136/135: Difference between revisions
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== Temperaments == | == Temperaments == | ||
[[Tempering out]] this comma in the full 17-limit results in the rank-6 '''diatismic''' temperament, or in the 2.3.5.17 subgroup, the rank-3 '''diatic''' temperament. | |||
Since 136/135 = ([[225/224]])⋅([[256/255]]), it would make sense to temper out both [[256/255]] ({{S|16}}) and [[289/288]] ({{S|17}}), thereby tempering diatic to [[srutal archagall]], which is equivalently described as "[[charic]] [[semitonic]]". This can be further restricted to the 2.3.17/5-subgroup {136/135}, called fiventeen, which is a rank-2 temperament generated by an octave and a perfect fifth. | |||
=== Fiventeen === | === Fiventeen === | ||
[[ | In fiventeen, [[17/15]] is equated with [[9/8]], so it implies a [[supersoft]] [[pentic]] [[pentad]] of [[~]]30:34:40:45:51. [[17edo]] makes a good tuning especially for its size, which gives a [[supersoft]] pentic scale 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 [[97edo]] (= 80 + 17) and [[114edo]] (= 97 + 17) 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 [[114edo]] (= 34 + 80) and even [[194edo|194bc-edo]] (= 80 + 114), 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 [[143edo]] (= 63 + 80) tunings are found in the optimal ET sequence for fiventeen. | ||
[[Subgroup]]: 2.3.17/5 | [[Subgroup]]: 2.3.17/5 | ||
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<nowiki>*</nowiki> [[optimal patent val]]: [[177edo|177]] | <nowiki>*</nowiki> [[optimal patent val]]: [[177edo|177]] | ||
== Etymology == | == Etymology == | ||
Revision as of 10:57, 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
Tempering out this comma in the full 17-limit results in the rank-6 diatismic temperament, or in the 2.3.5.17 subgroup, the rank-3 diatic temperament.
Since 136/135 = (225/224)⋅(256/255), it would make sense to temper out both 256/255 (S16) and 289/288 (S17), thereby tempering diatic to srutal archagall, which is equivalently described as "charic semitonic". This can be further restricted to the 2.3.17/5-subgroup {136/135}, called fiventeen, which is a rank-2 temperament generated by an octave and a perfect fifth.
Fiventeen
In fiventeen, 17/15 is equated with 9/8, so it implies a supersoft pentic pentad of ~30:34:40:45:51. 17edo makes a good tuning especially for its size, which gives a supersoft pentic scale 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 97edo (= 80 + 17) and 114edo (= 97 + 17) 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 114edo (= 34 + 80) and even 194bc-edo (= 80 + 114), 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 143edo (= 63 + 80) 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*
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.