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Schismatic family: Difference between revisions

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m unfortunately thats not how optimal ET sequences work. it's a defined thing by fumica's temperament evaluator. the standard ive been abiding by is that missing entries are to be discussed on a case-by-case basis
 
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{{interwiki
| en = Schismatic family
| de = Schismatische Temperaturen
| es =
| ja =
}}
{{Technical data page}}
{{Technical data page}}
The [[5-limit]] parent comma for the '''schismatic''' (or '''schismic''') '''family''' is the [[schisma]] of 32805/32768, which is the amount by which the [[Pythagorean comma]] exceeds the [[syntonic comma]] (81/80), or alternatively put, the difference between a [[5/4|just major third]] and a [[8192/6561|Pythagorean diminished fourth]].  
The [[5-limit]] parent comma for the '''schismatic''' (or '''schismic''') '''family''' is the [[schisma]] of 32805/32768, which is the amount by which the [[Pythagorean comma]] exceeds the [[syntonic comma]] (81/80), or alternatively put, the difference between a [[5/4|just major third]] and a [[8192/6561|Pythagorean diminished fourth]].  
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=== Overview to extensions ===
=== Overview to extensions ===
The second comma of the [[normal forms #Normal forms for commas|normal comma list]] defines which 7-limit family member we are looking at. [[#Garibaldi|Garibaldi]] adds [[garischisma|{{monzo| 25 -14 0 -1 }}]]. [[#Grackle|Grackle]] adds {{monzo| -44 26 0 1 }}. [[#Pontiac|Pontiac]] adds {{monzo| -59 39 0 -1 }}. [[#Schism|Schism]] adds [[64/63|{{monzo| 6 -2 0 -1 }}]]. Those all have a fifth as generator.  
The second comma of the [[normal forms #Normal forms for commas|normal comma list]] defines which 7-limit family member we are looking at. [[#Garibaldi|Garibaldi]] adds [[garischisma|{{monzo| 25 -14 0 -1 }}]], [[#Grackle|grackle]] adds {{monzo| -44 26 0 1 }}, [[#Pontiac|pontiac]] adds {{monzo| -59 39 0 -1 }}, and [[#Schism|schism]] adds [[64/63|{{monzo| 6 -2 0 -1 }}]]. Those all have a fifth as generator.  


[[#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:
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* ''[[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]].
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Garibaldi tempers out the [[garischisma]], equating the [[64/63|septimal comma]] with both the [[syntonic comma]] and the [[Pythagorean comma]]. The 7/4 is found at -14 fifths, represented by the double-diminished octave (C–C𝄫), or down-minor seventh (C-vB♭) with the down-arrow representing the comma step. It necessitates a sharper fifth than pure. Its [[S-expression]]-based comma list is {[[5120/5103|S8/S9]], [[225/224|S15]]}.  
Garibaldi tempers out the [[garischisma]], equating the [[64/63|septimal comma]] with both the [[syntonic comma]] and the [[Pythagorean comma]]. The 7/4 is found at -14 fifths, represented by the double-diminished octave (C–C𝄫), or down-minor seventh (C-vB♭) with the down-arrow representing the comma step. It necessitates a sharper fifth than pure. Its [[S-expression]]-based comma list is {[[5120/5103|S8/S9]], [[225/224|S15]]}.  
[[147edo]] (from 94+53) is a [[patent val]] tuning with 5 and 7 very close to equally out-of-tune in opposite directions s.t only 7/5 and 10/7 are inconsistent in the 9-odd-limit, so might be considered for (EG) a 41-note subset ([[12L 29s]]).


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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* [[WE]]: ~2 = 1200.0989{{c}}, ~3/2 = 701.8145{{c}}
* [[WE]]: ~2 = 1200.0989{{c}}, ~3/2 = 701.8145{{c}}
: [[error map]]: {{val| +0.099 -0.042 -0.138 -0.038 }}
: [[error map]]: {{val| +0.099 -0.042 -0.138 -0.038 }}
* [[CWE]]: ~2 = 1200.0000{{c]}, ~3/2 = 701.7579{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.7579{{c}}
: error map: {{val| 0.000 -0.197 -0.377 -0.268 }}
: error map: {{val| 0.000 -0.197 -0.377 -0.268 }}


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* [[WE]]: ~2 = 1199.7974{{c}}, ~3/2 = 701.1210{{c}}
* [[WE]]: ~2 = 1199.7974{{c}}, ~3/2 = 701.1210{{c}}
: [[error map]]: {{val| -0.203 -1.037 +3.300 -1.618 }}
: [[error map]]: {{val| -0.203 -1.037 +3.300 -1.618 }}
* [[CWE]]: ~2 = 1200.0000{{c]}, ~3/2 = 701.2465{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.2465{{c}}
: error map: {{val| 0.000 -0.709 +3.715 -1.234 }}
: error map: {{val| 0.000 -0.709 +3.715 -1.234 }}


Line 1,075: Line 1,071:


Badness (Sintel): 2.01
Badness (Sintel): 2.01
== Quasipyth ==
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
[[Comma list]]: 32805/32768, 390625/387072
{{Mapping|legend=1| 1 0 15 109 | 0 1 -8 -67 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.2569{{c}}, ~3/2 = 702.1149{{c}}
: [[error map]]: {{val| +0.2569 +0.4168 -1.4342 +0.2685 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9615{{c}}
: error map: {{val| 0.0000 +0.0065 -2.0054 -0.2437 }}
{{Optimal ET sequence|legend=1| 53, 147d, 200, 253, 306c, 559c }}
[[Badness]] (Sintel): 5.04
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 385/384, 19712/19683, 78125/77616
Mapping: {{mapping| 1 0 15 109 -117 | 0 1 -8 -67 76 }}
Optimal tunings:
* WE: ~2 = 1200.3283{{c}}, ~3/2 = 702.1636{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.9713{{c}}
{{Optimal ET sequence|legend=0| 53, 200, 253, 559ce }}
Badness (Sintel): 3.83
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 325/324, 385/384, 2200/2197, 19712/19683
Mapping: {{mapping| 1 0 15 109 -117 -28 | 0 1 -8 -67 76 20 }}
Optimal tunings:
* WE: ~2 = 1200.3229{{c}}, ~3/2 = 702.1603{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.9714{{c}}
{{Optimal ET sequence|legend=0| 53, 200, 253, 559ce }}
Badness (Sintel): 2.13


== Schism ==
== Schism ==
Line 1,318: Line 1,363:


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.9362{c}}, ~11/9 = 351.0061{{c}}
* WE: ~2 = 1199.9362{{c}}, ~11/9 = 351.0061{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 351.0247{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 351.0247{{c}}


Line 1,426: Line 1,471:


== Term ==
== 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
[[Subgroup]]: 2.3.5.7
Line 1,579: Line 1,624:


=== 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 1,726: Line 1,771:


== Tertiaschis ==
== Tertiaschis ==
Named by [[Xenllium]] in 2021, tertiaschis may be described as {{nowrap| 94 & 159 }}. It has a [[~]][[11/10]] generator, sharing the same 2.3.5.11 subgroup with [[#Squirrel|squirrel]], but tempers out 1071875/1062882 for prime 7.  
Named by [[Xenllium]] in 2021, tertiaschis may be described as {{nowrap| 94 & 159 }}. It has a [[~]][[11/10]] generator, sharing the same 2.3.5.11 subgroup with [[#Squirrel|squirrel]], but tempers out 1071875/1062882 for prime 7. See also [[daemotertiaschis]] which is made using every other generator of tertiaschis.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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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 2,695: Line 2,740:


Badness (Sintel): 1.46
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 ==
== 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) ===
=== 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
Line 2,782: Line 2,761:


==== 2.3.5.13.19 subgroup ====
==== 2.3.5.13.19 subgroup ====
Subgroup: 2.3.5.13
Subgroup: 2.3.5.13.19


Comma list: 325/324, 361/360, 513/512
Comma list: 325/324, 361/360, 513/512
Line 2,838: Line 2,817:
: ''See also: [[No-elevens subgroup temperaments #Garibaldia]] and [[No-elevens subgroup temperaments #Pontia|#Pontia]]''
: ''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
[[Subgroup]]: 2.3.5.19
Line 2,950: Line 2,929:
Badness (Sintel): 1.17
Badness (Sintel): 1.17


[[Category:Schismatic family| ]] <!-- main article -->
[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Schismatic family| ]] <!-- main article -->
[[Category:Catalogs of rank-2 temperaments]]
[[Category:Rank 2]]
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