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Aberschismic temperaments: Difference between revisions

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{{Technical data page}}
{{Technical data page}}
This is a collection of [[rank-2 temperament|rank-2]] [[regular temperament|temperaments]] [[tempering out]] the [[hemifamity comma]] ({{monzo|legend=1| 10 -6 1 -1 }}, [[ratio]]: 5120/5103). These temperaments divide an exact or approximate septimal quartertone, [[36/35]] into two equal steps, each representing [[81/80]][[~]][[64/63]], the syntonic comma or the septimal comma. Therefore, classical and septimal intervals are found by the same [[chain of fifths]] inflected by the syntonic~septimal comma to the opposite sides. In addition we may identify [[10/7]] by the augmented fourth and [[50/49]] by the [[Pythagorean comma]].  
This is a collection of [[rank-2 temperament|rank-2]] '''aberschismic temperaments''', which [[tempering out|temper out]] the [[aberschisma]] ({{monzo|legend=1| 10 -6 1 -1 }}, [[ratio]]: 5120/5103). These temperaments divide an exact or approximate septimal quartertone, [[36/35]] into two equal steps, each representing [[81/80]][[~]][[64/63]], the syntonic comma or the septimal comma. Therefore, classical and septimal intervals are found by the same [[chain of fifths]] inflected by the syntonic~septimal comma to the opposite sides. In addition we may identify [[10/7]] by the augmented fourth and [[50/49]] by the [[Pythagorean comma]].  


Temperaments belonging to this category and generated by the fifth are dominant, garibaldi, kwai, and leapday. Dominant has 5/4 mapped to M3. Garibaldi has 5/4 mapped to d4. Kwai has 5/4 mapped to 4A7. Undecental has 5/4 mapped to 5d7. Leapday has 5/4 mapped to 3A1.  
Temperaments belonging to this category and generated by the fifth are dominant, garibaldi, kwai, undecental, and leapday. Dominant has 5/4 mapped to M3. Garibaldi has 5/4 mapped to d4. Kwai has 5/4 mapped to 4A7. Undecental has 5/4 mapped to 5d7. Leapday has 5/4 mapped to 3A1.  


Diaschismic is generated by the fifth with a semi-octave period. Hemififths has the fifth sliced into two and 5/4 mapped to the hemififth + Pyth. comma. Hemidromeda has the fourth sliced into two and 5/4 mapped to the hemifourth + 3d4. Rodan has the fifth sliced into three as does slendric. Alphatrimot has the twelfth sliced into three as does alphatricot. Monkey has the fifth sliced into four as does tetracot. Buzzard has the twelfth sliced into four as does vulture. Misty is generated by the fifth with a 1/3-octave period. Supers has the fifth sliced into three with a semi-octave period. Undim is generated by the fifth with a 1/4-octave period. Quinticosiennic and quintakwai have the fourth sliced into five. Amity has the eleventh sliced into five. Countercata has the twelfth sliced into six as does hanson. Warrior has the 6th harmonic sliced into seven as does sensi. Finally, alphaquarter has the fourth sliced into nine as does escapade.  
Diaschismic is generated by the fifth with a semi-octave period. Hemififths has the fifth sliced into two and 5/4 mapped to the hemififth + Pyth. comma. Hemidromeda has the fourth sliced into two and 5/4 mapped to the hemifourth + 3d4. Rodan has the fifth sliced into three as does slendric. Alphatrimot has the twelfth sliced into three as does alphatricot. Monkey has the fifth sliced into four as does tetracot. Buzzard has the twelfth sliced into four as does vulture. Misty is generated by the fifth with a 1/3-octave period. Supers has the fifth sliced into three with a semi-octave period. Undim is generated by the fifth with a 1/4-octave period. Quinticosiennic and quintakwai have the fourth sliced into five. Amity has the eleventh sliced into five. Countercata has the twelfth sliced into six as does hanson. Warrior has the 6th harmonic sliced into seven as does sensi. Finally, alphaquarter has the fourth sliced into nine as does escapade.  
Line 9: Line 9:
* [[Dominant (temperament)|Dominant]] (+36/35) → [[Meantone family #Dominant|Meantone family]]
* [[Dominant (temperament)|Dominant]] (+36/35) → [[Meantone family #Dominant|Meantone family]]
* [[Garibaldi]] (+225/224) → [[Schismatic family #Garibaldi|Schismatic family]]
* [[Garibaldi]] (+225/224) → [[Schismatic family #Garibaldi|Schismatic family]]
* ''[[Kwai]]'' (+16875/16807) → [[Mirkwai clan #Kwai|Mirkwai clan]]
* [[Leapday]] (+686/675) → [[Sengic temperaments #Leapday|Sengic temperaments]]
* [[Diaschismic]] (+126/125) → [[Diaschismic family #Septimal diaschismic|Diaschismic family]]
* [[Diaschismic]] (+126/125) → [[Diaschismic family #Septimal diaschismic|Diaschismic family]]
* [[Hemififths]] (+2401/2400) → [[Breedsmic temperaments #Hemififths|Breedsmic temperaments]]
* [[Hemififths]] (+2401/2400) → [[Breedsmic temperaments #Hemififths|Breedsmic temperaments]]
* [[Rodan]] (+245/243) → [[Gamelismic clan #Rodan|Gamelismic clan]]
* [[Rodan]] (+245/243) → [[Gamelismic clan #Rodan|Gamelismic clan]]
* ''[[Alphatrimot]]'' (+2430/2401) → [[Alphatricot family #Alphatrimot|Alphatricot family]]
* ''[[Alphatrimot]]'' (+2430/2401) → [[Alphatricot family #Alphatrimot|Alphatricot family]]
* ''[[Monkey]]'' (+875/864) → [[Tetracot family #Monkey|Tetracot family]]
* [[Misty]] (+3136/3125) → [[Misty family #Misty|Misty family]]
* [[Monkey]] (+875/864) → [[Tetracot family #Monkey|Tetracot family]]
* [[Buzzard]] (+1728/1715) → [[Buzzardsmic clan #Buzzard|Buzzardsmic clan]]
* [[Buzzard]] (+1728/1715) → [[Buzzardsmic clan #Buzzard|Buzzardsmic clan]]
* [[Misty]] (+3136/3125) → [[Misty family #Misty|Misty family]]
* ''[[Supers]]'' (+118098/117649) → [[Stearnsmic clan #Supers|Stearnsmic clan]]
* ''[[Undim]]'' (+390625/388962) → [[Undim family #Septimal undim|Undim family]]
* ''[[Undim]]'' (+390625/388962) → [[Undim family #Septimal undim|Undim family]]
* ''[[Quinticosiennic]]'' (+395136/390625) → [[Quintaleap family #Quinticosiennic|Quintaleap family]]
* ''[[Quinticosiennic]]'' (+395136/390625) → [[Quintaleap family #Quinticosiennic|Quintaleap family]]
Line 23: Line 22:
* [[Amity]] (+4375/4374) → [[Amity family #Septimal amity|Amity family]]
* [[Amity]] (+4375/4374) → [[Amity family #Septimal amity|Amity family]]
* ''[[Countercata]]'' (+15625/15552) → [[Kleismic family #Countercata|Kleismic family]]
* ''[[Countercata]]'' (+15625/15552) → [[Kleismic family #Countercata|Kleismic family]]
* ''[[Abergravity]]'' (+177147/175000) → [[Gravity family #Abergravity|Gravity family]]
* ''[[Supers]]'' (+118098/117649) → [[Stearnsmic clan #Supers|Stearnsmic clan]]
* ''[[Warrior]]'' (+78732/78125) → [[Sensipent family #Warrior|Sensipent family]]
* ''[[Warrior]]'' (+78732/78125) → [[Sensipent family #Warrior|Sensipent family]]
* ''[[Alphaquarter]]'' (+29360128/29296875) → [[Escapade family #Alphaquarter|Escapade family]]
* ''[[Alphaquarter]]'' (+29360128/29296875) → [[Escapade family #Alphaquarter|Escapade family]]


Considered below are septiquarter, ketchup, undecental, leapday, mystery, hemidromeda, countriton, artoneutral, quanic and jorgensen, in the order of increasing [[TE logflat badness]].  
Considered below are septiquarter, kwai, ketchup, undecental, mystery, hemidromeda, countriton, artoneutral, quanic and jorgensen, in the order of increasing [[TE logflat badness]].  


== Septiquarter ==
== Septiquarter ==
Line 53: Line 54:
Comma list: 5120/5103, 9801/9800, 14641/14580
Comma list: 5120/5103, 9801/9800, 14641/14580


Mapping: {{mapping| 2 -8 -56 12 -25 | 0 7 38 -4 20 }}
{{Mapping|legend=0| 2 -8 -56 12 -25 | 0 7 38 -4 20 }}


Optimal tunings:  
Optimal tunings:  
Line 68: Line 69:
Comma list: 352/351, 847/845, 1716/1715, 14641/14580
Comma list: 352/351, 847/845, 1716/1715, 14641/14580


Mapping: {{mapping| 2 -8 -56 12 -25 9 | 0 7 38 -4 20 -1 }}
{{Mapping|legend=0| 2 -8 -56 12 -25 9 | 0 7 38 -4 20 -1 }}


Optimal tunings:  
Optimal tunings:  
Line 78: Line 79:
Badness (Sintel): 1.44
Badness (Sintel): 1.44


== Ketchup ==
== Kwai ==
Ketchup may be described as the {{nowrap| 46 & 94 }} temperament. It has a semi-octave period and a generator for a syntonic~septimal comma, four of which plus a period gives the perfect fifth; its ploidacot is diploid gamma-tetracot. [[140edo]] is an obvious tuning for this temperament.  
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Kwai]].''
 
Named by [[Gene Ward Smith]] in 2004 for its "bridgeability"<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_10766.html Yahoo! Tuning Group | ''Kwai'']</ref>, kwai is generated by a [[3/2|perfect fifth]], and can be described as {{nowrap| 41 & 70 }}.  


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


[[Comma list]]: 5120/5103, 1071875/1062882
[[Comma list]]: 5120/5103, 16875/16807


{{Mapping|legend=1| 2 3 4 6 | 0 4 15 -9 }}
{{Mapping|legend=1| 1 0 -50 -40 | 0 1 33 27 }}
: mapping generators: ~1225/864, ~64/63
: mapping generators: ~2, ~3


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~1225/864 = 599.9685{{c}}, ~64/63 = 25.7181{{c}}
* [[WE]]: ~2 = 1199.7337{{c}}, ~3/2 = 702.4600{{c}}
: [[error map]]: {{val| -0.063 +0.823 -0.668 -0.478 }}
: [[error map]]: {{val| -0.266 +0.239 -0.607 +1.055 }}
* [[CWE]]: ~1225/864 = 600.0000{{c}}, ~64/63 = 25.7181{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.6085{{c}}
: error map: {{val| 0.000 +0.917 -0.543 -0.288 }}
: error map: {{val| 0.000 +0.653 -0.234 +1.603 }}


{{Optimal ET sequence|legend=1| 46, 94, 140 }}
{{Optimal ET sequence|legend=1| 41, 111, 152, 345, 497d }}


[[Badness]] (Sintel): 2.14
[[Badness]] (Sintel): 1.38


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


Comma list: 385/384, 1331/1323, 2200/2187
Comma list: 540/539, 1375/1372, 5120/5103


Mapping: {{mapping| 2 3 4 6 7 | 0 4 15 -9 -2 }}
{{Mapping|legend=0| 1 0 -50 -40 32 | 0 1 33 27 -18 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~99/70 = 600.0678{{c}}, ~64/63 = 25.6963{{c}}
* WE: ~2 = 1199.6672{{c}}, ~3/2 = 702.4282{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~64/63 = 25.6956{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6189{{c}}


{{Optimal ET sequence|legend=0| 46, 94, 140 }}
{{Optimal ET sequence|legend=0| 41, 111, 152, 497de, 649dde }}


Badness (Sintel): 1.31
Badness (Sintel): 0.867


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


Comma list: 325/324, 352/351, 385/384, 1331/1323
Comma list: 352/351, 540/539, 729/728, 1375/1372


Mapping: {{mapping| 2 3 4 6 7 8 | 0 4 15 -9 -2 -14 }}
{{Mapping|legend=0| 1 0 -50 -40 32 27 | 0 1 33 27 -18 -21 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~99/70 = 600.0612{{c}}, ~66/65 = 25.7000{{c}}
* WE: ~2 = 1199.4772{{c}}, ~3/2 = 702.3379{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~66/65 = 25.6978{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6409{{c}}


{{Optimal ET sequence|legend=0| 46, 94, 140 }}
{{Optimal ET sequence|legend=0| 41, 111, 152f, 415dff }}


Badness (Sintel): 1.03
Badness (Sintel): 1.01


=== 17-limit ===
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 289/288, 325/324, 352/351, 385/384, 561/560
Comma list: 256/255, 352/351, 540/539, 715/714, 1089/1088


Mapping: {{mapping| 2 3 4 6 7 8 8 | 0 4 15 -9 -2 -14 4 }}
{{Mapping|legend=0| 1 0 -50 -40 32 27 58 | 0 1 33 27 -18 -21 -34 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~17/12 = 600.0896{{c}}, ~66/65 = 25.7048{{c}}
* WE: ~2 = 1199.3537{{c}}, ~3/2 = 702.2850{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~66/65 = 25.7017{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6589{{c}}


{{Optimal ET sequence|legend=0| 46, 94, 140 }}
{{Optimal ET sequence|legend=0| 41, 70, 111, 152fg, 263dfg }}


Badness (Sintel): 0.845
Badness (Sintel): 1.12


=== 19-limit ===
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 190/189, 209/208, 289/288, 352/351, 385/384, 561/560
Comma list: 256/255, 352/351, 400/399, 456/455, 715/714, 847/845


Mapping: {{mapping| 2 3 4 6 7 8 8 9 | 0 4 15 -9 -2 -14 4 -12 }}
{{Mapping|legend=0| 1 0 -50 -40 32 27 58 -56 | 0 1 33 27 -18 -21 -34 38 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~17/12 = 600.1639{{c}}, ~66/65 = 25.6669{{c}}
* WE: ~2 = 1199.3401{{c}}, ~3/2 = 702.2705{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~66/65 = 25.6597{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6548{{c}}


{{Optimal ET sequence|legend=0| 46, 94, 140h }}
{{Optimal ET sequence|legend=0| 41, 70h, 111, 152fg, 263dfgh }}


Badness (Sintel): 1.11
Badness (Sintel): 1.03


=== 23-limit ===
==== Hemikwai ====
Subgroup: 2.3.5.7.11.13.17.19.23
Subgroup: 2.3.5.7.11.13


Comma list: 190/189, 209/208, 253/252, 289/288, 323/322, 352/351, 385/384
Comma list: 540/539, 676/675, 1375/1372, 5120/5103


Mapping: {{mapping| 2 3 4 6 7 8 8 9 9 | 0 4 15 -9 -2 -14 4 -12 1 }}
{{Mapping|legend=0| 1 0 -50 -40 32 -51 | 0 2 66 54 -36 69 }}
: mapping generators: ~2, ~26/15


Optimal tunings:  
Optimal tunings:  
* WE: ~17/12 = 600.1777{{c}}, ~66/65 = 25.6682{{c}}
* WE: ~2 = 1199.6968{{c}}, ~26/15 = 951.0740{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~66/65 = 25.6605{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.3123{{c}}


{{Optimal ET sequence|legend=0| 46, 94, 140h }}
{{Optimal ET sequence|legend=0| 82, 111, 193, 304d }}


Badness (Sintel): 1.00
Badness (Sintel): 1.82


== Undecental ==
===== 17-limit =====
Undecental adds the triwellisma to the comma list and may be described as the {{nowrap| 29 & 70 }} temperament. 5/4 is mapped to the quintuple-diminished seventh or equivalently the perfect fourth minus three [[diesis (scale theory)|dieses]]. [[99edo|58\99]] is an almost perfect generator, just as the name suggests. Another interesting tuning choice is the argent fifth, {{nowrap| 2<sup>(2 - sqrt (2))</sup> }}.
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 442/441, 540/539, 676/675, 715/714, 5120/5103
 
{{Mapping|legend=0| 1 0 -50 -40 32 -51 -30 | 0 2 66 54 -36 69 43 }}


[[Subgroup]]: 2.3.5.7
Optimal tunings:  
* WE: ~2 = 1199.6861{{c}}, ~26/15 = 951.0654{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.3120{{c}}


[[Comma list]]: 5120/5103, 235298/234375
{{Optimal ET sequence|legend=0| 82, 111, 193, 304d }}


{{Mapping|legend=1| 1 0 61 71 | 0 1 -37 -43 }}
Badness (Sintel): 1.31
: mapping generators: ~2, ~3


[[Optimal tuning]]s:
===== 19-limit =====
* [[WE]]: ~2 = 1199.6543{{c}}, ~3/2 = 702.8370{{c}}
Subgroup: 2.3.5.7.11.13.17.19
: [[error map]]: {{val| -0.346 +0.536 +0.423 -0.494 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.0465{{c}}
: error map: {{val| 0.000 +1.092 +0.966 +0.175 }}


{{Optimal ET sequence|legend=1| 29, 70, 99, 722bc, 821bc, 920bc, 1019bc }}
Comma list: 400/399, 442/441, 540/539, 676/675, 715/714, 1445/1444


[[Badness]] (Sintel): 2.39
{{Mapping|legend=0| 1 0 -50 -40 32 -51 -30 -56 | 0 2 66 54 -36 69 43 76 }}


== Leapday ==
Optimal tunings:
{{Main| Leapday }}
* WE: ~2 = 1199.6718{{c}}, ~26/15 = 951.0526{{c}}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Leapday]].''
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.3103{{c}}


Leapday tempers out [[686/675]], the senga, in addition to the hemifamity comma, and may be described as the {{nowrap| 29 & 46 }} temperament. It extends [[leapfrog]], such that [[7/4]] is found by 15 generators up, as a double-augmented fifth (a major sixth and a diesis). 5/4 is found by a tritone above that, as a triple-augmented unison (a minor third and two dieses). [[46edo]] itself is an excellent tuning for this.
{{Optimal ET sequence|legend=0| 82, 111, 193, 304dh }}


Leapday is more notable in the higher limits than the lower, as it nails the 13-limit pretty well from identifying [[14/11]] by a major third and [[13/11]] by a minor third, tempering out not only [[352/351]] and [[364/363]] but [[91/90]], [[121/120]], [[169/168]] and [[196/195]]. It can be further extended to include the [[17/1|17th]] and [[23/1|23rd]] [[harmonic]]s. Adding 17 would fix the valid diamond monotone tuning to 46edo, however.  
Badness (Sintel): 1.16


Leapday has an alternative extension called [[porwell temperaments #Polypyth|polypyth]], which tempers out the same 5-limit comma as leapday, but with the porwell ([[6144/6125]]) rather than the hemifamity comma tempered out.
== Ketchup ==
Ketchup may be described as the {{nowrap| 46 & 94 }} temperament. It has a semi-octave period and a generator for a syntonic~septimal comma, four of which plus a period gives the perfect fifth; its [[ploidacot]] is diploid gamma-tetracot. [[140edo]] is an obvious tuning for this temperament.  


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


[[Comma list]]: 686/675, 5120/5103
[[Comma list]]: 5120/5103, 1071875/1062882


{{Mapping|legend=1| 1 0 -31 -21 | 0 1 21 15 }}
{{Mapping|legend=1| 2 3 4 6 | 0 4 15 -9 }}
: mapping generators: ~2, ~3
: mapping generators: ~1225/864, ~64/63


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.7167{{c}}, ~3/2 = 704.0971{{c}}
* [[WE]]: ~1225/864 = 599.9685{{c}}, ~64/63 = 25.7181{{c}}
: [[error map]]: {{val| -0.283 +1.859 +2.559 -5.669 }}
: [[error map]]: {{val| -0.063 +0.823 -0.668 -0.478 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.2504{{c}}
* [[CWE]]: ~1225/864 = 600.0000{{c}}, ~64/63 = 25.7181{{c}}
: error map: {{val| 0.000 +2.295 +2.945 -5.070 }}
: error map: {{val| 0.000 +0.917 -0.543 -0.288 }}


{{Optimal ET sequence|legend=1| 17c, 29, 46 }}
{{Optimal ET sequence|legend=1| 46, 94, 140 }}


[[Badness]] (Sintel): 2.43
[[Badness]] (Sintel): 2.14


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


Comma list: 121/120, 441/440, 686/675
Comma list: 385/384, 1331/1323, 2200/2187


Mapping: {{mapping| 1 0 -31 -21 -14 | 0 1 21 15 11 }}
{{Mapping|legend=0| 2 3 4 6 7 | 0 4 15 -9 -2 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.0731{{c}}, ~3/2 = 704.2933{{c}}
* WE: ~99/70 = 600.0678{{c}}, ~64/63 = 25.6963{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2538{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~64/63 = 25.6956{{c}}


{{Optimal ET sequence|legend=0| 17c, 29, 46 }}
{{Optimal ET sequence|legend=0| 46, 94, 140 }}


Badness (Sintel): 1.28
Badness (Sintel): 1.31


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


Comma list: 91/90, 121/120, 169/168, 352/351
Comma list: 325/324, 352/351, 385/384, 1331/1323


Mapping: {{mapping| 1 0 -31 -21 -14 -9 | 0 1 21 15 11 8 }}
{{Mapping|legend=0| 2 3 4 6 7 8 | 0 4 15 -9 -2 -14 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.4758{{c}}, ~3/2 = 704.4930{{c}}
* WE: ~99/70 = 600.0612{{c}}, ~66/65 = 25.7000{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2346{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~66/65 = 25.6978{{c}}


{{Optimal ET sequence|legend=0| 17c, 29, 46, 121def }}
{{Optimal ET sequence|legend=0| 46, 94, 140 }}


Badness (Sintel): 1.02
Badness (Sintel): 1.03


=== 17-limit ===
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 91/90, 121/120, 136/135, 154/153, 169/168
Comma list: 289/288, 325/324, 352/351, 385/384, 442/441


Mapping: {{mapping| 1 0 -31 -21 -14 -9 -34 | 0 1 21 15 11 8 24 }}
{{Mapping|legend=0| 2 3 4 6 7 8 8 | 0 4 15 -9 -2 -14 4 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.4818{{c}}, ~3/2 = 704.5121{{c}}
* WE: ~17/12 = 600.0896{{c}}, ~66/65 = 25.7048{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2507{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~66/65 = 25.7017{{c}}


{{Optimal ET sequence|legend=0| 17cg, 29g, 46, 121defg }}
{{Optimal ET sequence|legend=0| 46, 94, 140 }}


Badness (Sintel): 0.910
Badness (Sintel): 0.845


==== 19-limit ====
=== 2.3.5.7.11.13.17.23 subgroup ===
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17.23


Comma list: 91/90, 121/120, 133/132, 136/135, 154/153, 169/168
Comma list: 253/252, 289/288, 325/324, 352/351, 385/384, 391/390


Mapping: {{mapping| 1 0 -31 -21 -14 -9 -34 9 | 0 1 21 15 11 8 24 -3 }}
{{Mapping|legend=0| 2 3 4 6 7 8 8 9 | 0 4 15 -9 -2 -14 4 1 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1201.0192{{c}}, ~3/2 = 704.7333{{c}}
* WE: ~17/12 = 600.1139{{c}}, ~66/65 = 25.7053{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.1680{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~66/65 = 25.7013{{c}}


{{Optimal ET sequence|legend=0| 17cg, 29g, 46, 75dfgh, 121defgh }}
{{Optimal ET sequence|legend=0| 46, 94, 140 }}


Badness (Sintel): 1.06
Badness (Sintel): 0.772


===== 23-limit =====
== Undecental ==
Subgroup: 2.3.5.7.11.13.17.19.23
Undecental adds the triwellisma to the comma list and may be described as the {{nowrap| 29 & 70 }} temperament. 5/4 is mapped to the quintuple-diminished seventh or equivalently the perfect fourth minus three [[diesis (scale theory)|dieses]]. [[99edo|58\99]] is an almost perfect generator, just as the name suggests. Another interesting tuning choice is the argent fifth, {{nowrap| 2<sup>(2 - sqrt (2))</sup> }}.  


Comma list: 91/90, 121/120, 133/132, 136/135, 154/153, 161/160, 169/168
[[Subgroup]]: 2.3.5.7


Mapping: {{mapping| 1 0 -31 -21 -14 -9 -34 9 -5 | 0 1 21 15 11 8 24 -3 6 }}
[[Comma list]]: 5120/5103, 235298/234375


Optimal tunings:
{{Mapping|legend=1| 1 0 61 71 | 0 1 -37 -43 }}
* WE: ~2 = 1200.9738{{c}}, ~3/2 = 704.7120{{c}}
: mapping generators: ~2, ~3
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.1695{{c}}


{{Optimal ET sequence|legend=0| 17cg, 29g, 46, 75dfgh, 121defgh }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.6543{{c}}, ~3/2 = 702.8370{{c}}
: [[error map]]: {{val| -0.346 +0.536 +0.423 -0.494 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.0465{{c}}
: error map: {{val| 0.000 +1.092 +0.966 +0.175 }}


Badness (Sintel): 1.01
{{Optimal ET sequence|legend=1| 29, 70, 99, 722bc, 821bc, 920bc, 1019bc }}


==== Leapling ====
[[Badness]] (Sintel): 2.39
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 77/76, 91/90, 121/120, 136/135, 153/152, 169/168
 
Mapping: {{mapping| 1 0 -31 -21 -14 -9 -34 -37 | 0 1 21 15 11 8 24 26 }}
 
Optimal tunings:
* WE: ~2 = 1200.4745{{c}}, ~3/2 = 704.4016{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.1442{{c}}
 
{{Optimal ET sequence|legend=0| 17cgh, 29g, 46h, 75dfg }}
 
Badness (Sintel): 1.16
 
===== 23-limit =====
Subgroup: 2.3.5.7.11.13.17.19.23
 
Comma list: 77/76, 91/90, 115/114, 121/120, 136/135, 153/152, 161/160
 
Mapping: {{mapping| 1 0 -31 -21 -14 -9 -34 -37 -5 | 0 1 21 15 11 8 24 26 6 }}
 
Optimal tunings:
* WE: ~2 = 1200.5425{{c}}, ~3/2 = 704.4319{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.1349{{c}}
 
{{Optimal ET sequence|legend=0| 17cgh, 29g, 46h, 75dfg }}
 
Badness (Sintel): 1.15


== Mystery ==
== Mystery ==
Line 353: Line 335:
Comma list: 441/440, 896/891, 3388/3375
Comma list: 441/440, 896/891, 3388/3375


Mapping: {{mapping| 29 46 0 14 33 | 0 0 1 1 1 }}
{{Mapping|legend=0| 29 46 0 14 33 | 0 0 1 1 1 }}


Optimal tunings:  
Optimal tunings:  
Line 368: Line 350:
Comma list: 196/195, 352/351, 364/363, 676/675
Comma list: 196/195, 352/351, 364/363, 676/675


Mapping: {{mapping| 29 46 0 14 33 40 | 0 0 1 1 1 1 }}
{{Mapping|legend=0| 29 46 0 14 33 40 | 0 0 1 1 1 1 }}


Optimal tunings:  
Optimal tunings:  
Line 379: Line 361:


== Hemidromeda ==
== Hemidromeda ==
Hemidromeda may be described as the {{nowrap| 29 & 111 }} temperament. The name ''hemidromeda'' comes from "hemi-" (Ancient Greek for "one half") and ''[[andromeda]]'', because the generator is 1/2 of andromeda's perfect twelfth (~3/1, about 1902.4 cents); the ploidacot for this temperament is alpha-dicot.  
Hemidromeda may be described as the {{nowrap| 29 & 111 }} temperament. Named by [[Xenllium]] in 2023, ''hemidromeda'' comes from ''hemi-'' (Ancient Greek for "one half") and ''[[andromeda]]'', because the generator is 1/2 of andromeda's perfect twelfth (~3/1, about 1902.4 cents); the ploidacot for this temperament is alpha-dicot.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 403: Line 385:
Comma list: 1331/1323, 1375/1372, 5120/5103
Comma list: 1331/1323, 1375/1372, 5120/5103


Mapping: {{mapping| 1 0 38 48 32 | 0 2 -45 -57 -36 }}
{{Mapping|legend=0| 1 0 38 48 32 | 0 2 -45 -57 -36 }}


Optimal tunings:  
Optimal tunings:  
Line 418: Line 400:
Comma list: 352/351, 676/675, 847/845, 1331/1323
Comma list: 352/351, 676/675, 847/845, 1331/1323


Mapping: {{mapping| 1 0 38 48 32 37 | 0 2 -45 -57 -36 -42 }}
{{Mapping|legend=0| 1 0 38 48 32 37 | 0 2 -45 -57 -36 -42 }}


Optimal tunings:  
Optimal tunings:  
Line 433: Line 415:
Comma list: 352/351, 442/441, 561/560, 676/675, 715/714
Comma list: 352/351, 442/441, 561/560, 676/675, 715/714


Mapping: {{mapping| 1 0 38 48 32 37 58 | 0 2 -45 -57 -36 -42 -68 }}
{{Mapping|legend=0| 1 0 38 48 32 37 58 | 0 2 -45 -57 -36 -42 -68 }}


Optimal tunings:  
Optimal tunings:  
Line 448: Line 430:
Comma list: 286/285, 352/351, 363/361, 442/441, 476/475, 561/560
Comma list: 286/285, 352/351, 363/361, 442/441, 476/475, 561/560


Mapping: {{mapping| 1 0 38 48 32 37 58 32 | 0 2 -45 -57 -36 -42 -68 -35 }}
{{Mapping|legend=0| 1 0 38 48 32 37 58 32 | 0 2 -45 -57 -36 -42 -68 -35 }}


Optimal tunings:  
Optimal tunings:  
Line 463: Line 445:
Comma list: 253/252, 286/285, 352/351, 363/361, 391/390, 442/441, 460/459
Comma list: 253/252, 286/285, 352/351, 363/361, 391/390, 442/441, 460/459


Mapping: {{mapping| 1 0 38 48 32 37 58 32 18 | 0 2 -45 -57 -36 -42 -68 -35 -17 }}
{{Mapping|legend=0| 1 0 38 48 32 37 58 32 18 | 0 2 -45 -57 -36 -42 -68 -35 -17 }}


Optimal tunings:  
Optimal tunings:  
Line 477: Line 459:


Countriton may be described as the {{nowrap| 51c & 53 }} temperament. It splits the [[24/1|24th harmonic]] into nine tritone generators; its ploidacot is thus delta-enneacot. Among the possible tunings are [[157edo]] and [[210edo]], as well as [[104edo]] in the 104c val.  
Countriton may be described as the {{nowrap| 51c & 53 }} temperament. It splits the [[24/1|24th harmonic]] into nine tritone generators; its ploidacot is thus delta-enneacot. Among the possible tunings are [[157edo]] and [[210edo]], as well as [[104edo]] in the 104c val.  
Countriton was named by [[Xenllium]] in 2022 as a counterpart of [[untriton]].


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 500: Line 484:
Comma list: 176/175, 5120/5103, 41503/41472
Comma list: 176/175, 5120/5103, 41503/41472


Mapping: {{mapping| 1 -3 -15 13 -21 | 0 9 34 -20 48 }}
{{Mapping|legend=0| 1 -3 -15 13 -21 | 0 9 34 -20 48 }}


Optimal tunings:  
Optimal tunings:  
Line 515: Line 499:
Comma list: 176/175, 351/350, 847/845, 2197/2187
Comma list: 176/175, 351/350, 847/845, 2197/2187


Mapping: {{mapping| 1 -3 -15 13 -21 -7 | 0 9 34 -20 48 21 }}
{{Mapping|legend=0| 1 -3 -15 13 -21 -7 | 0 9 34 -20 48 21 }}


Optimal tunings:  
Optimal tunings:  
Line 527: Line 511:
== Artoneutral ==
== Artoneutral ==
Artoneutral can be described as the {{nowrap| 87 & 94 }} temperament. It is generated by an artoneutral third of ~11/9 (or a tendoneutral sixth of ~18/11), nine of which make the [[12/1|12th harmonic]]; its ploidacot is thus beta-enneacot. [[181edo]] may be recommended as a tuning.  
Artoneutral can be described as the {{nowrap| 87 & 94 }} temperament. It is generated by an artoneutral third of ~11/9 (or a tendoneutral sixth of ~18/11), nine of which make the [[12/1|12th harmonic]]; its ploidacot is thus beta-enneacot. [[181edo]] may be recommended as a tuning.  
Artoneutral was named by [[Flora Canou]] in 2023 for its generator's quality.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 550: Line 536:
Comma list: 385/384, 2200/2187, 4000/3993
Comma list: 385/384, 2200/2187, 4000/3993


Mapping: {{mapping| 1 -1 -4 12 -2 | 0 9 22 -32 19 }}
{{Mapping|legend=0| 1 -1 -4 12 -2 | 0 9 22 -32 19 }}


Optimal tunings:  
Optimal tunings:  
Line 565: Line 551:
Comma list: 325/324, 352/351, 385/384, 1575/1573
Comma list: 325/324, 352/351, 385/384, 1575/1573


Mapping: {{mapping| 1 -1 -4 12 -2 6 | 0 9 22 -32 19 -8 }}
{{Mapping|legend=0| 1 -1 -4 12 -2 6 | 0 9 22 -32 19 -8 }}


Optimal tunings:  
Optimal tunings:  
Line 580: Line 566:
Comma list: 325/324, 352/351, 375/374, 385/384, 595/594
Comma list: 325/324, 352/351, 375/374, 385/384, 595/594


Mapping: {{mapping| 1 -1 -4 12 -2 6 -12 | 0 9 22 -32 19 -8 56 }}
{{Mapping|legend=0| 1 -1 -4 12 -2 6 -12 | 0 9 22 -32 19 -8 56 }}


Optimal tunings:  
Optimal tunings:  
Line 595: Line 581:
Comma list: 325/324, 352/351, 375/374, 385/384, 400/399, 595/594
Comma list: 325/324, 352/351, 375/374, 385/384, 400/399, 595/594


Mapping: {{mapping| 1 -1 -4 12 -2 6 -12 -15 | 0 9 22 -32 19 -8 56 67 }}
{{Mapping|legend=0| 1 -1 -4 12 -2 6 -12 -15 | 0 9 22 -32 19 -8 56 67 }}


Optimal tunings:  
Optimal tunings:  
Line 610: Line 596:
Comma list: 300/299, 325/324, 352/351, 375/374, 385/384, 400/399, 484/483
Comma list: 300/299, 325/324, 352/351, 375/374, 385/384, 400/399, 484/483


Mapping: {{mapping| 1 -1 -4 12 -2 6 -12 -15 -13 | 0 9 22 -32 19 -8 56 67 61 }}
{{Mapping|legend=0| 1 -1 -4 12 -2 6 -12 -15 -13 | 0 9 22 -32 19 -8 56 67 61 }}


Optimal tunings:  
Optimal tunings:  
Line 645: Line 631:
Comma list: 540/539, 1331/1323, 5120/5103
Comma list: 540/539, 1331/1323, 5120/5103


Mapping: {{mapping| 1 1 -4 0 1 | 0 5 54 24 21 }}
{{Mapping|legend=0| 1 1 -4 0 1 | 0 5 54 24 21 }}


Optimal tunings:  
Optimal tunings:  
Line 660: Line 646:
Comma list: 352/351, 540/539, 729/728, 1331/1323
Comma list: 352/351, 540/539, 729/728, 1331/1323


Mapping: {{mapping| 1 1 -4 0 1 3 | 0 5 54 24 21 6 }}
{{Mapping|legend=0| 1 1 -4 0 1 3 | 0 5 54 24 21 6 }}


Optimal tunings:  
Optimal tunings:  
Line 675: Line 661:
Comma list: 352/351, 442/441, 540/539, 715/714, 847/845
Comma list: 352/351, 442/441, 540/539, 715/714, 847/845


Mapping: {{mapping| 1 1 -4 0 1 3 -2 | 0 5 54 24 21 6 52 }}
{{Mapping|legend=0| 1 1 -4 0 1 3 -2 | 0 5 54 24 21 6 52 }}


Optimal tunings:  
Optimal tunings:  
Line 690: Line 676:
Comma list: 352/351, 400/399, 442/441, 456/455, 495/494, 715/714
Comma list: 352/351, 400/399, 442/441, 456/455, 495/494, 715/714


Mapping: {{mapping| 1 1 -4 0 1 3 -2 -5 | 0 5 54 24 21 6 52 79 }}
{{Mapping|legend=0| 1 1 -4 0 1 3 -2 -5 | 0 5 54 24 21 6 52 79 }}


Optimal tunings:  
Optimal tunings:  
Line 703: Line 689:
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Jorgensen]].''
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Jorgensen]].''


Jorgensen tempers out the [[linus comma]] in addition to the hemifamity comma, and may be described as the {{nowrap| 70 & 140 }} temperament, with a 70th-octave period. Its ploidacot is 70-ploid acot.  
Jorgensen tempers out the [[linus comma]] in addition to the aberschisma, and may be described as the {{nowrap| 70 & 140 }} temperament, with a 70th-octave period. Its ploidacot is 70-ploid acot.  


It is the natural 7-limit extension of the 5-limit temperament tempering out the 70-comma, named by [[Mike Battaglia]] in 2012 for historical interests<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_103982.html Yahoo! Tuning Group | ''Jorgensen Temperament'']</ref>.  
It is the natural 7-limit extension of the 5-limit temperament tempering out the 70-comma, named by [[Mike Battaglia]] in 2012 for historical interests<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_103982.html Yahoo! Tuning Group | ''Jorgensen Temperament'']</ref>.  
Line 726: Line 712:
== References ==
== References ==


[[Category:Aberschismic temperaments| ]] <!-- main article -->
[[Category:Aberschismic| ]] <!-- key article -->
[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Hemifamity temperaments| ]] <!-- main article -->
[[Category:Catalogs of rank-2 temperaments]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Aberschismic_temperaments"