136/135: Difference between revisions

(10 intermediate revisions by 3 users not shown)

Line 1:

{{Infobox Interval

| Name = diatisma, fiventeen comma

| Name = diatisma, diatic comma, fiventeen comma

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

| Comma = yes

}}

'''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 between~~ [[16/15]] and [[18/17]] ~~and therefore~~ the difference between [[17/16]] * [[16/15]] = [[17~~/15~~]] ~~and~~ [[~~18/17~~]] * [[17/16]] = [[9/8]] (~~as the two~~ 17/16~~'s cancel~~).

'''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]]) 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.

[[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

Line 16:

: sval mapping generators: ~2, ~3

[[Optimal tuning]] ([[Tp tuning|subgroup CTE]]): ~2 = 1\1, ~3/2 = 704.1088

[[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

Line 27:

Line 29:

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

[[Optimal tuning]] ([[Tp tuning|subgroup CTE]]): ~2 = 1\1, ~3/2 = 704.1088, ~5/4 = 387.8544

[[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 }}

Line 53:

Line 55:

: 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 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 }}*

Line 68:

Line 70:

* [[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 = diatisma, fiventeen comma
+| Name = diatisma, diatic comma, fiventeen comma
 | Color name = 17og2, Sogu 2nd, <br>Sogu comma
 | Comma = yes
 }}
-'''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 between [[16/15]] and [[18/17]] and therefore the difference between [[17/16]] * [[16/15]] = [[17/15]] and [[18/17]] * [[17/16]] = [[9/8]] (as the two 17/16's cancel).
+'''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]]) 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.
+[[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
@@ Line 16: / Line 16: @@
 : sval mapping generators: ~2, ~3
-[[Optimal tuning]] ([[Tp tuning|subgroup CTE]]): ~2 = 1\1, ~3/2 = 704.1088
+[[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 }}
@@ Line 27: / Line 29: @@
 : sval mapping generators: ~2, ~3, ~5
-[[Optimal tuning]] ([[Tp tuning|subgroup CTE]]): ~2 = 1\1, ~3/2 = 704.1088, ~5/4 = 387.8544
+[[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 }}
@@ Line 53: / Line 55: @@
 : 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 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 }}*
@@ Line 68: / Line 70: @@
 * [[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.}}