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Aberschismic temperaments: Difference between revisions

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{{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]].


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. 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.
Commas: 1728/1715, 5120/5103


[[POTE_tuning|POTE generator]]: ~320/243 = 475.636
Temperaments discussed elsewhere are:
* [[Dominant (temperament)|Dominant]] (+36/35) → [[Meantone family #Dominant|Meantone family]]
* [[Garibaldi]] (+225/224) → [[Schismatic family #Garibaldi|Schismatic family]]
* [[Leapday]] (+686/675) → [[Sengic temperaments #Leapday|Sengic temperaments]]
* [[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]]


Map: [<1 0 -6 4|, <0 4 21 -3|]
Considered below are septiquarter, kwai, ketchup, undecental, mystery, hemidromeda, countriton, artoneutral, quanic and jorgensen, in the order of increasing [[TE logflat badness]].


Wedgie: <<4 21 -3 24 -16 -66||
== 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.


EDOs: 48, 53, 111, 164d, 275d
[[Subgroup]]: 2.3.5.7


Badness: 0.0480
[[Comma list]]: 5120/5103, 420175/419904


==11-limit==
{{Mapping|legend=1| 1 -4 -28 6 | 0 7 38 -4 }}
Commas: 176/175, 540/539, 5120/5103
: mapping generators: ~2, ~243/140


[[POTE_tuning|POTE generator]]: ~320/243 = 475.700
[[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 }}


Map: [<1 0 -6 4 -12|, <0 4 21 -3 39|]
{{Optimal ET sequence|legend=1| 94, 99, 292, 391, 881bd, 1272bcd }}


EDOs: 53, 58, 111, 280cd, 391cd
[[Badness]] (Sintel): 1.36


Badness: 0.0345
=== Semiseptiquarter ===
Subgroup: 2.3.5.7.11


==13-limit==
Comma list: 5120/5103, 9801/9800, 14641/14580
Commas: 176/175, 351/350, 540/539, 676/675


[[POTE_tuning|POTE generator]]: ~320/243 = 475.697
Mapping: {{mapping| 2 -8 -56 12 -25 | 0 7 38 -4 20 }}


Map: [<1 0 -6 4 -12 -7|, <0 4 21 -3 39 27|]
Optimal tunings:  
* WE: ~99/70 = 599.8953{{c}}, ~210/121 = 957.3819{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~210/121 = 957.5449{{c}}


EDOs: 53, 58, 111, 280cdf, 391cdf
{{Optimal ET sequence|legend=0| 94, 198, 292, 490 }}


Badness: 0.0188
Badness (Sintel): 2.12


==17-limit==
==== 13-limit ====
Commas: 176/175, 256/255, 351/350, 442/441, 540/539
Subgroup: 2.3.5.7.11.13


[[POTE_tuning|POTE generator]]: ~320/243 = 475.692
Comma list: 352/351, 847/845, 1716/1715, 14641/14580


Map: [<1 0 -6 4 -12 -7 14|, <0 4 21 -3 39 27 -25|]
Mapping: {{mapping| 2 -8 -56 12 -25 9 | 0 7 38 -4 20 -1 }}


EDOs: 53, 58, 111, 321cdfg
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: 0.0184
{{Optimal ET sequence|legend=0| 94, 198, 490f }}


==19-limit==
Badness (Sintel): 1.44
Commas: 176/175, 256/255, 286/285, 324/323, 351/350, 540/539


[[POTE_tuning|POTE generator]]: ~320/243 = 475.679
== Kwai ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Kwai]].''


Map: [<1 0 -6 4 -12 -7 14 -12|, <0 4 21 -3 39 27 -25 41|]
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 }}.


EDOs: 53, 58h, 111
[[Subgroup]]: 2.3.5.7


Badness: 0.0156
[[Comma list]]: 5120/5103, 16875/16807


==Buteo==
{{Mapping|legend=1| 1 0 -50 -40 | 0 1 33 27 }}
Commas: 99/98, 385/384, 2200/2187
: mapping generators: ~2, ~3


POTE generator: ~21/16 = 475.436
[[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 }}


Map: [&lt;1 0 -6 4 9|, &lt;0 4 21 -3 -14|]
{{Optimal ET sequence|legend=1| 41, 111, 152, 345, 497d }}


EDOs: 5, 48, 53
[[Badness]] (Sintel): 1.38


Badness: 0.0602
=== 11-limit ===
Subgroup: 2.3.5.7.11


===13-limit===
Comma list: 540/539, 1375/1372, 5120/5103
Commas: 99/98, 275/273, 385/384, 572/567


POTE generator: ~21/16 = 475.464
Mapping: {{mapping| 1 0 -50 -40 32 | 0 1 33 27 -18 }}


Map: [&lt;1 0 -6 4 9 -7|, &lt;0 4 21 -3 -14 27|]
Optimal tunings:  
* WE: ~2 = 1199.6672{{c}}, ~3/2 = 702.4282{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6189{{c}}


EDOs: 5, 53
{{Optimal ET sequence|legend=0| 41, 111, 152, 497de, 649dde }}


Badness: 0.0390
Badness (Sintel): 0.867


=Undecental=
==== 13-limit ====
Commas: 5120/5103, 235298/234375
Subgroup: 2.3.5.7.11.13


[[POTE_tuning|POTE generator]]: ~3/2 = 703.039
Comma list: 352/351, 540/539, 729/728, 1375/1372


Map: [&lt;1 0 61 71|, &lt;0 1 -37 -43|]
Mapping: {{mapping| 1 0 -50 -40 32 27 | 0 1 33 27 -18 -21 }}


Wedgie: &lt;&lt;1 -37 -43 -61 -71 4||
Optimal tunings:  
* WE: ~2 = 1199.4772{{c}}, ~3/2 = 702.3379{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6409{{c}}


EDOs: 12, 29, 70, 99
{{Optimal ET sequence|legend=0| 41, 111, 152f, 415dff }}


=Leapday=
Badness (Sintel): 1.01
Comma: 10737418240/10460353203


POTE generator: ~3/2 = 704.179
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


Map: [&lt;1 0 -31|, &lt;0 1 21|]
Comma list: 256/255, 352/351, 540/539, 715/714, 1089/1088


EDOs: 29, 46, 121, 167, 455bc, 622bc
Mapping: {{mapping| 1 0 -50 -40 32 27 58 | 0 1 33 27 -18 -21 -34 }}


Badness: 0.5232
Optimal tunings:  
* WE: ~2 = 1199.3537{{c}}, ~3/2 = 702.2850{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6589{{c}}


==7-limit==
{{Optimal ET sequence|legend=0| 41, 70, 111, 152fg, 263dfg }}
Commas: 686/675, 5120/5103


[[POTE_tuning|POTE generator]]: ~3/2 = 704.263
Badness (Sintel): 1.12


Map: [&lt;1 0 -31 -21|, &lt;0 1 21 15|]
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


Wedgie: &lt;&lt;1 21 15 31 21 -24||
Comma list: 256/255, 352/351, 400/399, 456/455, 715/714, 847/845


EDOs: 29, 46, 305
Mapping: {{mapping| 1 0 -50 -40 32 27 58 -56 | 0 1 33 27 -18 -21 -34 38 }}


Badness: 0.0961
Optimal tunings:  
* WE: ~2 = 1199.3401{{c}}, ~3/2 = 702.2705{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6548{{c}}


==11-limit==
{{Optimal ET sequence|legend=0| 41, 70h, 111, 152fg, 263dfgh }}
Commas: 121/120, 441/440, 686/675


[[POTE_tuning|POTE generator]]: ~3/2 = 704.250
Badness (Sintel): 1.03


Map: [&lt;1 0 -31 -21 -14|, &lt;0 1 21 15 11|]
==== Hemikwai ====
Subgroup: 2.3.5.7.11.13


EDOs: 29, 46, 259
Comma list: 540/539, 676/675, 1375/1372, 5120/5103


Badness: 0.0386
Mapping: {{mapping| 1 0 -50 -40 32 -51 | 0 2 66 54 -36 69 }}
: mapping generators: ~2, ~26/15


==13-limit==
Optimal tunings:
Commas: 91/90, 121/120, 169/168, 441/440
* WE: ~2 = 1199.6968{{c}}, ~26/15 = 951.0740{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.3123{{c}}


[[POTE_tuning|POTE generator]]: ~3/2 = 704.214
{{Optimal ET sequence|legend=0| 82, 111, 193, 304d }}


Map: [&lt;1 0 -31 -21 -14 -9|, &lt;0 1 21 15 11 8|]
Badness (Sintel): 1.82


EDOs: 29, 46, 167, 213, 380
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


Badness: 0.0247
Comma list: 442/441, 540/539, 676/675, 715/714, 5120/5103


==17-limit==
Mapping: {{mapping| 1 0 -50 -40 32 -51 -30 | 0 2 66 54 -36 69 43 }}
Commas: 91/90, 121/120, 136/135, 154/153, 169/168


[[POTE_tuning|POTE generator]]: ~3/2 = 704.229
Optimal tunings:
* WE: ~2 = 1199.6861{{c}}, ~26/15 = 951.0654{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.3120{{c}}


[&lt;1 0 -31 -21 -14 -9 -34|, &lt;0 1 21 15 11 8 24|]
{{Optimal ET sequence|legend=0| 82, 111, 193, 304d }}


EDOs: 29g, 46, 121defg, 167defg, 213defg
Badness (Sintel): 1.31


Badness: 0.0179
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


==19-limit==
Comma list: 400/399, 442/441, 540/539, 676/675, 715/714, 1445/1444
Commas: 91/90, 121/120, 133/132, 136/135, 154/153, 169/168


[[POTE_tuning|POTE generator]]: ~3/2 = 704.135
Mapping: {{mapping| 1 0 -50 -40 32 -51 -30 -56 | 0 2 66 54 -36 69 43 76 }}


Map: [&lt;1 0 -31 -21 -14 -9 -34 9|, &lt;0 1 21 15 11 8 24 -3|]
Optimal tunings:  
* WE: ~2 = 1199.6718{{c}}, ~26/15 = 951.0526{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.3103{{c}}


EDOs: 29g, 46, 75dfgh, 121defgh
{{Optimal ET sequence|legend=0| 82, 111, 193, 304dh }}


Badness: 0.0174
Badness (Sintel): 1.16


===Leapling===
== Ketchup ==
Commas: 77/76, 91/90, 121/120, 136/135, 153/152, 169/168
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.


[[POTE_tuning|POTE generator]]: ~3/2 = 704.123
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 0 -31 -21 -14 -9 -34 -37|, &lt;0 1 21 15 11 8 24 26|]
[[Comma list]]: 5120/5103, 1071875/1062882


EDOs: 29g, 46h, 75dfg
{{Mapping|legend=1| 2 3 4 6 | 0 4 15 -9 }}
: mapping generators: ~1225/864, ~64/63


Badness: 0.0191
[[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 }}


=Mystery=
{{Optimal ET sequence|legend=1| 46, 94, 140 }}
Commas: 5120/5103, 50421/50000


[[POTE_tuning|POTE generator]]: ~5/4 = 388.646
[[Badness]] (Sintel): 2.14


Map: [&lt;29 46 0 14|, &lt;0 0 1 1|]
=== 11-limit ===
Subgroup: 2.3.5.7.11


Wedgie: &lt;&lt;0 29 29 46 46 -14||
Comma list: 385/384, 1331/1323, 2200/2187


EDOs: 29, 58, 87, 145
Mapping: {{mapping| 2 3 4 6 7 | 0 4 15 -9 -2 }}


Badness: 0.1037
Optimal tunings:  
* WE: ~99/70 = 600.0678{{c}}, ~64/63 = 25.6963{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~64/63 = 25.6956{{c}}


==11-limit==
{{Optimal ET sequence|legend=0| 46, 94, 140 }}
Commas: 441/440, 896/891, 3388/3375


[[POTE_tuning|POTE generator]]: ~5/4 = 388.460
Badness (Sintel): 1.31


Map: [&lt;29 46 0 14 33|, &lt;0 0 1 1 1|]
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 325/324, 352/351, 385/384, 1331/1323
 
Mapping: {{mapping| 2 3 4 6 7 8 | 0 4 15 -9 -2 -14 }}
 
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| 46, 94, 140 }}
 
Badness (Sintel): 1.03


EDOs: 29, 58, 87, 145
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Badness: 0.0343
Comma list: 289/288, 325/324, 352/351, 385/384, 442/441


==13-limit==
Mapping: {{mapping| 2 3 4 6 7 8 8 | 0 4 15 -9 -2 -14 4 }}
Commas: 196/195, 352/351, 364/363, 676/675


[[POTE_tuning|POTE generator]]: ~5/4 = 388.354
Optimal tunings:  
* WE: ~17/12 = 600.0896{{c}}, ~66/65 = 25.7048{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~66/65 = 25.7017{{c}}


Map: [&lt;29 46 0 14 33 40|, &lt;0 0 1 1 1 1|]
{{Optimal ET sequence|legend=0| 46, 94, 140 }}


EDOs: 29, 58, 87, 145, 232, 377
Badness (Sintel): 0.845


Badness: 0.0186
=== 2.3.5.7.11.13.17.23 subgroup ===
Subgroup: 2.3.5.7.11.13.17.23


=Quanic=
Comma list: 253/252, 289/288, 325/324, 352/351, 385/384, 391/390
Commas: 5120/5103, 5832000/5764801


POTE generator: ~160/147 = 140.493
Mapping: {{mapping| 2 3 4 6 7 8 8 9 | 0 4 15 -9 -2 -14 4 1 }}


Map: [&lt;1 1 -4 0|, &lt;0 5 54 24|]
Optimal tunings:  
* WE: ~17/12 = 600.1139{{c}}, ~66/65 = 25.7053{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~66/65 = 25.7013{{c}}


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


Badness: 0.1795
Badness (Sintel): 0.772


==11-limit==
== Undecental ==
Commas: 540/539, 1331/1323, 5120/5103
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> }}.


POTE generator: ~88/81 = 140.489
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 1 -4 0 1|, &lt;0 5 54 24 21|]
[[Comma list]]: 5120/5103, 235298/234375


EDOs: 94, 111, 205
{{Mapping|legend=1| 1 0 61 71 | 0 1 -37 -43 }}
: mapping generators: ~2, ~3


Badness: 0.0587
[[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 }}


==13-limit==
{{Optimal ET sequence|legend=1| 29, 70, 99, 722bc, 821bc, 920bc, 1019bc }}
Commas: 352/351, 540/539, 729/728, 1331/1323


POTE generator: ~13/12 = 140.496
[[Badness]] (Sintel): 2.39


Map: [&lt;1 1 -4 0 1 3|, &lt;0 5 54 24 21 6|]
== Mystery ==
{{Main| Mystery }}
: ''For the 5-limit version, see [[29th-octave temperaments #Mystery]].''


EDOs: 94, 111, 205
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.


Badness: 0.0325
[[Subgroup]]: 2.3.5.7


==17-limit==
[[Comma list]]: 5120/5103, 50421/50000
Commas: 352/351, 442/441, 540/539, 715/714, 847/845


POTE generator: ~13/12 = 140.497
{{Mapping|legend=1| 29 46 0 14 | 0 0 1 1 }}
: mapping generators: ~50/49, ~5


Map: [&lt;1 1 -4 0 1 3 -2|, &lt;0 5 54 24 21 6 52|]
[[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 }}


EDOs: 94, 111, 205
{{Optimal ET sequence|legend=1| 29, 58, 87, 145 }}


Badness: 0.0211
[[Badness]] (Sintel): 2.63


==19-limit==
=== 11-limit ===
Commas: 352/351, 400/399, 442/441, 456/455, 495/494, 715/714
Subgroup: 2.3.5.7.11


POTE generator: ~13/12 = 140.496
Comma list: 441/440, 896/891, 3388/3375


Map: [&lt;1 1 -4 0 1 3 -2 -5|, &lt;0 5 54 24 21 6 52 79|]
Mapping: {{mapping| 29 46 0 14 33 | 0 0 1 1 1 }}


EDOs: 94, 111, 205
Optimal tunings:  
* WE: ~45/44 = 41.3637{{c}}, ~5/4 = 388.3136{{c}}
* CWE: ~45/44 = 41.3793{{c}}, ~5/4 = 388.0598{{c}}


Badness: 0.0173
{{Optimal ET sequence|legend=0| 29, 58, 87, 145 }}


=Supers=
Badness (Sintel): 1.13
Commas: 5120/5103, 118098/117649


POTE generator: ~9/7 = 434.218
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Map: [&lt;2 1 -12 2|, &lt;0 3 23 5|]
Comma list: 196/195, 352/351, 364/363, 676/675


Wedgie: &lt;&lt;6 46 10 59 -1 -106||
Mapping: {{mapping| 29 46 0 14 33 40 | 0 0 1 1 1 1 }}


EDOs: 58, 94, 152
Optimal tunings:  
* WE: ~45/44 = 41.3623{{c}}, ~5/4 = 388.1942{{c}}
* CWE: ~40/39 = 41.3793{{c}}, ~5/4 = 387.9017{{c}}


Badness: 92.748
{{Optimal ET sequence|legend=0| 29, 58, 87, 145, 232 }}


==11-limit==
Badness (Sintel): 0.768
Commas: 540/539, 4000/3993, 5120/5103


POTE generator: ~9/7 = 434.217
== 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.  


Map: [&lt;2 1 -12 2 -9|, &lt;0 3 23 5 22|]
[[Subgroup]]: 2.3.5.7


EDOs: 58, 94, 152
[[Comma list]]: 5120/5103, 52734375/52706752


Badness: 0.0282
{{Mapping|legend=1| 1 0 38 48 | 0 2 -45 -57 }}
: mapping generator: ~2, ~12500/7203


==13-limit==
[[Optimal tuning]]s:
Commas: 352/351, 540/539, 729/728, 1575/1573
* [[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 }}


POTE generator: ~9/7 = 434.221
{{Optimal ET sequence|legend=1| 29, 82cd, 111, 140, 251, 391, 1424bbcdd }}


Map: [&lt;2 1 -12 2 -9 -2|, &lt;0 3 23 5 22 13|]
[[Badness]] (Sintel): 2.93


EDOs: 58, 94, 152f
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.0216
Comma list: 1331/1323, 1375/1372, 5120/5103


=Alphaquarter=
Mapping: {{mapping| 1 0 38 48 32 | 0 2 -45 -57 -36 }}
Commas: 5120/5103, 29360128/29296875


POTE generator: ~16128/15625 = 55.243
Optimal tunings:  
* WE: ~2 = 1199.8767{{c}}, ~400/231 = 951.3065{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~400/231 = 951.4063{{c}}


Map: [&lt;1 2 2 0|, &lt;0 -9 7 61|]
{{Optimal ET sequence|legend=0| 29, 82cd, 111, 140, 251, 391e }}


Wedgie: &lt;&lt;9 -7 -61 -32 -122 -122||
Badness (Sintel): 2.01


EDOs: 87, 152, 239, 391
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Badness: 0.1166
Comma list: 352/351, 676/675, 847/845, 1331/1323


==11-limit==
Mapping: {{mapping| 1 0 38 48 32 37 | 0 2 -45 -57 -36 -42 }}
Commas: 3025/3024, 4000/3993, 5120/5103


POTE generator: ~33/32 = 55.243
Optimal tunings:  
* WE: ~2 = 1199.8753{{c}}, ~26/15 = 951.3054{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.4064{{c}}


Map: [&lt;1 2 2 0 3|, &lt;0 -9 7 61 10|]
{{Optimal ET sequence|legend=0| 29, 82cdf, 111, 140, 251, 391e }}


EDOs: 87, 152, 239, 391
Badness (Sintel): 1.18


Badness: 0.0296
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


= Septiquarter =
Comma list: 352/351, 442/441, 561/560, 676/675, 715/714
Commas: 5120/5103, 420175/419904


POTE generator: ~147/128 = 242.453
Mapping: {{mapping| 1 0 38 48 32 37 58 | 0 2 -45 -57 -36 -42 -68 }}


Map: [&lt;1 3 10 2|, &lt;0 -7 -38 4|]
Optimal tunings:  
* WE: ~2 = 1199.8770{{c}}, ~26/15 = 951.3039{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.4035{{c}}


Wedgie: &lt;&lt;7 38 -4 44 -26 -116||
{{Optimal ET sequence|legend=0| 29g, 82cdfg, 111, 140, 251, 391e }}


EDOs: {{EDOs|94, 99, 292, 391, 881bd, 1272bcd}}
Badness (Sintel): 0.971


Badness: 0.0538
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


== Semiseptiquarter ==
Comma list: 286/285, 352/351, 363/361, 442/441, 476/475, 561/560
Commas: 5120/5103, 9801/9800, 14641/14580


POTE generators: ~121/105 = 242.4511
Mapping: {{mapping| 1 0 38 48 32 37 58 32 | 0 2 -45 -57 -36 -42 -68 -35 }}


Map: [<2 6 20 4 15|, <0 -7 -38 4 -20|]
Optimal tunings:  
* WE: ~2 = 1199.7534{{c}}, ~26/15 = 951.2024{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.4020{{c}}


EDOs: {{EDOs|94, 198, 292, 490}}
{{Optimal ET sequence|legend=0| 29g, 82cdfgh, 111, 140 }}


Badness: 0.0642
Badness (Sintel): 1.01
 
=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23
 
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 }}
 
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=0| 29g, 82cdfgh, 111, 140 }}
 
Badness (Sintel): 1.10
 
== Countriton ==
: ''For the 5-limit version, see [[Schismic–Mercator equivalence continuum #Countritonic]].''
 
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
 
[[Comma list]]: 5120/5103, 7558272/7503125
 
{{Mapping|legend=1| 1 -3 -15 13 | 0 9 34 -20 }}
: mapping generators: ~2, ~1225/864
 
[[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| 51c, 53, 157, 210, 473cdd }}
 
[[Badness]] (Sintel): 3.32
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 176/175, 5120/5103, 41503/41472
 
Mapping: {{mapping| 1 -3 -15 13 -21 | 0 9 34 -20 48 }}
 
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| 51ce, 53, 104c, 157 }}
 
Badness (Sintel): 2.80


=== 13-limit ===
=== 13-limit ===
Commas: 352/351, 847/845, 1716/1715, 14641/14580
Subgroup: 2.3.5.7.11.13


POTE generators: ~121/105 = 242.4448
Comma list: 176/175, 351/350, 847/845, 2197/2187


Map: [<2 6 20 4 15 7|, <0 -7 -38 4 -20 1|]
Mapping: {{mapping| 1 -3 -15 13 -21 -7 | 0 9 34 -20 48 21 }}


EDOs: {{EDOs|94, 198, 490f}}
Optimal tunings:  
* WE: ~2 = 1199.5944{{c}}, ~77/54 = 611.2491{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~77/54 = 611.4506{{c}}


Badness: 0.0348
{{Optimal ET sequence|legend=0| 51ce, 53, 104c, 157 }}


= Tricot =
Badness (Sintel): 1.75
{{see also|Tricot family}}


The generator for tricot temperament is the real cube root of third harmonic, 3<sup>1/3</sup>, tuned between 63/44 and 13/9. Tricot temperament can be described as 53&amp;70 temperament (also known as ''trimot''), tempering out the [[Tricot|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.
== 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.  


== 5-limit ==
Artoneutral was named by [[Flora Canou]] in 2023 for its generator's quality.
Comma: {{monzo | 39 -29 3 }} = 68719476736000/68630377364883


POTE generator: ~59049/40960 = 634.012
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 0 -13|, &lt;0 3 29|]
[[Comma list]]: 5120/5103, 3828125/3779136


Wedgie: &lt;&lt;3 29 39||
{{Mapping|legend=1| 1 -1 -4 12 | 0 9 22 -32 }}
: mapping generators: ~2, ~128/105


EDOs: {{EDOs| 53, 229, 282, 335, 388, 441, 1376, 1817, 2258 }}
[[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 }}


Badness: 0.0461
{{Optimal ET sequence|legend=1| 87, 94, 181 }}


== 7-limit ==
[[Badness]] (Sintel): 3.98
Commas: 2430/2401, 5120/5103


POTE generator: ~81/56 = 634.026
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 0 -13 -3|, &lt;0 3 29 11|]
Comma list: 385/384, 2200/2187, 4000/3993


Wedgie: &lt;&lt;3 29 11 39 9 -56||
Mapping: {{mapping| 1 -1 -4 12 -2 | 0 9 22 -32 19 }}


EDOs: {{EDOs| 17c, 36c, 53, 229dd, 282dd }}
Optimal tunings:  
* WE: ~2 = 1200.1668{{c}}, ~11/9 = 344.8027{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 344.7557{{c}}


Badness: 0.1001
{{Optimal ET sequence|legend=0| 87, 181 }}


== 11-limit ==
Badness (Sintel): 1.52
Commas: 99/98, 121/120, 5120/5103


POTE generator: ~63/44 = 634.027
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 325/324, 352/351, 385/384, 1575/1573
 
Mapping: {{mapping| 1 -1 -4 12 -2 6 | 0 9 22 -32 19 -8 }}
 
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 }}
 
Badness (Sintel): 1.08
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
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 }}
 
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 }}
 
Badness (Sintel): 1.16
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
 
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 }}
 
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 }}
 
Badness (Sintel): 1.19
 
=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23
 
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 }}
 
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 }}
 
Badness (Sintel): 1.17


Map: [&lt;1 0 -13 -3 -5|, &lt;0 3 29 11 16|]
== 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.


EDOs: {{EDOs| 17c, 36ce, 53, 176de }}
[[Subgroup]]: 2.3.5.7


Badness: 0.0561
[[Comma list]]: 5120/5103, 5832000/5764801


== 13-limit ==
{{Mapping|legend=1| 1 1 -4 0 | 0 5 54 24 }}
Commas: 99/98, 121/120, 169/168, 352/351
: mapping generators: ~2, ~160/147


POTE generator: ~13/9 = 634.012
[[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 }}


Map: [&lt;1 0 -13 -3 -5 0|, &lt;0 3 29 11 16 7|]
{{Optimal ET sequence|legend=1| 94, 111, 205 }}


EDOs: {{EDOs| 17c, 36ce, 53 }}
[[Badness]] (Sintel): 4.54


Badness: 0.0321
=== 11-limit ===
Subgroup: 2.3.5.7.11


=Ketchup=
Comma list: 540/539, 1331/1323, 5120/5103
Commas: 5120/5103, 1071875/1062882


POTE generator: ~64/63 = ~81/80 = 25.719
Mapping: {{mapping| 1 1 -4 0 1 | 0 5 54 24 21 }}


Map: [&lt;2 3 4 6|, &lt;0 4 15 -9|]
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| 94, 111, 205 }}
 
Badness (Sintel): 1.94
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


EDOS: 46, 94, 140
Comma list: 352/351, 540/539, 729/728, 1331/1323


Badness: 0.0845
Mapping: {{mapping| 1 1 -4 0 1 3 | 0 5 54 24 21 6 }}


==11-limit==
Optimal tunings:
Commas: 385/384, 1331/1323, 2200/2187
* 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 = 25.693
{{Optimal ET sequence|legend=0| 94, 111, 205 }}


Map: [&lt;2 3 4 6 7|, &lt;0 4 15 -9 -2|]
Badness (Sintel): 1.34


EDOs: 46, 94, 140
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Badness: 0.0396
Comma list: 352/351, 442/441, 540/539, 715/714, 847/845


==13-limit==
Mapping: {{mapping| 1 1 -4 0 1 3 -2 | 0 5 54 24 21 6 52 }}
Commas: 325/324, 352/351, 847/845, 1331/1323


POTE generator: ~55/54 = ~64/63 = ~81/80 = 25.697
Optimal tunings:  
* WE: ~2 = 1199.6699{{c}}, ~13/12 = 140.4586{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.4920{{c}}


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


EDOs: 46, 94, 140
Badness (Sintel): 1.08


Badness: 0.0248
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


==17-limit==
Comma list: 352/351, 400/399, 442/441, 456/455, 495/494, 715/714
Commas: 289/288, 325/324, 352/351, 385/384, 561/560


POTE generator: ~55/54 = ~64/63 = ~81/80 = 25.701
Mapping: {{mapping| 1 1 -4 0 1 3 -2 -5 | 0 5 54 24 21 6 52 79 }}


Map: [&lt;2 3 4 6 7 8 8|, &lt;0 4 15 -9 -2 -14 4|]
Optimal tunings:  
* WE: ~2 = 1199.6745{{c}}, ~13/12 = 140.4574{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.4908{{c}}


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


Badness: 0.0166
Badness (Sintel): 1.05


==19-limit==
== Jorgensen ==
Commas: 190/189, 209/208, 289/288, 352/351, 385/384, 561/560
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Jorgensen]].''


POTE generator: ~55/54 = ~64/63 = ~81/80 = 25.660
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.  


Map: [&lt;2 3 4 6 7 8 8 9|, &lt;0 4 15 -9 -2 -14 4 -12|]
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>.


EDOs: 46, 94, 140h, 234eh
[[Subgroup]]: 2.3.5.7


Badness: 0.0182
[[Comma list]]: 5120/5103, 578509309952/576650390625


==23-limit==
{{Mapping|legend=1| 70 111 0 34 | 0 0 1 1 }}
Commas: 190/189, 209/208, 253/252, 289/288, 323/322, 352/351, 385/384
: mapping generators: ~50421/50000, ~5


POTE generator: ~55/54 = ~64/63 = ~81/80 = 25.661
[[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 }}


Map: [&lt;2 3 4 6 7 8 8 9 9|, &lt;0 4 15 -9 -2 -14 4 -12 1|]
{{Optimal ET sequence|legend=1| 70, 140, 350, 490 }}


EDOs: 46, 94, 140h, 234ehi
[[Badness]] (Sintel): 5.40


Badness: 0.0140
== References ==


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