136/135: Difference between revisions
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'''136/135''', the ''diatisma'' or ''fiventeen'' | '''136/135''', the ''diatisma'' or ''fiventeen comma'', is a [[17-limit]] [[small comma]]. It is equal to ([[32/27]])/([[20/17]]) and therefore ([[51/40]])/([[81/64]]). It is also [[1/2-square-particular|trivially]] the difference between [[17/15]] and [[9/8]] and between [[16/15]] and [[18/17]]. | ||
== Temperaments == | == Temperaments == | ||
Revision as of 01:23, 9 March 2024
| Interval information |
fiventeen comma
Sogu comma
reduced
136/135, the diatisma or fiventeen comma, is a 17-limit small comma. It is equal to (32/27)/(20/17) and therefore (51/40)/(81/64). It is also trivially the difference between 17/15 and 9/8 and between 16/15 and 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) 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
Mapping: [⟨1 0 -3], ⟨0 1 3]]
CTE generator: ~3 = 1904.109 ¢
Patent val EDO tunings with 20/17 and 3/2 off by less than 25% relative error (contorted in brackets): 5, 12, 17, 22, 29, (34,) 39, 46, (51,) 56, 63, (68,) 80
See also: Srutal archagall for the rank 2 temperament tempering out {S16, S17}.
Diatic
Subgroup: 2.3.5.17
Mapping: [⟨1 0 0 -3], ⟨0 1 0 3], ⟨0 0 1 1]]
CTE generators: ~3 = 1904.109 ¢, ~5 = 2787.854 ¢
Patent val EDO tunings with 20/17, 3/2 and 5/4 off by less than 25% relative error (contorted in brackets): 12, 22, 34, 46, 56, (68,) 80
See also: Srutal archagall for the rank 2 temperament tempering out {S16, S17}.
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: 17-limit
Mapping: [same as diatic with added trivial entries for primes 7, 11 and 13]
CTE generators: [same as diatic with purely tuned 7, 11 and 13 added]
EDO tunings with less than 33% relative error for all primes in the no-7's no-11's 17-limit: 10, 24, 34, 44, 46, 56, 80, 114
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".