136/135: Difference between revisions

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{{Infobox Interval

| Name = ~~diatonisma~~

| Name = diatisma, 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'' 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]].

== 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]]) 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.

Subgroup: 2.3.17/5

Mapping: {{mapping| 1 0 -3 | 0 1 3 }}

[[CTE]] generator: ~3 = 1904.109{{cent}}

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

See also: [[Srutal archagall]] for the rank 2 temperament tempering out {[[256/255|S16]], [[289/288|S17]]}.

=== Diatic ===

Subgroup: 2.3.5.17

Mapping: {{mapping| 1 0 0 -3 | 0 1 0 3 | 0 0 1 1 }}

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

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

See also: [[Srutal archagall]] for the rank 2 temperament tempering out {[[256/255|S16]], [[289/288|S17]]}.

=== 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.

Subgroup: [[17-limit]]

Mapping: [same as diatic with added trivial entries for primes 7, 11 and 13]

[[CTE]] generators: [same as diatic with purely tuned 7, 11 and 13 added]

EDO tunings with less than 33% [[relative error]] for all primes in the [[17-limit]]: 46, 56

== 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]]

@@ Line 1: / Line 1: @@
 {{Infobox Interval
-| Name = diatonisma
+| Name = diatisma, 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'' 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]].
+== 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]]) 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.
+Subgroup: 2.3.17/5
+Mapping: {{mapping| 1 0 -3 | 0 1 3 }}
+[[CTE]] generator: ~3 = 1904.109{{cent}}
+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
+See also: [[Srutal archagall]] for the rank 2 temperament tempering out {[[256/255|S16]], [[289/288|S17]]}.
+=== Diatic ===
+Subgroup: 2.3.5.17
+Mapping: {{mapping| 1 0 0 -3 | 0 1 0 3 | 0 0 1 1 }}
+[[CTE]] generators: ~3 = 1904.109{{cent}}, ~5 = 2787.854{{cent}}
+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
+See also: [[Srutal archagall]] for the rank 2 temperament tempering out {[[256/255|S16]], [[289/288|S17]]}.
+=== 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.
+Subgroup: [[17-limit]]
+Mapping: [same as diatic with added trivial entries for primes 7, 11 and 13]
+[[CTE]] generators: [same as diatic with purely tuned 7, 11 and 13 added]
+EDO tunings with less than 33% [[relative error]] for all primes in the [[17-limit]]: 46, 56
 == 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]]