Harmonisma: Difference between revisions

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== 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, or in the 2.3.7.11.13 subgroup, the rank-4 '''harmonismatic''' temperament. Tuning either temperament with very high accuracy implies observing an interval corresponding to 352/351~364/363, the major–minor minthma or gentle comma. Adding it to a [[3/2|perfect fifth]] leads 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. Equal temperaments with such tunings are generally very precise, but [[224edo]], [[270edo]] and [[311edo]] offer manageable compromises. If lower accuracy is desired still, then 352/351 and 364/363 have to be tempered out, leading to [[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.

The harmonisma 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.

The [[secorian comma]] is made up of ([[896/891]])⋅(352/351) or in other words (352/351)⋅(364/363)⋅(352/351), and is thus a harmonisma smaller than (352/351)3. In 29-HTT, each 13/11 is thus a third of a harmonisma or 0.054 cents wider than just. Secor's HTT fifth of 703.579 cents, or precisely (504/13)1/9 or wide by 1/9 of a secorian comma, would thus need to be 1/9 harmonisma larger at 703.597 cents to produce a just 13/11, a temperament of the fifth by (352/351)1/3.

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Here, for example 33/32 is 4/3 vs. 11/8; 91/88 is 22/13 vs. 7/4; 121/117 is 13/11 vs. 11/9; and 28/27 is 9/8 vs. 7/6.

=== Harmonismatic ===

Harmonismatic is a natural detemperament of parapyth where the last generator can be taken to be 352/351~364/363. Extensions of this temperament to the whole 13-limit include [[kalimera]] and [[alfheim]].

[[Subgroup]]: 2.3.7.11.13

[[Comma list]]: 10648/10647

{{Mapping|legend=3| 1 0 0 1 3 | 0 1 0 0 −1 | 0 0 1 1 1 | 0 0 0 -2 -3 }}

: mapping generators: ~2, ~3, ~7, ~44/39

[[Optimal tuning]]:

* [[WE]]: ~2 = 1199.9974{{c}}, ~3/2 = 701.9619{{c}}, ~7/4 = 968.8378{{c}}, ~44/39 = 208.7715{{c}}

: [[error map]]: {{val| -0.003 +0.004 +0.007 -0.031 +0.024 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9615{{c}}, ~7/4 = 968.8365{{c}}, ~44/39 = 208.7728{{c}}

: error map: {{val| 0.000 +0.007 +0.011 -0.027 +0.029 }}

{{Optimal ET sequence|legend=1| 12f, 16, 17, 41, 46, 58, 87, 103, 104, 149, 166, 207, 270, 431, 477, 701, 971, 1012, 1178, 1282, 1489, 1506, 1983, 7455, 7662, 9645, 11628, 13134, 15117, 17100 }}

[[Badness]] (Sintel): 0.0382

=== Harmonismic ===

[[Subgroup]]: 2.3.5.7.11.13

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[[Optimal tuning]]s:

* [[WE]]: ~2 = 1199.9974{{c}}, ~3/2 = 701.9619{{c}}, ~5/4 = 386.3189{{c}}, ~7/4 = 968.8378{{c}}, ~44/39 = 208.7715{{c}}

: [[error map]]: {{val| -0.003 +0.004 -0.000 +0.007 -0.031 +0.024 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9615{{c}}, ~5/4 = 386.3168{{c}}, ~7/4 = 968.8365{{c}}, ~44/39 = 208.7728{{c}}

: error map: {{val| 0.000 +0.007 +0.003 +0.011 -0.027 +0.029 }}

{{Optimal ET sequence|legend=1| 41, 46, 58, 80, 87, 103, 121, 149, 161, 183, 190, 224, 270, 494, 684, 764, 954, 1178, 1236, 1448, 1506, 2190, 2684, 4190, 4771, 6691, 6961, 7455, 9645, 17100 }}

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== Etymology ==

The harmonisma was named by [[Margo Schulter]] in 2002 in honor of the [[harmonia|harmoniai]] of [[Kathleen Schlesinger]].

The harmonisma was named by [[Margo Schulter]] in 2002 in honor of the [[harmonia|harmoniai]] of [[Kathleen Schlesinger]]. The minimal-prime-subgroup temperament is not called ''harmonic'' temperament to avoid possible confusion.

[[Category:Harmonismic]]

[[Category:Commas named for other reasons]]

@@ Line 10: / Line 10: @@
 == 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, or in the 2.3.7.11.13 subgroup, the rank-4 '''harmonismatic''' temperament. Tuning either temperament with very high accuracy implies observing an interval corresponding to 352/351~364/363, the major–minor minthma or gentle comma. Adding it to a [[3/2|perfect fifth]] leads 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. Equal temperaments with such tunings are generally very precise, but [[224edo]], [[270edo]] and [[311edo]] offer manageable compromises. If lower accuracy is desired still, then 352/351 and 364/363 have to be tempered out, leading to [[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.
+The harmonisma 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.
 The [[secorian comma]] is made up of ([[896/891]])⋅(352/351) or in other words (352/351)⋅(364/363)⋅(352/351), and is thus a harmonisma smaller than (352/351)<sup>3</sup>. In 29-HTT, each 13/11 is thus a third of a harmonisma or 0.054 cents wider than just. Secor's HTT fifth of 703.579 cents, or precisely (504/13)<sup>1/9</sup> or wide by 1/9 of a secorian comma, would thus need to be 1/9 harmonisma larger at 703.597 cents to produce a just 13/11, a temperament of the fifth by (352/351)<sup>1/3</sup>.
@@ Line 24: / Line 24: @@
 Here, for example 33/32 is 4/3 vs. 11/8; 91/88 is 22/13 vs. 7/4; 121/117 is 13/11 vs. 11/9; and 28/27 is 9/8 vs. 7/6.
+=== Harmonismatic ===
+Harmonismatic is a natural detemperament of parapyth where the last generator can be taken to be 352/351~364/363. Extensions of this temperament to the whole 13-limit include [[kalimera]] and [[alfheim]].
+[[Subgroup]]: 2.3.7.11.13
+[[Comma list]]: 10648/10647
+{{Mapping|legend=3| 1 0 0 1 3 | 0 1 0 0 −1 | 0 0 1 1 1 | 0 0 0 -2 -3 }}
+: mapping generators: ~2, ~3, ~7, ~44/39
+[[Optimal tuning]]:
+* [[WE]]: ~2 = 1199.9974{{c}}, ~3/2 = 701.9619{{c}}, ~7/4 = 968.8378{{c}}, ~44/39 = 208.7715{{c}}
+: [[error map]]: {{val| -0.003 +0.004 +0.007 -0.031 +0.024 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9615{{c}}, ~7/4 = 968.8365{{c}}, ~44/39 = 208.7728{{c}}
+: error map: {{val| 0.000 +0.007 +0.011 -0.027 +0.029 }}
+{{Optimal ET sequence|legend=1| 12f, 16, 17, 41, 46, 58, 87, 103, 104, 149, 166, 207, 270, 431, 477, 701, 971, 1012, 1178, 1282, 1489, 1506, 1983, 7455, 7662, 9645, 11628, 13134, 15117, 17100 }}
+[[Badness]] (Sintel): 0.0382
+=== Harmonismic ===
 [[Subgroup]]: 2.3.5.7.11.13
@@ Line 43: / Line 64: @@
 [[Optimal tuning]]s:
 * [[WE]]: ~2 = 1199.9974{{c}}, ~3/2 = 701.9619{{c}}, ~5/4 = 386.3189{{c}}, ~7/4 = 968.8378{{c}}, ~44/39 = 208.7715{{c}}
+: [[error map]]: {{val| -0.003 +0.004 -0.000 +0.007 -0.031 +0.024 }}
 * [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9615{{c}}, ~5/4 = 386.3168{{c}}, ~7/4 = 968.8365{{c}}, ~44/39 = 208.7728{{c}}
+: error map: {{val| 0.000 +0.007 +0.003 +0.011 -0.027 +0.029 }}
 {{Optimal ET sequence|legend=1| 41, 46, 58, 80, 87, 103, 121, 149, 161, 183, 190, 224, 270, 494, 684, 764, 954, 1178, 1236, 1448, 1506, 2190, 2684, 4190, 4771, 6691, 6961, 7455, 9645, 17100 }}
@@ Line 50: / Line 73: @@
 == Etymology ==
-The harmonisma was named by [[Margo Schulter]] in 2002 in honor of the [[harmonia|harmoniai]] of [[Kathleen Schlesinger]].
+The harmonisma was named by [[Margo Schulter]] in 2002 in honor of the [[harmonia|harmoniai]] of [[Kathleen Schlesinger]]. The minimal-prime-subgroup temperament is not called ''harmonic'' temperament to avoid possible confusion.
 [[Category:Harmonismic]]
 [[Category:Commas named for other reasons]]

Harmonisma: Difference between revisions

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