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 temperaments where this comma is tempered with very high accuracy 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 temperaments where this comma is tempered with very high accuracy 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 George Secor's 29-HTT 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 x 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 [[George Secor]]'s 29-HTT 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 secorian comma is made up of (896/891 x 352/351) or in other words (352/351 x 364/363 x 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.

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.~~273c~~) and 91/88 (58.~~036c~~) at 364/363 ~~apart~~ (4.~~763c~~); 91/88 and 121/117 at 10648/10647 ~~apart~~ (0.~~163c~~); 121/117 and 28/27 at 62.~~961c~~ at 364/363 (4.~~763c~~) apart; and the smallest and largest intervals among these represented by the parapyth spacing, 33/32 and 28/27 at 896/891 (9.~~688c~~) apart. Thus 896/891 = (352/351 x 364/363), and also (364/363 x 10648/10647 x 364/363). A difference of 352/351 (4.~~925c~~) 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¢) 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.

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.

~~[[User:Mschulter1325|Mschulter1325]] 00:03, 10 November 2022 (UTC)~~

== Etymology ==

@@ Line 7: / Line 7: @@
 == 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 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 temperaments where this comma is tempered with very high accuracy 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 George Secor's 29-HTT 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 x 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 [[George Secor]]'s 29-HTT 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 secorian comma is made up of (896/891 x 352/351) or in other words (352/351 x 364/363 x 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)<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.
+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 (53.273c) and 91/88 (58.036c) at 364/363 apart (4.763c); 91/88 and 121/117 at 10648/10647 apart (0.163c); 121/117 and 28/27 at 62.961c at 364/363 (4.763c) apart; and the smallest and largest intervals among these represented by the parapyth spacing, 33/32 and 28/27 at 896/891 (9.688c) apart. Thus 896/891 = (352/351 x 364/363), and also (364/363 x 10648/10647 x 364/363). A difference of 352/351 (4.925c) 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¢) 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.
 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.
-[[User:Mschulter1325|Mschulter1325]] 00:03, 10 November 2022 (UTC)
 == Etymology ==