136/135: Difference between revisions

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[[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

[[Subgroup]]: 2.3.17/5

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

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

: sval mapping generators: ~2, ~3

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

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

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

=== Diatic ===

Subgroup: 2.3.5.17

[[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}}~~

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

~~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~~

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

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 }}

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. 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]].

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

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

[[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

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

[[Optimal tuning]] ([[Tp tuning|subgroup CTE]]): ~2 = 1\1, ~3/2 = 704.1088, ~5/4 = 387.8544, ~7/4, ~11/8, ~13/8

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

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

~~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~~

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

== Etymology ==

@@ Line 10: / Line 10: @@
 [[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
+[[Subgroup]]: 2.3.17/5
-Mapping: {{mapping| 1 0 -3 | 0 1 3 }}
+{{Mapping|legend=2| 1 0 -3 | 0 1 3 }}
-[[CTE]] generator: ~3 = 1904.109{{cent}}
+: sval mapping generators: ~2, ~3
-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]] ([[Tp tuning|subgroup CTE]]): ~2 = 1\1, ~3/2 = 704.1088
-See also: [[Srutal archagall]] for the rank 2 temperament tempering out {[[256/255|S16]], [[289/288|S17]]}.
+{{Optimal ET sequence|legend=1| 5, 12, 17, 46, 63, 143 }}
+See also: [[Srutal archagall]] for the rank-2 temperament tempering out {[[256/255|S16]], [[289/288|S17]]}.
 === Diatic ===
-Subgroup: 2.3.5.17
+[[Subgroup]]: 2.3.5.17
-Mapping: {{mapping| 1 0 0 -3 | 0 1 0 3 | 0 0 1 1 }}
+{{Mapping|legend=2| 1 0 0 -3 | 0 1 0 3 | 0 0 1 1 }}
-[[CTE]] generators: ~3 = 1904.109{{cent}}, ~5 = 2787.854{{cent}}
+: sval mapping generators: ~2, ~3, ~5
-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
+[[Optimal tuning]] ([[Tp tuning|subgroup CTE]]): ~2 = 1\1, ~3/2 = 704.1088, ~5/4 = 387.8544
-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 }}
+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. 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]].
+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
-Subgroup: [[17-limit]]
+[[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
-Mapping: [same as diatic with added trivial entries for primes 7, 11 and 13]
+[[Optimal tuning]] ([[Tp tuning|subgroup CTE]]): ~2 = 1\1, ~3/2 = 704.1088, ~5/4 = 387.8544, ~7/4, ~11/8, ~13/8
-[[CTE]] generators: [same as diatic with purely tuned 7, 11 and 13 added]
+{{Optimal ET sequence|legend=1| 22, 27eg, 29g, 34d, 39dfg, 41g, 46, 58, 80, 104c, 114e, 126(f), 136ef, 148d, 167g, 216bdef }}*
-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
+<nowiki>*</nowiki> [[optimal patent val]]: [[177edo|177]]
 == Etymology ==