Editing Equal temperament (section)

==Other equal temperaments==<!--[[10 equal temperament]], etc. redirect directly here-->
{{See also|Sonido 13}}

=== Five-, seven-, and nine-tone temperaments in ethnomusicology ===<!--[[7 equal temperament]] and [[5 equal temperament]] redirect directly here-->
[[File:7-tet scale on C.png|thumb|300px|Approximation of {{nobr|7 {{sc|tet}}}}]]

Five- and seven-tone equal temperament (''{{nobr|5 {{sc|TET}}}}'' {{audio|5-tet scale on C.mid|Play}} and ''7 {{sc|TET}}''{{audio|7-tet scale on C.mid|Play}} ), with 240&nbsp;cent {{Audio|1 step in 5-et on C.mid|Play}} and 171&nbsp;cent {{Audio|1 step in 7-et on C.mid|Play}} steps, respectively, are fairly common.

{{nobr|5 {{sc|TET}}}} and {{nobr|7 {{sc|TET}}}} mark the endpoints of the [[syntonic temperament]]'s valid tuning range, as shown in [[#Figure 1|Figure&nbsp;1]].
* In {{nobr|5 {{sc|TET}},}} the tempered perfect fifth is 720&nbsp;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&nbsp;cents.
* In {{nobr|7 {{sc|TET}},}} the tempered perfect fifth is 686&nbsp;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&nbsp;cents each).

====5&nbsp;tone and 9&nbsp;tone equal temperament====
According to [[Jaap Kunst|Kunst]] (1949), Indonesian [[gamelan]]s are tuned to {{nobr|5 {{sc|TET}},}} but according to [[Mantle Hood|Hood]] (1966) and [[Colin McPhee|McPhee]] (1966) their tuning varies widely, and according to [[Michael Tenzer|Tenzer]] (2000) they contain [[pseudo-octave|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 {{nobr|9 {{sc|TET}}}} (133-cent steps {{Audio|Semitone Maximus on C.mid|Play}}).<ref>{{harvp|Surjodiningrat|Sudarjana|Susanto|1972}}</ref>

====7-tone equal temperament====
A [[Thai music|Thai]] xylophone measured by Morton in 1974 "varied only plus or minus 5&nbsp;cents" from {{nobr|7 {{sc|TET}}}}.<ref>{{harvp|Morton|1980}}</ref> 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."<ref>{{cite book |last=Morton |first=David |year=1980 |title=The Music of Thailand |series=Musics of Many Cultures |page=70 |editor-last=May |editor-first=Elizabeth |isbn=0-520-04778-8 }}</ref> {{audio|Thai pentatonic scale mode 1.mid|Play}}

A South American Indian scale from a pre-instrumental culture measured by Boiles in 1969 featured 175&nbsp;cent seven-tone equal temperament, which stretches the octave slightly, as with instrumental gamelan music.<ref>{{harvp|Boiles|1969}}</ref>

[[Music of China|Chinese music]] has traditionally used {{nobr|7 {{sc|TET}}}}.{{efn|
'Hepta-equal temperament' in our folk music has always been a controversial issue.<!-- mostly from Google translate, please verify --><ref>{{cite web |url=http://www.wanfangdata.com.cn/qikan/periodical.Articles/ZHONGUOYY/ZHON2004/0404/040425.htm |script-title=zh:有关"七平均律"新文献著作的发现 |language=zh |trans-title=Findings of new literatures concerning the hepta – equal temperament |archive-url=https://web.archive.org/web/20071027064731/http://www.wanfangdata.com.cn/qikan/periodical.Articles/ZHONGUOYY/ZHON2004/0404/040425.htm |archive-date=2007-10-27}}</ref>
}}{{efn|From the flute for two thousand years of the production process, and the Japanese shakuhachi remaining in the production of Sui and Tang Dynasties and the actual temperament, identification of people using the so-called 'Seven Laws' at least two thousand years of history; and decided that this law system associated with the flute law.<!-- from Google translate, please verify --><ref>{{cite web |url=http://scholar.ilib.cn/Abstract.aspx?A=xhyyxyxb200102005 |script-title=zh:七平均律"琐谈--兼及旧式均孔曲笛制作与转调 |language=zh |trans-title=abstract of ''About "Seven- equal- tuning System"''  |access-date=2007-06-25 |archive-url=https://web.archive.org/web/20070930155436/http://scholar.ilib.cn/Abstract.aspx?A=xhyyxyxb200102005 |archive-date=2007-09-30 |url-status=dead }}</ref>
}}

=== Various equal temperaments ===
{{more citations needed section|date=March 2020}}
[[File:16-tet scale on C.png|400px|thumb|[[Easley Blackwood, Jr.|Easley Blackwood]]'s notation system for 16&nbsp;equal temperament: Intervals are notated similarly to those they approximate and there are fewer [[enharmonic]] equivalents.<ref>{{cite book |first=Myles Leigh |last=Skinner |year=2007 |title=Toward a Quarter-Tone Syntax: Analyses of selected works by Blackwood, Haba, Ives, and Wyschnegradsky |page=55 |isbn=9780542998478}}</ref> {{audio|16-tet scale on C.mid|Play}}]]

[[File:Equal temperaments comparison diagram.svg|thumb|Comparison of equal temperaments from 9 to 25<ref>{{harvp|Sethares|2005|p=58}}</ref>{{efn|name=Sethares}}]]

; [[19 equal temperament|19&nbsp;EDO]]: Many instruments have been built using [[19 equal temperament|19&nbsp;EDO]] tuning. Equivalent to {{nobr|{{sfrac| 1 | 3 }} comma}} meantone, it has a slightly flatter perfect fifth (at 695&nbsp;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&nbsp;EDO being 232&nbsp;EDO. Its [[perfect fourth]] (at 505&nbsp;cents), is seven cents sharper than just intonation's and five cents sharper than 12&nbsp;EDO's.

; [[22 equal temperament|22&nbsp;EDO]]: [[22 equal temperament|22&nbsp;EDO]] is one of the most accurate EDOs to represent "superpythagprean" temperament (where 7:4 and 16:9 are the same interval). The perfect fifth is tuned sharp, resulting in four fifths and three fourths reaching supermajor thirds (9/7) and subminor thirds (7/6). One step closer to each other are the classical major and minor thirds (5/4 and 6/5).

; [[23 equal temperament|23&nbsp;EDO]]: [[23 equal temperament|23&nbsp;EDO]] is the largest EDO that fails to approximate the 3rd, 5th, 7th, and 11th&nbsp;harmonics (3:2, 5:4, 7:4, 11:8) within 20&nbsp;cents, but it does approximate some ratios between them (such as the 6:5 minor third) very well, making it attractive to microtonalists seeking unusual harmonic territory.

; [[Quarter tone|24&nbsp;EDO]]: [[Quarter tone|24&nbsp;EDO]], the [[quarter tone scale|quarter-tone scale]], is particularly popular, as it represents a convenient access point for composers conditioned on standard Western 12&nbsp;EDO pitch and notation practices who are also interested in microtonality. Because 24&nbsp;EDO contains all the pitches of 12&nbsp;EDO, musicians employ the additional colors without losing any tactics available in 12&nbsp;tone harmony. That 24 is a multiple of 12 also makes 24&nbsp;EDO easy to achieve instrumentally by employing two traditional 12&nbsp;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&nbsp;tone notation. Various composers, including [[Charles Ives]], experimented with music for quarter-tone pianos. 24&nbsp;EDO also approximates the 11th and 13th harmonics very well, unlike 12&nbsp;EDO.

; 26&nbsp;EDO: 26 is the denominator of a convergent to log<sub>2</sub>(7), tuning the 7th harmonic (7:4) with less than half a cent of error. Although it is a meantone temperament, it is a very flat one, with four of its perfect fifths producing a major third 17&nbsp;cents flat (equated with the 11:9 neutral third). 26&nbsp;EDO has two minor thirds and two minor sixths and could be an alternate temperament for [[Close and open harmony|barbershop harmony]].

; 27&nbsp;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]].

; [[58 equal temperament|29&nbsp;EDO]]: [[58 equal temperament|29]] is the lowest number of equal divisions of the octave whose perfect fifth is closer to just than in 12&nbsp;EDO, in which the fifth is 1.5&nbsp;cents sharp instead of 2&nbsp;cents flat. Its classic major third is roughly as inaccurate as 12&nbsp;EDO, but is tuned 14&nbsp;cents flat rather than 14&nbsp;cents sharp. It also tunes the 7th, 11th, and 13th&nbsp;harmonics flat by roughly the same amount, allowing 29&nbsp;EDO to match intervals such as 7:5, 11:7, and 13:11 very accurately. Cutting all 29 intervals in half produces [[58 equal temperament|58 EDO]], which allows for lower errors for some just tones.

; [[31 equal temperament|31&nbsp;EDO]]: [[31 equal temperament|31&nbsp;EDO]] was advocated by [[Christiaan Huygens]] and [[Adriaan Fokker]] and represents a rectification of [[quarter-comma meantone]] into an equal temperament. 31&nbsp;EDO does not have as accurate a perfect fifth as 12&nbsp;EDO (like 19&nbsp;EDO), but its major thirds and minor sixths are less than 1&nbsp;cent away from just. It also provides good matches for harmonics up to 11, of which the seventh harmonic is particularly accurate.

; [[34 equal temperament|34&nbsp;EDO]]: [[34 equal temperament|34&nbsp;EDO]] gives slightly lower total combined errors of approximation to 3:2, 5:4, 6:5, and their inversions than 31&nbsp;EDO does, despite having a slightly less accurate fit for 5:4. 34&nbsp;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&nbsp;cent tritone, since 34 is an even number.

; [[41 equal temperament|41&nbsp;EDO]]: [[41 equal temperament|41]] is the next EDO with a better perfect fifth than 29&nbsp;EDO and 12&nbsp;EDO. Its classical major third is also more accurate, at only six cents flat. It is not a meantone temperament, so it distinguishes 10:9 and 9:8, along with the classic and Pythagorean major thirds, unlike 31&nbsp;EDO. It is more accurate in the 13&nbsp;limit than 31&nbsp;EDO.

; 46&nbsp;EDO: 46&nbsp;EDO provides major thirds and perfect fifths that are both slightly sharp of just, and many{{who|date=September 2024}} say that this gives major triads a characteristic bright sound. The prime harmonics up to 17 are all within 6&nbsp;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 equal temperament|53&nbsp;EDO]]: [[53 equal temperament|53&nbsp;EDO]] has only had occasional use, but is better at approximating the traditional [[just intonation|just]] consonances than 12, 19 or 31&nbsp;EDO. Its extremely accurate [[perfect fifth]]s make it equivalent to an extended [[Pythagorean tuning]], as 53 is the denominator of a convergent to log<sub>2</sub>(3). With its accurate cycle of fifths and multi-purpose comma step, 53&nbsp;EDO has been used in [[Turkish music]] theory. It is not a meantone temperament, which put good thirds within easy reach by stacking fifths; instead, like all [[schismatic temperament|schismatic temperaments]], the very consonant thirds are represented by a Pythagorean diminished fourth (C-F{{music|b}}), reached by stacking eight perfect fourths. It also tempers out the [[kleisma]], allowing its fifth to be reached by a stack of six minor thirds (6:5).

; [[58 equal temperament|58&nbsp;EDO]]: [[58 equal temperament]] is a duplication of 29&nbsp;EDO, which it contains as an embedded temperament. Like 29&nbsp;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 equal temperament|72&nbsp;EDO]]: [[72 equal temperament|72&nbsp;EDO]] approximates many [[just intonation]] intervals well, providing near-just equivalents to the 3rd, 5th, 7th, and 11th harmonics. 72&nbsp;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). As it is a multiple of 12, 72&nbsp;EDO can be considered an extension of 12&nbsp;EDO, containing six copies of 12&nbsp;EDO starting on different pitches, three copies of 24&nbsp;EDO, and two copies of 36&nbsp;EDO.

; [[96 equal temperament|96&nbsp;EDO]]: [[96 equal temperament|96&nbsp;EDO]] approximates all intervals within 6.25&nbsp;cents, which is barely distinguishable. As an eightfold multiple of 12, it can be used fully like the common 12&nbsp;EDO. It has been advocated by several composers, especially [[Julián Carrillo]].<ref>{{cite web |last1=Monzo |first1=Joe |title=Equal-temperament |url=http://tonalsoft.com/enc/e/equal-temperament.aspx#edo-table |website=Tonalsoft Encyclopedia of Microtonal Music Theory |publisher=Joe Monzo |access-date=26 February 2019 |date=2005}}</ref>

Other equal divisions of the octave that have found occasional use include [[13 equal temperament|13&nbsp;EDO]], [[15 equal temperament|15&nbsp;EDO]], [[17 equal temperament|17&nbsp;EDO]], and 55&nbsp;EDO.

2, 5, 12, 41, 53, 306, 665 and 15601 are [[denominator]]s of first [[convergent (continued fraction)|convergents]] of log{{sub|2}}(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 intonation|just]] twelfths/fifths than in any equal temperament with fewer tones.<ref>{{cite web |title=665&nbsp;edo |website=xenoharmonic (microtonal wiki) |url=http://xenharmonic.wikispaces.com/665edo |access-date=2014-06-18 |archive-date=2015-11-18 |archive-url=https://web.archive.org/web/20151118233400/http://xenharmonic.wikispaces.com/665edo |url-status=dead }}</ref><ref>{{cite web |title=convergents log2(3), 10 |publisher=[[WolframAlpha]] |url=http://www.wolframalpha.com/input/?i=convergents%28log2%283%29%2C+10%29 |access-date=2014-06-18}}</ref>

1, 2, 3, 5, 7, 12, 29, 41, 53, 200,&nbsp;... {{OEIS|A060528}} 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.{{efn|
OEIS sequences that contain divisions of the octave that provide improving approximations of just intervals:
: {{OEIS|A060528}} — 3:2
: {{OEIS|A054540}} — 3:2 and 4:3, 5:4 and 8:5, 6:5 and 5:3
: {{OEIS|A060525}} — 3:2 and 4:3, 5:4 and 8:5
: {{OEIS|A060526}} — 3:2 and 4:3, 5:4 and 8:5, 7:4 and 8:7
: {{OEIS|A060527}} — 3:2 and 4:3, 5:4 and 8:5, 7:4 and 8:7, 16:11 and 11:8
: {{OEIS|A060233}} — 4:3 and 3:2, 5:4 and 8:5, 6:5 and 5:3, 7:4 and 8:7, 16:11 and 11:8, 16:13 and 13:8
: {{OEIS|A061920}} — 3:2 and 4:3, 5:4 and 8:5, 6:5 and 5:3, 9:8 and 16:9, 10:9 and 9:5, 16:15 and 15:8, 45:32 and 64:45
: {{OEIS|A061921}} — 3:2 and 4:3, 5:4 and 8:5, 6:5 and 5:3, 9:8 and 16:9, 10:9 and 9:5, 16:15 and 15:8, 45:32 and 64:45, 27:20 and 40:27, 32:27 and 27:16, 81:64 and 128:81, 256:243 and 243:128
: {{OEIS|A061918}} — 5:4 and 8:5
: {{OEIS|A061919}} — 6:5 and 5:3
: {{OEIS|A060529}} — 6:5 and 5:3, 7:5 and 10:7, 7:6 and 12:7
: {{OEIS|A061416}} — 11:8 and 16:11
}}

=== Equal temperaments of non-octave intervals ===
The equal-tempered version of the [[Bohlen–Pierce scale]] consists of the ratio 3:1 (1902&nbsp;cents) conventionally a [[perfect fifth]] plus an [[octave]] (that is, a perfect twelfth), called in this theory a [[tritave]] ({{Audio|Tritave on C.mid|play}}), and split into 13&nbsp;equal parts. This provides a very close match to [[just intonation|justly tuned]] ratios consisting only of odd numbers. Each step is 146.3 cents ({{Audio|BP scale et.mid|play}}), or <math display=inline>\sqrt[13]{3}</math>.

[[Wendy Carlos]] created three unusual equal temperaments after a thorough study of the properties of possible temperaments with step size between 30 and 120&nbsp;cents. These were called ''[[alpha scale|alpha]]'', ''[[beta scale|beta]]'', and ''[[gamma scale|gamma]]''. They can be considered equal divisions of the perfect fifth. Each of them provides a very good approximation of several just intervals.<ref>{{cite web |last1=Carlos |first1=Wendy |title=Three Asymmetric Divisions of the Octave |url=http://www.wendycarlos.com/resources/pitch.html |website=wendycarlos.com |publisher=Serendip LLC |access-date=2016-09-01}}</ref> Their step sizes:
* ''alpha'': <math display=inline>\sqrt[9]{\frac{3}{2}}</math> (78.0&nbsp;cents) {{audio|Alpha scale step on C.mid|Play}}
* ''beta'': <math display=inline>\sqrt[11]{\frac{3}{2}}</math> (63.8&nbsp;cents) {{audio|Beta scale step on C.mid|Play}}
* ''gamma'': <math display=inline>\sqrt[20]{\frac{3}{2}}</math> (35.1&nbsp;cents) {{audio|Gamma scale step on C.mid|Play}}
Alpha and beta may be heard on the title track of Carlos's 1986 album ''[[Beauty in the Beast]]''.

=== Equal temperament with a non-integral number of notes per octave ===
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:

* ''11 steps × 63.45 cents ≈ 698.95 cents,''
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:

* ''6 steps × 63.45 cents ≈ 380.70 cents,''
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.

=== Proportions between semitone and whole tone ===
{{More citations needed section|date=August 2017}}
In this section, ''semitone'' and ''whole tone'' may not have their usual 12&nbsp;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 {{mvar|s}}, and the number of steps in a tone be {{mvar|t}}.

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, {{sc|'''C'''}}, {{sc|'''D'''}}, {{sc|'''E'''}}, {{sc|'''F'''}}, and {{sc|'''F'''}}{{music|#}} are in ascending order if they preserve their usual relationships to {{sc|'''C'''}}). That is, fixing {{mvar|q}} to a proper fraction in the relationship {{nobr|{{math|''q t'' {{=}} ''s''}} }} also defines a unique family of one equal temperament and its multiples that fulfil this relationship.

For example, where {{mvar|k}} is an integer, {{nobr|12{{mvar|k}} {{sc|EDO}} }} sets {{nobr|{{math|''q'' {{=}} {{sfrac|1|2}}}},}} {{nobr|19 {{mvar|k}} {{sc|EDO}} }} sets {{nobr|{{math|''q'' {{=}} {{sfrac|1|3}}}},}} and {{nobr|31 {{mvar|k}} {{sc|EDO}} }} sets {{nobr|{{math|''q'' {{=}} {{sfrac| 2 | 5 }} }}.}} 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&nbsp;{{sc|EDO}}, the half-sharps and half-flats are not in the circle of fifths generated starting from {{sc|'''C'''}}.) The extreme cases are {{nobr|5 {{mvar|k}} {{sc|EDO}},}} where {{nobr|{{math|''q'' {{=}} 0}} }} and the semitone becomes a unison, and {{nobr|7 {{mvar|k}} {{sc|EDO}} }}, where {{nobr|{{math|''q'' {{=}} 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 {{nobr|{{math| 7 ''t'' − 2 ''s''}} steps}} and the perfect fifth into {{nobr|{{math| 4 ''t'' − ''s'' }} steps.}} If there are notes outside the circle of fifths, one must then multiply these results by {{mvar|n}}, the number of nonoverlapping circles of fifths required to generate all the notes (e.g., two in 24&nbsp;{{sc|EDO}}, six in 72&nbsp;{{sc|EDO}}). (One must take the small semitone for this purpose: 19&nbsp;{{sc|EDO}} has two semitones, one being {{sfrac| 1 | 3 }} tone and the other being {{sfrac| 2 | 3 }}. Similarly, 31&nbsp;{{sc|EDO}} has two semitones, one being {{sfrac| 2 | 5 }} tone and the other being {{sfrac| 3 | 5 }}).

The smallest of these families is {{nobr|12 {{mvar|k}} {{sc|EDO}},}} and in particular, 12&nbsp;{{sc|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&nbsp;{{sc|EDO}} has become the most commonly used equal temperament. (Another reason is that 12&nbsp;EDO is the smallest equal temperament to closely approximate 5&nbsp;limit harmony, the next-smallest being 19&nbsp;EDO.)

Each choice of fraction {{mvar|q}} for the relationship results in exactly one equal temperament family, but the converse is not true: 47&nbsp;{{sc|EDO}} has two different semitones, where one is {{sfrac| 1 | 7 }} tone and the other is {{sfrac| 8 | 9 }}, which are not complements of each other like in 19&nbsp;{{sc|EDO}} ({{sfrac| 1 | 3 }} and {{sfrac| 2 | 3 }}). Taking each semitone results in a different choice of perfect fifth.