Шаблон:Sc, Шаблон:Sc, Шаблон:Sc, Шаблон:Sc, and Шаблон:ScШаблон:Music are in ascending order if they preserve their usual relationships to Шаблон:Sc). That is, fixing Шаблон:Mvar to a proper fraction in the relationship Шаблон:Math also defines a unique family of one equal temperament and its multiples that fulfil this relationship.

For example, where Шаблон:Mvar is an integer, 12Шаблон:Mvar Шаблон:Sc sets Шаблон:Math, 19 Шаблон:Mvar Шаблон:Sc sets Шаблон:Math, and 31 Шаблон:Mvar Шаблон:Sc sets Шаблон:Math. The smallest multiples in these families (e.g. 12, 19 and 31 above) has the additional property of having no notes outside the circle of fifths. (This is not true in general; in 24 Шаблон:Sc, the half-sharps and half-flats are not in the circle of fifths generated starting from Шаблон:Sc.) The extreme cases are 5 Шаблон:Mvar Шаблон:Sc, where Шаблон:Math and the semitone becomes a unison, and 7 Шаблон:Mvar Шаблон:Sc , where Шаблон:Math 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 Шаблон:Math steps and the perfect fifth into Шаблон:Math steps. If there are notes outside the circle of fifths, one must then multiply these results by Шаблон:Mvar, the number of nonoverlapping circles of fifths required to generate all the notes (e.g., two in 24 Шаблон:Sc, six in 72 Шаблон:Sc). (One must take the small semitone for this purpose: 19 Шаблон:Sc has two semitones, one being Шаблон:Sfrac tone and the other being Шаблон:Sfrac. Similarly, 31 Шаблон:Sc has two semitones, one being Шаблон:Sfrac tone and the other being Шаблон:Sfrac).

The smallest of these families is 12 Шаблон:Mvar Шаблон:Sc, and in particular, 12 Шаблон:Sc 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 Шаблон:Sc 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 fraction Шаблон:Mvar for the relationship results in exactly one equal temperament family, but the converse is not true: 47 Шаблон:Sc has two different semitones, where one is Шаблон:Sfrac tone and the other is Шаблон:Sfrac, which are not complements of each other like in 19 Шаблон:Sc (Шаблон:Sfrac and Шаблон:Sfrac). Taking each semitone results in a different choice of perfect fifth.

Related tuning systems

Regular diatonic tunings

Шаблон:Anchor

Файл:Rank-2 temperaments with the generator close to a fifth and period an octave.jpg
Figure 1: The regular diatonic tunings continuum, which include many notable "equal temperament" tunings.[35]

The diatonic tuning in 12 tone equal temperament (12 Шаблон:Sc) can be generalized to any regular diatonic tuning dividing the octave as a sequence of steps  Шаблон:Mvar  (or some circular shift or "rotation" of it). To be called a regular diatonic tuning, each of the two semitones (Шаблон:Mvar) must be smaller than either of the tones (greater tone, Шаблон:Mvar, and lesser tone, Шаблон:Mvar). The comma Шаблон:Mvar is implicit as the size ratio between the greater and lesser tones: Expressed as frequencies Шаблон:Math , or as cents Шаблон:Math .

The notes in a regular diatonic tuning are connected in a cycle of three perfect fifths  Шаблон:Mvar , interrupted by a grave fifth  Шаблон:Mvar  (grave means "flat by a comma"), another sequence of two perfect fifths, and another grave fifth, and then it repeats indefinitely, flattening by two commas with every transition from natural to sharp pitches (or single sharps to double sharps), and reciprocally sharpening by two commas with every transition from natural pitches to flattened pitches (or flats to double flats). If left unmodified, the two grave fifths in each octave are the source of "wolf" intervals.

Since the comma, Шаблон:Mvar, expands the lesser tone  Шаблон:Mvar , into the greater tone,  Шаблон:Mvar , just intonation  Шаблон:Mvar  can be broken up into a sequence  Шаблон:Mvar , (or a circular shift of it) of diatonic semitones Шаблон:Mvar, chromatic semitones Шаблон:Mvar, and commas  Шаблон:Mvar . Various equal temperaments alter the interval sizes, usually breaking apart the three commas and then redistributing their parts into the seven diatonic semitones Шаблон:Mvar, or into the five chromatic semitones Шаблон:Mvar, or into both Шаблон:Mvar and Шаблон:Mvar, with some fixed proportion for each type of semitone.

The sequence of intervals Шаблон:Mvar, Шаблон:Mvar, and Шаблон:Mvar can be repeatedly appended to itself into a greater spiral 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.

Morphing diatonic tunings into EDO

An equal temperament can be created if the sizes of the major and minor tones (Шаблон:Mvar, Шаблон:Mvar) are altered to be the same (say, by setting Шаблон:Math, with the others expanded to still fill out the octave), and both semitones ([[diatonic semitone|Шаблон:Mvar]] and Шаблон:Mvar) the same size, then twelve equal semitones, two per tone, result. In [[12 equal temperament|12 Шаблон:Sc]], the semitone, Шаблон:Mvar, is exactly half the size of the same-size whole tones Шаблон:Mvar = Шаблон:Mvar.

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 obtains [[7 equal temperament|7 Шаблон:Sc]] in the limit as the size of Шаблон:Mvar and Шаблон:Mvar tend to zero, with the octave kept fixed, and 5 Шаблон:Sc in the limit as Шаблон:Mvar and Шаблон:Mvar tend to zero; 12 Шаблон:Sc is of course, the case  Шаблон:Mvar  and  Шаблон:Math . For instance:

[[5 equal temperament|5 Шаблон:Sc]] and [[7 equal temperament|7 Шаблон:Sc]]
There are two extreme cases that bracket this framework: When Шаблон:Mvar and Шаблон:Mvar reduce to zero with the octave size kept fixed, the result is Шаблон:Mvar , a 5 tone equal temperament. As the Шаблон:Mvar gets larger (and absorbs the space formerly used for the comma Шаблон:Mvar), eventually the steps are all the same size, Шаблон:Mvar , and the result is seven-tone equal temperament. These two extremes are not included as "regular" diatonic tunings.
[[19 equal temperament|19 Шаблон:Sc]]
If the diatonic semitone is set double the size of the chromatic semitone, i.e.  Шаблон:Mvar (in cents) and  Шаблон:Math , the result is [[19 equal temperament|19 Шаблон:Sc]], with one step for the chromatic semitone Шаблон:Mvar, two steps for the diatonic semitone Шаблон:Mvar, three steps for the tones Шаблон:Mvar = Шаблон:Mvar, and the total number of steps Шаблон:Math 19 steps. The imbedded 12 tone sub-system closely approximates the historically important Шаблон:Sfrac comma meantone system.
[[31 equal temperament|31 Шаблон:Sc]]
If the chromatic semitone is two-thirds the size of the diatonic semitone, i.e. Шаблон:Math , with Шаблон:Math , the result is [[31 equal temperament|31 Шаблон:Sc]], with two steps for the chromatic semitone, three steps for the diatonic semitone, and five steps for the tone, where Шаблон:Math 31 steps. The imbedded 12 tone sub-system closely approximates the historically important [[quarter comma meantone|Шаблон:Sfrac comma meantone]].
[[53 equal temperament|53 Шаблон:Sc]]
If the chromatic semitone is made the same size as three commas,  Шаблон:Math  (in cents, in frequency  Шаблон:Math ) the diatonic the same as five commas,  Шаблон:Math , that makes the lesser tone eight commas Шаблон:Math , and the greater tone nine,  Шаблон:Math . Hence Шаблон:Math for 53 steps of one comma each. The comma size / step size is  Шаблон:Math ¢ exactly, or  Шаблон:Math ¢  Шаблон:Math ¢ , the syntonic comma. It is an exceedingly close approximation to just intonation, and is still in use for classical Turkish music theory.

See also

Шаблон:Div col begin

Шаблон:Div col end

Footnotes

Шаблон:Notelist

References

Шаблон:Reflist

Sources

Шаблон:Refbegin

Шаблон:Refend

Further reading

  • Шаблон:Cite book
    — A foundational work on acoustics and the perception of sound. Especially the material in Appendix XX: Additions by the translator, pages 430–556, (pdf pages 451–577) (see also wiki article On Sensations of Tone)

External links

Шаблон:Atonality Шаблон:Microtonal music Шаблон:Musical tuning

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  1. ↑ Шаблон:Cite web
  2. ↑ Шаблон:Cite book
  3. ↑ Шаблон:Cite journal
  4. ↑ 4,0 4,1 Шаблон:Harvp
  5. ↑ Шаблон:Cite journal
  6. ↑ Шаблон:Cite book
  7. ↑ 7,0 7,1 Шаблон:Cite book
  8. ↑ 8,0 8,1 Шаблон:Cite book
  9. ↑ Шаблон:Harvp
  10. ↑ Шаблон:Cite journal
  11. ↑ Шаблон:Cite book
  12. ↑ Шаблон:Cite web
  13. ↑ Шаблон:Harvp
  14. ↑ Шаблон:Cite book — reduced version of the original Шаблон:Harvp.
  15. ↑ Шаблон:Cite book
  16. ↑ Шаблон:Cite book
  17. ↑ Шаблон:Cite web
  18. ↑ Шаблон:Cite book
  19. ↑ Шаблон:Cite web
  20. ↑ Шаблон:Cite book
  21. ↑ Шаблон:Cite book
  22. ↑ Шаблон:Cite book
  23. ↑ Шаблон:Cite book
  24. ↑ Шаблон:Cite web
  25. ↑ Шаблон:Harvp
  26. ↑ Шаблон:Harvp
  27. ↑ Шаблон:Cite book
  28. ↑ Шаблон:Harvp
  29. ↑ Шаблон:Cite book
  30. ↑ Шаблон:Harvp
  31. ↑ Шаблон:Cite web
  32. ↑ Шаблон:Cite web
  33. ↑ Шаблон:Cite web
  34. ↑ Шаблон:Cite web
  35. ↑ Шаблон:Cite journal Online: Шаблон:ISSN
Источник — http://wikihandbk.com/ruwiki/index.php?title=Английская_Википедия:Equal_temperament&oldid=14557688