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== Temperaments == | == Temperaments == | ||
Tempering out this comma in the 17-limit results in the rank-6 '''sextantonismic temperament''', or in the 2.3.5.13.17 subgroup, the rank-4 '''sextantonic temperament'''. In either case [[26/25]] is split into two equal parts, each representing 51/50~52/51, and enables the [[sextantonismic chords]]. | Tempering out this comma in the 17-limit results in the rank-6 '''sextantonismic temperament''', or in the 2.3.5.13.17 subgroup, the rank-4 '''sextantonic temperament'''. In either case [[26/25]] is split into two equal parts, each representing 51/50~52/51, and enables the [[sextantonismic chords]]. | ||
=== Sextantonismic === | |||
[[Subgroup]]: 2.3.5.7.11.13.17 | |||
[[Mapping]]: <br> | |||
{| class="right-all" | |||
|- | |||
| [⟨ || 1 || 0 || 0 || 0 || 0 || 1 || 2 || ], | |||
|- | |||
| ⟨ || 0 || 1 || 0 || 0 || 0 || 0 || -1 || ], | |||
|- | |||
| ⟨ || 0 || 0 || 1 || 0 || 0 || 0 || 1 || ], | |||
|- | |||
| ⟨ || 0 || 0 || 0 || 1 || 0 || 0 || 0 || ], | |||
|- | |||
| ⟨ || 0 || 0 || 0 || 0 || 1 || 0 || 0 || ], | |||
|- | |||
| ⟨ || 0 || 0 || 0 || 0 || 0 || 2 || 1 || ]] | |||
|} | |||
: mapping generators: ~2, ~3, ~5, ~7, ~11, ~51/20 | |||
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.9252, ~5/4 = 386.3777, ~7/4, ~11/8, ~51/40 = 420.3045 | |||
{{Optimal ET sequence|legend=1| 17cg, 22f, 26, 29g, 31, 38df, 43, 46, 60e, 65d, 68, 72, 103, 111, 140, 171, 183, 217, 243e, 282, 311, 354, 400, 422, 460, 494, 742, 814, 954, 1236, 1696, 2190g, 4069g }}<nowiki>*</nowiki> | |||
<nowiki>*</nowiki> [[optimal patent val]]: [[2932edo|2932]] | |||
=== Sextantonic === | |||
[[Subgroup]]: 2.3.5.7.11.13.17 | |||
{{Mapping|legend=2| 1 0 0 2 1 | 0 1 0 0 -1 | 0 0 0 0 1 | 0 0 0 2 1 }} | |||
: sval mapping generators: ~2, ~3, ~5, ~51/20 | |||
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.9252, ~5/4 = 386.3777, ~51/40 = 420.3045 | |||
{{Optimal ET sequence|legend=1| 12, 25, 26, 31, 34, 72, 106, 137, 171, 183, 217, 277, 354, 388, 460, 494, 677, 814, 848 }} | |||
== Etymology == | == Etymology == | ||
Revision as of 13:34, 13 August 2023
| Interval information |
Sosothugugu comma
reduced
2601/2600, the sextantonisma, is a 17-limit (also 2.3.5.13.17 subgroup) superparticular comma measuring about 0.67 cents. It may be properly described as the septendecimal sixth-tones comma, since it is the difference between 51/50 and 52/51, the two 17-limit sixth-tones. Another prominent identity is the little gap between 18/13 and a stack of two 20/17's.
Commatic relations
In terms of commas, it is the difference between the following pairs:
- 289/288 and 325/324
- 561/560 and 715/714
- 833/832 and 1225/1224
- 1156/1155 and 2080/2079
- 1275/1274 and 2500/2499
- 1701/1700 and 4914/4913
- 2401/2400 and 31213/31212
Temperaments
Tempering out this comma in the 17-limit results in the rank-6 sextantonismic temperament, or in the 2.3.5.13.17 subgroup, the rank-4 sextantonic temperament. In either case 26/25 is split into two equal parts, each representing 51/50~52/51, and enables the sextantonismic chords.
Sextantonismic
Subgroup: 2.3.5.7.11.13.17
| [⟨ | 1 | 0 | 0 | 0 | 0 | 1 | 2 | ], |
| ⟨ | 0 | 1 | 0 | 0 | 0 | 0 | -1 | ], |
| ⟨ | 0 | 0 | 1 | 0 | 0 | 0 | 1 | ], |
| ⟨ | 0 | 0 | 0 | 1 | 0 | 0 | 0 | ], |
| ⟨ | 0 | 0 | 0 | 0 | 1 | 0 | 0 | ], |
| ⟨ | 0 | 0 | 0 | 0 | 0 | 2 | 1 | ]] |
- mapping generators: ~2, ~3, ~5, ~7, ~11, ~51/20
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.9252, ~5/4 = 386.3777, ~7/4, ~11/8, ~51/40 = 420.3045
Optimal ET sequence: 17cg, 22f, 26, 29g, 31, 38df, 43, 46, 60e, 65d, 68, 72, 103, 111, 140, 171, 183, 217, 243e, 282, 311, 354, 400, 422, 460, 494, 742, 814, 954, 1236, 1696, 2190g, 4069g*
Sextantonic
Subgroup: 2.3.5.7.11.13.17
Sval mapping: [⟨1 0 0 2 1], ⟨0 1 0 0 -1], ⟨0 0 0 0 1], ⟨0 0 0 2 1]]
- sval mapping generators: ~2, ~3, ~5, ~51/20
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.9252, ~5/4 = 386.3777, ~51/40 = 420.3045
Optimal ET sequence: 12, 25, 26, 31, 34, 72, 106, 137, 171, 183, 217, 277, 354, 388, 460, 494, 677, 814, 848
Etymology
The sextantonisma was named by Flora Canou in 2023. It is a contraction of septendecimal sixth-tones comma into a single word consisting of Latin sextans ("sixth") and tonus ("tone").