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 a no-5's [[13-limit]] [[~~unnoticeable~~ 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]]).

== Temperaments ==

Tempering out this comma in the full 13-limit gives the rank-5 '''harmonismic temperament'''. Equal ~~temperaments~~ where this comma is tempered 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]] (9/8 × 14/13), or a diminished seventh (-9 fifths) at [[104/63]], which exceeds three 13/11 thirds by a harmonisma. ~~The~~ 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 32768/19683 by 28672/28431.

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 [[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.

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.

Another manifestation of the harmonisma in 29-HTT is the tuning of 11/9 at 347.353 cents, a third of a harmonisma narrow. Here 7/4 is just, with the spacing between the relevant chains of fifths in a subset of 29-HTT which served as a prototype for parapyth temperament at 58.090 cents, as compared with the [[91/88]] spacing which would produce a just 7/4 if the regular major sixth were at a just 22/13. But it is a third of a harmonisma narrow, and the spacing consequently a third of a harmonisma greater.

Another manifestation of the harmonisma in 29-HTT is the tuning of 11/9 at 347.353 cents, a third of a harmonisma narrow. Here [[7/4]] is just, with the spacing between the relevant chains of fifths in a subset of 29-HTT which served as a prototype for parapyth temperament at 58.090 cents, as compared with the [[91/88]] spacing which would produce a just 7/4 if the regular major sixth were at a just [[22/13]]. But it is a third of a harmonisma narrow, and the spacing consequently a third of a harmonisma greater.

As it happens, the difference between 11/9 and 13/11 is [[121/117]], a harmonisma greater than 91/88 (e.g. 22/13 vs. 7/4). Since the tempered 13/11 is a third of a harmonisma greater than just, and the spacing a third of a harmonisma greater than 91/88, this leaves 1/3 of the harmonisma difference between 91/88 and 121/117 unaccounted for, the amount by which 11/9 is narrow.

In parapyth, generally, the spacing can represent four ratios, whose differences show the four commas tempered out: 33/32 (53.~~273¢~~) and 91/88 (58.~~036¢~~) at 364/363 (4.~~763¢~~) apart ; 91/88 and 121/117 at 10648/10647 (0.~~163¢~~) apart; 121/117 and 28/27 (62.~~961¢~~) at 364/363 (4.~~763¢~~) apart; and the smallest and largest intervals among these represented by the parapyth spacing, 33/32 and 28/27 at 896/891 (9.~~688¢~~) apart. Thus 896/891 = (352/351 × 364/363), and also (364/363 × 10648/10647 × 364/363). A difference of 352/351 (4.~~925¢~~) or (364/363 x 10648/10647) occurs between 121/117 and 33/32, and between 28/27 and 91/88.

In parapyth, generally, the spacing can represent four ratios, whose differences show the four commas tempered out: 33/32 (53.273{{c}}) and 91/88 (58.036{{c}}) at 364/363 (4.763{{c}}) apart; 91/88 and 121/117 at 10648/10647 (0.163{{c}}) apart; 121/117 and 28/27 (62.961{{c}}) at 364/363 (4.763{{c}}) apart; and the smallest and largest intervals among these represented by the parapyth spacing, 33/32 and 28/27 at 896/891 (9.688{{c}}) apart. Thus 896/891 = (352/351)⋅(364/363), and also (364/363)⋅(10648/10647)⋅(364/363). A difference of 352/351 (4.925{{c}}) or (364/363)⋅(10648/10647) occurs between 121/117 and 33/32, and between 28/27 and 91/88.

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.

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| ⟨ || 0 || 0 || 0 || 0 || -2 || -3 || ]]

|}

: mapping generators: ~2, ~3, ~5, ~7, ~44/39

: ~~mapping generators~~: ~2, ~3, ~5, ~7, ~44/39

[[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}}

* [[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}}

{{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 }}

[[Badness]] (Sintel): 0.207

== Etymology ==

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[[Category:Harmonismic]]

[[Category:Commas named for other reasons]]

@@ Line 5: / Line 5: @@
 | Comma = yes
 }}
+'''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 a no-5's [[13-limit]] [[unnoticeable 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]]).
 == Temperaments ==
-Tempering out this comma in the full 13-limit gives the rank-5 '''harmonismic temperament'''. Equal temperaments where this comma is tempered 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]] (9/8 × 14/13), or a diminished seventh (-9 fifths) at [[104/63]], which exceeds three 13/11 thirds by a harmonisma. The 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 32768/19683 by 28672/28431.
+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 [[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>.
+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>.
-Another manifestation of the harmonisma in 29-HTT is the tuning of 11/9 at 347.353 cents, a third of a harmonisma narrow. Here 7/4 is just, with the spacing between the relevant chains of fifths in a subset of 29-HTT which served as a prototype for parapyth temperament at 58.090 cents, as compared with the [[91/88]] spacing which would produce a just 7/4 if the regular major sixth were at a just 22/13. But it is a third of a harmonisma narrow, and the spacing consequently a third of a harmonisma greater.
+Another manifestation of the harmonisma in 29-HTT is the tuning of 11/9 at 347.353 cents, a third of a harmonisma narrow. Here [[7/4]] is just, with the spacing between the relevant chains of fifths in a subset of 29-HTT which served as a prototype for parapyth temperament at 58.090 cents, as compared with the [[91/88]] spacing which would produce a just 7/4 if the regular major sixth were at a just [[22/13]]. But it is a third of a harmonisma narrow, and the spacing consequently a third of a harmonisma greater.
 As it happens, the difference between 11/9 and 13/11 is [[121/117]], a harmonisma greater than 91/88 (e.g. 22/13 vs. 7/4). Since the tempered 13/11 is a third of a harmonisma greater than just, and the spacing a third of a harmonisma greater than 91/88, this leaves 1/3 of the harmonisma difference between 91/88 and 121/117 unaccounted for, the amount by which 11/9 is narrow.
-In parapyth, generally, the spacing can represent four ratios, whose differences show the four commas tempered out: 33/32 (53.273¢) and 91/88 (58.036¢) at 364/363 (4.763¢) apart ; 91/88 and 121/117 at 10648/10647 (0.163¢) apart; 121/117 and 28/27 (62.961¢) at 364/363 (4.763¢) apart; and the smallest and largest intervals among these represented by the parapyth spacing, 33/32 and 28/27 at 896/891 (9.688¢) apart. Thus 896/891 = (352/351 × 364/363), and also (364/363 × 10648/10647 × 364/363). A difference of 352/351 (4.925¢) or (364/363 x 10648/10647) occurs between 121/117 and 33/32, and between 28/27 and 91/88.
+In parapyth, generally, the spacing can represent four ratios, whose differences show the four commas tempered out: 33/32 (53.273{{c}}) and 91/88 (58.036{{c}}) at 364/363 (4.763{{c}}) apart; 91/88 and 121/117 at 10648/10647 (0.163{{c}}) apart; 121/117 and 28/27 (62.961{{c}}) at 364/363 (4.763{{c}}) apart; and the smallest and largest intervals among these represented by the parapyth spacing, 33/32 and 28/27 at 896/891 (9.688{{c}}) apart. Thus 896/891 = (352/351)⋅(364/363), and also (364/363)⋅(10648/10647)⋅(364/363). A difference of 352/351 (4.925{{c}}) or (364/363)⋅(10648/10647) occurs between 121/117 and 33/32, and between 28/27 and 91/88.
 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.
@@ Line 38: / Line 37: @@
 | ⟨ || 0 || 0 || 0 || 0 || -2 || -3 || ]]
 |}
+: mapping generators: ~2, ~3, ~5, ~7, ~44/39
-: mapping generators: ~2, ~3, ~5, ~7, ~44/39
+[[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}}
+* [[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}}
 {{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 }}
+[[Badness]] (Sintel): 0.207
 == Etymology ==
@@ Line 47: / Line 51: @@
 [[Category:Harmonismic]]
+[[Category:Commas named for other reasons]]