22edo: Difference between revisions

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== Theory ==

The 22edo system is the third equal division, after 12 and 19, which is capable of approximating the [[5-limit]] to within a [[Tenney–Euclidean temperament measures #TE error|Tenney–Euclidean error]] of 4{{c}} per octave. Moreover, ~~there is more to~~ it ~~than~~ just the 5-limit; unlike 12 or 19, it is able to approximate the [[7-limit|7-]] and [[11-limit]] to within 3 cents/oct of error~~. While [[31edo]] does much better, 22edo still allows the use of these higher-limit harmonies~~, and in fact 22 is the smallest equal division of the octave to represent the [[11-odd-limit]] [[consistent]]ly.

The 22edo system is the third equal division, after 12 and 19, which is capable of approximating the [[5-limit]] to within a [[Tenney–Euclidean temperament measures #TE error|Tenney–Euclidean error]] of 4{{c}} per octave. Moreover, it does well beyond just the 5-limit; unlike 12 or 19, it is able to approximate the [[7-limit|7-]] and [[11-limit]] to within 3 cents/oct of error, and in fact 22 is the smallest equal division of the octave to represent the [[11-odd-limit]] [[consistent]]ly, though [[31edo]] is more accurate.

Possibly the most striking characteristic of 22edo to those not used to it is that it does '''not''' temper out [[81/80]] (~~[[S-expression|S9]],~~ the syntonic comma), and instead maps it to one step. The net effect is that 22 allows, and to some extent even forces, the exploration of less familiar musical territory; yet it is small enough that it can be used in live performances with suitably designed instruments, like 22-tone guitars.

Possibly the most striking characteristic of 22edo to those not used to it is that it does ''not'' temper out [[81/80]] (the syntonic comma), and instead maps it to one step. The net effect is that 22 allows, and to some extent even forces, the exploration of less familiar musical territory; yet it is small enough that it can be used in live performances with suitably designed instruments, like 22-tone guitars.

22edo's approximation to the 7th harmonic is about 13 cents sharp, somewhat similar to 12edo's approximation to the 5th harmonic. Because of this and the sharp fifth, 22edo tempers out [[64/63]], equating the pythagorean minor seventh with [[7/4]], and supporting [[superpyth]]. In that manner, 22edo can be thought of as widening the gap of [[49/48]] between septimal intervals like [[7/6]] and [[8/7]] to a full quarter-tone. However, the opposite effect consequentially occurs in the 5-limit: 5/4 is ~~narrow~~, and as a result, the interval of 6/5 is significantly sharp of just intonation, with [[25/24]] narrowed to a quarter tone. An important reason for this contrast is that 22edo tempers out [[50/49]], ~~equating~~ the [[7/5]] and [[10/7]] tritones. Reasonably, [[36/35]] is tempered to the ~~same~~ 1-step interval.

22edo's approximation to the 7th harmonic is about 13 cents sharp, somewhat similar to 12edo's approximation to the 5th harmonic. Because of this and the sharp fifth, 22edo tempers out [[64/63]], equating the pythagorean minor seventh with [[7/4]], and supporting [[superpyth]]. In that manner, 22edo can be thought of as widening the gap of [[49/48]] between septimal intervals like [[7/6]] and [[8/7]] to a full quarter-tone. However, the opposite effect consequentially occurs in the 5-limit: 5/4 is flat, and as a result, the interval of 6/5 is significantly sharp of just intonation, with [[25/24]] narrowed to a quarter tone. An important reason for this contrast is that 22edo tempers out [[50/49]], so the [[7/5]] and [[10/7]] tritones are equated, and 5/4 and 7/4 are seperated by a semioctave, as well as 6/5 and [[12/7]]. Reasonably, [[36/35]] is also tempered to the 1-step interval as is 25/24 and 49/48.

22edo's approximation of the 11-limit is somewhat contentious: ~~while~~ it represents 11/8 well (about 5-6 cents flat) and maps 14/11 to a supermajor third (albeit inaccurately sharp), it lacks a [[neutral third]] ~~(unless~~ the ~~sharp ~6/5 is considered to be one)~~, ~~meaning that~~ 11-limit harmony that is dependent upon neutral intervals does not work very well. This is partially because of its fifth, which is about 7 cents sharp, but also because 22edo's step is too large to include 5 categories of seconds and thirds (subminor, minor, neutral, major, and supermajor, which 31edo all includes).

22edo's approximation of the 11-limit is somewhat contentious: While it represents 11/8 well (about 5-6 cents flat) and maps 14/11 to a supermajor third (albeit inaccurately sharp), it lacks a [[neutral third]] dividing the perfect fifth in two, which means 11-limit harmony that is dependent upon neutral intervals does not work very well. This is partially because of its fifth, which is about 7 cents sharp, but also because 22edo's step is too large to include 5 categories of seconds and thirds (subminor, minor, neutral, major, and supermajor, which 31edo all includes).

Since 22edo's fifth is sharp of just by approximately one-quarter of the septimal comma ([[64/63]]), and since it tunes the septimal supermajor third ([[9/7]]) almost exactly just, it can be treated, for all practical purposes, as an extended "quarter-comma superpyth", in the same way that 31edo can be treated as an extended [[quarter-comma meantone]].

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 == Theory ==
-The 22edo system is the third equal division, after 12 and 19, which is capable of approximating the [[5-limit]] to within a [[Tenney–Euclidean temperament measures #TE error|Tenney–Euclidean error]] of 4{{c}} per octave. Moreover, there is more to it than just the 5-limit; unlike 12 or 19, it is able to approximate the [[7-limit|7-]] and [[11-limit]] to within 3 cents/oct of error. While [[31edo]] does much better, 22edo still allows the use of these higher-limit harmonies, and in fact 22 is the smallest equal division of the octave to represent the [[11-odd-limit]] [[consistent]]ly.
+The 22edo system is the third equal division, after 12 and 19, which is capable of approximating the [[5-limit]] to within a [[Tenney–Euclidean temperament measures #TE error|Tenney–Euclidean error]] of 4{{c}} per octave. Moreover, it does well beyond just the 5-limit; unlike 12 or 19, it is able to approximate the [[7-limit|7-]] and [[11-limit]] to within 3 cents/oct of error, and in fact 22 is the smallest equal division of the octave to represent the [[11-odd-limit]] [[consistent]]ly, though [[31edo]] is more accurate.
-Possibly the most striking characteristic of 22edo to those not used to it is that it does '''not''' temper out [[81/80]] ([[S-expression|S9]], the syntonic comma), and instead maps it to one step. The net effect is that 22 allows, and to some extent even forces, the exploration of less familiar musical territory; yet it is small enough that it can be used in live performances with suitably designed instruments, like 22-tone guitars.
+Possibly the most striking characteristic of 22edo to those not used to it is that it does ''not'' temper out [[81/80]] (the syntonic comma), and instead maps it to one step. The net effect is that 22 allows, and to some extent even forces, the exploration of less familiar musical territory; yet it is small enough that it can be used in live performances with suitably designed instruments, like 22-tone guitars.
-edo's approximation to the 7th harmonic is about 13 cents sharp, somewhat similar to 12edo's approximation to the 5th harmonic. Because of this and the sharp fifth, 22edo tempers out [[64/63]], equating the pythagorean minor seventh with [[7/4]], and supporting [[superpyth]]. In that manner, 22edo can be thought of as widening the gap of [[49/48]] between septimal intervals like [[7/6]] and [[8/7]] to a full quarter-tone. However, the opposite effect consequentially occurs in the 5-limit: 5/4 is narrow, and as a result, the interval of 6/5 is significantly sharp of just intonation, with [[25/24]] narrowed to a quarter tone. An important reason for this contrast is that 22edo tempers out [[50/49]], equating the [[7/5]] and [[10/7]] tritones. Reasonably, [[36/35]] is tempered to the same 1-step interval.
+edo's approximation to the 7th harmonic is about 13 cents sharp, somewhat similar to 12edo's approximation to the 5th harmonic. Because of this and the sharp fifth, 22edo tempers out [[64/63]], equating the pythagorean minor seventh with [[7/4]], and supporting [[superpyth]]. In that manner, 22edo can be thought of as widening the gap of [[49/48]] between septimal intervals like [[7/6]] and [[8/7]] to a full quarter-tone. However, the opposite effect consequentially occurs in the 5-limit: 5/4 is flat, and as a result, the interval of 6/5 is significantly sharp of just intonation, with [[25/24]] narrowed to a quarter tone. An important reason for this contrast is that 22edo tempers out [[50/49]], so the [[7/5]] and [[10/7]] tritones are equated, and 5/4 and 7/4 are seperated by a semioctave, as well as 6/5 and [[12/7]]. Reasonably, [[36/35]] is also tempered to the 1-step interval as is 25/24 and 49/48.
-edo's approximation of the 11-limit is somewhat contentious: while it represents 11/8 well (about 5-6 cents flat) and maps 14/11 to a supermajor third (albeit inaccurately sharp), it lacks a [[neutral third]] (unless the sharp ~6/5 is considered to be one), meaning that 11-limit harmony that is dependent upon neutral intervals does not work very well. This is partially because of its fifth, which is about 7 cents sharp, but also because 22edo's step is too large to include 5 categories of seconds and thirds (subminor, minor, neutral, major, and supermajor, which 31edo all includes).
+edo's approximation of the 11-limit is somewhat contentious: While it represents 11/8 well (about 5-6 cents flat) and maps 14/11 to a supermajor third (albeit inaccurately sharp), it lacks a [[neutral third]] dividing the perfect fifth in two, which means 11-limit harmony that is dependent upon neutral intervals does not work very well. This is partially because of its fifth, which is about 7 cents sharp, but also because 22edo's step is too large to include 5 categories of seconds and thirds (subminor, minor, neutral, major, and supermajor, which 31edo all includes).
 Since 22edo's fifth is sharp of just by approximately one-quarter of the septimal comma ([[64/63]]), and since it tunes the septimal supermajor third ([[9/7]]) almost exactly just, it can be treated, for all practical purposes, as an extended "quarter-comma superpyth", in the same way that 31edo can be treated as an extended [[quarter-comma meantone]].