136/135: Difference between revisions
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'''136/135''', the '''diatisma''', '''diatic comma''' or '''fiventeen comma''', is a [[17-limit]] [[small comma]]. It is equal to ([[32/27]])/([[20/17]]) and also ([[51/40]])/([[81/64]]). It is also | '''136/135''', the '''diatisma''', '''diatic comma''' or '''fiventeen comma''', is a [[17-limit]] [[small comma]]. It is equal to ([[32/27]])/([[20/17]]) and also ([[51/40]])/([[81/64]]). It is also the difference between [[16/15]] and [[18/17]] and therefore the difference between ([[17/16]])([[16/15]]) = [[17/15]] and ([[18/17]])([[17/16]]) = [[9/8]] as the two 17/16's cancel. | ||
== Temperaments == | == Temperaments == | ||
Revision as of 11:35, 27 May 2024
| Interval information |
diatic comma,
fiventeen comma
Sogu comma
reduced
136/135, the diatisma, diatic comma or fiventeen comma, is a 17-limit small comma. It is equal to (32/27)/(20/17) and also (51/40)/(81/64). It is also the difference between 16/15 and 18/17 and therefore the difference between (17/16)(16/15) = 17/15 and (18/17)(17/16) = 9/8 as the two 17/16's cancel.
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) although 80edo might be preferred for a more accurate 51/40 and it and 46edo might be preferred for more accurate fifths. The same is true of the related rank 3 temperament diatic, described below.
Subgroup: 2.3.17/5
Sval mapping: [⟨1 0 -3], ⟨0 1 3]]
- sval mapping generators: ~2, ~3
Optimal tuning (subgroup CTE): ~2 = 1\1, ~3/2 = 704.1088
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".