Dicot family: Difference between revisions

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In the 11-limit, we have the identity 25/24 = ([[45/44]])⋅([[55/54]]), so it makes sense to temper out all of them. This leads to the very natural subgroup temperament where [[11/9]]~[[27/22]] is mapped to the neutral third. As such, this is also the path that most of the septimal extensions take to get their 11-limit versions.

An alternative identity 25/24 = ([[33/32]])⋅([[100/99]]) ~~also makes sense to temper~~ out ~~in the 11-limit, leading~~ to ~~some undecimal extensions as~~ eudicot.

An alternative identity is 25/24 = ([[33/32]])⋅([[100/99]]), and tempering out these commas leads to eudicot (named by [[Xenllium]] in 2026).

=== 2.3.5.11 ~~dicot~~ ===

=== 2.3.5.11 subgroup ===

Subgroup: 2.3.5.11

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Badness (Sintel): 0.536

=== ~~2.3.5.11 eudicot~~ ===

=== Eudicot ===

Subgroup: 2.3.5.11

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* CWE: ~2 = 1200.000{{c}}, ~5/4 = 346.056{{c}}

Badness (Sintel): 0.493

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== Sidi ==

Sidi tempers out 245/243, and splits 5/2~12/5 in two. Its ploidacot is beta-tetracot. This relates it to [[squares]], to which it can be used as a simpler alternative. 14edo in the 14c val can be used as a tuning, in which case it is identical to squares, however.

Sidi tempers out [[245/243]], and splits [[5/2]][[~]][[12/5]] in two. Its [[ploidacot]] is beta-tetracot. This relates it to [[squares]], to which it can be used as a simpler alternative. 14edo in the 14c val can be used as a tuning, in which case it is identical to squares, however.

[[Subgroup]]: 2.3.5.7

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== Sida ==

~~Sida~~ is described as {{nowrap|3 & 14c}} temperment, and tempers out 1323/1280 and 4000/3969. Its ploidacot is beta-tetracot.

Named by [[Xenllium]] in 2026, sida is described as the {{nowrap| 3 & 14c }} temperment, and tempers out [[1323/1280]] and [[4000/3969]]. Its [[ploidacot]] is beta-tetracot, the same as [[#Sidi|sidi]].

[[Subgroup]]: 2.3.5.7

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[[Comma list]]: 25/24, 1323/1280

: mapping generators: ~2, ~32/21

: mapping generators: ~2, ~21~~/16~~

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1209.021{{c}}, ~21~~/16~~ = ~~430~~.~~723~~{{c}}

* [[WE]]: ~2 = 1209.021{{c}}, ~32/21 = 778.298{{c}}

: [[error map]]: {{val| +9.021 +2.216 -20.696 -6.188 }}

* [[CWE]]: ~2 = 1200.000{{c}}, ~21~~/16~~ = ~~427~~.~~215~~{{c}}

* [[CWE]]: ~2 = 1200.000{{c}}, ~32/21 = 772.785{{c}}

: error map: {{val| 0.000 -10.816 -40.744 -32.749 }}

[[Badness]] (Sintel): 2.12

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Optimal tunings:

* WE: ~2 = 1209.621{{c}}, ~14/11 = ~~431~~.~~245~~{{c}}

* WE: ~2 = 1209.621{{c}}, ~14/11 = 772.376{{c}}

* CWE: ~2 = 1200.000{{c}}, ~14/11 = ~~427~~.~~753~~{{c}}

* CWE: ~2 = 1200.000{{c}}, ~14/11 = 772.247{{c}}

Badness (Sintel): 1.54

@@ Line 46: / Line 46: @@
 In the 11-limit, we have the identity 25/24 = ([[45/44]])⋅([[55/54]]), so it makes sense to temper out all of them. This leads to the very natural subgroup temperament where [[11/9]]~[[27/22]] is mapped to the neutral third. As such, this is also the path that most of the septimal extensions take to get their 11-limit versions.
-An alternative identity 25/24 = ([[33/32]])⋅([[100/99]]) also makes sense to temper out in the 11-limit, leading to some undecimal extensions as eudicot.
+An alternative identity is 25/24 = ([[33/32]])⋅([[100/99]]), and tempering out these commas leads to eudicot (named by [[Xenllium]] in 2026).
-=== 2.3.5.11 dicot ===
+=== 2.3.5.11 subgroup ===
 Subgroup: 2.3.5.11
@@ Line 82: / Line 82: @@
 Badness (Sintel): 0.536
-=== 2.3.5.11 eudicot ===
+=== Eudicot ===
 Subgroup: 2.3.5.11
@@ Line 95: / Line 95: @@
 * CWE: ~2 = 1200.000{{c}}, ~5/4 = 346.056{{c}}
-{{Optimal ET sequence|legend=0| 3, 4, 7 }}
+{{Optimal ET sequence|legend=0| 3, 4, 7, 45cceee }}
 Badness (Sintel): 0.493
@@ Line 461: / Line 461: @@
 == Sidi ==
-Sidi tempers out 245/243, and splits 5/2~12/5 in two. Its ploidacot is beta-tetracot. This relates it to [[squares]], to which it can be used as a simpler alternative. 14edo in the 14c val can be used as a tuning, in which case it is identical to squares, however.
+Sidi tempers out [[245/243]], and splits [[5/2]][[~]][[12/5]] in two. Its [[ploidacot]] is beta-tetracot. This relates it to [[squares]], to which it can be used as a simpler alternative. 14edo in the 14c val can be used as a tuning, in which case it is identical to squares, however.
 [[Subgroup]]: 2.3.5.7
@@ Line 497: / Line 497: @@
 == Sida ==
-Sida is described as {{nowrap|3 & 14c}} temperment, and tempers out 1323/1280 and 4000/3969. Its ploidacot is beta-tetracot.
+Named by [[Xenllium]] in 2026, sida is described as the {{nowrap| 3 & 14c }} temperment, and tempers out [[1323/1280]] and [[4000/3969]]. Its [[ploidacot]] is beta-tetracot, the same as [[#Sidi|sidi]].
 [[Subgroup]]: 2.3.5.7
@@ Line 503: / Line 503: @@
 [[Comma list]]: 25/24, 1323/1280
-{{Mapping|legend=1| 1 3 3 1 | 0 -4 -2 5 }}
+{{Mapping|legend=1| 1 -1 1 6 | 0 4 2 -5 }}
+: mapping generators: ~2, ~32/21
-: mapping generators: ~2, ~21/16
 [[Optimal tuning]]s:
-* [[WE]]: ~2 = 1209.021{{c}}, ~21/16 = 430.723{{c}}
+* [[WE]]: ~2 = 1209.021{{c}}, ~32/21 = 778.298{{c}}
 : [[error map]]: {{val| +9.021 +2.216 -20.696 -6.188 }}
-* [[CWE]]: ~2 = 1200.000{{c}}, ~21/16 = 427.215{{c}}
+* [[CWE]]: ~2 = 1200.000{{c}}, ~32/21 = 772.785{{c}}
 : error map: {{val| 0.000 -10.816 -40.744 -32.749 }}
-{{Optimal ET sequence|legend=1| 3, …, 11c, 14c }}
+{{Optimal ET sequence|legend=1| 3, 11c, 14c, 45ccdd }}
 [[Badness]] (Sintel): 2.12
@@ Line 525: / Line 524: @@
 Optimal tunings:
-* WE: ~2 = 1209.621{{c}}, ~14/11 = 431.245{{c}}
+* WE: ~2 = 1209.621{{c}}, ~14/11 = 772.376{{c}}
-* CWE: ~2 = 1200.000{{c}}, ~14/11 = 427.753{{c}}
+* CWE: ~2 = 1200.000{{c}}, ~14/11 = 772.247{{c}}
-{{Optimal ET sequence|legend=0| 3, …, 11c, 14c }}
+{{Optimal ET sequence|legend=0| 3, 11c, 14c }}
 Badness (Sintel): 1.54