Schismatic family: Difference between revisions

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== Term ==

Term tempers out the [[landscape comma]], mapping [[63/50]] to the 1/3-octave period. It can be described as {{nowrap| 12 & 171 }}, and is the unique temperament that ~~equates a~~ syntonic~Pythagorean ~~comma~~ with a stack of three [[marvel comma]]s. A [[septimal comma]] is then found as a stack of four marvel commas. In ~~some~~ 7-limit adaptive-tuning practice, the marvel comma corresponds to a melodic unit called a [[kleisma]], with three kleismas making a comma, so this temperament may be useful for modeling that. [[171edo]] makes for an excellent tuning.

Term tempers out the [[landscape comma]], mapping [[63/50]] to the 1/3-octave period. It can be described as {{nowrap| 12 & 171 }}, and is the unique temperament that tempers together the syntonic and Pythagorean commas and equates it with a stack of three [[marvel comma]]s. A [[septimal comma]] is then found as a stack of four marvel commas. In certain 7-limit adaptive-tuning practice, the marvel comma corresponds to a melodic unit called a [[kleisma #As an interval region|kleisma]], with three kleismas making a comma, so this temperament may be useful for modeling that. [[171edo]] makes for an excellent tuning.

[[Subgroup]]: 2.3.5.7

Line 2,744:

Badness (Sintel): 1.46

~~== Tridecafifths ==~~

Named by [[Eliora]] in 2023, tridecafifths may be described as the {{nowrap| 89 & 200 }} temperament. It divides the [[3/2|perfect fifth]] into thirteen quartertones, so its [[ploidacot]] is 13-cot. [[289edo]] gives a highly recommendable tuning.

~~[[Subgroup]]: 2.3.5.7~~

~~[[Comma list]]: 32805/32768, {{monzo| -14 -1 -9 13 }}~~

~~{{Mapping|legend=1| 1 1 7 6 | 0 13 -104 -71 }}~~

~~: mapping generators: ~2, ~1323/1280~~

~~[[Optimal tuning]]s:~~

* [[WE]]: ~2 = 1200.1431{{c}}, ~1323/1280 = 53.9838{{c}}

~~: [[error map]]: {{val| +0.143 -0.023 +0.375 -0.816 }}~~

* [[CWE]]: ~2 = 1200.0000{{c}}, ~1323/1280 = 53.9764{{c}}

~~: error map: {{val| 0.000 -0.261 -0.221 -0.421 }}~~

~~{{Optimal ET sequence|legend=1| 89, 200, 289 }}~~

~~[[Badness]] (Sintel): 10.9~~

~~=== 11-limit ===~~

~~Subgroup: 2.3.5.7.11~~

~~Comma list: 441/440, 32805/32768, 55296000/55240493~~

~~Mapping: {{mapping| 1 1 7 6 4 | 0 13 -104 -71 -12 }}~~

~~Optimal tunings:~~

* WE: ~2 = 1200.0311{{c}}, ~33/32 = 53.9766{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~33/32 = 53.9750{{c}}

~~{{Optimal ET sequence|legend=0| 89, 200, 289 }}~~

~~Badness (Sintel): 4.23~~

== Subgroup extensions ==

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

Proposed by [[Eufalesio]] in 2026, maqamschismic is equivalent to the no-7 [[cassandra]]. 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 [[maqam]]at (especially the Turkish version), as outlined by [[Ozan Yarman]]. 41edo and 53edo are simplest, but 94edo is more optimized. It is only slightly worse than the no-7 [[helenus]].

~~Subgroup: 2.3.5.11~~

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

~~Subgroup-val mapping: {{mapping| 1 0 15 -33 | 0 1 -8 23 }}~~

~~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 }}~~

~~Badness (Sintel): 1.34~~

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

~~Subgroup: 2.3.5.11.13~~

~~Comma list: 325/324, 352/351, 4125/4096~~

~~Subgroup-val mapping: {{mapping| 1 0 15 -33 -28 | 0 1 -8 23 20 }}~~

~~Optimal tunings:~~

* WE: ~2 = 1200.4565{{c}} ~3/2 = 702.3057{{c}}

* CWE: 2 = 1200.0000{{c}}, ~3/2 = 702.0485{{c}}

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

~~Badness (Sintel): 0.862~~

=== 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]].

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

Subgroup: 2.3.5.13

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

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]].

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]]. However, the dyadic tuning sensitivity of [[19/16]] suggests using tunings like [[65edo]] and [[77edo]] to optimize in favour of prime 19 (especially the minor triad ~16:19:24 which is equated with the Pythagorean minor triad), as [[171edo]] is already arguably undertempered for it despite being the optimal patent val.

[[Subgroup]]: 2.3.5.19

@@ Line 1,475: / Line 1,475: @@
 == Term ==
-Term tempers out the [[landscape comma]], mapping [[63/50]] to the 1/3-octave period. It can be described as {{nowrap| 12 & 171 }}, and is the unique temperament that equates a syntonic~Pythagorean comma with a stack of three [[marvel comma]]s. A [[septimal comma]] is then found as a stack of four marvel commas. In some 7-limit adaptive-tuning practice, the marvel comma corresponds to a melodic unit called a [[kleisma]], with three kleismas making a comma, so this temperament may be useful for modeling that. [[171edo]] makes for an excellent tuning.
+Term tempers out the [[landscape comma]], mapping [[63/50]] to the 1/3-octave period. It can be described as {{nowrap| 12 & 171 }}, and is the unique temperament that tempers together the syntonic and Pythagorean commas and equates it with a stack of three [[marvel comma]]s. A [[septimal comma]] is then found as a stack of four marvel commas. In certain 7-limit adaptive-tuning practice, the marvel comma corresponds to a melodic unit called a [[kleisma #As an interval region|kleisma]], with three kleismas making a comma, so this temperament may be useful for modeling that. [[171edo]] makes for an excellent tuning.
 [[Subgroup]]: 2.3.5.7
@@ Line 2,744: / Line 2,744: @@
 Badness (Sintel): 1.46
-== Tridecafifths ==
-Named by [[Eliora]] in 2023, tridecafifths may be described as the {{nowrap| 89 & 200 }} temperament. It divides the [[3/2|perfect fifth]] into thirteen quartertones, so its [[ploidacot]] is 13-cot. [[289edo]] gives a highly recommendable tuning.
-[[Subgroup]]: 2.3.5.7
-[[Comma list]]: 32805/32768, {{monzo| -14 -1 -9 13 }}
-{{Mapping|legend=1| 1 1 7 6 | 0 13 -104 -71 }}
-: mapping generators: ~2, ~1323/1280
-[[Optimal tuning]]s:
-* [[WE]]: ~2 = 1200.1431{{c}}, ~1323/1280 = 53.9838{{c}}
-: [[error map]]: {{val| +0.143 -0.023 +0.375 -0.816 }}
-* [[CWE]]: ~2 = 1200.0000{{c}}, ~1323/1280 = 53.9764{{c}}
-: error map: {{val| 0.000 -0.261 -0.221 -0.421 }}
-{{Optimal ET sequence|legend=1| 89, 200, 289 }}
-[[Badness]] (Sintel): 10.9
-=== 11-limit ===
-Subgroup: 2.3.5.7.11
-Comma list: 441/440, 32805/32768, 55296000/55240493
-Mapping: {{mapping| 1 1 7 6 4 | 0 13 -104 -71 -12 }}
-Optimal tunings:
-* WE: ~2 = 1200.0311{{c}}, ~33/32 = 53.9766{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~33/32 = 53.9750{{c}}
-{{Optimal ET sequence|legend=0| 89, 200, 289 }}
-Badness (Sintel): 4.23
 == Subgroup extensions ==
-=== Maqamschismic (2.3.5.11) ===
-Proposed by [[Eufalesio]] in 2026, maqamschismic is equivalent to the no-7 [[cassandra]]. 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 [[maqam]]at (especially the Turkish version), as outlined by [[Ozan Yarman]]. 41edo and 53edo are simplest, but 94edo is more optimized. It is only slightly worse than the no-7 [[helenus]].
-Subgroup: 2.3.5.11
-Comma list: 2200/2187, 4125/4096
-Subgroup-val mapping: {{mapping| 1 0 15 -33 | 0 1 -8 23 }}
-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 }}
-Badness (Sintel): 1.34
-==== 2.3.5.11.13 subgroup ====
-Subgroup: 2.3.5.11.13
-Comma list: 325/324, 352/351, 4125/4096
-Subgroup-val mapping: {{mapping| 1 0 15 -33 -28 | 0 1 -8 23 20 }}
-Optimal tunings:
-* WE: ~2 = 1200.4565{{c}} ~3/2 = 702.3057{{c}}
-* CWE: 2 = 1200.0000{{c}}, ~3/2 = 702.0485{{c}}
-{{Optimal ET sequence|legend=0| 12e, …, 41, 53, 94, 147e }}
-Badness (Sintel): 0.862
 === 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]].
+Proposed by [[Eufalesio]] in 2026, tridecaschismic adds the [[325/324|marveltwin comma]] to the comma list, or equivalently, the [[tridecapyth comma]]. It benefits from a fifth that is just, or practically indistinguishable from just, like in 53edo. It is one of the lowest badness schismic extensions. It is also equivalent to the 2.3.5.13 [[restriction]] of 13-limit [[cassandra]].
 Subgroup: 2.3.5.13
@@ Line 2,887: / Line 2,821: @@
 : ''See also: [[No-elevens subgroup temperaments #Garibaldia]] and [[No-elevens subgroup temperaments #Pontia|#Pontia]]''
-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]].
+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]]. However, the dyadic tuning sensitivity of [[19/16]] suggests using tunings like [[65edo]] and [[77edo]] to optimize in favour of prime 19 (especially the minor triad ~16:19:24 which is equated with the Pythagorean minor triad), as [[171edo]] is already arguably undertempered for it despite being the optimal patent val.
 [[Subgroup]]: 2.3.5.19