273/272

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]\text{P1}^{7,13}_{17}[/math]
Special properties	superparticular, reduced
Tenney height (log2 nd)	16.1802
Weil height (log2 max(n, d))	16.1855
Wilson height (sopfr(nd))	48
Harmonic entropy (Shannon, [math]\sqrt{nd}[/math])	~1.56714 bits
Comma size	small
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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.

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

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. However, that is not all, as according to Scott Dakota, tannic is actually of similar overall utility to marvel temperament.

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 tuning (CTE): ~2 = 1\1, ~3/2 = 701.5638, ~5/4, ~7/4 = 967.5985, ~11/8, ~13/8 = 838.3951

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

Prototannic

Subgroup: 2.3.7.13.17

Comma list: 273/272

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 tuning (CTE): ~2 = 1\1, ~3/2 = 701.5638, ~7/4 = 967.5985, ~13/8 = 838.3951

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

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 logging standard that a comma name "X-isma" should not be associated to a temperament "X-ic" if "X-ic" is a weak extension of the temperament induced by the comma, since 17-limit tannic is a weak extension of 2.3.7.13.17[273/272] (see above). Prototannisma has thus been proposed as an alternative name.