136/135: Difference between revisions
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'''136/135''', the '''diatisma''', '''diatic comma''' or '''fiventeen comma''', is a [[17-limit]] [[ | '''136/135''', the '''diatisma''', '''diatic comma''' or '''fiventeen comma''', is a [[small comma|small]] [[17-limit]] [[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)). | ||
== Temperaments == | == Temperaments == | ||
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{{Mapping|legend=2| 1 0 -3 | 0 1 3 }} | {{Mapping|legend=2| 1 0 -3 | 0 1 3 }} | ||
: mapping generators: ~2, ~3 | |||
: | |||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[Tp tuning|subgroup]] [[CEE]]: 2 = | * [[Tp tuning|subgroup]] [[CEE]]: ~2 = 1200.000{{c}}, ~3/2 = 705.440{{c}} | ||
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = | * [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.1088{{c}} | ||
{{Optimal ET sequence|legend=1| 5, 12, 17, 46, 63, 143 }} | {{Optimal ET sequence|legend=1| 5, 12, 17, 46, 63, 143 }} | ||
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{{Mapping|legend=2| 1 0 0 -3 | 0 1 0 3 | 0 0 1 1 }} | {{Mapping|legend=2| 1 0 0 -3 | 0 1 0 3 | 0 0 1 1 }} | ||
: mapping generators: ~2, ~3, ~5 | |||
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 704.1088{{c}}, ~5/4 = 387.8544{{c}} | |||
[[Optimal tuning]] ( | |||
{{Optimal ET sequence|legend=1| 10, 12, 22, 34, 80, 114, 194bc }} | {{Optimal ET sequence|legend=1| 10, 12, 22, 34, 80, 114, 194bc }} | ||
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| ⟨ || 0 || 0 || 0 || 0 || 0 || 1 || 0 || ]] | | ⟨ || 0 || 0 || 0 || 0 || 0 || 1 || 0 || ]] | ||
|} | |} | ||
: | : mapping generators: ~2, ~3, ~5, ~7, ~11, ~13 | ||
[[Optimal tuning]] ( | [[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 704.1088{{c}}, ~5/4 = 387.8544{{c}}, ~7/4, ~11/8, ~13/8 | ||
{{Optimal ET sequence|legend=1| 22, 27eg, 29g, 34d, 39dfg, 41g, 46, 58, 80, 104c, 114e, 126(f), 136ef, 148d, 167g, 216bdef }}* | {{Optimal ET sequence|legend=1| 22, 27eg, 29g, 34d, 39dfg, 41g, 46, 58, 80, 104c, 114e, 126(f), 136ef, 148d, 167g, 216bdef }}* | ||
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=== Srutal archagall === | === Srutal archagall === | ||
[[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 { | [[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]]} | ||
== Etymology == | == Etymology == | ||
The name was formerly ''diatonisma'', suggested by [[ | The name of this comma was formerly ''diatonisma'', suggested by [[Xenllium]] in 2023, but this name would imply a problematic "diatonic" subgroup temperament. Therefore ''diatisma'', a shortenage of ''diatonisma'', and ''fiventeenisma'' a portmanteau of ''five'' and ''seventeen'' for its relation to a chord involving primes 5 and 17, were proposed by [[Godtone]] in 2024. The name ''fiventeen'' was soon given to the rank-2 2.3.17/5-subgroup temperament, and hence the name ''fiventeenisma'' became just ''fiventeen comma''. | ||
== See also == | == See also == | ||
* [[List of superparticular intervals]] | * [[List of superparticular intervals]] | ||
[[Category:Commas | [[Category:Diatismic]] | ||
[[Category:Commas named for their regular temperament properties]] | |||
Revision as of 10:39, 21 March 2026
| Interval information |
diatic comma,
fiventeen comma
Sogu comma
reduced
136/135, the diatisma, diatic comma or fiventeen comma, is a small 17-limit 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
Subgroup-val mapping: [⟨1 0 -3], ⟨0 1 3]]
- mapping generators: ~2, ~3
Optimal ET sequence: 5, 12, 17, 46, 63, 143
Diatic
Subgroup: 2.3.5.17
Subgroup-val mapping: [⟨1 0 0 -3], ⟨0 1 0 3], ⟨0 0 1 1]]
- mapping generators: ~2, ~3, ~5
Optimal tuning (CTE): ~2 = 1200.0000 ¢, ~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 | ]] |
- mapping generators: ~2, ~3, ~5, ~7, ~11, ~13
Optimal tuning (CTE): ~2 = 1200.0000 ¢, ~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 of this comma was formerly diatonisma, suggested by Xenllium in 2023, but this name would imply a problematic "diatonic" subgroup temperament. Therefore diatisma, a shortenage of diatonisma, and fiventeenisma a portmanteau of five and seventeen for its relation to a chord involving primes 5 and 17, were proposed by Godtone in 2024. The name fiventeen was soon given to the rank-2 2.3.17/5-subgroup temperament, and hence the name fiventeenisma became just fiventeen comma.