273/272

273/272, the tannisma, or the prototannisma, is a small 17-limit (also 2.3.7.13.17-subgroup) comma with a value of roughly 6.35 cents. It forms the difference between 21/17 and 16/13, and the difference between 39/32 and 17/14, as well as the difference between 17/13 and 21/16.

Interval information

Ratio	273/272
Factorization	2-4 × 3 × 7 × 13 × 17-1
Monzo	[-4 1 0 1 0 1 -1⟩
Size in cents	6.35316¢
Names	tannisma, prototannisma
Color name	17u3oz1, suthozo 1sn, Suthozo comma
FJS name	[math]\displaystyle{ \text{P1}^{7,13}_{17} }[/math]
Special properties	superparticular, reduced
Tenney norm (log2 nd)	16.1802
Weil norm (log2 max(n, d))	16.1855
Wilson norm (sopfr(nd))	48
Comma size	small
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Commatic relations

This comma is the difference between the following superparticular pairs:

• 35/34 and 40/39
• 52/51 and 64/63
• 65/64 and 85/84
• 91/90 and 136/135
• 154/153 and 352/351
• 169/168 and 442/441
• 221/220 and 1156/1155
• 225/224 and 1275/1274
• 256/255 and 4096/4095

It factors into the following superparticular pairs:

• 441/440 and 715/714
• 385/384 and 936/935
• 375/374 and 1001/1000
• 364/363 and 1089/1088
• 351/350 and 1225/1224
• 325/324 and 1701/1700
• 289/288 and 4914/4913

It also factors neatly into three superparticular commas: 715/714, 833/832, and 936/935.

Temperaments

Tempering out this comma in the 17-limit leads to the rank-6 prototannismic temperament, or in the 2.3.7.13.17 subgroup, the rank-4 prototannic temperament, both characterized by the equivalences introduced above, and lead to a type of essentially tempered chords called prototannismic chords. The prototannic temperament has a notable weak extension to the full 17-limit called tannic. According to Scott Dakota, tannic is of similar overall utility to marvel.

Prototannic

Subgroup: 2.3.7.13.17

Comma list: 273/272

Subgroup-val mapping: [⟨1 0 0 0 -4], ⟨0 1 0 0 1], ⟨0 0 1 0 1], ⟨0 0 0 1 1]]

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

Optimal tunings:

• WE: ~2 = 1200.4469 ¢, ~3/2 = 701.2262 ¢, ~7/4 = 967.0489 ¢, ~13/8 = 837.6531 ¢
• CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.2150 ¢, ~7/4 = 967.0685 ¢, ~13/8 = 837.7810 ¢

Optimal ET sequence: 10, 17g, 24, 26, 31, 36, 113, 149

Badness (Sintel): 0.113

Prototannismic

Subgroup: 2.3.5.7.11.13.17

Comma list: 273/272

Mapping:

[⟨	1	0	0	0	0	0	-4	],
⟨	0	1	0	0	0	0	1	],
⟨	0	0	1	0	0	0	0	],
⟨	0	0	0	1	0	0	1	],
⟨	0	0	0	0	1	0	0	],
⟨	0	0	0	0	0	1	1	]]

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

Optimal tunings:

• WE: ~2 = 1200.4469 ¢, ~3/2 = 701.2262 ¢, ~5/4 = 385.4186 ¢, ~7/4 = 967.0489 ¢, ~11/8 = 549.9753 ¢, ~13/8 = 837.6531 ¢
• CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.2150 ¢, ~5/4 = 385.7085 ¢, ~7/4 = 967.0685 ¢, ~11/8 = 550.4163 ¢, ~13/8 = 837.7810 ¢

Optimal ET sequence: 17cg, 19eg, 22, 26, 27eg, 29g, 31, 41, 46, 72, 103, 130g, 149, 159, 221ef, 262df, 308def, 334cdf

Badness (Sintel): 0.678

Etymology

This comma was named by Scott Dakota, who also devised tannic temperament, as the tannisma no later than 2017. The name tannisma does not adhere to the wiki comma naming standard as it would induce tannic as a temperament that tempers out the comma alone in its minimal prime subgroup, 2.3.7.13.17 {273/272}, for which Scott's 17-limit tannic is a weak extension and cannot share the same time. Prototannisma has thus been proposed as an alternative name.

See also

• List of superparticular intervals