Harmonisma: Difference between revisions

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'''10648/10647''', the '''harmonisma''', is an [[unnoticeable comma|unnoticeable]] no-5's [[13-limit]] [[comma]] of about 0.~~1626~~ [[cent]]s. It is ~~equal to~~ (([[16/13]])⋅([[11/9]]))/(([[14/11]])⋅([[13/11]])). In terms of other commas, it is ([[352/351]])/([[364/363]]), ([[3025/3024]])/([[4225/4224]]), ([[4096/4095]])/([[6656/6655]]), or ([[9801/9800]])/([[123201/123200]]).

'''10648/10647''', the '''harmonisma''', is an [[unnoticeable comma|unnoticeable]] [[2.3.7.11.13 subgroup|no-5's]] [[13-limit]] [[comma]] of about 0.163 [[cent]]s. It is the amount by which a stack of two [[44/39]] major-minthmic whole tones exceeds [[14/11]], as well as the difference between two sharp fifths: (([[16/13]])⋅([[11/9]]))/((14/11)⋅([[13/11]])).

In terms of other commas, it is equal to ([[352/351]])/([[364/363]]), ([[3025/3024]])/([[4225/4224]]), ([[4096/4095]])/([[6656/6655]]), and ([[9801/9800]])/([[123201/123200]]).

== Temperaments ==

[[Tempering out]] this comma in the full 13-limit gives the rank-5 '''harmonismic ~~temperament~~'''. [[Equal temperament]]s where this comma is tempered out with very high accuracy, such as [[764edo]], will have an interval corresponding to a "sharp fifth" of (ideally) 706.7 to 706.9 cents, corresponding to the range of fifths from (13/11)⋅(14/11) = [[182/121]] on the lower end and (11/9)⋅(16/13) = [[176/117]] on the higher end, and this interval is not mapped to [[3/2]]. However, such temperaments are generally very precise, so [[224edo]], [[270edo]] and [[311edo]] offer slightly more manageable tunings. For less accurate temperaments still, 10648/10647 is notable as a comma of [[parapyth]].

[[Tempering out]] this comma in the full 13-limit gives the rank-5 '''harmonismic''' temperament. [[Equal temperament]]s where this comma is tempered out with very high accuracy, such as [[764edo]], will have an interval corresponding to a "sharp fifth" of (ideally) 706.7 to 706.9 cents, corresponding to the range of fifths from (13/11)⋅(14/11) = [[182/121]] on the lower end and (11/9)⋅(16/13) = [[176/117]] on the higher end, and this interval is not mapped to [[3/2]]. However, such temperaments are generally very precise, so [[224edo]], [[270edo]] and [[311edo]] offer slightly more manageable tunings. For less accurate temperaments still, 10648/10647 is notable as a comma of [[parapyth]].

The harmonisma, 10648/10647, plays a striking role in [[Secor29htt|George Secor's 29-tone high tolerance temperament]] of 1975, the first temperament in the High Tolerance Temperament family. In this tuning, the fifth at 703.579 cents produces an augmented second (+9 fifths) at a just [[63/52]] (equal to ([[9/8]])⋅([[14/13]])), or a diminished seventh (-9 fifths) at [[104/63]], which exceeds three 13/11 thirds by a harmonisma. 63/52 exceeds the Pythagorean augmented second, [[19683/16384]] (a [[32805/32768]] schisma larger than [[6/5]]), by the [[secorian comma]], 28672/28431. Likewise 104/63 is narrower than the Pythagorean diminished seventh [[32768/19683]] by 28672/28431.

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-'''10648/10647''', the '''harmonisma''', is an [[unnoticeable comma|unnoticeable]] no-5's [[13-limit]] [[comma]] of about 0.1626 [[cent]]s. It is equal to (([[16/13]])⋅([[11/9]]))/(([[14/11]])⋅([[13/11]])). In terms of other commas, it is ([[352/351]])/([[364/363]]), ([[3025/3024]])/([[4225/4224]]), ([[4096/4095]])/([[6656/6655]]), or ([[9801/9800]])/([[123201/123200]]).
+'''10648/10647''', the '''harmonisma''', is an [[unnoticeable comma|unnoticeable]] [[2.3.7.11.13 subgroup|no-5's]] [[13-limit]] [[comma]] of about 0.163 [[cent]]s. It is the amount by which a stack of two [[44/39]] major-minthmic whole tones exceeds [[14/11]], as well as the difference between two sharp fifths: (([[16/13]])⋅([[11/9]]))/((14/11)⋅([[13/11]])).
+In terms of other commas, it is equal to ([[352/351]])/([[364/363]]), ([[3025/3024]])/([[4225/4224]]), ([[4096/4095]])/([[6656/6655]]), and ([[9801/9800]])/([[123201/123200]]).
 == Temperaments ==
-[[Tempering out]] this comma in the full 13-limit gives the rank-5 '''harmonismic temperament'''. [[Equal temperament]]s where this comma is tempered out with very high accuracy, such as [[764edo]], will have an interval corresponding to a "sharp fifth" of (ideally) 706.7 to 706.9 cents, corresponding to the range of fifths from (13/11)⋅(14/11) = [[182/121]] on the lower end and (11/9)⋅(16/13) = [[176/117]] on the higher end, and this interval is not mapped to [[3/2]]. However, such temperaments are generally very precise, so [[224edo]], [[270edo]] and [[311edo]] offer slightly more manageable tunings. For less accurate temperaments still, 10648/10647 is notable as a comma of [[parapyth]].
+[[Tempering out]] this comma in the full 13-limit gives the rank-5 '''harmonismic''' temperament. [[Equal temperament]]s where this comma is tempered out with very high accuracy, such as [[764edo]], will have an interval corresponding to a "sharp fifth" of (ideally) 706.7 to 706.9 cents, corresponding to the range of fifths from (13/11)⋅(14/11) = [[182/121]] on the lower end and (11/9)⋅(16/13) = [[176/117]] on the higher end, and this interval is not mapped to [[3/2]]. However, such temperaments are generally very precise, so [[224edo]], [[270edo]] and [[311edo]] offer slightly more manageable tunings. For less accurate temperaments still, 10648/10647 is notable as a comma of [[parapyth]].
 The harmonisma, 10648/10647, plays a striking role in [[Secor29htt|George Secor's 29-tone high tolerance temperament]] of 1975, the first temperament in the High Tolerance Temperament family. In this tuning, the fifth at 703.579 cents produces an augmented second (+9 fifths) at a just [[63/52]] (equal to ([[9/8]])⋅([[14/13]])), or a diminished seventh (-9 fifths) at [[104/63]], which exceeds three 13/11 thirds by a harmonisma. 63/52 exceeds the Pythagorean augmented second, [[19683/16384]] (a [[32805/32768]] schisma larger than [[6/5]]), by the [[secorian comma]], 28672/28431. Likewise 104/63 is narrower than the Pythagorean diminished seventh [[32768/19683]] by 28672/28431.