136/135: Difference between revisions

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You're not supposed to "recommend" one optimization method over another. Ideally we wanna keep CTE and CWE here but CEE could stay for now
Move fiventeen to subgroup temps
 
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'''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)).  
'''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 ==
=== Fiventeen ===
[[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.  
[[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
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.  
 
{{Mapping|legend=2| 1 0 -3 | 0 1 3 }}
 
: sval mapping generators: ~2, ~3
 
[[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 }}


=== Diatic ===
=== Diatic ===
Line 26: Line 15:


{{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


: sval mapping generators: ~2, ~3, ~5
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.2838{{c}}, ~3/2 = 704.4600{{c}}, ~5/4 = 389.0228{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.5286{{c}}, ~5/4 = 388.6162{{c}}


[[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 }}
[[Badness]] (Sintel): 0.139


=== 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. 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]].
The only edo tuning that has less than 25% [[relative error]] for all primes in the [[17-limit]] tempering out 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 for a good and more accurate tuning. Alternatively, if you do not care as much about prime 11, [[68edo]] makes for a great tuning.


[[Subgroup]]: 2.3.5.7.11.13.17
[[Subgroup]]: 2.3.5.7.11.13.17
Line 53: Line 45:
| ⟨ || 0 || 0 || 0 || 0 || 0 || 1 || 0 || ]]
| ⟨ || 0 || 0 || 0 || 0 || 0 || 1 || 0 || ]]
|}
|}
: sval mapping generators: ~2, ~3, ~5, ~7, ~11, ~13
: 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]]s:
* [[WE]]: ~2 = 1199.2838{{c}}, ~3/2 = 704.4600{{c}}, ~5/4 = 389.0228{{c}}, ~7/4 = 970.2512{{c}}, ~11/8 = 553.4578{{c}}, ~13/8 = 842.6669{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.5286{{c}}, ~5/4 = 388.6162{{c}}, ~7/4 = 969.9161{{c}}, ~11/8 = 552.6614{{c}}, ~13/8 = 841.9647{{c}}


{{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 }} *


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


=== Srutal archagall ===
[[Badness]] (Sintel): 1.15
[[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 [[User:Xenllium]] in 2023, but this name has [[comma naming|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".
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 ==
* [[Small comma]]
* [[List of superparticular intervals]]
* [[List of superparticular intervals]]
[[Category:Diatismic]]
[[Category:Commas named for their regular temperament properties]]

Latest revision as of 12:08, 21 March 2026

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 norm (log2 nd) 14.1643
Weil norm (log2 max(n, d)) 14.1749
Wilson norm (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 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.

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 tunings:

  • WE: ~2 = 1199.2838 ¢, ~3/2 = 704.4600 ¢, ~5/4 = 389.0228 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.5286 ¢, ~5/4 = 388.6162 ¢

Optimal ET sequence: 10, 12, 22, 34, 80, 114, 194bc

Badness (Sintel): 0.139

Diatismic

The only edo tuning that has less than 25% relative error for all primes in the 17-limit tempering out 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 for a good and more accurate tuning. Alternatively, if you do not care as much about prime 11, 68edo makes for a great tuning.

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

Optimal tunings:

  • WE: ~2 = 1199.2838 ¢, ~3/2 = 704.4600 ¢, ~5/4 = 389.0228 ¢, ~7/4 = 970.2512 ¢, ~11/8 = 553.4578 ¢, ~13/8 = 842.6669 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.5286 ¢, ~5/4 = 388.6162 ¢, ~7/4 = 969.9161 ¢, ~11/8 = 552.6614 ¢, ~13/8 = 841.9647 ¢

Optimal ET sequence: 22, 27eg, 29g, 34d, 39dfg, 41g, 46, 58, 80, 104c, 114e, 126(f), 136ef, 148d, 167g, 216bdef *

* optimal patent val: 177

Badness (Sintel): 1.15

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.

See also

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