936/935

Interval information

Ratio	936/935
Factorization	23 × 32 × 5-1 × 11-1 × 13 × 17-1
Monzo	[3 2 -1 0 -1 1 -1⟩
Size in cents	1.850598¢
Names	ainisma, ainic comma
Color name	17u3o1ug1, sutholugu unison
FJS name	[math]\text{P1}^{13}_{5,11,17}[/math]
Special properties	superparticular, reduced
Tenney height (log2 nd)	19.7392
Weil height (log2 max(n, d))	19.7407
Wilson height (sopfr(nd))	58
Harmonic entropy (Shannon, [math]\sqrt{nd}[/math])	~1.23106 bits
Comma size	unnoticeable
S-expression	S52 × S53 × S54
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936/935, the ainisma or ainic comma, is a 17-limit unnoticeable comma with a size of roughly 1.85 cents. It arises as the amount by which a stack consisting of 18/17 and 13/11 exceeds 5/4, as well as the amount by which a stack consisting of 10/9 and 17/16 falls short of 13/11. Moreover, it is also the interval that differentiates the tannisma (273/272) from the keenanisma (385/384), and the septendecimal kleisma (256/255) from the minthma (352/351). Thus, tempering out this comma is a good way to extend minthmic and gentle harmonies to the 17-limit, as well as a good way to bring keenanismic and tannismic harmonies together. See #Commatic relations for more.

Commatic relations

This comma is the difference between the following superparticular pairs:

• 78/77 and 85/84
• 144/143 and 170/169
• 221/220 and 289/288
• 256/255 and 352/351
• 273/272 and 385/384
• 351/350 and 561/560
• 364/363 and 595/594
• 375/374 and 625/624
• 441/440 and 833/832
• 540/539 and 1275/1274
• 676/675 and 2431/2430
• 715/714 and 3025/3024

It factors into the following superparticular pairs:

• 1716/1715 and 2058/2057
• 1701/1700 and 2080/2079
• 1156/1155 and 4914/4913
• 1089/1088 and 6656/6655
• 1001/1000 and 14400/14399

Temperaments

When tempered out in the full 17-limit, the resulting rank-6 temperament is called the ainismic temperament, or in the 2.3.5.11.13.17 subgroup, the rank-5 ainic temperament. Both are characterized by the presence of essentially tempered chords called ainismic chords.

Ainic

Subgroup: 2.3.5.11.13.17

Comma list: 936/935

Sval mapping:

[⟨	1	0	0	0	0	3	],
⟨	0	1	0	0	0	2	],
⟨	0	0	1	0	0	-1	],
⟨	0	0	0	1	0	-1	],
⟨	0	0	0	0	1	1	]]

sval mapping generators: ~2, ~3, ~5, ~11, ~13

Optimal tunings:

• CTE: ~2 = 1\1, ~3/2 = 701.7942, ~5/4 = 386.4863, ~11/8 = 551.7011, ~13/8 = 840.0893
• CWE: ~2 = 1\1, ~3/2 = 701.8763, ~5/4 = 386.5772, ~11/8 = 551.8274, ~13/8 = 840.2857

Optimal ET sequence: 22, 24, 31, 34, 46, 53, 58, 65, 72, 87, 111, 137, 183, 304, 320, 354, 400, 407, 441, 537, 552g, 624

Ainismic

Subgroup: 2.3.5.7.11.13.17

Comma list: 936/935

Mapping:

[⟨	1	0	0	0	0	0	3	],
⟨	0	1	0	0	0	0	2	],
⟨	0	0	1	0	0	0	-1	],
⟨	0	0	0	1	0	0	0	],
⟨	0	0	0	0	1	0	-1	],
⟨	0	0	0	0	0	1	1	]]

mapping generators: ~2, ~3, ~5, ~7, ~11, ~13

Optimal tunings:

• CTE: ~2 = 1\1, ~3/2 = 701.7942, ~5/4 = 386.4863, ~7/4 = 968.8259, ~11/8 = 551.7011, ~13/8 = 840.0893
• CWE: ~2 = 1\1, ~3/2 = 701.8763, ~5/4 = 386.5773, ~7/4 = 968.9509, ~11/8 = 551.8275, ~13/8 = 840.2857

Optimal ET sequence: 22, 26, 31, 38df, 41, 46, 58, 72, 103, 111, 121, 149, 159, 171, 183, 217, 224, 270, 354, 400, 441, 552g, 624, 1393ceegg, 1576cdegg

Etymology

This comma was named by Aura in 2020. Its names come from the Ancient Greek word aînos ("tale", "story" or "fable"), which is fitting due to the comma serving as a viable 17-limit extension to minthmic temperaments, among others. Funny enough, this same Greek word is the source of the Ancient Greek word aínigma ("riddle"), from which we ultimately get our word "enigma", and this is also fitting due to the sheer difficulty that was involved in the initial process of working out both the name and the uses of this comma in a short span of time.

See also

• Unnoticeable comma
• List of superparticular intervals