Schismatic family: Difference between revisions

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== Subgroup extensions ==

~~=== Maqamschismic (2.3.5.11) ===~~

~~Proposed by [[Eufalesio]] in 2026, maqamschismic is the no-7 [[restriction]] of 13-limit [[cassandra]]; it is unclear what relation this temperament has with maqam.~~

~~Subgroup:~~ 2.3.5.11

=== Tridecaschismic (2.3.5.13) ===

Proposed by [[Eufalesio]] in 2026, tridecaschismic adds the [[325/324|marveltwin comma]] to the comma list, benefitting from a fifth that is just, or practically indistinguishable from just, like in 53edo. It is has among the lowest badness of schismic extensions. It is also equivalent to the 2.3.5.13 [[restriction]] of 13-limit [[cassandra]].

~~Comma list~~: ~~2200/2187, 4125/4096~~

Subgroup: 2.3.5.13

Mapping: ~~{{mapping| 1~~ 1 7 -~~10 | 0~~ 1 -8 ~~23 }}~~

Comma list: 32805/32768, 325/234

: Mapping: [⟨1 1 7 -8], ⟨0 1 -8 20]]

Optimal tunings:

* WE: ~2 = 1200.5458{{c}} ~3/2 = 702.4021{{c}}

* CWE: 2 = 1200.0000{{c}}, ~3/2 = 702.0906{{c}}

{{Optimal ET sequence|legend=0| ~~12e~~, …, 41, 53, 94, ~~147e~~, ~~241ce~~, ~~335ce~~ }}

* WE: ~2 = 1200.333 ¢ ~3/2 = 702.109 ¢

* CWE: 2 = 1200 ¢, ~3/2 = 701.919 ¢

{{Optimal ET sequence|legend=0| 12, …, 41, 53, 65f, 94, 118f, 147, 200}}

Badness (Sintel): 0.582

~~Badness~~ (~~Sintel~~)~~: 1~~.34

==== Maqamschismic (2.3.5.11.13 subgroup) ====

The 2.3.5.11.13 subgroup adds [[352/351]] to the comma list and tempers 11/9 ~ 39/32 together (and 16/13 ~ 27/22), providing a very simple framework for tuning [[Middle-Eastern music#Level 2|''maqamāt'']], specially Turkish ''maqāmāt'', as outlined by [[Ozan Yarman|Ozar Yarman]]. It is equivalent to the no-7 [[cassandra]]. 41edo and 53edo are simplest, but 94edo is more optimized. It is only slightly worse than the no-7 [[Schismatic family#13-limit 3|helenus]].

~~==== 2.3.5.11.13 subgroup ====~~

Subgroup: 2.3.5.11.13

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: ''See also: [[No-elevens subgroup temperaments #Garibaldia]] and [[No-elevens subgroup temperaments #Pontia|#Pontia]]''

~~The [[S~~-~~expression]]-based comma list~~ of ~~this temperament is {[[1216/1215|S16/S18]], [[361/360|S19]], ([[513/512|S15/S20]])}~~. Note that despite prime [[19/1|19]] being optimized by a flatter fifth, the fifth in optimal tunings of nestoria is generally not flatter than the fifth in optimal schismic due to its optimization considering intervals like [[19/10]] and [[19/15]].

Nestoria is notable for having one of the lowest-badness subgroup extensions of schismic. Note that despite prime [[19/1|19]] being optimized by a flatter fifth, the fifth in optimal tunings of nestoria is generally not flatter than the fifth in optimal schismic due to its optimization considering intervals like [[19/10]] and [[19/15]]. Much like tridecaschismic,

[[Subgroup]]: 2.3.5.19

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[[Badness]] (Sintel): 0.126

==== 2.3.5.13.19 subgroup ====

Subgroup: 2.3.5.13

Comma list: 1216/1215, 325/234, 513/512

: Mapping: [⟨1 1 7 -8], ⟨0 1 -8 20]]

Optimal tunings:

* WE: ~2 = 1200.424 ¢ ~3/2 = 702.151 ¢

* CWE: 2 = 1200 ¢, ~3/2 = 701.906¢

Badness (Sintel): 0.354

=== Taylor (2.3.5.13) ===

@@ Line 2,732: / Line 2,732: @@
 == Subgroup extensions ==
-=== Maqamschismic (2.3.5.11) ===
-Proposed by [[Eufalesio]] in 2026, maqamschismic is the no-7 [[restriction]] of 13-limit [[cassandra]]; it is unclear what relation this temperament has with maqam.
-Subgroup: 2.3.5.11
+=== Tridecaschismic (2.3.5.13) ===
+Proposed by [[Eufalesio]] in 2026, tridecaschismic adds the [[325/324|marveltwin comma]] to the comma list, benefitting from a fifth that is just, or practically indistinguishable from just, like in 53edo. It is has among the lowest badness of schismic extensions. It is also equivalent to  the 2.3.5.13 [[restriction]] of 13-limit [[cassandra]].
-Comma list: 2200/2187, 4125/4096
+Subgroup: 2.3.5.13
-Mapping: {{mapping| 1 1 7 -10 | 0 1 -8 23 }}
+Comma list: 32805/32768, 325/234
+: Mapping: [⟨1 1 7 -8], ⟨0 1 -8 20]]
 Optimal tunings:
-* WE: ~2 = 1200.5458{{c}} ~3/2 = 702.4021{{c}}
-* CWE: 2 = 1200.0000{{c}}, ~3/2 = 702.0906{{c}}
-{{Optimal ET sequence|legend=0| 12e, …, 41, 53, 94, 147e, 241ce, 335ce }}
+* WE: ~2 = 1200.333 ¢ ~3/2 = 702.109 ¢
+* CWE: 2 = 1200 ¢, ~3/2 = 701.919 ¢
+{{Optimal ET sequence|legend=0| 12, …, 41, 53, 65f, 94, 118f, 147, 200}}
+Badness (Sintel): 0.582
-Badness (Sintel): 1.34
+==== Maqamschismic (2.3.5.11.13 subgroup) ====
+The 2.3.5.11.13 subgroup adds [[352/351]] to the comma list and tempers 11/9 ~ 39/32 together (and 16/13 ~ 27/22), providing a very simple framework for tuning [[Middle-Eastern music#Level 2|''maqamāt'']], specially Turkish ''maqāmāt'', as outlined by [[Ozan Yarman|Ozar Yarman]]. It is equivalent to the no-7 [[cassandra]]. 41edo and 53edo are simplest, but 94edo is more optimized. It is only slightly worse than the no-7 [[Schismatic family#13-limit 3|helenus]].
-==== 2.3.5.11.13 subgroup ====
 Subgroup: 2.3.5.11.13
@@ Line 2,806: / Line 2,810: @@
 : ''See also: [[No-elevens subgroup temperaments #Garibaldia]] and [[No-elevens subgroup temperaments #Pontia|#Pontia]]''
-The [[S-expression]]-based comma list of this temperament is {[[1216/1215|S16/S18]], [[361/360|S19]], ([[513/512|S15/S20]])}. Note that despite prime [[19/1|19]] being optimized by a flatter fifth, the fifth in optimal tunings of nestoria is generally not flatter than the fifth in optimal schismic due to its optimization considering intervals like [[19/10]] and [[19/15]].
+Nestoria is notable for having one of the lowest-badness subgroup extensions of schismic. Note that despite prime [[19/1|19]] being optimized by a flatter fifth, the fifth in optimal tunings of nestoria is generally not flatter than the fifth in optimal schismic due to its optimization considering intervals like [[19/10]] and [[19/15]]. Much like tridecaschismic,
 [[Subgroup]]: 2.3.5.19
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 [[Badness]] (Sintel): 0.126
+==== 2.3.5.13.19 subgroup ====
+Subgroup: 2.3.5.13
+Comma list: 1216/1215, 325/234, 513/512
+: Mapping: [⟨1 1 7 -8], ⟨0 1 -8 20]]
+Optimal tunings:
+* WE: ~2 = 1200.424 ¢ ~3/2 = 702.151 ¢
+* CWE: 2 = 1200 ¢, ~3/2 = 701.906¢
+{{Optimal ET sequence|legend=0| 12, …, 41, 53, 94, 147, 200h}}
+Badness (Sintel): 0.354
 === Taylor (2.3.5.13) ===