136/135: Difference between revisions

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{{Infobox Interval

| Name = ~~diatonisma~~

| Name = diatisma, diatic comma, fiventeen comma

| Color name = 17og2, Sogu 2nd, <br>Sogu comma

| Comma = yes

}}

'''136/135''', the '''~~diatonisma~~''', is a [[17-limit]] [[~~small~~ comma]]. It is the ~~difference between~~ [[9/8]] ~~(Pythagorean whole tone~~) and [[17/15]] (~~septendecimal whole tone~~), as ~~well as between~~ [[16/15]] and [[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 ==

[[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 ===

[[Subgroup]]: 2.3.5.17

: 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 ===

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]]: <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 || ]]

|}

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

[[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 }} *

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

[[Badness]] (Sintel): 1.15

== Etymology ==

The name ''diatonisma'' ~~was named~~ by [[~~User:Xenllium|~~Xenllium]] in 2023. ~~It refers~~ to ~~the~~ [[~~5L 2s|diatonic scale~~]], ~~especially [[Pythagorean tuning]]~~ and ~~[[superpyth]]~~.

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 ==

* [[Small comma]]

* [[List of superparticular intervals]]

[[Category:Diatismic]]

[[Category:Commas named for their regular temperament properties]]

@@ Line 1: / Line 1: @@
 {{Infobox Interval
-| Name = diatonisma
+| Name = diatisma, diatic comma, fiventeen comma
 | Color name = 17og2, Sogu 2nd, <br>Sogu comma
 | Comma = yes
 }}
-'''136/135''', the '''diatonisma''', is a [[17-limit]] [[small comma]]. It is the difference between [[9/8]] (Pythagorean whole tone) and [[17/15]] (septendecimal whole tone), as well as between [[16/15]] and [[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 ==
+[[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 ===
+[[Subgroup]]: 2.3.5.17
+{{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 ===
+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]]: <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 || ]]
+|}
+: mapping generators: ~2, ~3, ~5, ~7, ~11, ~13
+[[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 }} *
+<nowiki>*</nowiki> [[optimal patent val]]: [[177edo|177]]
+[[Badness]] (Sintel): 1.15
 == Etymology ==
-The name ''diatonisma'' was named by [[User:Xenllium|Xenllium]] in 2023. It refers to the [[5L 2s|diatonic scale]], especially [[Pythagorean tuning]] and [[superpyth]].
+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 ==
-* [[Small comma]]
 * [[List of superparticular intervals]]
+[[Category:Diatismic]]
+[[Category:Commas named for their regular temperament properties]]