22edo: Difference between revisions

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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]] 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).

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). Because 22edo does not contain "neutral" intervals, [[11/9]] is mapped to the minor third and [[12/11]] is mapped to the major second, inflating [[243/242]] to a full step.

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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 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]] 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).
+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). Because 22edo does not contain "neutral" intervals, [[11/9]] is mapped to the minor third and [[12/11]] is mapped to the major second, inflating [[243/242]] to a full step.
 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]].