2080/2079

Interval information

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]\displaystyle{ \text{P1}^{5,13}_{7,11} }[/math]
Special properties	superparticular, reduced
Tenney norm (log2 nd)	22.044
Weil norm (log2 max(n, d))	22.0447
Wilson norm (sopfr(nd))	55
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 an unnoticeable 13-limit 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:

• 896/891 and 196/195
• 5120/5103 and 352/351
• 352/351 and 847/845

Or as a relation in the four formal commas defined by Functional Just System:

• ((64/63)(1053/1024)) / ((81/80)(33/32)).

It factors into the following superparticular intervals:

• 4096/4095 and 4225/4224
• 3025/3024 and 6656/6655

In Sagittal notation, it is the default comma represented by two minas or six tinas.

Temperaments

Tempering out this comma in the 13-limit defines the rank-5 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 13/11 and 80/63 fifth complements of each other, as well as 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

Mapping:

[⟨	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 tunings:

WE: ~2 = 1199.9519 ¢, ~3/2 = 702.0756 ¢, ~5/4 = 386.3580 ¢, ~7/4 = 968.9979 ¢, ~11/8 = 551.5774 ¢

CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.0843 ¢, ~5/4 = 386.3184 ¢, ~7/4 = 968.9934 ¢, ~11/8 = 551.5497 ¢

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 *

* optimal patent val: 3044

Badness (Sintel): 0.149

Etymology

This comma was named by Margo Schulter in 2012, when it went by ibn Sina's comma. 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.

See also

• List of superparticular intervals