10648/10647

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Interval information
Ratio 10648/10647
Factorization 23 × 3-2 × 7-1 × 113 × 13-2
Monzo [3 -2 0 -1 3 -2
Size in cents 0.16259536¢
Name harmonisma
FJS name [math]\text{m}{-2}^{11,11,11}_{7,13,13}[/math]
Special properties superparticular,
reduced
Tenney height (log2 n⋅d) 26.7565
Weil height (max(n, d)) 10648
Benedetti height (n⋅d) 113369256
Harmonic entropy
(Shannon, [math]\sqrt{n\cdot d}[/math])
~2.39761 bits
Comma size unnoticeable
open this interval in xen-calc

10648/10647, the harmonisma, is a no-5's 13-limit unnoticeable comma of about 0.1626 cents. 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 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 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.

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.

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.

Subgroup: 2.3.5.7.11.13

Mapping:
[1 0 0 0 1 3]
0 1 0 0 0 -1]
0 0 1 0 0 0]
0 0 0 1 1 1]
0 0 0 0 -2 -3]]

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

Optimal GPV sequence41, 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

Etymology

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