273/272
| Ratio | 273/272 |
| Factorization | 2-4 × 3 × 7 × 13 × 17-1 |
| Monzo | [-4 1 0 1 0 1 -1⟩ |
| Size in cents | 6.3531596¢ |
| 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{n\cdot d}[/math]) |
~2.56887 bits |
| Comma size | small |
| open this interval in xen-calc | |
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[1].
Prototannismic
Subgroup: 2.3.5.7.11.13.17
Comma list: 273/272
| [⟨ | 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.