2080/2079
| Ratio | 2080/2079 |
| Factorization | 25 × 3-3 × 5 × 7-1 × 11-1 × 13 |
| Monzo | [5 -3 1 -1 -1 1⟩ |
| Size in cents | 0.8325242¢ |
| Names | ibnsinma, sinaisma |
| Color name | 3o1ury1, tholuruyo 1sn, Tholuruyo comma |
| FJS name | [math]\text{P1}^{5,13}_{7,11}[/math] |
| Special properties | superparticular, reduced |
| Tenney height (log2 nd) | 22.044 |
| Weil height (log2 max(n, d)) | 22.0447 |
| Wilson height (sopfr (nd)) | 55 |
| Harmonic entropy (Shannon, [math]\sqrt{n\cdot d}[/math]) |
~2.40044 bits |
| Comma size | unnoticeable |
| S-expressions | S64 × S65, S78 × S79 × S80 |
| open this interval in xen-calc | |
2080/2079, the ibnsinma, otherwise known as the sinaisma, is a 13-limit unnoticeable comma measuring about 0.83 cents. It is the amount by which a stack consisting of 11/10 and 14/13 falls short of Pythagorean minor third, 32/27.
Commatic relations
This comma identifies itself as the difference between eight pairs of 13-limit superparticular ratios:
- 64/63 and 66/65
- 78/77 and 81/80
- 100/99 and 105/104
- 325/324 and 385/384
- 364/363 and 441/440
- 540/539 and 729/728
- 676/675 and 1001/1000
- 1716/1715 and 9801/9800
Not to mention some nonsuperparticular but useful ratios:
Or as a relation in the four formal commas defined by Functional Just System:
It factors into the following superparticular intervals:
In Sagittal notation, it is the default comma represented by two minas or six tinas.
Temperaments
By tempering it out is defined the ibnsinmic or sinaismic temperament, which enables the ibnsinmic a.k.a. sinaismic chords, the essentially tempered chords in the 21-odd-limit. Another consequence is that it makes fifth complements of 13/11 and 80/63, and of 40/33 and 26/21. You may find a list of good equal temperaments that support this temperament below.
Subgroup: 2.3.5.7.11.13
| [⟨ | 1 | 0 | 0 | 0 | 0 | -5 | ], |
| ⟨ | 0 | 1 | 0 | 0 | 0 | 3 | ], |
| ⟨ | 0 | 0 | 1 | 0 | 0 | -1 | ], |
| ⟨ | 0 | 0 | 0 | 1 | 0 | 1 | ], |
| ⟨ | 0 | 0 | 0 | 0 | 1 | 1 | ]] |
- mapping generators: ~2, ~3, ~5, ~7, ~11
Optimal ET sequence: 12f, 14cf, 15, 17c, 19, 22f, 26, 29, 31f, 39df, 41, 46, 53, 58, 72, 87, 111, 130, 183, 198, 224, 270, 494, 764, 935, 1075, 1205, 1699, 2280, 2774e, 3326de, 3596de, 3907bdee, 4401bdee, 4671bde *
Etymology
This comma was named by Margo Schulter in 2012, when it went by ibn Sina's comma[1]. It was contracted to avicennma and later changed to ibnsinma by Gene Ward Smith, all essentially meaning the same. Note that avicennma refers to Avicenna's enharmonic diesis (525/512) today. In 2023, Flora Canou proposed sinaisma as another spelling, reflecting the fact that the interval region arising from the same source is called sinaic.