136/135: Difference between revisions

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{{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 [[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 ==

~~=== Fiventeen ===~~

[[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.

[[~~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

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.

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

=== Diatic ===

[[Subgroup]]: 2.3.5.17

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

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

~~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~~

[[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}}

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

{{Optimal ET sequence|legend=1| 10, 12, 22, 34, 80, 114, 194bc }}

~~=== Diatic ===~~

[[Badness]] (Sintel): 0.139

~~Subgroup~~: 2.~~3.5.17~~

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

=== 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.

[[~~CTE~~]] ~~generators~~: ~3 ~~= 1904~~.~~109{{cent}}, ~~~5 ~~= 2787~~.~~854{{cent}}~~

[[Subgroup]]: 2.3.5.7.11.13.17

~~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~~

[[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

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

[[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}}

=~~== Diatismic ===~~

{{Optimal ET sequence|legend=1| 22, 27eg, 29g, 34d, 39dfg, 41g, 46, 58, 80, 104c, 114e, 126(f), 136ef, 148d, 167g, 216bdef }} *

~~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~~: [[~~17-limit~~]]

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

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

[[Badness]] (Sintel): 1.15

~~[[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 no-7's no-11's~~ [[~~17-limit~~]]: ~~10, 24, 34, 44, 46, 56, 80, 114~~

== Etymology ==

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"~~.

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 = 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 [[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 ==
-=== Fiventeen ===
+[[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.
-[[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
+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.
-Mapping: {{mapping| 1 0 -3 | 0 1 3 }}
+=== Diatic ===
+[[Subgroup]]: 2.3.5.17
-[[CTE]] generator: ~3 = 1904.109{{cent}}
+{{Mapping|legend=2| 1 0 0 -3 | 0 1 0 3 | 0 0 1 1 }}
+: mapping generators: ~2, ~3, ~5
-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
+[[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}}
-See also: [[Srutal archagall]] for the rank 2 temperament tempering out {[[256/255|S16]], [[289/288|S17]]}.
+{{Optimal ET sequence|legend=1| 10, 12, 22, 34, 80, 114, 194bc }}
-=== Diatic ===
+[[Badness]] (Sintel): 0.139
-Subgroup: 2.3.5.17
-Mapping: {{mapping| 1 0 0 -3 | 0 1 0 3 | 0 0 1 1 }}
+=== 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.
-[[CTE]] generators: ~3 = 1904.109{{cent}}, ~5 = 2787.854{{cent}}
+[[Subgroup]]: 2.3.5.7.11.13.17
-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
+[[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
-See also: [[Srutal archagall]] for the rank 2 temperament tempering out {[[256/255|S16]], [[289/288|S17]]}.
+[[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}}
-=== Diatismic ===
+{{Optimal ET sequence|legend=1| 22, 27eg, 29g, 34d, 39dfg, 41g, 46, 58, 80, 104c, 114e, 126(f), 136ef, 148d, 167g, 216bdef }} *
-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: [[17-limit]]
+<nowiki>*</nowiki> [[optimal patent val]]: [[177edo|177]]
-Mapping: [same as diatic with added trivial entries for primes 7, 11 and 13]
+[[Badness]] (Sintel): 1.15
-[[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 no-7's no-11's [[17-limit]]: 10, 24, 34, 44, 46, 56, 80, 114
 == Etymology ==
-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".
+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]]