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 == | ||
[[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. | ||
{{ | |||
[[ | |||
{ | |||
=== Diatic === | === Diatic === | ||
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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]]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 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 | 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 | ||
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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]]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]] | ||
[[Badness]] (Sintel): 1.15 | |||
[[ | |||
== 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:Diatismic]] | |||
[[Category:Commas named for their regular temperament properties]] | |||
Latest revision as of 12:08, 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.
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
- 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
| [⟨ | 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
- 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 *
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.