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Aberschismic temperaments: Difference between revisions

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The hemifamity temperaments temper out the hemifamity comma, {{monzo| 10 -6 1 -1 }} = [[5120/5103]], dividing an exact or approximate septimal diesis, {{monzo| 2 2 -1 -1 }} = [[36/35]] into two equal steps.  
{{Technical data page}}
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]].  


Belonging to it and considered below are buzzard, undecental, leapday, mystery, quanic and ketchup. Other hemifamity temperaments are [[Meantone family #Dominant|dominant]], [[Schismatic family #Garibaldi|garibaldi]], [[Breedsmic temperaments #Hemififths|hemififths]], [[Ragismic microtemperaments #Amity|amity]], [[Hemimean clan #Misty|misty]], [[Gamelismic clan #Rodan|rodan]], [[Kleismic family #Countercata|countercata]] and [[Mirkwai clan #Kwai|kwai]].
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.  


= Buzzard =
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.
{{see also| Vulture family }}


Subgroup: 2.3.5.7
Temperaments discussed elsewhere are:
* [[Dominant (temperament)|Dominant]] (+36/35) → [[Meantone family #Dominant|Meantone family]]
* [[Garibaldi]] (+225/224) → [[Schismatic family #Garibaldi|Schismatic family]]
* [[Diaschismic]] (+126/125) → [[Diaschismic family #Septimal diaschismic|Diaschismic family]]
* [[Hemififths]] (+2401/2400) → [[Breedsmic temperaments #Hemififths|Breedsmic temperaments]]
* [[Rodan]] (+245/243) → [[Gamelismic clan #Rodan|Gamelismic clan]]
* ''[[Alphatrimot]]'' (+2430/2401) → [[Alphatricot family #Alphatrimot|Alphatricot 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]]
* ''[[Undim]]'' (+390625/388962) → [[Undim family #Septimal undim|Undim family]]
* ''[[Quinticosiennic]]'' (+395136/390625) → [[Quintaleap family #Quinticosiennic|Quintaleap family]]
* ''[[Quintakwai]]'' (+9765625/9680832) → [[Quindromeda family #Quintakwai|Quindromeda family]]
* [[Amity]] (+4375/4374) → [[Amity family #Septimal amity|Amity 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]]
* ''[[Alphaquarter]]'' (+29360128/29296875) → [[Escapade family #Alphaquarter|Escapade family]]


[[Comma list]]: 1728/1715, 5120/5103
Considered below are septiquarter, kwai, ketchup, undecental, leapday, mystery, hemidromeda, countriton, artoneutral, quanic and jorgensen, in the order of increasing [[TE logflat badness]].


[[Mapping]]: [{{val| 1 0 -6 4 }}, {{val| 0 4 21 -3 }}]
== 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.


{{Multival|legend=1| 4 21 -3 24 -16 -66 }}
[[Subgroup]]: 2.3.5.7


[[POTE generator]]: ~21/16 = 475.636
[[Comma list]]: 5120/5103, 420175/419904
 
{{Mapping|legend=1| 1 -4 -28 6 | 0 7 38 -4 }}
: mapping generators: ~2, ~243/140


{{Val list|legend=1| 5, 43c, 48, 53, 111, 164d, 275d }}
[[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 }}


[[Badness]]: 0.047963
{{Optimal ET sequence|legend=1| 94, 99, 292, 391, 881bd, 1272bcd }}


== 11-limit ==
[[Badness]] (Sintel): 1.36


=== Semiseptiquarter ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 176/175, 540/539, 5120/5103
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 }}


Mapping: [{{val| 1 0 -6 4 -12 }}, {{val| 0 4 21 -3 39 }}]
Badness (Sintel): 2.12


POTE generator: ~21/16 = 475.700
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Vals: {{Val list| 53, 58, 111, 280cd, 391cd }}
Comma list: 352/351, 847/845, 1716/1715, 14641/14580


Badness: 0.034484
Mapping: {{mapping| 2 -8 -56 12 -25 9 | 0 7 38 -4 20 -1 }}


=== 13-limit ===
Optimal tunings:
* WE: ~99/70 = 599.8565{{c}}, ~210/121 = 957.3261{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~210/121 = 957.5508{{c}}
 
{{Optimal ET sequence|legend=0| 94, 198, 490f }}


Subgroup: 2.3.5.7.11.13
Badness (Sintel): 1.44


Comma list: 176/175, 351/350, 540/539, 676/675
== Kwai ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Kwai]].''


Mapping: [{{val| 1 0 -6 4 -12 -7 }}, {{val| 0 4 21 -3 39 27 }}]
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 }}.


POTE generator: ~21/16 = 475.697
[[Subgroup]]: 2.3.5.7


Vals: {{Val list| 53, 58, 111, 280cdf, 391cdf }}
[[Comma list]]: 5120/5103, 16875/16807


Badness: 0.018842
{{Mapping|legend=1| 1 0 -50 -40 | 0 1 33 27 }}
: mapping generators: ~2, ~3


=== 17-limit ===
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.7337{{c}}, ~3/2 = 702.4600{{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 }}


Subgroup: 2.3.5.7.11.13.17
{{Optimal ET sequence|legend=1| 41, 111, 152, 345, 497d }}


Comma list: 176/175, 256/255, 351/350, 442/441, 540/539
[[Badness]] (Sintel): 1.38


Mapping: [{{val| 1 0 -6 4 -12 -7 14 }}, {{val| 0 4 21 -3 39 27 -25 }}]
=== 11-limit ===
Subgroup: 2.3.5.7.11


POTE generator: ~21/16 = 475.692
Comma list: 540/539, 1375/1372, 5120/5103


Vals: {{Val list| 53, 58, 111, 321cdfg }}
Mapping: {{mapping| 1 0 -50 -40 32 | 0 1 33 27 -18 }}


Badness: 0.018403
Optimal tunings:  
* WE: ~2 = 1199.6672{{c}}, ~3/2 = 702.4282{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6189{{c}}


=== 19-limit ===
{{Optimal ET sequence|legend=0| 41, 111, 152, 497de, 649dde }}


Subgroup: 2.3.5.7.11.13.17.19
Badness (Sintel): 0.867


Comma list: 176/175, 256/255, 286/285, 324/323, 351/350, 540/539
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Mapping: [{{val| 1 0 -6 4 -12 -7 14 -12 }}, {{val| 0 4 21 -3 39 27 -25 41 }}]
Comma list: 352/351, 540/539, 729/728, 1375/1372


POTE generator: ~21/16 = 475.679
Mapping: {{mapping| 1 0 -50 -40 32 27 | 0 1 33 27 -18 -21 }}


Vals: {{Val list| 53, 58h, 111 }}
Optimal tunings:  
* WE: ~2 = 1199.4772{{c}}, ~3/2 = 702.3379{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6409{{c}}


Badness: 0.015649
{{Optimal ET sequence|legend=0| 41, 111, 152f, 415dff }}


== Buteo ==
Badness (Sintel): 1.01


Subgroup: 2.3.5.7.11
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


Comma list: 99/98, 385/384, 2200/2187
Comma list: 256/255, 352/351, 540/539, 715/714, 1089/1088


Mapping: [{{val| 1 0 -6 4 9 }}, {{val| 0 4 21 -3 -14 }}]
Mapping: {{mapping| 1 0 -50 -40 32 27 58 | 0 1 33 27 -18 -21 -34 }}


POTE generator: ~21/16 = 475.436
Optimal tunings:  
* WE: ~2 = 1199.3537{{c}}, ~3/2 = 702.2850{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6589{{c}}


Vals: {{Val list| 5, 48, 53 }}
{{Optimal ET sequence|legend=0| 41, 70, 111, 152fg, 263dfg }}


Badness: 0.060238
Badness (Sintel): 1.12


=== 13-limit ===
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


Subgroup: 2.3.5.7.11.13
Comma list: 256/255, 352/351, 400/399, 456/455, 715/714, 847/845


Comma list: 99/98, 275/273, 385/384, 572/567
Mapping: {{mapping| 1 0 -50 -40 32 27 58 -56 | 0 1 33 27 -18 -21 -34 38 }}


Mapping: [{{val| 1 0 -6 4 9 -7 }}, {{val| 0 4 21 -3 -14 27 }}]
Optimal tunings:  
* WE: ~2 = 1199.3401{{c}}, ~3/2 = 702.2705{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6548{{c}}


POTE generator: ~21/16 = 475.464
{{Optimal ET sequence|legend=0| 41, 70h, 111, 152fg, 263dfgh }}


Vals: {{Val list| 5, 48f, 53 }}
Badness (Sintel): 1.03


Badness: 0.039854
==== Hemikwai ====
Subgroup: 2.3.5.7.11.13


= Undecental =
Comma list: 540/539, 676/675, 1375/1372, 5120/5103


Subgroup: 2.3.5.7
Mapping: {{mapping| 1 0 -50 -40 32 -51 | 0 2 66 54 -36 69 }}
: mapping generators: ~2, ~26/15


[[Comma list]]: 5120/5103, 235298/234375
Optimal tunings:  
* WE: ~2 = 1199.6968{{c}}, ~26/15 = 951.0740{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.3123{{c}}


[[Mapping]]: [{{val| 1 0 61 71 }}, {{val| 0 1 -37 -43 }}]
{{Optimal ET sequence|legend=0| 82, 111, 193, 304d }}


{{Multival|legend=1| 1 -37 -43 -61 -71 4 }}
Badness (Sintel): 1.82


[[POTE generator]]: ~3/2 = 703.039
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


{{Val list|legend=1| 29, 70, 99, 722bc, 821bc, 920bc, 1019bc, 1118bbcc, 1217bbcc, 1316bbccd }}
Comma list: 442/441, 540/539, 676/675, 715/714, 5120/5103


[[Badness]]: 0.094603
Mapping: {{mapping| 1 0 -50 -40 32 -51 -30 | 0 2 66 54 -36 69 43 }}


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


Subgroup: 2.3.5
{{Optimal ET sequence|legend=0| 82, 111, 193, 304d }}


[[Comma list]]: 10737418240/10460353203
Badness (Sintel): 1.31


[[Mapping]]: [{{val| 1 0 -31 }}, {{val| 0 1 21 }}]
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


[[POTE generator]]: ~3/2 = 704.179
Comma list: 400/399, 442/441, 540/539, 676/675, 715/714, 1445/1444


{{Val list|legend=1| 29, 46, 121, 167, 455bc, 622bbc }}
Mapping: {{mapping| 1 0 -50 -40 32 -51 -30 -56 | 0 2 66 54 -36 69 43 76 }}


[[Badness]]: 0.523182
Optimal tunings:  
* WE: ~2 = 1199.6718{{c}}, ~26/15 = 951.0526{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.3103{{c}}


== 7-limit ==
{{Optimal ET sequence|legend=0| 82, 111, 193, 304dh }}


Subgroup: 2.3.5.7
Badness (Sintel): 1.16


[[Comma list]]: 686/675, 5120/5103
== 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.


[[Mapping]]: [{{val| 1 0 -31 -21 }}, {{val| 0 1 21 15 }}]
[[Subgroup]]: 2.3.5.7


{{Multival|legend=1| 1 21 15 31 21 -24 }}
[[Comma list]]: 5120/5103, 1071875/1062882


[[POTE generator]]: ~3/2 = 704.263
{{Mapping|legend=1| 2 3 4 6 | 0 4 15 -9 }}
: mapping generators: ~1225/864, ~64/63


{{Val list|legend=1| 17c, 29, 46, 167d, 213d, 259cdd, 305bcdd }}
[[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 }}


[[Badness]]: 0.096123
{{Optimal ET sequence|legend=1| 46, 94, 140 }}


== 11-limit ==
[[Badness]] (Sintel): 2.14


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


POTE generator: ~3/2 = 704.250
Optimal tunings:  
* WE: ~99/70 = 600.0678{{c}}, ~64/63 = 25.6963{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~64/63 = 25.6956{{c}}


Vals: {{Val list| 17c, 29, 46, 167de, 213de, 259cdde }}
{{Optimal ET sequence|legend=0| 46, 94, 140 }}


Badness: 0.038624
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: [{{val| 1 0 -31 -21 -14 -9 }}, {{val| 0 1 21 15 11 8 }}]


POTE generator: ~3/2 = 704.214
Mapping: {{mapping| 2 3 4 6 7 8 | 0 4 15 -9 -2 -14 }}


Vals: {{Val list| 17c, 29, 46, 121def, 167def, 213deff }}
Optimal tunings:  
* WE: ~99/70 = 600.0612{{c}}, ~66/65 = 25.7000{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~66/65 = 25.6978{{c}}


Badness: 0.024732
{{Optimal ET sequence|legend=0| 46, 94, 140 }}


== 17-limit ==
Badness (Sintel): 1.03


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


POTE generator: ~3/2 = 704.229
Optimal tunings:  
* WE: ~17/12 = 600.0896{{c}}, ~66/65 = 25.7048{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~66/65 = 25.7017{{c}}


Vals: {{Val list| 17cg, 29g, 46, 121defg, 167defg, 213deffg }}
{{Optimal ET sequence|legend=0| 46, 94, 140 }}


Badness: 0.017863
Badness (Sintel): 0.845


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


Subgroup: 2.3.5.7.11.13.17.19
Comma list: 253/252, 289/288, 325/324, 352/351, 385/384, 391/390


Comma list: 91/90, 121/120, 133/132, 136/135, 154/153, 169/168
Mapping: {{mapping| 2 3 4 6 7 8 8 9 | 0 4 15 -9 -2 -14 4 1 }}


Mapping: [{{val| 1 0 -31 -21 -14 -9 -34 9 }}, {{val| 0 1 21 15 11 8 24 -3 }}]
Optimal tunings:  
* WE: ~17/12 = 600.1139{{c}}, ~66/65 = 25.7053{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~66/65 = 25.7013{{c}}


POTE generator: ~3/2 = 704.135
{{Optimal ET sequence|legend=0| 46, 94, 140 }}


Vals: {{Val list| 17cg, 29g, 46, 75dfgh, 121defgh }}
Badness (Sintel): 0.772


Badness: 0.017356
== 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> }}.  


=== Leapling ===
[[Subgroup]]: 2.3.5.7


Subgroup: 2.3.5.7.11.13.17.19
[[Comma list]]: 5120/5103, 235298/234375


Comma list: 77/76, 91/90, 121/120, 136/135, 153/152, 169/168
{{Mapping|legend=1| 1 0 61 71 | 0 1 -37 -43 }}
: mapping generators: ~2, ~3


Mapping: [{{val| 1 0 -31 -21 -14 -9 -34 -37 }}, {{val| 0 1 21 15 11 8 24 26 }}]
[[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 }}


POTE generator: ~3/2 = 704.123
{{Optimal ET sequence|legend=1| 29, 70, 99, 722bc, 821bc, 920bc, 1019bc }}


Vals: {{Val list| 17cgh, 29g, 46h, 75dfg, 121defghh }}
[[Badness]] (Sintel): 2.39


Badness: 0.019065
== Leapday ==
{{Main| Leapday }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Leapday]].''


= Mystery =
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.


Subgroup: 2.3.5.7
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.  


[[Comma list]]: 5120/5103, 50421/50000
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.


[[Mapping]]: [{{val| 29 46 0 14 }}, {{val| 0 0 1 1 }}]
[[Subgroup]]: 2.3.5.7


{{Multival|legend=1| 0 29 29 46 46 -14 }}
[[Comma list]]: 686/675, 5120/5103


[[POTE generator]]: ~5/4 = 388.646
{{Mapping|legend=1| 1 0 -31 -21 | 0 1 21 15 }}
: mapping generators: ~2, ~3


{{Val list|legend=1| 29, 58, 87, 145 }}
[[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 }}


[[Badness]]: 0.1037
{{Optimal ET sequence|legend=1| 17c, 29, 46 }}


== 11-limit ==
[[Badness]] (Sintel): 2.43


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


Comma list: 441/440, 896/891, 3388/3375
Comma list: 121/120, 441/440, 686/675
 
Mapping: {{mapping| 1 0 -31 -21 -14 | 0 1 21 15 11 }}


Mapping: [{{val| 29 46 0 14 33 }}, {{val| 0 0 1 1 1 }}]
Optimal tunings:  
* WE: ~2 = 1200.0731{{c}}, ~3/2 = 704.2933{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2538{{c}}


POTE generator: ~5/4 = 388.460
{{Optimal ET sequence|legend=0| 17c, 29, 46 }}


Vals: {{Val list| 29, 58, 87, 145 }}
Badness (Sintel): 1.28


Badness: 0.0343
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


== 13-limit ==
Comma list: 91/90, 121/120, 169/168, 352/351


Subgroup: 2.3.5.7.11.13
Mapping: {{mapping| 1 0 -31 -21 -14 -9 | 0 1 21 15 11 8 }}


Comma list: 196/195, 352/351, 364/363, 676/675
Optimal tunings:  
* WE: ~2 = 1200.4758{{c}}, ~3/2 = 704.4930{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2346{{c}}


Mapping: [{{val| 29 46 0 14 33 40 }}, {{val| 0 0 1 1 1 1 }}]
{{Optimal ET sequence|legend=0| 17c, 29, 46, 121def }}


POTE generator: ~5/4 = 388.354
Badness (Sintel): 1.02


Vals: {{Val list| 29, 58, 87, 145, 232, 377 }}
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Badness: 0.0186
Comma list: 91/90, 121/120, 136/135, 154/153, 169/168


= Quanic =
Mapping: {{mapping| 1 0 -31 -21 -14 -9 -34 | 0 1 21 15 11 8 24 }}


Subgroup: 2.3.5.7
Optimal tunings:  
* WE: ~2 = 1200.4818{{c}}, ~3/2 = 704.5121{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2507{{c}}


[[Comma list]]: 5120/5103, 5832000/5764801
{{Optimal ET sequence|legend=0| 17cg, 29g, 46, 121defg }}


[[Mapping]]: [{{val| 1 1 -4 0 }}, {{val| 0 5 54 24 }}]
Badness (Sintel): 0.910


[[POTE generator]]: ~160/147 = 140.493
=== 2.3.5.7.11.13.17.23 subgroup ===
Subgroup: 2.3.5.7.11.13.17.23


{{Val list|legend=1| 94, 111, 205 }}
Comma list: 91/90, 121/120, 136/135, 154/153, 161/160, 169/168


[[Badness]]: 0.1795
Mapping: {{mapping| 1 0 -31 -21 -14 -9 -34 -5 | 0 1 21 15 11 8 24 6 }}


== 11-limit ==
Optimal tunings:
* WE: ~2 = 1200.5169{{c}}, ~3/2 = 704.5279{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2450{{c}}


Subgroup: 2.3.5.7.11
{{Optimal ET sequence|legend=0| 17cg, 29g, 46, 121defg }}


Comma list: 540/539, 1331/1323, 5120/5103
Badness (Sintel): 0.872


Mapping: [{{val| 1 1 -4 0 1 }}, {{val| 0 5 54 24 21 }}]
== Mystery ==
{{Main| Mystery }}
: ''For the 5-limit version, see [[29th-octave temperaments #Mystery]].''


POTE generator: ~88/81 = 140.489
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.  


Vals: {{Val list| 94, 111, 205 }}
[[Subgroup]]: 2.3.5.7


Badness: 0.0587
[[Comma list]]: 5120/5103, 50421/50000


== 13-limit ==
{{Mapping|legend=1| 29 46 0 14 | 0 0 1 1 }}
: mapping generators: ~50/49, ~5


Subgroup: 2.3.5.7.11.13
[[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 }}


Comma list: 352/351, 540/539, 729/728, 1331/1323
{{Optimal ET sequence|legend=1| 29, 58, 87, 145 }}


Mapping: [{{val| 1 1 -4 0 1 3 }}, {{val| 0 5 54 24 21 6 }}]
[[Badness]] (Sintel): 2.63


POTE generator: ~13/12 = 140.496
=== 11-limit ===
Subgroup: 2.3.5.7.11


Vals: {{Val list| 94, 111, 205 }}
Comma list: 441/440, 896/891, 3388/3375


Badness: 0.0325
Mapping: {{mapping| 29 46 0 14 33 | 0 0 1 1 1 }}


== 17-limit ==
Optimal tunings:
* WE: ~45/44 = 41.3637{{c}}, ~5/4 = 388.3136{{c}}
* CWE: ~45/44 = 41.3793{{c}}, ~5/4 = 388.0598{{c}}


Subgroup: 2.3.5.7.11.13.17
{{Optimal ET sequence|legend=0| 29, 58, 87, 145 }}


Comma list: 352/351, 442/441, 540/539, 715/714, 847/845
Badness (Sintel): 1.13


Mapping: [{{val| 1 1 -4 0 1 3 -2 }}, {{val| 0 5 54 24 21 6 52 }}]
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


POTE generator: ~13/12 = 140.497
Comma list: 196/195, 352/351, 364/363, 676/675


Vals: {{Val list| 94, 111, 205 }}
Mapping: {{mapping| 29 46 0 14 33 40 | 0 0 1 1 1 1 }}


Badness: 0.0211
Optimal tunings:  
* WE: ~45/44 = 41.3623{{c}}, ~5/4 = 388.1942{{c}}
* CWE: ~40/39 = 41.3793{{c}}, ~5/4 = 387.9017{{c}}


== 19-limit ==
{{Optimal ET sequence|legend=0| 29, 58, 87, 145, 232 }}


Subgroup: 2.3.5.7.11.13.17.19
Badness (Sintel): 0.768


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


Mapping: [{{val| 1 1 -4 0 1 3 -2 -5 }}, {{val| 0 5 54 24 21 6 52 79 }}]
[[Subgroup]]: 2.3.5.7


POTE generator: ~13/12 = 140.496
[[Comma list]]: 5120/5103, 52734375/52706752


Vals: {{Val list| 94, 111, 205 }}
{{Mapping|legend=1| 1 0 38 48 | 0 2 -45 -57 }}
: mapping generator: ~2, ~12500/7203


Badness: 0.0173
[[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 }}


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


Subgroup: 2.3.5.7
[[Badness]] (Sintel): 2.93


[[Comma list]]: 5120/5103, 118098/117649
=== 11-limit ===
Subgroup: 2.3.5.7.11


[[Mapping]]: [{{val| 2 1 -12 2 }}, {{val| 0 3 23 5 }}]
Comma list: 1331/1323, 1375/1372, 5120/5103


{{Multival|legend=1| 6 46 10 59 -1 -106 }}
Mapping: {{mapping| 1 0 38 48 32 | 0 2 -45 -57 -36 }}


[[POTE generator]]: ~9/7 = 434.218
Optimal tunings:  
* WE: ~2 = 1199.8767{{c}}, ~400/231 = 951.3065{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~400/231 = 951.4063{{c}}


{{Val list|legend=1| 58, 94, 152 }}
{{Optimal ET sequence|legend=0| 29, 82cd, 111, 140, 251, 391e }}


[[Badness]]: 0.092748
Badness (Sintel): 2.01


== 11-limit ==
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Subgroup: 2.3.5.7.11
Comma list: 352/351, 676/675, 847/845, 1331/1323


Comma list: 540/539, 4000/3993, 5120/5103
Mapping: {{mapping| 1 0 38 48 32 37 | 0 2 -45 -57 -36 -42 }}


Mapping: [{{val| 2 1 -12 2 -9 }}, {{val| 0 3 23 5 22 }}]
Optimal tunings:  
* WE: ~2 = 1199.8753{{c}}, ~26/15 = 951.3054{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.4064{{c}}


POTE generator: ~9/7 = 434.217
{{Optimal ET sequence|legend=0| 29, 82cdf, 111, 140, 251, 391e }}


Vals: {{Val list| 58, 94, 152 }}
Badness (Sintel): 1.18


Badness: 0.028240
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


== 13-limit ==
Comma list: 352/351, 442/441, 561/560, 676/675, 715/714


Subgroup: 2.3.5.7.11.13
Mapping: {{mapping| 1 0 38 48 32 37 58 | 0 2 -45 -57 -36 -42 -68 }}


Comma list: 352/351, 540/539, 729/728, 1575/1573
Optimal tunings:  
* WE: ~2 = 1199.8770{{c}}, ~26/15 = 951.3039{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.4035{{c}}


Mapping: [{{val| 2 1 -12 2 -9 -2 }}, {{val| 0 3 23 5 22 13 }}]
{{Optimal ET sequence|legend=0| 29g, 82cdfg, 111, 140, 251, 391e }}


POTE generator: ~9/7 = 434.221
Badness (Sintel): 0.971


Vals: {{Val list| 58, 94, 152f }}
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


Badness: 0.021645
Comma list: 286/285, 352/351, 363/361, 442/441, 476/475, 561/560


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


Subgroup: 2.3.5.7.11.13.17
Optimal tunings:  
* WE: ~2 = 1199.7534{{c}}, ~26/15 = 951.2024{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.4020{{c}}


Comma list: 170/169, 289/288, 352/351, 442/441, 561/560
{{Optimal ET sequence|legend=0| 29g, 82cdfgh, 111, 140 }}


Mapping: [{{val| 2 1 -12 2 -9 -2 6 }}, {{val| 0 3 23 5 22 13 3 }}]
Badness (Sintel): 1.01


POTE generator: ~9/7 = 434.181
=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23


Vals: {{Val list| 58, 94, 152f }}
Comma list: 253/252, 286/285, 352/351, 363/361, 391/390, 442/441, 460/459


Badness: 0.021316
Mapping: {{mapping| 1 0 38 48 32 37 58 32 18 | 0 2 -45 -57 -36 -42 -68 -35 -17 }}


= Alphaquarter =
Optimal tunings:
* WE: ~2 = 1199.9128{{c}}, ~26/15 = 951.3371{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.4076{{c}}


Subgroup: 2.3.5.7
{{Optimal ET sequence|legend=0| 29g, 82cdfgh, 111, 140 }}


[[Comma list]]: 5120/5103, 29360128/29296875
Badness (Sintel): 1.10


[[Mapping]]: [{{val| 1 2 2 0 }}, {{val| 0 -9 7 61 }}]
== Countriton ==
: ''For the 5-limit version, see [[Schismic–Mercator equivalence continuum #Countritonic]].''


{{Multival|legend=1| 9 -7 -61 -32 -122 -122 }}
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.


[[POTE generator]]: ~16128/15625 = 55.243
Countriton was named by [[Xenllium]] in 2022 as a counterpart of [[untriton]].  


{{Val list|legend=1| 87, 152, 239, 391 }}
[[Subgroup]]: 2.3.5.7


[[Badness]]: 0.1166
[[Comma list]]: 5120/5103, 7558272/7503125


== 11-limit ==
{{Mapping|legend=1| 1 -3 -15 13 | 0 9 34 -20 }}
: mapping generators: ~2, ~1225/864


Subgroup: 2.3.5.7.11
[[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 }}


Comma list: 3025/3024, 4000/3993, 5120/5103
{{Optimal ET sequence|legend=1| 51c, 53, 157, 210, 473cdd }}


Mapping: [{{val| 1 2 2 0 3 }}, {{val| 0 -9 7 61 10 }}]
[[Badness]] (Sintel): 3.32


POTE generator: ~33/32 = 55.243
=== 11-limit ===
Subgroup: 2.3.5.7.11


Vals: {{Val list| 87, 152, 239, 391 }}
Comma list: 176/175, 5120/5103, 41503/41472


Badness: 0.0296
Mapping: {{mapping| 1 -3 -15 13 -21 | 0 9 34 -20 48 }}


= Septiquarter =
Optimal tunings:
* WE: ~2 = 1199.5178{{c}}, ~77/54 = 611.2097{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~77/54 = 611.4495{{c}}


Subgroup: 2.3.5.7
{{Optimal ET sequence|legend=0| 51ce, 53, 104c, 157 }}


[[Comma list]]: 5120/5103, 420175/419904
Badness (Sintel): 2.80


[[Mapping]]: [{{val| 1 3 10 2 }}, {{val| 0 -7 -38 4 }}]
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


{{Multival|legend=1| 7 38 -4 44 -26 -116 }}
Comma list: 176/175, 351/350, 847/845, 2197/2187


[[POTE generator]]: ~147/128 = 242.453
Mapping: {{mapping| 1 -3 -15 13 -21 -7 | 0 9 34 -20 48 21 }}


{{Val list|legend=1| 94, 99, 292, 391, 881bd, 1272bcd }}
Optimal tunings:
* WE: ~2 = 1199.5944{{c}}, ~77/54 = 611.2491{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~77/54 = 611.4506{{c}}


[[Badness]]: 0.0538
{{Optimal ET sequence|legend=0| 51ce, 53, 104c, 157 }}


== Semiseptiquarter ==
Badness (Sintel): 1.75


Subgroup: 2.3.5.7.11
== 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.  


Comma list: 5120/5103, 9801/9800, 14641/14580
Artoneutral was named by [[Flora Canou]] in 2023 for its generator's quality.


Mapping: [{{val| 2 6 20 4 15 }}, {{val| 0 -7 -38 4 -20 }}]
[[Subgroup]]: 2.3.5.7


POTE generators: ~121/105 = 242.4511
[[Comma list]]: 5120/5103, 3828125/3779136


Vals: {{Val list| 94, 198, 292, 490 }}
{{Mapping|legend=1| 1 -1 -4 12 | 0 9 22 -32 }}
: mapping generators: ~2, ~128/105


Badness: 0.0642
[[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 }}


=== 13-limit ===
{{Optimal ET sequence|legend=1| 87, 94, 181 }}


Subgroup: 2.3.5.7.11.13
[[Badness]] (Sintel): 3.98


Comma list: 352/351, 847/845, 1716/1715, 14641/14580
=== 11-limit ===
Subgroup: 2.3.5.7.11


Mapping: [{{val| 2 6 20 4 15 7 }}, {{val| 0 -7 -38 4 -20 1 }}]
Comma list: 385/384, 2200/2187, 4000/3993


POTE generators: ~121/105 = 242.4448
Mapping: {{mapping| 1 -1 -4 12 -2 | 0 9 22 -32 19 }}


Vals: {{Val list| 94, 198, 490f }}
Optimal tunings:  
* WE: ~2 = 1200.1668{{c}}, ~11/9 = 344.8027{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 344.7557{{c}}


Badness: 0.0348
{{Optimal ET sequence|legend=0| 87, 181 }}


= Tricot (aka Trimot) =
Badness (Sintel): 1.52
{{see also|Tricot family}}


The generator for tricot is the real cube root of third harmonic, 3<sup>1/3</sup>, tuned between 63/44 and 13/9. Tricot can be described as 53&amp;70 temperament (also known as ''trimot''), tempering out the [[tricot comma]], {{monzo| 39 -29 3 }} in the 5-limit, 2430/2401 (nuwell comma) and 5120/5103 in the 7-limit, 99/98 and 121/120 in the 11-limit, 169/168, 352/351, 640/637, and 729/728 in the 13-limit.
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Subgroup: 2.3.5.7
Comma list: 325/324, 352/351, 385/384, 1575/1573


[[Comma list]]: 2430/2401, 5120/5103
Mapping: {{mapping| 1 -1 -4 12 -2 6 | 0 9 22 -32 19 -8 }}


[[Mapping]]: [{{val| 1 0 -13 -3 }}, {{val| 0 3 29 11 }}]
Optimal tunings:  
* WE: ~2 = 1200.0662{{c}}, ~11/9 = 344.7804{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 344.7617{{c}}


{{Multival|legend=1| 3 29 11 39 9 -56 }}
{{Optimal ET sequence|legend=0| 87, 181 }}


[[POTE generator]]: ~81/56 = 634.026
Badness (Sintel): 1.08


{{Val list|legend=1| 17c, 36c, 53, 229dd, 282dd }}
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


[[Badness]]: 0.100127
Comma list: 325/324, 352/351, 375/374, 385/384, 595/594


== 11-limit ==
Mapping: {{mapping| 1 -1 -4 12 -2 6 -12 | 0 9 22 -32 19 -8 56 }}


Subgroup: 2.3.5.7.11
Optimal tunings:  
* WE: ~2 = 1200.0346{{c}}, ~11/9 = 344.7589{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 344.7492{{c}}


Comma list: 99/98, 121/120, 5120/5103
{{Optimal ET sequence|legend=0| 87, 94, 181 }}


Mapping: [{{val| 1 0 -13 -3 -5 }}, {{val| 0 3 29 11 16 }}]
Badness (Sintel): 1.16


POTE generator: ~63/44 = 634.027
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


Vals: {{Val list| 17c, 36ce, 53, 70, 123de }}
Comma list: 325/324, 352/351, 375/374, 385/384, 400/399, 595/594


Badness: 0.056134
Mapping: {{mapping| 1 -1 -4 12 -2 6 -12 -15 | 0 9 22 -32 19 -8 56 67 }}


== 13-limit ==
Optimal tunings:
* WE: ~2 = 1200.0282{{c}}, ~11/9 = 344.7532{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 344.7453{{c}}


Subgroup: 2.3.5.7.11.13
{{Optimal ET sequence|legend=0| 87, 94, 181 }}


Comma list: 99/98, 121/120, 169/168, 352/351
Badness (Sintel): 1.19


Mapping: [{{val| 1 0 -13 -3 -5 0 }}, {{val| 0 3 29 11 16 7 }}]
=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23


POTE generator: ~13/9 = 634.012
Comma list: 300/299, 325/324, 352/351, 375/374, 385/384, 400/399, 484/483


Vals: {{Val list| 17c, 36ce, 53, 70, 123de }}
Mapping: {{mapping| 1 -1 -4 12 -2 6 -12 -15 -13 | 0 9 22 -32 19 -8 56 67 61 }}


Badness: 0.032102
Optimal tunings:  
* WE: ~2 = 1200.0163{{c}}, ~11/9 = 344.7461{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 344.7416{{c}}


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


Subgroup: 2.3.5.7
Badness (Sintel): 1.17


[[Comma list]]: 5120/5103, 1071875/1062882
== 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.


[[Mapping]]: [{{val| 2 3 4 6 }}, {{val| 0 4 15 -9 }}]
[[Subgroup]]: 2.3.5.7


{{Multival|legend=1| 8 30 -18 29 -51 -126 }}
[[Comma list]]: 5120/5103, 5832000/5764801


[[POTE generator]]: ~64/63 = ~81/80 = 25.719
{{Mapping|legend=1| 1 1 -4 0 | 0 5 54 24 }}
: mapping generators: ~2, ~160/147


{{Val list|legend=1| 46, 94, 140 }}
[[Optimal tuning]]s:
* [[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 }}


[[Badness]]: 0.084538
{{Optimal ET sequence|legend=1| 94, 111, 205 }}


== 11-limit ==
[[Badness]] (Sintel): 4.54


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


POTE generator: ~55/54 = ~64/63 = ~81/80 = 25.693
Optimal tunings:  
* WE: ~2 = 1199.7834{{c}}, ~88/81 = 140.4635{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~88/81 = 140.4850{{c}}


Vals: {{Val list| 46, 94, 140 }}
{{Optimal ET sequence|legend=0| 94, 111, 205 }}


Badness: 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: 352/351, 540/539, 729/728, 1331/1323


Comma list: 325/324, 352/351, 847/845, 1331/1323
Mapping: {{mapping| 1 1 -4 0 1 3 | 0 5 54 24 21 6 }}


Mapping: [{{val| 2 3 4 6 7 8 }}, {{val| 0 4 15 -9 -2 -14 }}]
Optimal tunings:  
* WE: ~2 = 1199.6639{{c}}, ~13/12 = 140.4562{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.4904{{c}}


POTE generator: ~55/54 = ~64/63 = ~81/80 = 25.697
{{Optimal ET sequence|legend=0| 94, 111, 205 }}


Vals: {{Val list| 46, 94, 140 }}
Badness (Sintel): 1.34


Badness: 0.024824
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


== 17-limit ==
Comma list: 352/351, 442/441, 540/539, 715/714, 847/845


Subgroup: 2.3.5.7.11.13.17
Mapping: {{mapping| 1 1 -4 0 1 3 -2 | 0 5 54 24 21 6 52 }}


Comma list: 289/288, 325/324, 352/351, 385/384, 561/560
Optimal tunings:  
* WE: ~2 = 1199.6699{{c}}, ~13/12 = 140.4586{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.4920{{c}}


Mapping: [{{val| 2 3 4 6 7 8 8 }}, {{val| 0 4 15 -9 -2 -14 4 }}]
{{Optimal ET sequence|legend=0| 94, 111, 205 }}


POTE generator: ~55/54 = ~64/63 = ~81/80 = 25.701
Badness (Sintel): 1.08


Vals: {{Val list| 46, 94, 140 }}
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


Badness: 0.016591
Comma list: 352/351, 400/399, 442/441, 456/455, 495/494, 715/714


== 19-limit ==
Mapping: {{mapping| 1 1 -4 0 1 3 -2 -5 | 0 5 54 24 21 6 52 79 }}


Subgroup: 2.3.5.7.11.13.17.19
Optimal tunings:  
* WE: ~2 = 1199.6745{{c}}, ~13/12 = 140.4574{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.4908{{c}}


Comma list: 190/189, 209/208, 289/288, 352/351, 385/384, 561/560
{{Optimal ET sequence|legend=0| 94, 111, 205 }}


Mapping: [{{val| 2 3 4 6 7 8 8 9 }}, {{val| 0 4 15 -9 -2 -14 4 -12 }}]
Badness (Sintel): 1.05


POTE generator: ~55/54 = ~64/63 = ~81/80 = 25.660
== Jorgensen ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Jorgensen]].''


Vals: {{Val list| 46, 94, 140h, 234eh }}
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.


Badness: 0.018170
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>.  


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


Subgroup: 2.3.5.7.11.13.17.19.23
[[Comma list]]: 5120/5103, 578509309952/576650390625


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: [{{val| 2 3 4 6 7 8 8 9 9 }}, {{val| 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 }}


POTE generator: ~55/54 = ~64/63 = ~81/80 = 25.661
{{Optimal ET sequence|legend=1| 70, 140, 350, 490 }}


Vals: {{Val list| 46, 94, 140h, 234ehi }}
[[Badness]] (Sintel): 5.40


Badness: 0.014033
== References ==


[[Category:Regular temperament theory]]
[[Category:Temperament collections]]
[[Category:Temperament collection]]
[[Category:Aberschismic temperaments| ]] <!-- main article -->
[[Category:Hemifamity]]
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Aberschismic_temperaments"