Editing Equal temperament (section)

=== 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
}}