Harmonisma: Difference between revisions

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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 × 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]] (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]] × [[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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 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 × 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]] (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]] × [[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>.