136/135: Difference between revisions

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{{Infobox Interval
{{Infobox Interval
| Name = diatisma, fiventeen comma
| Name = diatisma, diatic comma, fiventeen comma
| Color name = 17og2, Sogu 2nd, <br>Sogu comma
| Color name = 17og2, Sogu 2nd, <br>Sogu comma
| Comma = yes
| Comma = yes
}}
}}
'''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]].
'''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


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


[[CTE]] generator: ~3 = 1904.109{{cent}}
: sval mapping generators: ~2, ~3


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
[[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


See also: [[Srutal archagall]] for the rank 2 temperament tempering out {[[256/255|S16]], [[289/288|S17]]}.
{{Optimal ET sequence|legend=1| 5, 12, 17, 46, 63, 143 }}


=== Diatic ===
=== Diatic ===
Subgroup: 2.3.5.17
[[Subgroup]]: 2.3.5.17


Mapping: {{mapping| 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 }}


[[CTE]] generators: ~3 = 1904.109{{cent}}, ~5 = 2787.854{{cent}}
: sval mapping generators: ~2, ~3, ~5


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


See also: [[Srutal archagall]] for the rank 2 temperament tempering out {[[256/255|S16]], [[289/288|S17]]}.
{{Optimal ET sequence|legend=1| 10, 12, 22, 34, 80, 114, 194bc }}


=== Diatismic ===
=== 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.
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]]
[[Subgroup]]: 2.3.5.7.11.13.17


Mapping: [same as diatic with added trivial entries for primes 7, 11 and 13]
[[Mapping]]: <br>
{| class="right-all"
|-
| [⟨ || 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


[[CTE]] generators: [same as diatic with purely tuned 7, 11 and 13 added]
[[Optimal tuning]] ([[Tp tuning|subgroup]] [[CTE]]): ~2 = 1\1, ~3/2 = 704.1088, ~5/4 = 387.8544, ~7/4, ~11/8, ~13/8


EDO tunings with less than 33% [[relative error]] for all primes in the [[17-limit]]: 46, 56
{{Optimal ET sequence|legend=1| 22, 27eg, 29g, 34d, 39dfg, 41g, 46, 58, 80, 104c, 114e, 126(f), 136ef, 148d, 167g, 216bdef }}*
 
<nowiki>*</nowiki> [[optimal patent val]]: [[177edo|177]]
 
=== 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 {S16 × S17 , [[24576/24565|S16/S17]]} = {[[256/255|S16]], [[289/288|S17]]}


== Etymology ==
== Etymology ==
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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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