Xenharmonic Wiki

Schismatic family: Difference between revisions

← Older edit
VisualWikitext
→Maqamschismic (2.3.5.13): 325/234 → 325/324 this might be a typo
Tags: Mobile edit Mobile web edit Advanced mobile edit
Godtone (talk | contribs)
autopatrolled, emailconfirmed
2,389 edits
m →Tridecaschismic (2.3.5.13): "thus also" feels weird to use if theres no explanation for how it follows
 
(14 intermediate revisions by 8 users not shown)
Line 37: Line 37:


=== 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:
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 552: Line 552:
* [[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 }}


Line 945: Line 945:
* [[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 1,075:


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 1,367:


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 1,579: 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 2,496: 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 2,732: Line 2,781:


== 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]].


=== Maqamschismic (2.3.5.13) ===
Subgroup: 2.3.5.11
Subgroup: 2.3.5.13
 
Comma list: 2200/2187, 4125/4096


Comma list: 32805/32768, 325/324{{See also|2.3.5.13 subgroup}}Mapping: {{mapping| 1 1 7 -8| 0 1 -8 20}}
Subgroup-val mapping: {{mapping| 1 0 15 -33 | 0 1 -8 23 }}


Optimal tunings:
Optimal tunings:
* WE: ~2 = 1200.5458{{c}} ~3/2 = 702.4021{{c}}
* CWE: 2 = 1200.0000{{c}}, ~3/2 = 702.0906{{c}}


* WE: ~2 = 1200.333{{c}} ~3/2 = 702.109{{c}}
{{Optimal ET sequence|legend=0| 12e, …, 41, 53, 94, 147e, 241ce, 335ce }}
* CWE: 2 = 1200{{c}}, ~3/2 = 701.919{{c}}


{{Optimal ET sequence|legend=0|12, 41, 53, 94, 65f, 147, 200}}
Badness (Sintel): 1.34
 
Badness (Sintel): 0.582


==== 2.3.5.11.13 subgroup ====
==== 2.3.5.11.13 subgroup ====
Subgroup: 2.3.5.11.13
Subgroup: 2.3.5.11.13


Comma list: 4125/4096, 325/324, 352/351
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, 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
 
Comma list: 325/324, 32805/32768


Mapping: {{mapping| 1 1 7 -10 -8| 0 1 -8 23 20}}
Subgroup-val mapping: {{mapping| 1 0 15 -28 | 0 1 -8 20 }}


Optimal tunings:
Optimal tunings:
* WE: ~2 = 1200.3326{{c}} ~3/2 = 702.1092{{c}}
* CWE: 2 = 1200.0000{{c}}, ~3/2 = 701.9189{{c}}
{{Optimal ET sequence|legend=0| 12, …, 41, 53, 412cf, 465cf, …, 783ccff, 836ccfff }}
Badness (Sintel): 0.582
==== 2.3.5.13.19 subgroup ====
Subgroup: 2.3.5.13.19
Comma list: 325/324, 361/360, 513/512
Subgroup-val mapping: {{mapping| 1 0 15 -28 9 | 0 1 -8 20 -3 }}


* WE: ~2 = 1200.457{{c}} ~3/2 = 702.306{{c}}
Optimal tunings:
* CWE: 2 = 1200{{c}}, ~3/2 = 702.048{{c}}
* WE: ~2 = 1200.4236{{c}}, ~3/2 = 702.1510{{c}}
* CWE: 2 = 1200.0000{{c}}, ~3/2 = 701.9064{{c}}


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


Badness (Sintel): 0.862
Badness (Sintel): 0.354


=== Photia (2.3.5.17) ===
=== Photia (2.3.5.17) ===
Line 2,805: Line 2,887:
: ''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]]''


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]].  


[[Subgroup]]: 2.3.5.19
[[Subgroup]]: 2.3.5.19
Retrieved from "https://en.xen.wiki/w/Schismatic_family"