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 the difference between [[16/15]] and [[18/17]] which is easily deduced from its S-expression of S16 * S17, and is therefore the difference between ([[17/16]])([[16/15]]) = [[17/15]] and ([[18/17]])([[17/16]]) = [[9/8]] as the two 17/16's cancel.
'''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)).  


== Temperaments ==
== Temperaments ==
=== Fiventeen ===
=== 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.
[[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
[[Subgroup]]: 2.3.17/5
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: sval mapping generators: ~2, ~3
: sval mapping generators: ~2, ~3


[[Optimal tuning]] ([[Tp tuning|subgroup CTE]]): ~2 = 1\1, ~3/2 = 704.1088
[[Optimal tuning]]s:
* [[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 }}
{{Optimal ET sequence|legend=1| 5, 12, 17, 46, 63, 143 }}
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: sval mapping generators: ~2, ~3, ~5
: sval mapping generators: ~2, ~3, ~5


[[Optimal tuning]] ([[Tp tuning|subgroup CTE]]): ~2 = 1\1, ~3/2 = 704.1088, ~5/4 = 387.8544
[[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 }}
{{Optimal ET sequence|legend=1| 10, 12, 22, 34, 80, 114, 194bc }}
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: sval mapping generators: ~2, ~3, ~5, ~7, ~11, ~13
: 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
[[Optimal tuning]] ([[Tp tuning|subgroup]] [[CTE]]): ~2 = 1\1, ~3/2 = 704.1088, ~5/4 = 387.8544, ~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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* [[Small comma]]
* [[Small comma]]
* [[List of superparticular intervals]]
* [[List of superparticular intervals]]
[[Category:Commas with unknown etymology]]
{{todo|improve readability|inline=1|comment=Rewrite the etymology section to be easier to parse and less vague.}}

Latest revision as of 00:30, 16 November 2024

Interval information
Ratio 136/135
Factorization 23 × 3-3 × 5-1 × 17
Monzo [3 -3 -1 0 0 0 1⟩
Size in cents 12.77669¢
Names diatisma,
diatic comma,
fiventeen comma
Color name 17og2, Sogu 2nd,
Sogu comma
FJS name [math]\displaystyle{ \text{d2}^{17}_{5} }[/math]
Special properties superparticular,
reduced
Tenney height (log2 nd) 14.1643
Weil height (log2 max(n, d)) 14.1749
Wilson height (sopfr(nd)) 37
Comma size small
S-expression S16 × S17
Open this interval in xen-calc

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 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

Sval mapping: [⟨1 0 -3], ⟨0 1 3]]

sval mapping generators: ~2, ~3

Optimal tunings:

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

Mapping:

[⟨ 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*

* optimal patent val: 177

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".

See also


Todo: improve readability

Rewrite the etymology section to be easier to parse and less vague.

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