22edo: Difference between revisions

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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]] (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 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.

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 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]] (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 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.