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 [[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 ==

=== 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 = [[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

: 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

=== Diatic ===

[[Subgroup]]: 2.3.5.17

: sval mapping generators: ~2, ~3, ~5

[[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 }}

=== 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]]: <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

[[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 }}*

<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 ==

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 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".

== See also ==

* [[Small comma]]

* [[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.}}

@@ 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 [[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 ==
+=== 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 = [[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
+{{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 ===
+[[Subgroup]]: 2.3.5.17
+{{Mapping|legend=2| 1 0 0 -3 | 0 1 0 3 | 0 0 1 1 }}
+: sval mapping generators: ~2, ~3, ~5
+[[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 }}
+=== 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]]: <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
+[[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 }}*
+<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 ==
-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 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".
 == See also ==
 * [[Small comma]]
 * [[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.}}