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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, 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.  
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.  
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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.  


Temperaments considered below are undecental, leapday, hemidromeda, mystery, quanic, septiquarter, countriton, artoneutral and ketchup. Discussed elsewhere are:
Temperaments discussed elsewhere are:
* [[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]]
* [[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]]
* [[Buzzard]] (+1728/1715) → [[Vulture family #Buzzard|Vulture family]]
* [[Misty]] (+3136/3125) → [[Misty family #Misty|Misty family]]
* [[Misty]] (+3136/3125) → [[Misty family #Misty|Misty family]]
* ''[[Supers]]'' (+118098/117649) → [[Stearnsmic clan #Supers|Stearnsmic clan]]
* [[Monkey]] (+875/864) → [[Tetracot family #Monkey|Tetracot family]]
* [[Buzzard]] (+1728/1715) → [[Buzzardsmic clan #Buzzard|Buzzardsmic 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]]
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* [[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]]


== Undecental ==
Considered below are septiquarter, kwai, ketchup, undecental, leapday, mystery, hemidromeda, countriton, artoneutral, quanic and jorgensen, in the order of increasing [[TE logflat badness]].
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> }}.  
 
== Septiquarter ==
Septiquarter tempers out [[420175/419904]] and may be described as the {{nowrap| 94 & 99 }} temperament. Its [[ploidacot]] is epsilon-heptacot. [[99edo]] makes for an excellent tuning, and [[292edo]] an even better one. [[94edo]] and [[104edo]] in the 104c val are also among the possibilities.  


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


[[Comma list]]: 5120/5103, 235298/234375
[[Comma list]]: 5120/5103, 420175/419904
 
{{Mapping|legend=1| 1 -4 -28 6 | 0 7 38 -4 }}
: mapping generators: ~2, ~243/140
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.7212{{c}}, ~243/140 = 957.3250{{c}}
: [[error map]]: {{val| -0.279 +0.435 -0.158 +0.201 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~243/140 = 957.5424{{c}}
: error map: {{val| 0.000 +0.842 +0.298 +1.004 }}
 
{{Optimal ET sequence|legend=1| 94, 99, 292, 391, 881bd, 1272bcd }}
 
[[Badness]] (Sintel): 1.36
 
=== Semiseptiquarter ===
Subgroup: 2.3.5.7.11
 
Comma list: 5120/5103, 9801/9800, 14641/14580
 
Mapping: {{mapping| 2 -8 -56 12 -25 | 0 7 38 -4 20 }}
 
Optimal tunings:
* WE: ~99/70 = 599.8953{{c}}, ~210/121 = 957.3819{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~210/121 = 957.5449{{c}}
 
{{Optimal ET sequence|legend=0| 94, 198, 292, 490 }}
 
Badness (Sintel): 2.12
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


{{Mapping|legend=1| 1 0 61 71 | 0 1 -37 -43 }}
Comma list: 352/351, 847/845, 1716/1715, 14641/14580
: mapping generators: ~2, ~3


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~3/2 = 703.039{{c}}
Mapping: {{mapping| 2 -8 -56 12 -25 9 | 0 7 38 -4 20 -1 }}


{{Optimal ET sequence|legend=1| 29, 70, 99, 722bc, 821bc, 920bc, 1019bc, 1118bbcc, 1217bbcc, 1316bbccd }}
Optimal tunings:
* WE: ~99/70 = 599.8565{{c}}, ~210/121 = 957.3261{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~210/121 = 957.5508{{c}}


[[Badness]] (Smith): 0.094603
{{Optimal ET sequence|legend=0| 94, 198, 490f }}


== Leapday ==
Badness (Sintel): 1.44
{{Main| Leapday }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Leapday]].''


Leapday tempers out the leapday comma, {{monzo| 31 -21 1 }}, in the 5-limit, mapping 5/4 to the triple-augmented unison or equivalently the minor third and two dieses. In the 7-limit it can be described as the {{nowrap| 29 & 46 }} temperament, which tempers out the hemifamity and [[686/675]] (senga), and extends [[leapfrog]].  
== Kwai ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Kwai]].''


It 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.
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]]: 686/675, 5120/5103
[[Comma list]]: 5120/5103, 16875/16807


{{Mapping|legend=1| 1 0 -31 -21 | 0 1 21 15 }}
{{Mapping|legend=1| 1 0 -50 -40 | 0 1 33 27 }}
: mapping generators: ~2, ~3
: mapping generators: ~2, ~3


Optimal tunings:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.2257{{c}}
* [[WE]]: ~2 = 1199.7337{{c}}, ~3/2 = 702.4600{{c}}
* [[POTE]]: ~2 = 1200.000{{c}}, ~3/2 = 704.263{{c}}
: [[error map]]: {{val| -0.266 +0.239 -0.607 +1.055 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.6085{{c}}
: error map: {{val| 0.000 +0.653 -0.234 +1.603 }}


{{Optimal ET sequence|legend=1| 17c, 29, 46 }}
{{Optimal ET sequence|legend=1| 41, 111, 152, 345, 497d }}


[[Badness]] (Smith): 0.096123
[[Badness]] (Sintel): 1.38


=== 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: 540/539, 1375/1372, 5120/5103


Mapping: {{mapping| 1 0 -31 -21 -14 | 0 1 21 15 11 }}
Mapping: {{mapping| 1 0 -50 -40 32 | 0 1 33 27 -18 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2625{{c}}
* WE: ~2 = 1199.6672{{c}}, ~3/2 = 702.4282{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 704.250{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6189{{c}}


{{Optimal ET sequence|legend=0| 17c, 29, 46 }}
{{Optimal ET sequence|legend=0| 41, 111, 152, 497de, 649dde }}


Badness (Smith): 0.038624
Badness (Sintel): 0.867


=== 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: 352/351, 540/539, 729/728, 1375/1372


Mapping: {{mapping| 1 0 -31 -21 -14 -9 | 0 1 21 15 11 8 }}
Mapping: {{mapping| 1 0 -50 -40 32 27 | 0 1 33 27 -18 -21 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2924{{c}}
* WE: ~2 = 1199.4772{{c}}, ~3/2 = 702.3379{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 704.214{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6409{{c}}


{{Optimal ET sequence|legend=0| 17c, 29, 46, 121def }}
{{Optimal ET sequence|legend=0| 41, 111, 152f, 415dff }}


Badness (Smith): 0.024732
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: 91/90, 121/120, 136/135, 154/153, 169/168
Comma list: 256/255, 352/351, 540/539, 715/714, 1089/1088


Mapping: {{mapping| 1 0 -31 -21 -14 -9 -34 | 0 1 21 15 11 8 24 }}
Mapping: {{mapping| 1 0 -50 -40 32 27 58 | 0 1 33 27 -18 -21 -34 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.0000{{c}}, ~3/2 = 704.3098{{c}}
* WE: ~2 = 1199.3537{{c}}, ~3/2 = 702.2850{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 704.229{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6589{{c}}


{{Optimal ET sequence|legend=0| 17cg, 29g, 46, 121defg }}
{{Optimal ET sequence|legend=0| 41, 70, 111, 152fg, 263dfg }}


Badness (Smith): 0.017863
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: 91/90, 121/120, 133/132, 136/135, 154/153, 169/168
Comma list: 256/255, 352/351, 400/399, 456/455, 715/714, 847/845


Mapping: {{mapping| 1 0 -31 -21 -14 -9 -34 9 | 0 1 21 15 11 8 24 -3 }}
Mapping: {{mapping| 1 0 -50 -40 32 27 58 -56 | 0 1 33 27 -18 -21 -34 38 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2990{{c}}
* WE: ~2 = 1199.3401{{c}}, ~3/2 = 702.2705{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 704.135{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6548{{c}}


{{Optimal ET sequence|legend=0| 17cg, 29g, 46, 75dfgh, 121defgh }}
{{Optimal ET sequence|legend=0| 41, 70h, 111, 152fg, 263dfgh }}


Badness (Smith): 0.017356
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: 91/90, 121/120, 133/132, 136/135, 154/153, 161/160, 169/168
Comma list: 540/539, 676/675, 1375/1372, 5120/5103


Mapping: {{mapping| 1 0 -31 -21 -14 -9 -34 9 -5 | 0 1 21 15 11 8 24 -3 6 }}
Mapping: {{mapping| 1 0 -50 -40 32 -51 | 0 2 66 54 -36 69 }}
: mapping generators: ~2, ~26/15


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.0000{{c}}, ~3/2 = 704.3035{{c}}
* WE: ~2 = 1199.6968{{c}}, ~26/15 = 951.0740{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 704.141{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.3123{{c}}


{{Optimal ET sequence|legend=0| 17cg, 29g, 46, 75dfgh, 121defgh }}
{{Optimal ET sequence|legend=0| 82, 111, 193, 304d }}


Badness (Smith): 0.014065
Badness (Sintel): 1.82


==== Leapling ====
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17


Comma list: 77/76, 91/90, 121/120, 136/135, 153/152, 169/168
Comma list: 442/441, 540/539, 676/675, 715/714, 5120/5103


Mapping: {{mapping| 1 0 -31 -21 -14 -9 -34 -37 | 0 1 21 15 11 8 24 26 }}
Mapping: {{mapping| 1 0 -50 -40 32 -51 -30 | 0 2 66 54 -36 69 43 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2037{{c}}
* WE: ~2 = 1199.6861{{c}}, ~26/15 = 951.0654{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 704.123{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.3120{{c}}


{{Optimal ET sequence|legend=0| 17cgh, 29g, 46h, 75dfg }}
{{Optimal ET sequence|legend=0| 82, 111, 193, 304d }}


Badness (Smith): 0.019065
Badness (Sintel): 1.31


===== 23-limit =====
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19.23
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 77/76, 91/90, 115/114, 121/120, 136/135, 153/152, 161/160
Comma list: 400/399, 442/441, 540/539, 676/675, 715/714, 1445/1444


Mapping: {{mapping| 1 0 -31 -21 -14 -9 -34 -37 -5 | 0 1 21 15 11 8 24 26 6 }}
Mapping: {{mapping| 1 0 -50 -40 32 -51 -30 -56 | 0 2 66 54 -36 69 43 76 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2082{{c}}
* WE: ~2 = 1199.6718{{c}}, ~26/15 = 951.0526{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 704.114{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.3103{{c}}


{{Optimal ET sequence|legend=0| 17cgh, 29g, 46h, 75dfg }}
{{Optimal ET sequence|legend=0| 82, 111, 193, 304dh }}


Badness (Smith): 0.016067
Badness (Sintel): 1.16


== Hemidromeda ==
== Ketchup ==
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).
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]]: 5120/5103, 52734375/52706752
[[Comma list]]: 5120/5103, 1071875/1062882


{{Mapping|legend=1| 1 0 38 48 | 0 2 -45 -57 }}
{{Mapping|legend=1| 2 3 4 6 | 0 4 15 -9 }}
: mapping generator: ~2, ~12500/7203
: mapping generators: ~1225/864, ~64/63


[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~7203/6250 = 248.581{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~1225/864 = 599.9685{{c}}, ~64/63 = 25.7181{{c}}
: [[error map]]: {{val| -0.063 +0.823 -0.668 -0.478 }}
* [[CWE]]: ~1225/864 = 600.0000{{c}}, ~64/63 = 25.7181{{c}}
: error map: {{val| 0.000 +0.917 -0.543 -0.288 }}


{{Optimal ET sequence|legend=1| 29, 82cd, 111, 140, 531, 671, 811b, 951b }}
{{Optimal ET sequence|legend=1| 46, 94, 140 }}


[[Badness]] (Smith): 0.115803
[[Badness]] (Sintel): 2.14


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


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


Mapping: {{mapping| 1 0 38 48 32 | 0 2 -45 -57 -36 }}
Mapping: {{mapping| 2 3 4 6 7 | 0 4 15 -9 -2 }}


Optimal tuning (CTE): ~2 = 1200.000{{c}}, ~405/352 = 248.589{{c}}
Optimal tunings:  
* WE: ~99/70 = 600.0678{{c}}, ~64/63 = 25.6963{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~64/63 = 25.6956{{c}}


{{Optimal ET sequence|legend=0| 29, 82cd, 111, 140, 251, 391e }}
{{Optimal ET sequence|legend=0| 46, 94, 140 }}


Badness (Smith): 0.060808
Badness (Sintel): 1.31


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


Comma list: 352/351, 676/675, 847/845, 1331/1323
Comma list: 325/324, 352/351, 385/384, 1331/1323


Mapping: {{mapping| 1 0 38 48 32 37 | 0 2 -45 -57 -36 -42 }}
Mapping: {{mapping| 2 3 4 6 7 8 | 0 4 15 -9 -2 -14 }}


Optimal tuning (CTE): ~2 = 1200.000{{c}}, ~15/13 = 248.588{{c}}
Optimal tunings:  
* WE: ~99/70 = 600.0612{{c}}, ~66/65 = 25.7000{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~66/65 = 25.6978{{c}}


{{Optimal ET sequence|legend=0| 29, 82cdf, 111, 140, 391e, 531e }}
{{Optimal ET sequence|legend=0| 46, 94, 140 }}


Badness (Smith): 0.028632
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: 352/351, 442/441, 561/560, 676/675, 715/714
Comma list: 289/288, 325/324, 352/351, 385/384, 442/441
 
Mapping: {{mapping| 2 3 4 6 7 8 8 | 0 4 15 -9 -2 -14 4 }}
 
Optimal tunings:
* WE: ~17/12 = 600.0896{{c}}, ~66/65 = 25.7048{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~66/65 = 25.7017{{c}}
 
{{Optimal ET sequence|legend=0| 46, 94, 140 }}
 
Badness (Sintel): 0.845
 
=== 2.3.5.7.11.13.17.23 subgroup ===
Subgroup: 2.3.5.7.11.13.17.23
 
Comma list: 253/252, 289/288, 325/324, 352/351, 385/384, 391/390
 
Mapping: {{mapping| 2 3 4 6 7 8 8 9 | 0 4 15 -9 -2 -14 4 1 }}
 
Optimal tunings:
* WE: ~17/12 = 600.1139{{c}}, ~66/65 = 25.7053{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~66/65 = 25.7013{{c}}
 
{{Optimal ET sequence|legend=0| 46, 94, 140 }}
 
Badness (Sintel): 0.772
 
== Undecental ==
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
 
[[Comma list]]: 5120/5103, 235298/234375
 
{{Mapping|legend=1| 1 0 61 71 | 0 1 -37 -43 }}
: mapping generators: ~2, ~3
 
[[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 }}
 
{{Optimal ET sequence|legend=1| 29, 70, 99, 722bc, 821bc, 920bc, 1019bc }}
 
[[Badness]] (Sintel): 2.39
 
== Leapday ==
{{Main| Leapday }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Leapday]].''
 
Leapday tempers out [[686/675]], the senga, in addition to the aberschisma, 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.
 
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.
 
Leapday has an alternative extension called [[porwell temperaments #Polypyth|polypyth]], which tempers out the same 5-limit comma as leapday, but with the porwell comma ([[6144/6125]]) rather than the aberschisma tempered out.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 686/675, 5120/5103
 
{{Mapping|legend=1| 1 0 -31 -21 | 0 1 21 15 }}
: mapping generators: ~2, ~3
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.7167{{c}}, ~3/2 = 704.0971{{c}}
: [[error map]]: {{val| -0.283 +1.859 +2.559 -5.669 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.2504{{c}}
: error map: {{val| 0.000 +2.295 +2.945 -5.070 }}
 
{{Optimal ET sequence|legend=1| 17c, 29, 46 }}
 
[[Badness]] (Sintel): 2.43
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 121/120, 441/440, 686/675
 
Mapping: {{mapping| 1 0 -31 -21 -14 | 0 1 21 15 11 }}
 
Optimal tunings:
* WE: ~2 = 1200.0731{{c}}, ~3/2 = 704.2933{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2538{{c}}
 
{{Optimal ET sequence|legend=0| 17c, 29, 46 }}
 
Badness (Sintel): 1.28
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 91/90, 121/120, 169/168, 352/351


Mapping: {{mapping| 1 0 38 48 32 37 58 | 0 2 -45 -57 -36 -42 -68 }}
Mapping: {{mapping| 1 0 -31 -21 -14 -9 | 0 1 21 15 11 8 }}


Optimal tuning (CTE): ~2 = 1200.000{{c}}, ~15/13 = 248.591{{c}}
Optimal tunings:
* WE: ~2 = 1200.4758{{c}}, ~3/2 = 704.4930{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2346{{c}}


{{Optimal ET sequence|legend=0| 29g, 82cdfg, 111, 140, 251, 391e }}
{{Optimal ET sequence|legend=0| 17c, 29, 46, 121def }}


Badness (Smith): 0.019054
Badness (Sintel): 1.02


=== 19-limit ===
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17


Comma list: 286/285, 352/351, 363/361, 442/441, 476/475, 561/560
Comma list: 91/90, 121/120, 136/135, 154/153, 169/168


Mapping: {{mapping| 1 0 38 48 32 37 58 32 | 0 2 -45 -57 -36 -42 -68 -35 }}
Mapping: {{mapping| 1 0 -31 -21 -14 -9 -34 | 0 1 21 15 11 8 24 }}


Optimal tuning (CTE): ~2 = 1200.000{{c}}, ~15/13 = 248.587{{c}}
Optimal tunings:
* WE: ~2 = 1200.4818{{c}}, ~3/2 = 704.5121{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2507{{c}}


{{Optimal ET sequence|legend=0| 29g, 82cdfgh, 111, 140, 391ehh, 531ehh }}
{{Optimal ET sequence|legend=0| 17cg, 29g, 46, 121defg }}


Badness (Smith): 0.016609
Badness (Sintel): 0.910


=== 23-limit ===
=== 2.3.5.7.11.13.17.23 subgroup ===
Subgroup: 2.3.5.7.11.13.17.19.23
Subgroup: 2.3.5.7.11.13.17.23


Comma list: 253/252, 286/285, 352/351, 363/361, 391/390, 442/441, 460/459
Comma list: 91/90, 121/120, 136/135, 154/153, 161/160, 169/168


Mapping: {{mapping| 1 0 38 48 32 37 58 32 18 | 0 2 -45 -57 -36 -42 -68 -35 -17 }}
Mapping: {{mapping| 1 0 -31 -21 -14 -9 -34 -5 | 0 1 21 15 11 8 24 6 }}


Optimal tuning (CTE): ~2 = 1200.000{{c}}, ~15/13 = 248.588{{c}}
Optimal tunings:
* WE: ~2 = 1200.5169{{c}}, ~3/2 = 704.5279{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2450{{c}}


{{Optimal ET sequence|legend=0| 29g, 82cdfgh, 111, 140, 391ehhi, 531ehhii }}
{{Optimal ET sequence|legend=0| 17cg, 29g, 46, 121defg }}


Badness (Smith): 0.015361
Badness (Sintel): 0.872


== Mystery ==
== Mystery ==
Line 255: Line 397:
: ''For the 5-limit version, see [[29th-octave temperaments #Mystery]].''
: ''For the 5-limit version, see [[29th-octave temperaments #Mystery]].''


Mystery tempers out [[50421/50000]] and may be described as the {{nowrap| 29 & 58 }} temperament. It has a 1\29 period and primes 5, 7, 11 and 13 are all reached by one generator step. [[145edo]] or [[232edo]] are good candidates for tunings.  
Mystery tempers out [[50421/50000]] and may be described as the {{nowrap| 29 & 58 }} temperament. It has a 1\29 period and primes 5, 7, 11 and 13 are all reached by one generator step; its ploidacot is 29-ploid acot. [[145edo]] or [[232edo]] are good candidates for tunings.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 264: Line 406:
: mapping generators: ~50/49, ~5
: mapping generators: ~50/49, ~5


[[Optimal tuning]] ([[POTE]]): ~50/49 = 41.379{{c}}, ~5/4 = 388.646{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~50/49 = 41.3652{{c}}, ~5/4 = 388.5128{{c}}
: [[error map]]: {{val| -0.410 +0.842 +1.378 -2.022 }}
* [[CWE]]: ~50/49 = 41.3793{{c}}, ~5/4 = 388.3030{{c}}
: error map: {{val| 0.000 +1.493 +1.989 -1.213 }}


{{Optimal ET sequence|legend=1| 29, 58, 87, 145 }}
{{Optimal ET sequence|legend=1| 29, 58, 87, 145 }}


[[Badness]] (Smith): 0.103734
[[Badness]] (Sintel): 2.63


=== 11-limit ===
=== 11-limit ===
Line 277: Line 423:
Mapping: {{mapping| 29 46 0 14 33 | 0 0 1 1 1 }}
Mapping: {{mapping| 29 46 0 14 33 | 0 0 1 1 1 }}


Optimal tuning (POTE): ~45/44 = 41.379{{c}}, ~5/4 = 388.460{{c}}
Optimal tunings:
* WE: ~45/44 = 41.3637{{c}}, ~5/4 = 388.3136{{c}}
* CWE: ~45/44 = 41.3793{{c}}, ~5/4 = 388.0598{{c}}


{{Optimal ET sequence|legend=0| 29, 58, 87, 145 }}
{{Optimal ET sequence|legend=0| 29, 58, 87, 145 }}


Badness (Smith): 0.034291
Badness (Sintel): 1.13


=== 13-limit ===
=== 13-limit ===
Line 290: Line 438:
Mapping: {{mapping| 29 46 0 14 33 40 | 0 0 1 1 1 1 }}
Mapping: {{mapping| 29 46 0 14 33 40 | 0 0 1 1 1 1 }}


Optimal tuning (POTE): ~45/44 = 41.379{{c}}, ~5/4 = 388.354{{c}}
Optimal tunings:
* WE: ~45/44 = 41.3623{{c}}, ~5/4 = 388.1942{{c}}
* CWE: ~40/39 = 41.3793{{c}}, ~5/4 = 387.9017{{c}}
 
{{Optimal ET sequence|legend=0| 29, 58, 87, 145, 232 }}


{{Optimal ET sequence|legend=0| 29, 58, 87, 145, 232, 377cef }}
Badness (Sintel): 0.768


Badness (Smith): 0.018591
== Hemidromeda ==
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.  


== Quanic ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 5120/5103, 5832000/5764801
[[Comma list]]: 5120/5103, 52734375/52706752


{{Mapping|legend=1| 1 1 -4 0 | 0 5 54 24 }}
{{Mapping|legend=1| 1 0 38 48 | 0 2 -45 -57 }}
: mapping generators: ~2, ~160/147
: mapping generator: ~2, ~12500/7203


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~160/147 = 140.493{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.7236{{c}}, ~12500/7203 = 951.1864{{c}}
: [[error map]]: {{val| -0.276 +0.418 -0.205 +0.282 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~12500/7203 = 951.4098{{c}}
: error map: {{val| 0.000 +0.865 +0.243 +0.813 }}


{{Optimal ET sequence|legend=1| 94, 111, 205 }}
{{Optimal ET sequence|legend=1| 29, 82cd, 111, 140, 251, 391, 1424bbcdd }}


[[Badness]] (Smith): 0.179475
[[Badness]] (Sintel): 2.93


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


Comma list: 540/539, 1331/1323, 5120/5103
Comma list: 1331/1323, 1375/1372, 5120/5103


Mapping: {{mapping| 1 1 -4 0 1 | 0 5 54 24 21 }}
Mapping: {{mapping| 1 0 38 48 32 | 0 2 -45 -57 -36 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~88/81 = 140.489{{c}}
Optimal tunings:
* WE: ~2 = 1199.8767{{c}}, ~400/231 = 951.3065{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~400/231 = 951.4063{{c}}


{{Optimal ET sequence|legend=0| 94, 111, 205 }}
{{Optimal ET sequence|legend=0| 29, 82cd, 111, 140, 251, 391e }}


Badness (Smith): 0.058678
Badness (Sintel): 2.01


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


Comma list: 352/351, 540/539, 729/728, 1331/1323
Comma list: 352/351, 676/675, 847/845, 1331/1323


Mapping: {{mapping| 1 1 -4 0 1 3 | 0 5 54 24 21 6 }}
Mapping: {{mapping| 1 0 38 48 32 37 | 0 2 -45 -57 -36 -42 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~13/12 = 140.496{{c}}
Optimal tunings:
* WE: ~2 = 1199.8753{{c}}, ~26/15 = 951.3054{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.4064{{c}}


{{Optimal ET sequence|legend=0| 94, 111, 205 }}
{{Optimal ET sequence|legend=0| 29, 82cdf, 111, 140, 251, 391e }}


Badness (Smith): 0.032481
Badness (Sintel): 1.18


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


Comma list: 352/351, 442/441, 540/539, 715/714, 847/845
Comma list: 352/351, 442/441, 561/560, 676/675, 715/714


Mapping: {{mapping| 1 1 -4 0 1 3 -2 | 0 5 54 24 21 6 52 }}
Mapping: {{mapping| 1 0 38 48 32 37 58 | 0 2 -45 -57 -36 -42 -68 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~13/12 = 140.497{{c}}
Optimal tunings:
* WE: ~2 = 1199.8770{{c}}, ~26/15 = 951.3039{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.4035{{c}}


{{Optimal ET sequence|legend=0| 94, 111, 205 }}
{{Optimal ET sequence|legend=0| 29g, 82cdfg, 111, 140, 251, 391e }}


Badness (Smith): 0.021112
Badness (Sintel): 0.971


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


Comma list: 352/351, 400/399, 442/441, 456/455, 495/494, 715/714
Comma list: 286/285, 352/351, 363/361, 442/441, 476/475, 561/560


Mapping: {{mapping| 1 1 -4 0 1 3 -2 -5 | 0 5 54 24 21 6 52 79 }}
Mapping: {{mapping| 1 0 38 48 32 37 58 32 | 0 2 -45 -57 -36 -42 -68 -35 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~13/12 = 140.496{{c}}
Optimal tunings:
* WE: ~2 = 1199.7534{{c}}, ~26/15 = 951.2024{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.4020{{c}}


{{Optimal ET sequence|legend=0| 94, 111, 205 }}
{{Optimal ET sequence|legend=0| 29g, 82cdfgh, 111, 140 }}


Badness (Smith): 0.017273
Badness (Sintel): 1.01


== Septiquarter ==
=== 23-limit ===
[[Subgroup]]: 2.3.5.7
Subgroup: 2.3.5.7.11.13.17.19.23


[[Comma list]]: 5120/5103, 420175/419904
Comma list: 253/252, 286/285, 352/351, 363/361, 391/390, 442/441, 460/459


{{Mapping|legend=1| 1 3 10 2 | 0 -7 -38 4 }}
Mapping: {{mapping| 1 0 38 48 32 37 58 32 18 | 0 2 -45 -57 -36 -42 -68 -35 -17 }}
: mapping generators: ~2, ~147/128


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~147/128 = 242.453{{c}}
Optimal tunings:
* WE: ~2 = 1199.9128{{c}}, ~26/15 = 951.3371{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.4076{{c}}


{{Optimal ET sequence|legend=1| 94, 99, 292, 391, 881bd, 1272bcd }}
{{Optimal ET sequence|legend=0| 29g, 82cdfgh, 111, 140 }}


[[Badness]] (Smith): 0.053760
Badness (Sintel): 1.10


=== Semiseptiquarter ===
== Countriton ==
Subgroup: 2.3.5.7.11
: ''For the 5-limit version, see [[Schismic–Mercator equivalence continuum #Countritonic]].''
 
Comma list: 5120/5103, 9801/9800, 14641/14580
 
Mapping: {{mapping| 2 6 20 4 15 | 0 -7 -38 4 -20 }}
 
Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~121/105 = 242.4511{{c}}
 
{{Optimal ET sequence|legend=0| 94, 198, 292, 490 }}
 
Badness (Smith): 0.064160


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


Comma list: 352/351, 847/845, 1716/1715, 14641/14580
Countriton was named by [[Xenllium]] in 2022 as a counterpart of [[untriton]].  
 
Mapping: {{mapping| 2 6 20 4 15 7 | 0 -7 -38 4 -20 1 }}
 
Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~121/105 = 242.4448{{c}}
 
{{Optimal ET sequence|legend=0| 94, 198, 490f }}
 
Badness (Smith): 0.034834
 
== Countriton ==
: ''For the 5-limit version, see [[Schismic–Mercator equivalence continuum #Countritonic]].''


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 409: Line 552:
[[Comma list]]: 5120/5103, 7558272/7503125
[[Comma list]]: 5120/5103, 7558272/7503125


{{Mapping|legend=1| 1 6 19 -7 | 0 -9 -34 20 }}
{{Mapping|legend=1| 1 -3 -15 13 | 0 9 34 -20 }}
: mapping generators: ~2, ~1728/1225
: mapping generators: ~2, ~1225/864


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~1728/1225 = 588.582{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.4179{{c}}, ~1225/864 = 611.1213{{c}}
: [[error map]]: {{val| -0.582 -0.117 +0.541 +1.181 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~1225/864 = 611.4120{{c}}
: error map: {{val| 0.000 +0.753 +1.695 +2.934 }}


{{Optimal ET sequence|legend=1| 53, 157, 210 }}
{{Optimal ET sequence|legend=1| 51c, 53, 157, 210, 473cdd }}


[[Badness]] (Smith): 0.131191
[[Badness]] (Sintel): 3.32


=== 11-limit ===
=== 11-limit ===
Line 423: Line 570:
Comma list: 176/175, 5120/5103, 41503/41472
Comma list: 176/175, 5120/5103, 41503/41472


Mapping: {{mapping| 1 6 19 -7 27 | 0 -9 -34 20 -48 }}
Mapping: {{mapping| 1 -3 -15 13 -21 | 0 9 34 -20 48 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~108/77 = 588.545{{c}}
Optimal tunings:
* WE: ~2 = 1199.5178{{c}}, ~77/54 = 611.2097{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~77/54 = 611.4495{{c}}


{{Optimal ET sequence|legend=0| 53, 104c, 157 }}
{{Optimal ET sequence|legend=0| 51ce, 53, 104c, 157 }}


Badness (Smith): 0.084782
Badness (Sintel): 2.80


=== 13-limit ===
=== 13-limit ===
Line 436: Line 585:
Comma list: 176/175, 351/350, 847/845, 2197/2187
Comma list: 176/175, 351/350, 847/845, 2197/2187


Mapping: {{mapping| 1 6 19 -7 27 14 | 0 -9 -34 20 -48 -21 }}
Mapping: {{mapping| 1 -3 -15 13 -21 -7 | 0 9 34 -20 48 21 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~108/77 = 588.544{{c}}
Optimal tunings:
* WE: ~2 = 1199.5944{{c}}, ~77/54 = 611.2491{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~77/54 = 611.4506{{c}}


{{Optimal ET sequence|legend=0| 53, 104c, 157 }}
{{Optimal ET sequence|legend=0| 51ce, 53, 104c, 157 }}


Badness (Smith): 0.042321
Badness (Sintel): 1.75


== Artoneutral ==
== Artoneutral ==
Artoneutral is generated by an artoneutral third of ~11/9 (or a tendoneutral sixth of ~18/11) and can be described as the {{nowrap| 87 & 94 }} temperament. [[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 451: Line 604:
[[Comma list]]: 5120/5103, 3828125/3779136
[[Comma list]]: 5120/5103, 3828125/3779136


{{Mapping|legend=1| 1 8 18 -20 | 0 -9 -22 32 }}
{{Mapping|legend=1| 1 -1 -4 12 | 0 9 22 -32 }}
: mapping generators: ~2, ~105/64
: mapping generators: ~2, ~128/105


[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~105/64 = 855.2452{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.1400{{c}}, ~128/105 = 344.7929{{c}}
: [[error map]]: {{val| +0.140 +1.041 -1.430 -0.518 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~128/105 = 344.7531{{c}}
: error map: {{val| 0.000 +0.823 -1.746 -0.925 }}


{{Optimal ET sequence|legend=1| 87, 94, 181 }}
{{Optimal ET sequence|legend=1| 87, 94, 181 }}


[[Badness]] (Smith): 0.157120
[[Badness]] (Sintel): 3.98


=== 11-limit ===
=== 11-limit ===
Line 465: Line 622:
Comma list: 385/384, 2200/2187, 4000/3993
Comma list: 385/384, 2200/2187, 4000/3993


Mapping: {{mapping| 1 8 18 -20 17 | 0 -9 -22 32 -19 }}
Mapping: {{mapping| 1 -1 -4 12 -2 | 0 9 22 -32 19 }}


Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~18/11 = 855.2397{{c}}
Optimal tunings:
* WE: ~2 = 1200.1668{{c}}, ~11/9 = 344.8027{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 344.7557{{c}}


{{Optimal ET sequence|legend=0| 87, 181 }}
{{Optimal ET sequence|legend=0| 87, 181 }}


Badness (Smith): 0.045920
Badness (Sintel): 1.52


=== 13-limit ===
=== 13-limit ===
Line 478: Line 637:
Comma list: 325/324, 352/351, 385/384, 1575/1573
Comma list: 325/324, 352/351, 385/384, 1575/1573


Mapping: {{mapping| 1 8 18 -20 17 -2 | 0 -9 -22 32 -19 8 }}
Mapping: {{mapping| 1 -1 -4 12 -2 6 | 0 9 22 -32 19 -8 }}


Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~18/11 = 855.2369{{c}}
Optimal tunings:
* WE: ~2 = 1200.0662{{c}}, ~11/9 = 344.7804{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 344.7617{{c}}


{{Optimal ET sequence|legend=0| 87, 181 }}
{{Optimal ET sequence|legend=0| 87, 181 }}


Badness (Smith): 0.026257
Badness (Sintel): 1.08


=== 17-limit ===
=== 17-limit ===
Line 491: Line 652:
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 8 18 -20 17 -2 44 | 0 -9 -22 32 -19 8 -56 }}
Mapping: {{mapping| 1 -1 -4 12 -2 6 -12 | 0 9 22 -32 19 -8 56 }}


Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~18/11 = 855.2495{{c}}
Optimal tunings:
* WE: ~2 = 1200.0346{{c}}, ~11/9 = 344.7589{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 344.7492{{c}}


{{Optimal ET sequence|legend=0| 87, 94, 181 }}
{{Optimal ET sequence|legend=0| 87, 94, 181 }}


Badness (Smith): 0.022749
Badness (Sintel): 1.16


=== 19-limit ===
=== 19-limit ===
Line 504: Line 667:
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 8 18 -20 17 -2 44 52 | 0 -9 -22 32 -19 8 -56 -67 }}
Mapping: {{mapping| 1 -1 -4 12 -2 6 -12 -15 | 0 9 22 -32 19 -8 56 67 }}


Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~18/11 = 855.2534{{c}}
Optimal tunings:
* WE: ~2 = 1200.0282{{c}}, ~11/9 = 344.7532{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 344.7453{{c}}


{{Optimal ET sequence|legend=0| 87, 94, 181 }}
{{Optimal ET sequence|legend=0| 87, 94, 181 }}


Badness (Smith): 0.019585
Badness (Sintel): 1.19


=== 23-limit ===
=== 23-limit ===
Line 517: Line 682:
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 8 18 -20 17 -2 44 52 48 | 0 -9 -22 32 -19 8 -56 -67 -61 }}
Mapping: {{mapping| 1 -1 -4 12 -2 6 -12 -15 -13 | 0 9 22 -32 19 -8 56 67 61 }}


Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~18/11 = 855.2576{{c}}
Optimal tunings:
* WE: ~2 = 1200.0163{{c}}, ~11/9 = 344.7461{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 344.7416{{c}}


{{Optimal ET sequence|legend=0| 87, 94, 181 }}
{{Optimal ET sequence|legend=0| 87, 94, 181 }}


Badness (Smith): 0.016332
Badness (Sintel): 1.17
 
== Quanic ==
Quanic may be described as the {{nowrap| 94 & 111 }} temperament. It splits the perfect fifth into five generators which in the 13-limit extension may be taken as ~13/12; its ploidacot is thus pentacot. [[205edo]] may be recommended as a tuning.  


== Ketchup ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 5120/5103, 1071875/1062882
[[Comma list]]: 5120/5103, 5832000/5764801


{{Mapping|legend=1| 2 3 4 6 | 0 4 15 -9 }}
{{Mapping|legend=1| 1 1 -4 0 | 0 5 54 24 }}
: mapping generators: ~2, ~160/147


[[Optimal tuning]] ([[POTE]]): ~1225/864 = 600.000{{c}}, ~64/63 = 25.719{{c}}
[[Optimal tuning]]s:
: mapping generators: ~1225/864, ~64/63
* [[WE]]: ~2 = 1199.6159{{c}}, ~160/147 = 140.4483{{c}}
: [[error map]]: {{val| -0.384 -0.098 -0.570 +1.933 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~160/147 = 140.4862{{c}}
: error map: {{val| 0.000 +0.476 -0.061 +2.842 }}


{{Optimal ET sequence|legend=1| 46, 94, 140 }}
{{Optimal ET sequence|legend=1| 94, 111, 205 }}


[[Badness]] (Smith): 0.084538
[[Badness]] (Sintel): 4.54


=== 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, 1331/1323, 5120/5103


Mapping: {{mapping| 2 3 4 6 7 | 0 4 15 -9 -2 }}
Mapping: {{mapping| 1 1 -4 0 1 | 0 5 54 24 21 }}


Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~64/63 = 25.693{{c}}
Optimal tunings:  
* WE: ~2 = 1199.7834{{c}}, ~88/81 = 140.4635{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~88/81 = 140.4850{{c}}


{{Optimal ET sequence|legend=0| 46, 94, 140 }}
{{Optimal ET sequence|legend=0| 94, 111, 205 }}


Badness (Smith): 0.039555
Badness (Sintel): 1.94


=== 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, 1331/1323


Mapping: {{mapping| 2 3 4 6 7 8 | 0 4 15 -9 -2 -14 }}
Mapping: {{mapping| 1 1 -4 0 1 3 | 0 5 54 24 21 6 }}


Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~66/65 = 25.697{{c}}
Optimal tunings:  
* WE: ~2 = 1199.6639{{c}}, ~13/12 = 140.4562{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.4904{{c}}


{{Optimal ET sequence|legend=0| 46, 94, 140 }}
{{Optimal ET sequence|legend=0| 94, 111, 205 }}


Badness (Smith): 0.024824
Badness (Sintel): 1.34


=== 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: 352/351, 442/441, 540/539, 715/714, 847/845


Mapping: {{mapping| 2 3 4 6 7 8 8 | 0 4 15 -9 -2 -14 4 }}
Mapping: {{mapping| 1 1 -4 0 1 3 -2 | 0 5 54 24 21 6 52 }}


Optimal tuning (POTE): ~17/12 = 600.000{{c}}, ~66/65 = 25.701{{c}}
Optimal tunings:  
* WE: ~2 = 1199.6699{{c}}, ~13/12 = 140.4586{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.4920{{c}}


{{Optimal ET sequence|legend=0| 46, 94, 140 }}
{{Optimal ET sequence|legend=0| 94, 111, 205 }}


Badness (Smith): 0.016591
Badness (Sintel): 1.08


=== 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: 352/351, 400/399, 442/441, 456/455, 495/494, 715/714


Mapping: {{mapping| 2 3 4 6 7 8 8 9 | 0 4 15 -9 -2 -14 4 -12 }}
Mapping: {{mapping| 1 1 -4 0 1 3 -2 -5 | 0 5 54 24 21 6 52 79 }}


Optimal tuning (POTE): ~17/12 = 600.000{{c}}, ~66/65 = 25.660{{c}}
Optimal tunings:  
* WE: ~2 = 1199.6745{{c}}, ~13/12 = 140.4574{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.4908{{c}}


{{Optimal ET sequence|legend=0| 46, 94, 140h, 234eh }}
{{Optimal ET sequence|legend=0| 94, 111, 205 }}
 
Badness (Sintel): 1.05
 
== Jorgensen ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Jorgensen]].''
 
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>.


Badness (Smith): 0.018170
[[Subgroup]]: 2.3.5.7


=== 23-limit ===
[[Comma list]]: 5120/5103, 578509309952/576650390625
Subgroup: 2.3.5.7.11.13.17.19.23


Comma list: 190/189, 209/208, 253/252, 289/288, 323/322, 352/351, 385/384
{{Mapping|legend=1| 70 111 0 34 | 0 0 1 1 }}
: mapping generators: ~50421/50000, ~5


Mapping: {{mapping| 2 3 4 6 7 8 8 9 9 | 0 4 15 -9 -2 -14 4 -12 1 }}
[[Optimal tuning]]s:  
* [[WE]]: ~50421/50000 = 17.1387{{c}}, ~5/4 = 386.8071{{c}}
: [[error map]]: {{val| -0.288 +0.445 -0.084 +0.121 }}
* [[CWE]]: ~50421/50000 = 17.1429{{c}}, ~5/4 = 386.6593{{c}}
: error map: {{val| 0.000 +0.902 +0.346 +0.690 }}


Optimal tuning (POTE): ~17/12 = 600.000{{c}}, ~66/65 = 25.661{{c}}
{{Optimal ET sequence|legend=1| 70, 140, 350, 490 }}


{{Optimal ET sequence|legend=0| 46, 94, 140h, 234ehi }}
[[Badness]] (Sintel): 5.40


Badness (Smith): 0.014033
== References ==


[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Pages with mostly numerical content]]
[[Category:Aberschismic temperaments| ]] <!-- main article -->
[[Category:Hemifamity temperaments| ]] <!-- main article -->
[[Category:Hemifamity| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Aberschismic_temperaments"