136/135
| Interval information |
diatic comma,
fiventeen comma
Sogu comma
reduced
(Shannon, [math]\sqrt{nd}[/math])
136/135, the diatisma, diatic comma or fiventeen comma, is a 17-limit small comma. It is the interval that separates 17/10 and 27/16 (or their octave complements 20/17 and 32/27) and that separates 30/17 and 16/9 (or their octave complements 17/15 and 9/8). It is also the difference between 16/15 and 18/17 with an S-expression of S16 × S17 or ((16/15)(17/16))/((17/16)(18/17)).
Temperaments
Fiventeen
17edo makes a good tuning (especially for its size) for the 2.3.17/5-subgroup {136/135} rank 2 temperament which implies a supersoft pentic pentad of ~30:34:40:45:51:60 (because as aforementioned 17/15 is equated with 9/8), corresponding approximately to a just 20/17 tuning, although 80edo might be preferred for an approximately just 51/40 to optimize plausibility slightly more, and 80 + 17 = 97edo and 97 + 17 = 114edo do even better in striking a balance between 80edo's more stable tuning and that having 20/17 more accurate (as in 17edo) is useful because of the more convincing suggestion of the two 15:17:20 chords present in the fiventeen pentad. The same is true of the related rank-3 temperament diatic, described below, for which the optimal ET sequence is much more characteristic of optimized tunings, finding 34edo, then 80edo, then 34 + 80 = 114edo and amazingly even 114 + 80 = 194bc-edo, though because of its focus on primes 5 and 17 it misses 97edo as a tuning, and slightly less optimized though still interesting 63edo and 63 + 80 = 143edo tunings are found in the optimal ET sequence for fiventeen.
Subgroup: 2.3.17/5
Sval mapping: [⟨1 0 -3], ⟨0 1 3]]
- sval mapping generators: ~2, ~3
Optimal ET sequence: 5, 12, 17, 46, 63, 143
Diatic
Subgroup: 2.3.5.17
Sval mapping: [⟨1 0 0 -3], ⟨0 1 0 3], ⟨0 0 1 1]]
- sval mapping generators: ~2, ~3, ~5
Optimal tuning (subgroup CTE): ~2 = 1\1, ~3/2 = 704.1088, ~5/4 = 387.8544
Optimal ET sequence: 10, 12, 22, 34, 80, 114, 194bc
Diatismic
The only edo tuning that has less than 25% relative error for all primes in the 17-limit tempering 136/135 is 46edo, which also tunes 20/17 with less than 25% relative error and 51/40 even more accurately. If you allow 7/4 to be sharper than 25% then 80edo makes a good and more accurate tuning that extends to the 23-limit. Alternatively, if you don't care (as much) about prime 11, 68edo makes a great tuning in the no-11's 19-limit and no-11's no-29's 31-limit.
Subgroup: 2.3.5.7.11.13.17
| [⟨ | 1 | 0 | 0 | 0 | 0 | 0 | -3 | ], |
| ⟨ | 0 | 1 | 0 | 0 | 0 | 0 | 3 | ], |
| ⟨ | 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 | 1 | 0 | ]] |
- sval mapping generators: ~2, ~3, ~5, ~7, ~11, ~13
Optimal tuning (subgroup CTE): ~2 = 1\1, ~3/2 = 704.1088, ~5/4 = 387.8544, ~7/4, ~11/8, ~13/8
Optimal ET sequence: 22, 27eg, 29g, 34d, 39dfg, 41g, 46, 58, 80, 104c, 114e, 126(f), 136ef, 148d, 167g, 216bdef*
Srutal archagall
Srutal archagall is an efficient rank-2 temperament tempering out both S16 and S17, which is equivalently described as charic semitonic due to the fact that {S16 × S17 , S16/S17} = {S16, S17}
Etymology
The name was formerly diatonisma, suggested by User:Xenllium in 2023, but this name has strong reasons against it due to implying an ambiguously-named "diatonic" subgroup temperament. Therefore fiventeenisma and diatisma were proposed. However, due to the need for a separate name for the rank 2 2.3.17/5 subgroup temperament and due to its relation to the chord (see Talk:136/135), the name "fiventeen" was given to the temperament and hence due to the lack of a need for "-ic/-ismic/-isma" (as that can apply to the already-short name of diatisma, itself a rename & shortenage of diatonisma) the name was shortened to just "fiventeen".
See also
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Todo: improve readability Rewrite the etymology section to be easier to parse and less vague. |