136/135: Difference between revisions

| Name = diatisma, diatic comma, fiventeen comma

'''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 [[256/255|S16]] × [[289/288|S17]] or ((16/15)(17/16))/((17/16)(18/17)).  

[[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 = [[194edo|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

{{Mapping|legend=2| 1 0 -3 | 0 1 3 }}

: sval mapping generators: ~2, ~3

* [[Tp tuning|subgroup]] [[CEE]]: 2 = 1\1, ~3/2 = 705.440

* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1\1, ~3/2 = 704.1088

{{Optimal ET sequence|legend=1| 5, 12, 17, 46, 63, 143 }}

[[Subgroup]]: 2.3.5.17

{{Mapping|legend=2| 1 0 0 -3 | 0 1 0 3 | 0 0 1 1 }}

: sval mapping generators: ~2, ~3, ~5

[[Optimal tuning]] ([[Tp tuning|subgroup]] [[CTE]]): ~2 = 1\1, ~3/2 = 704.1088, ~5/4 = 387.8544

{{Optimal ET sequence|legend=1| 10, 12, 22, 34, 80, 114, 194bc }}

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

[[Mapping]]: <br>

| [⟨ || 1 || 0 || 0 || 0 || 0 || 0 || -3 || ],

: sval mapping generators: ~2, ~3, ~5, ~7, ~11, ~13

[[Optimal tuning]] ([[Tp tuning|subgroup]] [[CTE]]): ~2 = 1\1, ~3/2 = 704.1088, ~5/4 = 387.8544, ~7/4, ~11/8, ~13/8

 

<nowiki>*</nowiki> [[optimal patent val]]: [[177edo|177]]

 

[[Srutal archagall]] is an efficient rank-2 temperament tempering out both [[256/255|S16]] and [[289/288|S17]], which is equivalently described as [[charic]] [[semitonic]] due to the fact that {S16 × S17 , [[24576/24565|S16/S17]]} = {[[256/255|S16]], [[289/288|S17]]}