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[[#Bischismic|Bischismic]] adds {{monzo| -69 40 0 2 }} and has a fifth generator with a half-octave period. [[#Salsa|Salsa]] adds [[parahemif comma|{{monzo| 15 -13 0 2 }}]] and has a hemififth generator. [[#Hemischis|Hemischis]] adds {{monzo| -34 25 0 -2 }} and has a hemitwelfth generator. [[Gamelismic clan #Guiron|Guiron]] adds [[1029/1024|{{monzo| -10 1 0 3 }}]], with an ~8/7 generator, three of which give the fifth. [[#Term|Term]] adds {{monzo| -94 54 0 3 }} with a 1/3-octave period. [[#Squirrel|Squirrel]], [[#Tertiaschis|tertiaschis]], and [[#Countertertiaschis|countertertiaschis]] each has a generator that is 1/3 of the fourth. [[#Quadrant|Quadrant]] adds {{monzo| -119 68 0 4 }} with a 1/4-octave period. [[#Kleischismic|Kleischismic]] adds {{monzo| 49 -38 0 4 }} with a half-octave period and also a bisect generator. [[#Sesquiquartififths|Sesquiquartififths]] adds {{monzo| -35 15 0 4 }} and slices the fifth in four.  
[[#Bischismic|Bischismic]] adds {{monzo| -69 40 0 2 }} and has a fifth generator with a half-octave period. [[#Salsa|Salsa]] adds [[parahemif comma|{{monzo| 15 -13 0 2 }}]] and has a hemififth generator. [[#Hemischis|Hemischis]] adds {{monzo| -34 25 0 -2 }} and has a hemitwelfth generator. [[Gamelismic clan #Guiron|Guiron]] adds [[1029/1024|{{monzo| -10 1 0 3 }}]], with an ~8/7 generator, three of which give the fifth. [[#Term|Term]] adds {{monzo| -94 54 0 3 }} with a 1/3-octave period. [[#Squirrel|Squirrel]], [[#Tertiaschis|tertiaschis]], and [[#Countertertiaschis|countertertiaschis]] each has a generator that is 1/3 of the fourth. [[#Quadrant|Quadrant]] adds {{monzo| -119 68 0 4 }} with a 1/4-octave period. [[#Kleischismic|Kleischismic]] adds {{monzo| 49 -38 0 4 }} with a half-octave period and also a bisect generator. [[#Sesquiquartififths|Sesquiquartififths]] adds {{monzo| -35 15 0 4 }} and slices the fifth in four.  


Temperaments involving larger splits include [[#Tsaharuk|tsaharuk]], [[#Quanharuk|quanharuk]], [[#Quintilipyth|quintilipyth]], [[#Quintaschis|quintaschis]], [[#Altinex|altinex]], [[Stearnsmic clan #Pogo|pogo]], [[#Sextilifourths|sextilifourths]], [[#Octant|octant]], [[#Nonant|nonant]], [[#Septant|septant]], [[#Septiquarschis|septiquarschis]], and [[#Tridecafifths|tridecafifths]]. Those split the schismic structure into five to thirteen parts.  
Temperaments involving larger splits include [[#Tsaharuk|tsaharuk]], [[#Quanharuk|quanharuk]], [[#Quintilipyth|quintilipyth]], [[#Quintaschis|quintaschis]], [[#Altinex|altinex]], [[Stearnsmic clan #Pogo|pogo]], [[#Sextilifourths|sextilifourths]], [[#Septant|septant]], [[#Octant|octant]], [[#Nonant|nonant]], [[#Septiquarschis|septiquarschis]], and [[#Tridecafifths|tridecafifths]]. Those split the schismic structure into five to thirteen parts.  


Temperaments discussed elsewhere include:
Temperaments discussed elsewhere include:
Line 47: Line 47:
* ''[[Pogo]]'' (+118098/117649) → [[Stearnsmic clan #Pogo|Stearnsmic clan]]
* ''[[Pogo]]'' (+118098/117649) → [[Stearnsmic clan #Pogo|Stearnsmic clan]]


Considered below are garibaldi, pontiac, grackle, schism, bischismic, kleischismic, salsa, hemischis, term, altinex, squirrel, tertiaschis, countertertiaschis, quadrant, sesquiquartififths, tsaharuk, quanharuk, quintilipyth, quintaschis, sextilifourths, octant, nonant, septant, septiquarschis, and tridecafifths.  
Considered below are garibaldi, pontiac, grackle, schism, bischismic, kleischismic, salsa, hemischis, term, altinex, squirrel, tertiaschis, countertertiaschis, quadrant, sesquiquartififths, tsaharuk, quanharuk, quintilipyth, quintaschis, sextilifourths, septant, octant, nonant, septiquarschis, and tridecafifths.  


The schismatic family boasts a variety of remarkable extensions to subgroups in high prime limits. These are listed at the bottom of this page, in [[#Subgroup extensions]].
The schismatic family boasts a variety of remarkable extensions to subgroups in high prime limits. These are listed at the bottom of this page, in [[#Subgroup extensions]].
Line 1,077: Line 1,077:


== Quasipyth ==
== Quasipyth ==
Quasipyth tempers out {{monzo|109 -67 0 -1}}, the [[nanisma]], as well as the [[catasyc comma]], 390625/387072. The 7/4 is found at −67 fifths, represented by the nonuple-diminished 13th.  
Named by [[Xenllium]] in 2026, quasipyth tempers out {{monzo| 109 -67 0 -1 }}, the [[nanisma]], as well as the [[catasyc comma]], 390625/387072. The 7/4 is found at −67 fifths, represented by the nonuple-diminished thirteenth.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 1,091: Line 1,091:
: error map: {{val| 0.0000 +0.0065 -2.0054 -0.2437 }}
: error map: {{val| 0.0000 +0.0065 -2.0054 -0.2437 }}


{{Optimal ET sequence|legend=1| 53, 147d, 200, 253, 306c }}
{{Optimal ET sequence|legend=1| 53, 147d, 200, 253, 306c, 559c }}


[[Badness]] (Sintel): 5.041
[[Badness]] (Sintel): 5.04


=== 11-limit ===
=== 11-limit ===
Line 1,106: Line 1,106:
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.9713{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.9713{{c}}


{{Optimal ET sequence|legend=0| 53, 147d, 200, 253, 306ce }}
{{Optimal ET sequence|legend=0| 53, 200, 253, 559ce }}


Badness (Sintel): 3.83
Badness (Sintel): 3.83
Line 1,121: Line 1,121:
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.9714{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.9714{{c}}


{{Optimal ET sequence|legend=0| 53, 147d, 200, 253, 306ce }}
{{Optimal ET sequence|legend=0| 53, 200, 253, 559ce }}


Badness (Sintel): 2.13
Badness (Sintel): 2.13
Line 1,628: Line 1,628:


=== Hemiterm ===
=== Hemiterm ===
The hemiterm temperament tempers out [[3025/3024]] (lehmerisma), and may be described as {{nowrap| 159 & 183 }}. Its ploidacot is triploid beta-dicot.  
The hemiterm temperament tempers out [[3025/3024]] (lehmerisma), and may be described as {{nowrap| 159 & 183 }}. Its ploidacot is triploid alpha-dicot.  


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 2,545: Line 2,545:
Badness (Sintel): 1.04
Badness (Sintel): 1.04


== Octant ==
== Septant ==
Octant may be described as the {{nowrap| 224 & 248 }} temperament. It has a period of 1/8 octave, and its [[ploidacot]] is octaploid monocot. In this temperament, [[12/11]], [[35/27]], and [[99/70]] are mapped to 1\8, 3\8, and 4\8 respectively.
Named by [[Xenllium]] in 2021, septant notably tempers out the [[akjaysma]] ({{monzo| 47 -7 -7 -7 }}) and may be described as the {{nowrap| 224 & 301 }} temperament. It has a period of 1/7 octave, and its [[ploidacot]] is heptaploid monocot.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 32805/32768, 2259436291848/2251875390625
[[Comma list]]: 32805/32768, 516560652/514714375


{{Mapping|legend=1| 8 0 120 -117 | 0 1 -8 11 }}
{{Mapping|legend=1| 7 0 105 -56 | 0 1 -8 7 }}
: mapping generators: ~42875/39366, ~3
: mapping generators: ~8575/7776, ~3


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~42875/39366 = 150.0048{{c}}, ~3/2 = 701.7356{{c}}
* [[WE]]: ~8575/7776 = 171.4303{{c}}, ~3/2 = 701.7091{{c}}
: [[error map]]: {{val| +0.039 -0.181 +0.071 +0.127 }}
: [[error map]]: {{val| +0.012 -0.234 +0.096 +0.265 }}
* [[CWE]]: ~42875/39366 = 150.0000{{c}}, ~3/2 = 701.7134{{c}}
* [[CWE]]: ~8575/7776 = 171.4286{{c}}, ~3/2 = 701.7022{{c}}
: error map: {{val| 0.000 -0.242 -0.021 +0.022 }}
: error map: {{val| 0.000 -0.253 +0.069 +0.232 }}


{{Optimal ET sequence|legend=1| 24, …, 224, 472, 696, 1168 }}
{{Optimal ET sequence|legend=1| 77, 147, 224, 301, 525, 826, 1351 }}


[[Badness]] (Sintel): 3.98
[[Badness]] (Sintel): 2.81


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 9801/9800, 32805/32768, 46656/46585
Comma list: 3025/3024, 24057/24010, 32805/32768


Mapping: {{mapping| 8 0 120 -117 15 | 0 1 -8 11 1 }}
Mapping: {{mapping| 7 0 105 -56 -120 | 0 1 -8 7 13 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~12/11 = 150.0010{{c}}, ~3/2 = 701.7177{{c}}
* WE: ~495/448 = 171.4334{{c}}, ~3/2 = 701.7387{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~3/2 = 701.7131{{c}}
* CWE: ~495/448 = 171.4286{{c}}, ~3/2 = 701.7198{{c}}


{{Optimal ET sequence|legend=0| 24, …, 224, 472, 696, 1168 }}
{{Optimal ET sequence|legend=0| 77, 147, 224, 301, 525 }}


Badness (Sintel): 1.48
Badness (Sintel): 1.46


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 729/728, 1575/1573, 2200/2197, 6656/6655
Comma list: 729/728, 1716/1715, 2200/2197, 3025/3024


Mapping: {{mapping| 8 0 120 -117 15 93 | 0 1 -8 11 1 -5 }}
Mapping: {{mapping| 7 0 105 -56 -120 37 | 0 1 -8 7 13 -1 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~12/11 = 149.9957{{c}}, ~3/2 = 701.7046{{c}}
* WE: ~495/448 = 171.4282{{c}}, ~3/2 = 701.7229{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~3/2 = 701.7247{{c}}
* CWE: ~495/448 = 171.4286{{c}}, ~3/2 = 701.7242{{c}}


{{Optimal ET sequence|legend=0| 24, 224, 472, 696 }}
{{Optimal ET sequence|legend=0| 77, 147, 224, 525, 1274f }}


Badness (Sintel): 1.26
Badness (Sintel): 1.02


== Nonant ==
== Octant ==
Named by [[Xenllium]] in 2023, nonant tempers out the [[septimal ennealimma]] ({{monzo| -11 -9 0 9 }}) and may be described as the {{nowrap| 36 & 171 }} temperament. It has a period of 1/9 octave, and its [[ploidacot]] is enneaploid monocot.  
Octant may be described as the {{nowrap| 224 & 248 }} temperament. It has a period of 1/8 octave, and its [[ploidacot]] is octaploid monocot. In this temperament, [[12/11]], [[35/27]], and [[99/70]] are mapped to 1\8, 3\8, and 4\8 respectively.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 32805/32768, 40353607/40310784
[[Comma list]]: 32805/32768, 2259436291848/2251875390625


{{Mapping|legend=1| 9 0 135 11 | 0 1 -8 1 }}
{{Mapping|legend=1| 8 0 120 -117 | 0 1 -8 11 }}
: mapping generators: ~2592/2401, ~3
: mapping generators: ~42875/39366, ~3


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2592/2401 = 133.3442{{c}}, ~3/2 = 701.8000{{c}}
* [[WE]]: ~42875/39366 = 150.0048{{c}}, ~3/2 = 701.7356{{c}}
: [[error map]]: {{val| +0.098 -0.057 -0.027 -0.141 }}
: [[error map]]: {{val| +0.039 -0.181 +0.071 +0.127 }}
* [[CWE]]: ~2592/2401 = 133.3333{{c}}, ~3/2 = 701.7384{{c}}
* [[CWE]]: ~42875/39366 = 150.0000{{c}}, ~3/2 = 701.7134{{c}}
: error map: {{val| 0.000 -0.217 -0.221 -0.421 }}
: error map: {{val| 0.000 -0.242 -0.021 +0.022 }}


{{Optimal ET sequence|legend=1| 36, 99c, 135, 171, 2772bd, 2943bdd, …, 5166bccddd, 5337bccddd }}
{{Optimal ET sequence|legend=1| 24, …, 224, 472, 696, 1168 }}


[[Badness]] (Sintel): 1.77
[[Badness]] (Sintel): 3.98


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 540/539, 32805/32768, 42875/42592
Comma list: 9801/9800, 32805/32768, 46656/46585


Mapping: {{mapping| 9 0 135 11 131 | 0 1 -8 1 -7 }}
Mapping: {{mapping| 8 0 120 -117 15 | 0 1 -8 11 1 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~242/225 = 133.3308{{c}}, ~3/2 = 701.8205{{c}}
* WE: ~12/11 = 150.0010{{c}}, ~3/2 = 701.7177{{c}}
* CWE: ~242/225 = 133.3333{{c}}, ~3/2 = 701.8351{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~3/2 = 701.7131{{c}}


{{Optimal ET sequence|legend=0| 36, 135, 171 }}
{{Optimal ET sequence|legend=0| 24, …, 224, 472, 696, 1168 }}


Badness (Sintel): 4.20
Badness (Sintel): 1.48


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 540/539, 729/728, 4096/4095, 16807/16731
Comma list: 729/728, 1575/1573, 2200/2197, 6656/6655


Mapping: {{mapping| 9 0 135 11 131 -38 | 0 1 -8 1 -7 5 }}
Mapping: {{mapping| 8 0 120 -117 15 93 | 0 1 -8 11 1 -5 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~242/225 = 133.3180{{c}}, ~3/2 = 701.6956{{c}}
* WE: ~12/11 = 149.9957{{c}}, ~3/2 = 701.7046{{c}}
* CWE: ~242/225 = 133.3333{{c}}, ~3/2 = 701.7800{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~3/2 = 701.7247{{c}}


{{Optimal ET sequence|legend=0| 36, 99cf, 135, 171 }}
{{Optimal ET sequence|legend=0| 24, 224, 472, 696 }}


Badness (Sintel): 3.15
Badness (Sintel): 1.26


== Septant ==
== Nonant ==
Named by [[Xenllium]] in 2021, septant notably tempers out the [[akjaysma]] ({{monzo| 47 -7 -7 -7 }}) and may be described as the {{nowrap| 224 & 301 }} temperament. It has a period of 1/7 octave, and its [[ploidacot]] is heptaploid monocot.  
Named by [[Xenllium]] in 2023, nonant tempers out the [[septimal ennealimma]] ({{monzo| -11 -9 0 9 }}) and may be described as the {{nowrap| 36 & 171 }} temperament. It has a period of 1/9 octave, and its [[ploidacot]] is enneaploid monocot.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 32805/32768, 516560652/514714375
[[Comma list]]: 32805/32768, 40353607/40310784


{{Mapping|legend=1| 7 0 105 -56 | 0 1 -8 7 }}
{{Mapping|legend=1| 9 0 135 11 | 0 1 -8 1 }}
: mapping generators: ~8575/7776, ~3
: mapping generators: ~2592/2401, ~3


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~8575/7776 = 171.4303{{c}}, ~3/2 = 701.7091{{c}}
* [[WE]]: ~2592/2401 = 133.3442{{c}}, ~3/2 = 701.8000{{c}}
: [[error map]]: {{val| +0.012 -0.234 +0.096 +0.265 }}
: [[error map]]: {{val| +0.098 -0.057 -0.027 -0.141 }}
* [[CWE]]: ~8575/7776 = 171.4286{{c}}, ~3/2 = 701.7022{{c}}
* [[CWE]]: ~2592/2401 = 133.3333{{c}}, ~3/2 = 701.7384{{c}}
: error map: {{val| 0.000 -0.253 +0.069 +0.232 }}
: error map: {{val| 0.000 -0.217 -0.221 -0.421 }}


{{Optimal ET sequence|legend=1| 77, 147, 224, 301, 525, 826, 1351 }}
{{Optimal ET sequence|legend=1| 36, 99c, 135, 171, 2772bd, 2943bdd, …, 5166bccddd, 5337bccddd }}


[[Badness]] (Sintel): 2.81
[[Badness]] (Sintel): 1.77


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 3025/3024, 24057/24010, 32805/32768
Comma list: 540/539, 32805/32768, 42875/42592


Mapping: {{mapping| 7 0 105 -56 -120 | 0 1 -8 7 13 }}
Mapping: {{mapping| 9 0 135 11 131 | 0 1 -8 1 -7 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~495/448 = 171.4334{{c}}, ~3/2 = 701.7387{{c}}
* WE: ~242/225 = 133.3308{{c}}, ~3/2 = 701.8205{{c}}
* CWE: ~495/448 = 171.4286{{c}}, ~3/2 = 701.7198{{c}}
* CWE: ~242/225 = 133.3333{{c}}, ~3/2 = 701.8351{{c}}


{{Optimal ET sequence|legend=0| 77, 147, 224, 301, 525 }}
{{Optimal ET sequence|legend=0| 36, 135, 171 }}


Badness (Sintel): 1.46
Badness (Sintel): 4.20


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 729/728, 1716/1715, 2200/2197, 3025/3024
Comma list: 540/539, 729/728, 4096/4095, 16807/16731


Mapping: {{mapping| 7 0 105 -56 -120 37 | 0 1 -8 7 13 -1 }}
Mapping: {{mapping| 9 0 135 11 131 -38 | 0 1 -8 1 -7 5 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~495/448 = 171.4282{{c}}, ~3/2 = 701.7229{{c}}
* WE: ~242/225 = 133.3180{{c}}, ~3/2 = 701.6956{{c}}
* CWE: ~495/448 = 171.4286{{c}}, ~3/2 = 701.7242{{c}}
* CWE: ~242/225 = 133.3333{{c}}, ~3/2 = 701.7800{{c}}


{{Optimal ET sequence|legend=0| 77, 147, 224, 525, 1274f }}
{{Optimal ET sequence|legend=0| 36, 99cf, 135, 171 }}


Badness (Sintel): 1.02
Badness (Sintel): 3.15


== Septiquarschis ==
== Septiquarschis ==
Line 2,814: Line 2,814:


=== Tridecaschismic (2.3.5.13) ===
=== 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
Subgroup: 2.3.5.13
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