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:
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.
* ''[[Dominant]]'', {36/35, 64/63} → [[Meantone family #Dominant|Meantone family]]
* [[Garibaldi]], {225/224, 3125/3087} → [[Schismatic family #Garibaldi|Schismatic family]]
* [[Rodan]], {245/243, 1029/1024} → [[Gamelismic clan #Rodan|Gamelismic clan]]
* [[Hemififths]], {2401/2400, 5120/5103} → [[Breedsmic temperaments #Hemififths|Breedsmic temperaments]]
* [[Misty]], {3136/3125, 5120/5103} → [[Misty family #Misty|Misty family]]
* [[Amity]], {4375/4374, 5120/5103} → [[Ragismic microtemperaments #Amity|Ragismic microtemperaments]] and [[Amity family #Septimal amity|Amity family]]
* ''[[Countercata]]'', {5120/5103, 15625/15552} → [[Kleismic family #Countercata|Kleismic family]]
* ''[[Kwai]]'', {5120/5103, 16875/16807} → [[Mirkwai clan #Kwai|Mirkwai clan]]
* ''[[Supers]]'', {5120/5103, 118098/117649} → [[Stearnsmic clan #Supers|Stearnsmic clan]]
* ''[[Quintakwai]]'', {5120/5103, 9765625/9680832} → [[Quindromeda family #Quintakwai|Quindromeda family]]


== 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


{{Val list|legend=1| 5, 43c, 48, 53, 111, 164d, 275d }}
{{Mapping|legend=1| 1 -4 -28 6 | 0 7 38 -4 }}
: mapping generators: ~2, ~243/140


[[Badness]]: 0.047963
[[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 }}


=== 11-limit ===
{{Optimal ET sequence|legend=1| 94, 99, 292, 391, 881bd, 1272bcd }}
 
[[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: [{{val| 1 0 -6 4 -12 }}, {{val| 0 4 21 -3 39 }}]
Mapping: {{mapping| 2 -8 -56 12 -25 | 0 7 38 -4 20 }}


POTE generator: ~21/16 = 475.700
Optimal tunings:  
* WE: ~99/70 = 599.8953{{c}}, ~210/121 = 957.3819{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~210/121 = 957.5449{{c}}


Vals: {{Val list| 53, 58, 111, 280cd, 391cd }}
{{Optimal ET sequence|legend=0| 94, 198, 292, 490 }}


Badness: 0.034484
Badness (Sintel): 2.12


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


Comma list: 176/175, 351/350, 540/539, 676/675
Comma list: 352/351, 847/845, 1716/1715, 14641/14580
 
Mapping: {{mapping| 2 -8 -56 12 -25 9 | 0 7 38 -4 20 -1 }}


Mapping: [{{val| 1 0 -6 4 -12 -7 }}, {{val| 0 4 21 -3 39 27 }}]
Optimal tunings:  
* WE: ~99/70 = 599.8565{{c}}, ~210/121 = 957.3261{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~210/121 = 957.5508{{c}}


POTE generator: ~21/16 = 475.697
{{Optimal ET sequence|legend=0| 94, 198, 490f }}


Vals: {{Val list| 53, 58, 111, 280cdf, 391cdf }}
Badness (Sintel): 1.44


Badness: 0.018842
== Kwai ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Kwai]].''


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


Comma list: 176/175, 256/255, 351/350, 442/441, 540/539
[[Subgroup]]: 2.3.5.7


Mapping: [{{val| 1 0 -6 4 -12 -7 14 }}, {{val| 0 4 21 -3 39 27 -25 }}]
[[Comma list]]: 5120/5103, 16875/16807


POTE generator: ~21/16 = 475.692
{{Mapping|legend=1| 1 0 -50 -40 | 0 1 33 27 }}
: mapping generators: ~2, ~3


Vals: {{Val list| 53, 58, 111, 321cdfg }}
[[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 }}


Badness: 0.018403
{{Optimal ET sequence|legend=1| 41, 111, 152, 345, 497d }}


==== 19-limit ====
[[Badness]] (Sintel): 1.38
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 176/175, 256/255, 286/285, 324/323, 351/350, 540/539
=== 11-limit ===
Subgroup: 2.3.5.7.11


Mapping: [{{val| 1 0 -6 4 -12 -7 14 -12 }}, {{val| 0 4 21 -3 39 27 -25 41 }}]
Comma list: 540/539, 1375/1372, 5120/5103


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


Vals: {{Val list| 53, 58h, 111 }}
Optimal tunings:  
* WE: ~2 = 1199.6672{{c}}, ~3/2 = 702.4282{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6189{{c}}


Badness: 0.015649
{{Optimal ET sequence|legend=0| 41, 111, 152, 497de, 649dde }}


=== Buteo ===
Badness (Sintel): 0.867
Subgroup: 2.3.5.7.11


Comma list: 99/98, 385/384, 2200/2187
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Mapping: [{{val| 1 0 -6 4 9 }}, {{val| 0 4 21 -3 -14 }}]
Comma list: 352/351, 540/539, 729/728, 1375/1372


POTE generator: ~21/16 = 475.436
Mapping: {{mapping| 1 0 -50 -40 32 27 | 0 1 33 27 -18 -21 }}


Vals: {{Val list| 5, 48, 53 }}
Optimal tunings:  
* WE: ~2 = 1199.4772{{c}}, ~3/2 = 702.3379{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6409{{c}}


Badness: 0.060238
{{Optimal ET sequence|legend=0| 41, 111, 152f, 415dff }}


==== 13-limit ====
Badness (Sintel): 1.01
Subgroup: 2.3.5.7.11.13


Comma list: 99/98, 275/273, 385/384, 572/567
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


Mapping: [{{val| 1 0 -6 4 9 -7 }}, {{val| 0 4 21 -3 -14 27 }}]
Comma list: 256/255, 352/351, 540/539, 715/714, 1089/1088


POTE generator: ~21/16 = 475.464
Mapping: {{mapping| 1 0 -50 -40 32 27 58 | 0 1 33 27 -18 -21 -34 }}


Vals: {{Val list| 5, 48f, 53 }}
Optimal tunings:  
* WE: ~2 = 1199.3537{{c}}, ~3/2 = 702.2850{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6589{{c}}


Badness: 0.039854
{{Optimal ET sequence|legend=0| 41, 70, 111, 152fg, 263dfg }}


== Undecental ==
Badness (Sintel): 1.12
Subgroup: 2.3.5.7


[[Comma list]]: 5120/5103, 235298/234375
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


[[Mapping]]: [{{val| 1 0 61 71 }}, {{val| 0 1 -37 -43 }}]
Comma list: 256/255, 352/351, 400/399, 456/455, 715/714, 847/845


{{Multival|legend=1| 1 -37 -43 -61 -71 4 }}
Mapping: {{mapping| 1 0 -50 -40 32 27 58 -56 | 0 1 33 27 -18 -21 -34 38 }}


[[POTE generator]]: ~3/2 = 703.039
Optimal tunings:  
* WE: ~2 = 1199.3401{{c}}, ~3/2 = 702.2705{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6548{{c}}


{{Val list|legend=1| 29, 70, 99, 722bc, 821bc, 920bc, 1019bc, 1118bbcc, 1217bbcc, 1316bbccd }}
{{Optimal ET sequence|legend=0| 41, 70h, 111, 152fg, 263dfgh }}


[[Badness]]: 0.094603
Badness (Sintel): 1.03


== Leapday ==
==== Hemikwai ====
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Leapday]].''
Subgroup: 2.3.5.7.11.13
: ''Not to be confused with calendar-based modes such as those in [[293edo]] or [[Irvic scale|Irvian mode]].''


[[Leapday]] tempers out {{monzo|31 -21 1}} (trisayo) in the 5-limit. This temperament can be described as 29&amp;46 temperament, which tempers out the hemifamity and 686/675 (senga). Alternative extension [[Porwell temperaments #Polypyth|polypyth]] (46&amp;121) tempers out the same 5-limit comma as the leapday, but with the porwell (6144/6125) rather than the hemifamity tempered out.
Comma list: 540/539, 676/675, 1375/1372, 5120/5103


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


Subgroup: 2.3.5.7
Optimal tunings:  
* WE: ~2 = 1199.6968{{c}}, ~26/15 = 951.0740{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.3123{{c}}


[[Comma list]]: 686/675, 5120/5103
{{Optimal ET sequence|legend=0| 82, 111, 193, 304d }}


[[Mapping]]: [{{val| 1 0 -31 -21 }}, {{val| 0 1 21 15 }}]
Badness (Sintel): 1.82


{{Multival|legend=1| 1 21 15 31 21 -24 }}
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


[[POTE generator]]: ~3/2 = 704.263
Comma list: 442/441, 540/539, 676/675, 715/714, 5120/5103


{{Val list|legend=1| 17c, 29, 46, 167d, 213d, 259cdd, 305bcdd }}
Mapping: {{mapping| 1 0 -50 -40 32 -51 -30 | 0 2 66 54 -36 69 43 }}


[[Badness]]: 0.096123
Optimal tunings:  
* WE: ~2 = 1199.6861{{c}}, ~26/15 = 951.0654{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.3120{{c}}


=== 11-limit ===
{{Optimal ET sequence|legend=0| 82, 111, 193, 304d }}
Subgroup: 2.3.5.7.11


Comma list: 121/120, 441/440, 686/675
Badness (Sintel): 1.31


Mapping: [{{val| 1 0 -31 -21 -14 }}, {{val| 0 1 21 15 11 }}]
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


POTE generator: ~3/2 = 704.250
Comma list: 400/399, 442/441, 540/539, 676/675, 715/714, 1445/1444


Vals: {{Val list| 17c, 29, 46, 167de, 213de, 259cdde }}
Mapping: {{mapping| 1 0 -50 -40 32 -51 -30 -56 | 0 2 66 54 -36 69 43 76 }}


Badness: 0.038624
Optimal tunings:  
* WE: ~2 = 1199.6718{{c}}, ~26/15 = 951.0526{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.3103{{c}}


=== 13-limit ===
{{Optimal ET sequence|legend=0| 82, 111, 193, 304dh }}
Subgroup: 2.3.5.7.11.13


Comma list: 91/90, 121/120, 169/168, 352/351
Badness (Sintel): 1.16


Mapping: [{{val| 1 0 -31 -21 -14 -9 }}, {{val| 0 1 21 15 11 8 }}]
== 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.


POTE generator: ~3/2 = 704.214
[[Subgroup]]: 2.3.5.7


Vals: {{Val list| 17c, 29, 46, 121def, 167def, 213deff }}
[[Comma list]]: 5120/5103, 1071875/1062882


Badness: 0.024732
{{Mapping|legend=1| 2 3 4 6 | 0 4 15 -9 }}
: mapping generators: ~1225/864, ~64/63


=== 17-limit ===
[[Optimal tuning]]s:
Subgroup: 2.3.5.7.11.13.17
* [[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 }}


Comma list: 91/90, 121/120, 136/135, 154/153, 169/168
{{Optimal ET sequence|legend=1| 46, 94, 140 }}


Mapping: [{{val| 1 0 -31 -21 -14 -9 -34 }}, {{val| 0 1 21 15 11 8 24 }}]
[[Badness]] (Sintel): 2.14


POTE generator: ~3/2 = 704.229
=== 11-limit ===
Subgroup: 2.3.5.7.11


Vals: {{Val list| 17cg, 29g, 46, 121defg, 167defg, 213deffg }}
Comma list: 385/384, 1331/1323, 2200/2187


Badness: 0.017863
Mapping: {{mapping| 2 3 4 6 7 | 0 4 15 -9 -2 }}


==== 19-limit ====
Optimal tunings:
Subgroup: 2.3.5.7.11.13.17.19
* WE: ~99/70 = 600.0678{{c}}, ~64/63 = 25.6963{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~64/63 = 25.6956{{c}}


Comma list: 91/90, 121/120, 133/132, 136/135, 154/153, 169/168
{{Optimal ET sequence|legend=0| 46, 94, 140 }}


Mapping: [{{val| 1 0 -31 -21 -14 -9 -34 9 }}, {{val| 0 1 21 15 11 8 24 -3 }}]
Badness (Sintel): 1.31


POTE generator: ~3/2 = 704.135
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Vals: {{Val list| 17cg, 29g, 46, 75dfgh, 121defgh }}
Comma list: 325/324, 352/351, 385/384, 1331/1323


Badness: 0.017356
Mapping: {{mapping| 2 3 4 6 7 8 | 0 4 15 -9 -2 -14 }}


==== Leapling ====
Optimal tunings:
Subgroup: 2.3.5.7.11.13.17.19
* WE: ~99/70 = 600.0612{{c}}, ~66/65 = 25.7000{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~66/65 = 25.6978{{c}}


Comma list: 77/76, 91/90, 121/120, 136/135, 153/152, 169/168
{{Optimal ET sequence|legend=0| 46, 94, 140 }}


Mapping: [{{val| 1 0 -31 -21 -14 -9 -34 -37 }}, {{val| 0 1 21 15 11 8 24 26 }}]
Badness (Sintel): 1.03


POTE generator: ~3/2 = 704.123
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Vals: {{Val list| 17cgh, 29g, 46h, 75dfg, 121defghh }}
Comma list: 289/288, 325/324, 352/351, 385/384, 442/441


Badness: 0.019065
Mapping: {{mapping| 2 3 4 6 7 8 8 | 0 4 15 -9 -2 -14 4 }}


== Mystery ==
Optimal tunings:
{{main| Mystery }}
* WE: ~17/12 = 600.0896{{c}}, ~66/65 = 25.7048{{c}}
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Mystery]].''
* CWE: ~17/12 = 600.0000{{c}}, ~66/65 = 25.7017{{c}}


Subgroup: 2.3.5.7
{{Optimal ET sequence|legend=0| 46, 94, 140 }}


[[Comma list]]: 5120/5103, 50421/50000
Badness (Sintel): 0.845


[[Mapping]]: [{{val| 29 46 0 14 }}, {{val| 0 0 1 1 }}]
=== 2.3.5.7.11.13.17.23 subgroup ===
Subgroup: 2.3.5.7.11.13.17.23


{{Multival|legend=1| 0 29 29 46 46 -14 }}
Comma list: 253/252, 289/288, 325/324, 352/351, 385/384, 391/390


[[POTE generator]]: ~5/4 = 388.646
Mapping: {{mapping| 2 3 4 6 7 8 8 9 | 0 4 15 -9 -2 -14 4 1 }}


{{Val list|legend=1| 29, 58, 87, 145 }}
Optimal tunings:
* WE: ~17/12 = 600.1139{{c}}, ~66/65 = 25.7053{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~66/65 = 25.7013{{c}}


[[Badness]]: 0.103734
{{Optimal ET sequence|legend=0| 46, 94, 140 }}


=== 11-limit ===
Badness (Sintel): 0.772
Subgroup: 2.3.5.7.11


Comma list: 441/440, 896/891, 3388/3375
== 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> }}.


Mapping: [{{val| 29 46 0 14 33 }}, {{val| 0 0 1 1 1 }}]
[[Subgroup]]: 2.3.5.7


POTE generator: ~5/4 = 388.460
[[Comma list]]: 5120/5103, 235298/234375


Vals: {{Val list| 29, 58, 87, 145 }}
{{Mapping|legend=1| 1 0 61 71 | 0 1 -37 -43 }}
: mapping generators: ~2, ~3


Badness: 0.034291
[[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 }}
Subgroup: 2.3.5.7.11.13


Comma list: 196/195, 352/351, 364/363, 676/675
[[Badness]] (Sintel): 2.39


Mapping: [{{val| 29 46 0 14 33 40 }}, {{val| 0 0 1 1 1 1 }}]
== Leapday ==
{{Main| Leapday }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Leapday]].''


POTE generator: ~5/4 = 388.354
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.  


Vals: {{Val list| 29, 58, 87, 145, 232, 377cef }}
Leapday is more notable in the higher limits than the lower, as it nails the 13-limit pretty well from identifying [[14/11]] by a major third and [[13/11]] by a minor third, tempering out not only [[352/351]] and [[364/363]] but [[91/90]], [[121/120]], [[169/168]] and [[196/195]]. It can be further extended to include the [[17/1|17th]] and [[23/1|23rd]] [[harmonic]]s. Adding 17 would fix the valid diamond monotone tuning to 46edo, however.


Badness: 0.018591
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.


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


[[Comma list]]: 5120/5103, 5832000/5764801
[[Comma list]]: 686/675, 5120/5103


[[Mapping]]: [{{val| 1 1 -4 0 }}, {{val| 0 5 54 24 }}]
{{Mapping|legend=1| 1 0 -31 -21 | 0 1 21 15 }}
: mapping generators: ~2, ~3


[[POTE generator]]: ~160/147 = 140.493
[[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 }}


{{Val list|legend=1| 94, 111, 205 }}
{{Optimal ET sequence|legend=1| 17c, 29, 46 }}


[[Badness]]: 0.179475
[[Badness]] (Sintel): 2.43


=== 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: 121/120, 441/440, 686/675


Mapping: [{{val| 1 1 -4 0 1 }}, {{val| 0 5 54 24 21 }}]
Mapping: {{mapping| 1 0 -31 -21 -14 | 0 1 21 15 11 }}


POTE generator: ~88/81 = 140.489
Optimal tunings:  
* WE: ~2 = 1200.0731{{c}}, ~3/2 = 704.2933{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2538{{c}}


Vals: {{Val list| 94, 111, 205 }}
{{Optimal ET sequence|legend=0| 17c, 29, 46 }}


Badness: 0.058678
Badness (Sintel): 1.28


=== 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: 91/90, 121/120, 169/168, 352/351


Mapping: [{{val| 1 1 -4 0 1 3 }}, {{val| 0 5 54 24 21 6 }}]
Mapping: {{mapping| 1 0 -31 -21 -14 -9 | 0 1 21 15 11 8 }}


POTE generator: ~13/12 = 140.496
Optimal tunings:  
* WE: ~2 = 1200.4758{{c}}, ~3/2 = 704.4930{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2346{{c}}


Vals: {{Val list| 94, 111, 205 }}
{{Optimal ET sequence|legend=0| 17c, 29, 46, 121def }}


Badness: 0.032481
Badness (Sintel): 1.02


=== 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: 91/90, 121/120, 136/135, 154/153, 169/168


Mapping: [{{val| 1 1 -4 0 1 3 -2 }}, {{val| 0 5 54 24 21 6 52 }}]
Mapping: {{mapping| 1 0 -31 -21 -14 -9 -34 | 0 1 21 15 11 8 24 }}


POTE generator: ~13/12 = 140.497
Optimal tunings:  
* WE: ~2 = 1200.4818{{c}}, ~3/2 = 704.5121{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2507{{c}}


Vals: {{Val list| 94, 111, 205 }}
{{Optimal ET sequence|legend=0| 17cg, 29g, 46, 121defg }}


Badness: 0.021112
Badness (Sintel): 0.910


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


Comma list: 352/351, 400/399, 442/441, 456/455, 495/494, 715/714
Comma list: 91/90, 121/120, 136/135, 154/153, 161/160, 169/168


Mapping: [{{val| 1 1 -4 0 1 3 -2 -5 }}, {{val| 0 5 54 24 21 6 52 79 }}]
Mapping: {{mapping| 1 0 -31 -21 -14 -9 -34 -5 | 0 1 21 15 11 8 24 6 }}


POTE generator: ~13/12 = 140.496
Optimal tunings:  
* WE: ~2 = 1200.5169{{c}}, ~3/2 = 704.5279{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2450{{c}}


Vals: {{Val list| 94, 111, 205 }}
{{Optimal ET sequence|legend=0| 17cg, 29g, 46, 121defg }}


Badness: 0.017273
Badness (Sintel): 0.872


== Alphaquarter ==
== Mystery ==
{{see also|Escapade family}}
{{Main| Mystery }}
: ''For the 5-limit version, see [[29th-octave temperaments #Mystery]].''


Subgroup: 2.3.5.7
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.  


[[Comma list]]: 5120/5103, 29360128/29296875
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val| 1 2 2 0 }}, {{val| 0 -9 7 61 }}]
[[Comma list]]: 5120/5103, 50421/50000


{{Multival|legend=1| 9 -7 -61 -32 -122 -122 }}
{{Mapping|legend=1| 29 46 0 14 | 0 0 1 1 }}
: mapping generators: ~50/49, ~5


[[POTE generator]]: ~16128/15625 = 55.243
[[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 }}


{{Val list|legend=1| 87, 152, 239, 391 }}
{{Optimal ET sequence|legend=1| 29, 58, 87, 145 }}


[[Badness]]: 0.116594
[[Badness]] (Sintel): 2.63


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


Comma list: 3025/3024, 4000/3993, 5120/5103
Comma list: 441/440, 896/891, 3388/3375
 
Mapping: {{mapping| 29 46 0 14 33 | 0 0 1 1 1 }}
 
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 }}
 
Badness (Sintel): 1.13
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 196/195, 352/351, 364/363, 676/675


Mapping: [{{val| 1 2 2 0 3 }}, {{val| 0 -9 7 61 10 }}]
Mapping: {{mapping| 29 46 0 14 33 40 | 0 0 1 1 1 1 }}


POTE generator: ~33/32 = 55.243
Optimal tunings:  
* WE: ~45/44 = 41.3623{{c}}, ~5/4 = 388.1942{{c}}
* CWE: ~40/39 = 41.3793{{c}}, ~5/4 = 387.9017{{c}}


Vals: {{Val list| 87, 152, 239, 391 }}
{{Optimal ET sequence|legend=0| 29, 58, 87, 145, 232 }}


Badness: 0.029638
Badness (Sintel): 0.768


== Septiquarter ==
== Hemidromeda ==
Subgroup: 2.3.5.7
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.  


[[Comma list]]: 5120/5103, 420175/419904
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val| 1 3 10 2 }}, {{val| 0 -7 -38 4 }}]
[[Comma list]]: 5120/5103, 52734375/52706752


{{Multival|legend=1| 7 38 -4 44 -26 -116 }}
{{Mapping|legend=1| 1 0 38 48 | 0 2 -45 -57 }}
: mapping generator: ~2, ~12500/7203


[[POTE generator]]: ~147/128 = 242.453
[[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 }}


{{Val list|legend=1| 94, 99, 292, 391, 881bd, 1272bcd }}
{{Optimal ET sequence|legend=1| 29, 82cd, 111, 140, 251, 391, 1424bbcdd }}


[[Badness]]: 0.053760
[[Badness]] (Sintel): 2.93


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


Comma list: 5120/5103, 9801/9800, 14641/14580
Comma list: 1331/1323, 1375/1372, 5120/5103


Mapping: [{{val| 2 6 20 4 15 }}, {{val| 0 -7 -38 4 -20 }}]
Mapping: {{mapping| 1 0 38 48 32 | 0 2 -45 -57 -36 }}


POTE generators: ~121/105 = 242.4511
Optimal tunings:  
* WE: ~2 = 1199.8767{{c}}, ~400/231 = 951.3065{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~400/231 = 951.4063{{c}}


Vals: {{Val list| 94, 198, 292, 490 }}
{{Optimal ET sequence|legend=0| 29, 82cd, 111, 140, 251, 391e }}


Badness: 0.064160
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, 847/845, 1716/1715, 14641/14580
Comma list: 352/351, 676/675, 847/845, 1331/1323


Mapping: [{{val| 2 6 20 4 15 7 }}, {{val| 0 -7 -38 4 -20 1 }}]
Mapping: {{mapping| 1 0 38 48 32 37 | 0 2 -45 -57 -36 -42 }}


POTE generators: ~121/105 = 242.4448
Optimal tunings:  
* WE: ~2 = 1199.8753{{c}}, ~26/15 = 951.3054{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.4064{{c}}


Vals: {{Val list| 94, 198, 490f }}
{{Optimal ET sequence|legend=0| 29, 82cdf, 111, 140, 251, 391e }}


Badness: 0.034834
Badness (Sintel): 1.18


== Tricot (aka trimot) ==
=== 17-limit ===
{{see also| Tricot family }}
Subgroup: 2.3.5.7.11.13.17


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.
Comma list: 352/351, 442/441, 561/560, 676/675, 715/714


Subgroup: 2.3.5.7
Mapping: {{mapping| 1 0 38 48 32 37 58 | 0 2 -45 -57 -36 -42 -68 }}


[[Comma list]]: 2430/2401, 5120/5103
Optimal tunings:  
* WE: ~2 = 1199.8770{{c}}, ~26/15 = 951.3039{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.4035{{c}}


[[Mapping]]: [{{val| 1 0 -13 -3 }}, {{val| 0 3 29 11 }}]
{{Optimal ET sequence|legend=0| 29g, 82cdfg, 111, 140, 251, 391e }}


{{Multival|legend=1| 3 29 11 39 9 -56 }}
Badness (Sintel): 0.971


[[POTE generator]]: ~81/56 = 634.026
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


{{Val list|legend=1| 17c, 36c, 53, 229dd, 282dd }}
Comma list: 286/285, 352/351, 363/361, 442/441, 476/475, 561/560


[[Badness]]: 0.100127
Mapping: {{mapping| 1 0 38 48 32 37 58 32 | 0 2 -45 -57 -36 -42 -68 -35 }}


=== 11-limit ===
Optimal tunings:
Subgroup: 2.3.5.7.11
* WE: ~2 = 1199.7534{{c}}, ~26/15 = 951.2024{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.4020{{c}}


Comma list: 99/98, 121/120, 5120/5103
{{Optimal ET sequence|legend=0| 29g, 82cdfgh, 111, 140 }}


Mapping: [{{val| 1 0 -13 -3 -5 }}, {{val| 0 3 29 11 16 }}]
Badness (Sintel): 1.01


POTE generator: ~63/44 = 634.027
=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23


Vals: {{Val list| 17c, 36ce, 53, 70, 123de }}
Comma list: 253/252, 286/285, 352/351, 363/361, 391/390, 442/441, 460/459


Badness: 0.056134
Mapping: {{mapping| 1 0 38 48 32 37 58 32 18 | 0 2 -45 -57 -36 -42 -68 -35 -17 }}


=== 13-limit ===
Optimal tunings:
Subgroup: 2.3.5.7.11.13
* WE: ~2 = 1199.9128{{c}}, ~26/15 = 951.3371{{c}}
 
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.4076{{c}}
Comma list: 99/98, 121/120, 169/168, 352/351


Mapping: [{{val| 1 0 -13 -3 -5 0 }}, {{val| 0 3 29 11 16 7 }}]
{{Optimal ET sequence|legend=0| 29g, 82cdfgh, 111, 140 }}


POTE generator: ~13/9 = 634.012
Badness (Sintel): 1.10


Vals: {{Val list| 17c, 36ce, 53, 70, 123de }}
== Countriton ==
: ''For the 5-limit version, see [[Schismic–Mercator equivalence continuum #Countritonic]].''


Badness: 0.032102
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.  


== Ketchup ==
Countriton was named by [[Xenllium]] in 2022 as a counterpart of [[untriton]].  
Subgroup: 2.3.5.7


[[Comma list]]: 5120/5103, 1071875/1062882
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val| 2 3 4 6 }}, {{val| 0 4 15 -9 }}]
[[Comma list]]: 5120/5103, 7558272/7503125


{{Multival|legend=1| 8 30 -18 29 -51 -126 }}
{{Mapping|legend=1| 1 -3 -15 13 | 0 9 34 -20 }}
: mapping generators: ~2, ~1225/864


[[POTE generator]]: ~64/63 = ~81/80 = 25.719
[[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 }}


{{Val list|legend=1| 46, 94, 140 }}
{{Optimal ET sequence|legend=1| 51c, 53, 157, 210, 473cdd }}


[[Badness]]: 0.084538
[[Badness]] (Sintel): 3.32


=== 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: 176/175, 5120/5103, 41503/41472


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


POTE generator: ~55/54 = ~64/63 = ~81/80 = 25.693
Optimal tunings:
* WE: ~2 = 1199.5178{{c}}, ~77/54 = 611.2097{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~77/54 = 611.4495{{c}}


Vals: {{Val list| 46, 94, 140 }}
{{Optimal ET sequence|legend=0| 51ce, 53, 104c, 157 }}


Badness: 0.039555
Badness (Sintel): 2.80


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


Comma list: 325/324, 352/351, 847/845, 1331/1323
Comma list: 176/175, 351/350, 847/845, 2197/2187
 
Mapping: [{{val| 2 3 4 6 7 8 }}, {{val| 0 4 15 -9 -2 -14 }}]
 
POTE generator: ~55/54 = ~64/63 = ~81/80 = 25.697


Vals: {{Val list| 46, 94, 140 }}
Mapping: {{mapping| 1 -3 -15 13 -21 -7 | 0 9 34 -20 48 21 }}


Badness: 0.024824
Optimal tunings:  
* WE: ~2 = 1199.5944{{c}}, ~77/54 = 611.2491{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~77/54 = 611.4506{{c}}


=== 17-limit ===
{{Optimal ET sequence|legend=0| 51ce, 53, 104c, 157 }}
Subgroup: 2.3.5.7.11.13.17


Comma list: 289/288, 325/324, 352/351, 385/384, 561/560
Badness (Sintel): 1.75


Mapping: [{{val| 2 3 4 6 7 8 8 }}, {{val| 0 4 15 -9 -2 -14 4 }}]
== 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.


POTE generator: ~55/54 = ~64/63 = ~81/80 = 25.701
Artoneutral was named by [[Flora Canou]] in 2023 for its generator's quality.  


Vals: {{Val list| 46, 94, 140 }}
[[Subgroup]]: 2.3.5.7


Badness: 0.016591
[[Comma list]]: 5120/5103, 3828125/3779136


=== 19-limit ===
{{Mapping|legend=1| 1 -1 -4 12 | 0 9 22 -32 }}
Subgroup: 2.3.5.7.11.13.17.19
: mapping generators: ~2, ~128/105


Comma list: 190/189, 209/208, 289/288, 352/351, 385/384, 561/560
[[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 }}


Mapping: [{{val| 2 3 4 6 7 8 8 9 }}, {{val| 0 4 15 -9 -2 -14 4 -12 }}]
{{Optimal ET sequence|legend=1| 87, 94, 181 }}


POTE generator: ~55/54 = ~64/63 = ~81/80 = 25.660
[[Badness]] (Sintel): 3.98


Vals: {{Val list| 46, 94, 140h, 234eh }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.018170
Comma list: 385/384, 2200/2187, 4000/3993


=== 23-limit ===
Mapping: {{mapping| 1 -1 -4 12 -2 | 0 9 22 -32 19 }}
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
Optimal tunings:  
* WE: ~2 = 1200.1668{{c}}, ~11/9 = 344.8027{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 344.7557{{c}}


Mapping: [{{val| 2 3 4 6 7 8 8 9 9 }}, {{val| 0 4 15 -9 -2 -14 4 -12 1 }}]
{{Optimal ET sequence|legend=0| 87, 181 }}


POTE generator: ~55/54 = ~64/63 = ~81/80 = 25.661
Badness (Sintel): 1.52


Vals: {{Val list| 46, 94, 140h, 234ehi }}
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Badness: 0.014033
Comma list: 325/324, 352/351, 385/384, 1575/1573


== Undim ==
Mapping: {{mapping| 1 -1 -4 12 -2 6 | 0 9 22 -32 19 -8 }}
Subgroup: 2.3.5


[[Comma list]]: {{monzo| 41 -20 -4 }} = 2199023255552/2179240250625
Optimal tunings:  
* WE: ~2 = 1200.0662{{c}}, ~11/9 = 344.7804{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 344.7617{{c}}


[[Mapping]]: [{{val| 4 0 41 }}, {{val| 0 1 -5 }}]
{{Optimal ET sequence|legend=0| 87, 181 }}


Mapping generators: ~1215/1024, ~3
Badness (Sintel): 1.08


[[POTE generator]]: ~3/2 = 702.736
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


{{Val list|legend=1| 12, 104, 116, 128, 140, 152, 620, 772, 924c, 1076bc, 1228bc }}
Comma list: 325/324, 352/351, 375/374, 385/384, 595/594


[[Badness]]: 0.241703
Mapping: {{mapping| 1 -1 -4 12 -2 6 -12 | 0 9 22 -32 19 -8 56 }}


=== 7-limit ===
Optimal tunings:
Subgroup: 2.3.5.7
* WE: ~2 = 1200.0346{{c}}, ~11/9 = 344.7589{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 344.7492{{c}}


[[Comma list]]: 5120/5103, 390625/388962
{{Optimal ET sequence|legend=0| 87, 94, 181 }}


[[Mapping]]: [{{val| 4 0 41 81 }}, {{val| 0 1 -5 -11 }}]
Badness (Sintel): 1.16


{{Multival|legend=1| 4 -20 -44 -41 -81 -46 }}
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


[[POTE generator]]: ~3/2 = 702.736
Comma list: 325/324, 352/351, 375/374, 385/384, 400/399, 595/594


{{Val list|legend=1| 12, 128, 140, 152, 292 }}
Mapping: {{mapping| 1 -1 -4 12 -2 6 -12 -15 | 0 9 22 -32 19 -8 56 67 }}


[[Badness]]: 0.062754
Optimal tunings:  
* WE: ~2 = 1200.0282{{c}}, ~11/9 = 344.7532{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 344.7453{{c}}


=== 11-limit ===
{{Optimal ET sequence|legend=0| 87, 94, 181 }}
Subgroup: 2.3.5.7.11


Comma list: 1375/1372, 5120/5103, 5632/5625
Badness (Sintel): 1.19


Mapping: [{{val| 4 0 41 81 128 }}, {{val| 0 1 -5 -11 -18 }}]
=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23


POTE generator: ~3/2 = 702.689
Comma list: 300/299, 325/324, 352/351, 375/374, 385/384, 400/399, 484/483


Vals: {{Val list| 12, 128e, 140, 152, 292, 444d, 596d }}
Mapping: {{mapping| 1 -1 -4 12 -2 6 -12 -15 -13 | 0 9 22 -32 19 -8 56 67 61 }}


Badness: 0.034837
Optimal tunings:  
* WE: ~2 = 1200.0163{{c}}, ~11/9 = 344.7461{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 344.7416{{c}}


== Quinticosiennic ==
{{Optimal ET sequence|legend=0| 87, 94, 181 }}
{{See also| 16ed5/2 #Regular temperaments }}


The ''quinticosiennic'' temperament (12&amp;145) tempers out the hemifamity comma (5120/5103) and 395136/390625 (trizo-aquadbigu) in the 7-limit; 441/440 (werckisma), 896/891 (pentacircle), and 78408/78125 (lolosepgu) in the 11-limit. The word "quinticosiennic" means 5 (quintuple) × 29 (είκοσι εννέα) = 145, and so named because 1/5 of [[29edo|29EDO]] fourth, i.e. 12\145, is a possible generator.
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.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 5120/5103, 395136/390625
[[Comma list]]: 5120/5103, 5832000/5764801


[[Mapping]]: [{{val|1 2 1 -1}}, {{val|0 -5 16 46}}]
{{Mapping|legend=1| 1 1 -4 0 | 0 5 54 24 }}
: mapping generators: ~2, ~160/147


{{Multival|legend=1|5 -16 -46 -37 -87 -62}}
[[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 }}


[[POTE generator]]: ~135/128 = 99.345
{{Optimal ET sequence|legend=1| 94, 111, 205 }}


{{Val list|legend=1| 12, 133, 145, 157, 302c, 459bcc }}
[[Badness]] (Sintel): 4.54
 
[[Badness]]: 0.158041


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


Comma list: 441/440, 896/891, 78408/78125
Comma list: 540/539, 1331/1323, 5120/5103


Mapping: [{{val|1 2 1 -1 -2}}, {{val|0 -5 16 46 66}}]
Mapping: {{mapping| 1 1 -4 0 1 | 0 5 54 24 21 }}


POTE generator: ~35/33 = 99.318
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| 12, 133, 145 }}
{{Optimal ET sequence|legend=0| 94, 111, 205 }}


Badness: 0.080674
Badness (Sintel): 1.94


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


Comma list: 196/195, 352/351, 364/363, 78408/78125
Comma list: 352/351, 540/539, 729/728, 1331/1323


Mapping: [{{val|1 2 1 -1 -2 -3}}, {{val|0 -5 16 46 66 81}}]
Mapping: {{mapping| 1 1 -4 0 1 3 | 0 5 54 24 21 6 }}


POTE generator: ~35/33 = 99.307
Optimal tunings:  
* WE: ~2 = 1199.6639{{c}}, ~13/12 = 140.4562{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.4904{{c}}


Vals: {{Val list| 12f, 133, 145 }}
{{Optimal ET sequence|legend=0| 94, 111, 205 }}


Badness: 0.052464
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: 196/195, 256/255, 352/351, 364/363, 3757/3750
Comma list: 352/351, 442/441, 540/539, 715/714, 847/845


Mapping: [{{val|1 2 1 -1 -2 -3 5}}, {{val|0 -5 16 46 66 81 -11}}]
Mapping: {{mapping| 1 1 -4 0 1 3 -2 | 0 5 54 24 21 6 52 }}


POTE generator: ~18/17 = 99.308
Optimal tunings:  
* WE: ~2 = 1199.6699{{c}}, ~13/12 = 140.4586{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.4920{{c}}


Vals: {{Val list| 12f, 133, 145 }}
{{Optimal ET sequence|legend=0| 94, 111, 205 }}


Badness: 0.037108
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: 196/195, 256/255, 352/351, 361/360, 364/363, 476/475
Comma list: 352/351, 400/399, 442/441, 456/455, 495/494, 715/714
 
Mapping: {{mapping| 1 1 -4 0 1 3 -2 -5 | 0 5 54 24 21 6 52 79 }}
 
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| 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>.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 5120/5103, 578509309952/576650390625
 
{{Mapping|legend=1| 70 111 0 34 | 0 0 1 1 }}
: mapping generators: ~50421/50000, ~5


Mapping: [{{val|1 2 1 -1 -2 -3 5 4}}, {{val|0 -5 16 46 66 81 -11 3}}]
[[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: ~18/17 = 99.303
{{Optimal ET sequence|legend=1| 70, 140, 350, 490 }}


Vals: {{Val list| 12f, 133, 145 }}
[[Badness]] (Sintel): 5.40


Badness: 0.028440
== References ==


[[Category:Regular temperament theory]]
[[Category:Temperament collections]]
[[Category:Temperament collection]]
[[Category:Aberschismic temperaments| ]] <!-- main article -->
[[Category:Hemifamity temperaments| ]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Rank 2]]

Latest revision as of 12:45, 6 June 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

This is a collection of rank-2 aberschismic temperaments, which temper out the aberschisma (monzo: [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.

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 discussed elsewhere are:

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

Septiquarter

Septiquarter tempers out 420175/419904 and may be described as the 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

Comma list: 5120/5103, 420175/419904

Mapping: [⟨1 -4 -28 6], ⟨0 7 38 -4]]

mapping generators: ~2, ~243/140

Optimal tunings:

  • WE: ~2 = 1199.7212 ¢, ~243/140 = 957.3250 ¢
error map: ⟨-0.279 +0.435 -0.158 +0.201]
  • CWE: ~2 = 1200.0000 ¢, ~243/140 = 957.5424 ¢
error map: ⟨0.000 +0.842 +0.298 +1.004]

Optimal ET sequence: 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: [⟨2 -8 -56 12 -25], ⟨0 7 38 -4 20]]

Optimal tunings:

  • WE: ~99/70 = 599.8953 ¢, ~210/121 = 957.3819 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~210/121 = 957.5449 ¢

Optimal ET sequence: 94, 198, 292, 490

Badness (Sintel): 2.12

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 847/845, 1716/1715, 14641/14580

Mapping: [⟨2 -8 -56 12 -25 9], ⟨0 7 38 -4 20 -1]]

Optimal tunings:

  • WE: ~99/70 = 599.8565 ¢, ~210/121 = 957.3261 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~210/121 = 957.5508 ¢

Optimal ET sequence: 94, 198, 490f

Badness (Sintel): 1.44

Kwai

For the 5-limit version, see Miscellaneous 5-limit temperaments #Kwai.

Named by Gene Ward Smith in 2004 for its "bridgeability"[1], kwai is generated by a perfect fifth, and can be described as 41 & 70.

Subgroup: 2.3.5.7

Comma list: 5120/5103, 16875/16807

Mapping: [⟨1 0 -50 -40], ⟨0 1 33 27]]

mapping generators: ~2, ~3

Optimal tunings:

  • WE: ~2 = 1199.7337 ¢, ~3/2 = 702.4600 ¢
error map: ⟨-0.266 +0.239 -0.607 +1.055]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.6085 ¢
error map: ⟨0.000 +0.653 -0.234 +1.603]

Optimal ET sequence: 41, 111, 152, 345, 497d

Badness (Sintel): 1.38

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 0 -50 -40 32], ⟨0 1 33 27 -18]]

Optimal tunings:

  • WE: ~2 = 1199.6672 ¢, ~3/2 = 702.4282 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.6189 ¢

Optimal ET sequence: 41, 111, 152, 497de, 649dde

Badness (Sintel): 0.867

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 540/539, 729/728, 1375/1372

Mapping: [⟨1 0 -50 -40 32 27], ⟨0 1 33 27 -18 -21]]

Optimal tunings:

  • WE: ~2 = 1199.4772 ¢, ~3/2 = 702.3379 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.6409 ¢

Optimal ET sequence: 41, 111, 152f, 415dff

Badness (Sintel): 1.01

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 256/255, 352/351, 540/539, 715/714, 1089/1088

Mapping: [⟨1 0 -50 -40 32 27 58], ⟨0 1 33 27 -18 -21 -34]]

Optimal tunings:

  • WE: ~2 = 1199.3537 ¢, ~3/2 = 702.2850 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.6589 ¢

Optimal ET sequence: 41, 70, 111, 152fg, 263dfg

Badness (Sintel): 1.12

19-limit

Subgroup: 2.3.5.7.11.13.17.19

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

Mapping: [⟨1 0 -50 -40 32 27 58 -56], ⟨0 1 33 27 -18 -21 -34 38]]

Optimal tunings:

  • WE: ~2 = 1199.3401 ¢, ~3/2 = 702.2705 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.6548 ¢

Optimal ET sequence: 41, 70h, 111, 152fg, 263dfgh

Badness (Sintel): 1.03

Hemikwai

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 0 -50 -40 32 -51], ⟨0 2 66 54 -36 69]]

mapping generators: ~2, ~26/15

Optimal tunings:

  • WE: ~2 = 1199.6968 ¢, ~26/15 = 951.0740 ¢
  • CWE: ~2 = 1200.0000 ¢, ~26/15 = 951.3123 ¢

Optimal ET sequence: 82, 111, 193, 304d

Badness (Sintel): 1.82

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 0 -50 -40 32 -51 -30], ⟨0 2 66 54 -36 69 43]]

Optimal tunings:

  • WE: ~2 = 1199.6861 ¢, ~26/15 = 951.0654 ¢
  • CWE: ~2 = 1200.0000 ¢, ~26/15 = 951.3120 ¢

Optimal ET sequence: 82, 111, 193, 304d

Badness (Sintel): 1.31

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 400/399, 442/441, 540/539, 676/675, 715/714, 1445/1444

Mapping: [⟨1 0 -50 -40 32 -51 -30 -56], ⟨0 2 66 54 -36 69 43 76]]

Optimal tunings:

  • WE: ~2 = 1199.6718 ¢, ~26/15 = 951.0526 ¢
  • CWE: ~2 = 1200.0000 ¢, ~26/15 = 951.3103 ¢

Optimal ET sequence: 82, 111, 193, 304dh

Badness (Sintel): 1.16

Ketchup

Ketchup may be described as the 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

Comma list: 5120/5103, 1071875/1062882

Mapping: [⟨2 3 4 6], ⟨0 4 15 -9]]

mapping generators: ~1225/864, ~64/63

Optimal tunings:

  • WE: ~1225/864 = 599.9685 ¢, ~64/63 = 25.7181 ¢
error map: ⟨-0.063 +0.823 -0.668 -0.478]
  • CWE: ~1225/864 = 600.0000 ¢, ~64/63 = 25.7181 ¢
error map: ⟨0.000 +0.917 -0.543 -0.288]

Optimal ET sequence: 46, 94, 140

Badness (Sintel): 2.14

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 1331/1323, 2200/2187

Mapping: [⟨2 3 4 6 7], ⟨0 4 15 -9 -2]]

Optimal tunings:

  • WE: ~99/70 = 600.0678 ¢, ~64/63 = 25.6963 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~64/63 = 25.6956 ¢

Optimal ET sequence: 46, 94, 140

Badness (Sintel): 1.31

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 352/351, 385/384, 1331/1323

Mapping: [⟨2 3 4 6 7 8], ⟨0 4 15 -9 -2 -14]]

Optimal tunings:

  • WE: ~99/70 = 600.0612 ¢, ~66/65 = 25.7000 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~66/65 = 25.6978 ¢

Optimal ET sequence: 46, 94, 140

Badness (Sintel): 1.03

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨2 3 4 6 7 8 8], ⟨0 4 15 -9 -2 -14 4]]

Optimal tunings:

  • WE: ~17/12 = 600.0896 ¢, ~66/65 = 25.7048 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~66/65 = 25.7017 ¢

Optimal ET sequence: 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: [⟨2 3 4 6 7 8 8 9], ⟨0 4 15 -9 -2 -14 4 1]]

Optimal tunings:

  • WE: ~17/12 = 600.1139 ¢, ~66/65 = 25.7053 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~66/65 = 25.7013 ¢

Optimal ET sequence: 46, 94, 140

Badness (Sintel): 0.772

Undecental

Undecental adds the triwellisma to the comma list and may be described as the 29 & 70 temperament. 5/4 is mapped to the quintuple-diminished seventh or equivalently the perfect fourth minus three dieses. 58\99 is an almost perfect generator, just as the name suggests. Another interesting tuning choice is the argent fifth, 2(2 - sqrt (2)).

Subgroup: 2.3.5.7

Comma list: 5120/5103, 235298/234375

Mapping: [⟨1 0 61 71], ⟨0 1 -37 -43]]

mapping generators: ~2, ~3

Optimal tunings:

  • WE: ~2 = 1199.6543 ¢, ~3/2 = 702.8370 ¢
error map: ⟨-0.346 +0.536 +0.423 -0.494]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 703.0465 ¢
error map: ⟨0.000 +1.092 +0.966 +0.175]

Optimal ET sequence: 29, 70, 99, 722bc, 821bc, 920bc, 1019bc

Badness (Sintel): 2.39

Leapday

Main article: 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 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 17th and 23rd harmonics. Adding 17 would fix the valid diamond monotone tuning to 46edo, however.

Leapday has an alternative extension called 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: [⟨1 0 -31 -21], ⟨0 1 21 15]]

mapping generators: ~2, ~3

Optimal tunings:

  • WE: ~2 = 1199.7167 ¢, ~3/2 = 704.0971 ¢
error map: ⟨-0.283 +1.859 +2.559 -5.669]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.2504 ¢
error map: ⟨0.000 +2.295 +2.945 -5.070]

Optimal ET sequence: 17c, 29, 46

Badness (Sintel): 2.43

11-limit

Subgroup: 2.3.5.7.11

Comma list: 121/120, 441/440, 686/675

Mapping: [⟨1 0 -31 -21 -14], ⟨0 1 21 15 11]]

Optimal tunings:

  • WE: ~2 = 1200.0731 ¢, ~3/2 = 704.2933 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.2538 ¢

Optimal ET sequence: 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: [⟨1 0 -31 -21 -14 -9], ⟨0 1 21 15 11 8]]

Optimal tunings:

  • WE: ~2 = 1200.4758 ¢, ~3/2 = 704.4930 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.2346 ¢

Optimal ET sequence: 17c, 29, 46, 121def

Badness (Sintel): 1.02

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 0 -31 -21 -14 -9 -34], ⟨0 1 21 15 11 8 24]]

Optimal tunings:

  • WE: ~2 = 1200.4818 ¢, ~3/2 = 704.5121 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.2507 ¢

Optimal ET sequence: 17cg, 29g, 46, 121defg

Badness (Sintel): 0.910

2.3.5.7.11.13.17.23 subgroup

Subgroup: 2.3.5.7.11.13.17.23

Comma list: 91/90, 121/120, 136/135, 154/153, 161/160, 169/168

Mapping: [⟨1 0 -31 -21 -14 -9 -34 -5], ⟨0 1 21 15 11 8 24 6]]

Optimal tunings:

  • WE: ~2 = 1200.5169 ¢, ~3/2 = 704.5279 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.2450 ¢

Optimal ET sequence: 17cg, 29g, 46, 121defg

Badness (Sintel): 0.872

Mystery

Main article: Mystery
For the 5-limit version, see 29th-octave temperaments #Mystery.

Mystery tempers out 50421/50000 and may be described as the 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

Comma list: 5120/5103, 50421/50000

Mapping: [⟨29 46 0 14], ⟨0 0 1 1]]

mapping generators: ~50/49, ~5

Optimal tunings:

  • WE: ~50/49 = 41.3652 ¢, ~5/4 = 388.5128 ¢
error map: ⟨-0.410 +0.842 +1.378 -2.022]
  • CWE: ~50/49 = 41.3793 ¢, ~5/4 = 388.3030 ¢
error map: ⟨0.000 +1.493 +1.989 -1.213]

Optimal ET sequence: 29, 58, 87, 145

Badness (Sintel): 2.63

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 896/891, 3388/3375

Mapping: [⟨29 46 0 14 33], ⟨0 0 1 1 1]]

Optimal tunings:

  • WE: ~45/44 = 41.3637 ¢, ~5/4 = 388.3136 ¢
  • CWE: ~45/44 = 41.3793 ¢, ~5/4 = 388.0598 ¢

Optimal ET sequence: 29, 58, 87, 145

Badness (Sintel): 1.13

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 352/351, 364/363, 676/675

Mapping: [⟨29 46 0 14 33 40], ⟨0 0 1 1 1 1]]

Optimal tunings:

  • WE: ~45/44 = 41.3623 ¢, ~5/4 = 388.1942 ¢
  • CWE: ~40/39 = 41.3793 ¢, ~5/4 = 387.9017 ¢

Optimal ET sequence: 29, 58, 87, 145, 232

Badness (Sintel): 0.768

Hemidromeda

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

Subgroup: 2.3.5.7

Comma list: 5120/5103, 52734375/52706752

Mapping: [⟨1 0 38 48], ⟨0 2 -45 -57]]

mapping generator: ~2, ~12500/7203

Optimal tunings:

  • WE: ~2 = 1199.7236 ¢, ~12500/7203 = 951.1864 ¢
error map: ⟨-0.276 +0.418 -0.205 +0.282]
  • CWE: ~2 = 1200.0000 ¢, ~12500/7203 = 951.4098 ¢
error map: ⟨0.000 +0.865 +0.243 +0.813]

Optimal ET sequence: 29, 82cd, 111, 140, 251, 391, 1424bbcdd

Badness (Sintel): 2.93

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 0 38 48 32], ⟨0 2 -45 -57 -36]]

Optimal tunings:

  • WE: ~2 = 1199.8767 ¢, ~400/231 = 951.3065 ¢
  • CWE: ~2 = 1200.0000 ¢, ~400/231 = 951.4063 ¢

Optimal ET sequence: 29, 82cd, 111, 140, 251, 391e

Badness (Sintel): 2.01

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 0 38 48 32 37], ⟨0 2 -45 -57 -36 -42]]

Optimal tunings:

  • WE: ~2 = 1199.8753 ¢, ~26/15 = 951.3054 ¢
  • CWE: ~2 = 1200.0000 ¢, ~26/15 = 951.4064 ¢

Optimal ET sequence: 29, 82cdf, 111, 140, 251, 391e

Badness (Sintel): 1.18

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 352/351, 442/441, 561/560, 676/675, 715/714

Mapping: [⟨1 0 38 48 32 37 58], ⟨0 2 -45 -57 -36 -42 -68]]

Optimal tunings:

  • WE: ~2 = 1199.8770 ¢, ~26/15 = 951.3039 ¢
  • CWE: ~2 = 1200.0000 ¢, ~26/15 = 951.4035 ¢

Optimal ET sequence: 29g, 82cdfg, 111, 140, 251, 391e

Badness (Sintel): 0.971

19-limit

Subgroup: 2.3.5.7.11.13.17.19

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

Mapping: [⟨1 0 38 48 32 37 58 32], ⟨0 2 -45 -57 -36 -42 -68 -35]]

Optimal tunings:

  • WE: ~2 = 1199.7534 ¢, ~26/15 = 951.2024 ¢
  • CWE: ~2 = 1200.0000 ¢, ~26/15 = 951.4020 ¢

Optimal ET sequence: 29g, 82cdfgh, 111, 140

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: [⟨1 0 38 48 32 37 58 32 18], ⟨0 2 -45 -57 -36 -42 -68 -35 -17]]

Optimal tunings:

  • WE: ~2 = 1199.9128 ¢, ~26/15 = 951.3371 ¢
  • CWE: ~2 = 1200.0000 ¢, ~26/15 = 951.4076 ¢

Optimal ET sequence: 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 51c & 53 temperament. It splits the 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: [⟨1 -3 -15 13], ⟨0 9 34 -20]]

mapping generators: ~2, ~1225/864

Optimal tunings:

  • WE: ~2 = 1199.4179 ¢, ~1225/864 = 611.1213 ¢
error map: ⟨-0.582 -0.117 +0.541 +1.181]
  • CWE: ~2 = 1200.0000 ¢, ~1225/864 = 611.4120 ¢
error map: ⟨0.000 +0.753 +1.695 +2.934]

Optimal ET sequence: 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: [⟨1 -3 -15 13 -21], ⟨0 9 34 -20 48]]

Optimal tunings:

  • WE: ~2 = 1199.5178 ¢, ~77/54 = 611.2097 ¢
  • CWE: ~2 = 1200.0000 ¢, ~77/54 = 611.4495 ¢

Optimal ET sequence: 51ce, 53, 104c, 157

Badness (Sintel): 2.80

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 176/175, 351/350, 847/845, 2197/2187

Mapping: [⟨1 -3 -15 13 -21 -7], ⟨0 9 34 -20 48 21]]

Optimal tunings:

  • WE: ~2 = 1199.5944 ¢, ~77/54 = 611.2491 ¢
  • CWE: ~2 = 1200.0000 ¢, ~77/54 = 611.4506 ¢

Optimal ET sequence: 51ce, 53, 104c, 157

Badness (Sintel): 1.75

Artoneutral

Artoneutral can be described as the 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 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

Comma list: 5120/5103, 3828125/3779136

Mapping: [⟨1 -1 -4 12], ⟨0 9 22 -32]]

mapping generators: ~2, ~128/105

Optimal tunings:

  • WE: ~2 = 1200.1400 ¢, ~128/105 = 344.7929 ¢
error map: ⟨+0.140 +1.041 -1.430 -0.518]
  • CWE: ~2 = 1200.0000 ¢, ~128/105 = 344.7531 ¢
error map: ⟨0.000 +0.823 -1.746 -0.925]

Optimal ET sequence: 87, 94, 181

Badness (Sintel): 3.98

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 2200/2187, 4000/3993

Mapping: [⟨1 -1 -4 12 -2], ⟨0 9 22 -32 19]]

Optimal tunings:

  • WE: ~2 = 1200.1668 ¢, ~11/9 = 344.8027 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 344.7557 ¢

Optimal ET sequence: 87, 181

Badness (Sintel): 1.52

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -1 -4 12 -2 6], ⟨0 9 22 -32 19 -8]]

Optimal tunings:

  • WE: ~2 = 1200.0662 ¢, ~11/9 = 344.7804 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 344.7617 ¢

Optimal ET sequence: 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: [⟨1 -1 -4 12 -2 6 -12], ⟨0 9 22 -32 19 -8 56]]

Optimal tunings:

  • WE: ~2 = 1200.0346 ¢, ~11/9 = 344.7589 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 344.7492 ¢

Optimal ET sequence: 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: [⟨1 -1 -4 12 -2 6 -12 -15], ⟨0 9 22 -32 19 -8 56 67]]

Optimal tunings:

  • WE: ~2 = 1200.0282 ¢, ~11/9 = 344.7532 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 344.7453 ¢

Optimal ET sequence: 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: [⟨1 -1 -4 12 -2 6 -12 -15 -13], ⟨0 9 22 -32 19 -8 56 67 61]]

Optimal tunings:

  • WE: ~2 = 1200.0163 ¢, ~11/9 = 344.7461 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 344.7416 ¢

Optimal ET sequence: 87, 94, 181

Badness (Sintel): 1.17

Quanic

Quanic may be described as the 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.

Subgroup: 2.3.5.7

Comma list: 5120/5103, 5832000/5764801

Mapping: [⟨1 1 -4 0], ⟨0 5 54 24]]

mapping generators: ~2, ~160/147

Optimal tunings:

  • WE: ~2 = 1199.6159 ¢, ~160/147 = 140.4483 ¢
error map: ⟨-0.384 -0.098 -0.570 +1.933]
  • CWE: ~2 = 1200.0000 ¢, ~160/147 = 140.4862 ¢
error map: ⟨0.000 +0.476 -0.061 +2.842]

Optimal ET sequence: 94, 111, 205

Badness (Sintel): 4.54

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 1 -4 0 1], ⟨0 5 54 24 21]]

Optimal tunings:

  • WE: ~2 = 1199.7834 ¢, ~88/81 = 140.4635 ¢
  • CWE: ~2 = 1200.0000 ¢, ~88/81 = 140.4850 ¢

Optimal ET sequence: 94, 111, 205

Badness (Sintel): 1.94

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 540/539, 729/728, 1331/1323

Mapping: [⟨1 1 -4 0 1 3], ⟨0 5 54 24 21 6]]

Optimal tunings:

  • WE: ~2 = 1199.6639 ¢, ~13/12 = 140.4562 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/12 = 140.4904 ¢

Optimal ET sequence: 94, 111, 205

Badness (Sintel): 1.34

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 1 -4 0 1 3 -2], ⟨0 5 54 24 21 6 52]]

Optimal tunings:

  • WE: ~2 = 1199.6699 ¢, ~13/12 = 140.4586 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/12 = 140.4920 ¢

Optimal ET sequence: 94, 111, 205

Badness (Sintel): 1.08

19-limit

Subgroup: 2.3.5.7.11.13.17.19

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

Mapping: [⟨1 1 -4 0 1 3 -2 -5], ⟨0 5 54 24 21 6 52 79]]

Optimal tunings:

  • WE: ~2 = 1199.6745 ¢, ~13/12 = 140.4574 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/12 = 140.4908 ¢

Optimal ET sequence: 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 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[2].

Subgroup: 2.3.5.7

Comma list: 5120/5103, 578509309952/576650390625

Mapping: [⟨70 111 0 34], ⟨0 0 1 1]]

mapping generators: ~50421/50000, ~5

Optimal tunings:

  • WE: ~50421/50000 = 17.1387 ¢, ~5/4 = 386.8071 ¢
error map: ⟨-0.288 +0.445 -0.084 +0.121]
  • CWE: ~50421/50000 = 17.1429 ¢, ~5/4 = 386.6593 ¢
error map: ⟨0.000 +0.902 +0.346 +0.690]

Optimal ET sequence: 70, 140, 350, 490

Badness (Sintel): 5.40

References

  1. ↑ Yahoo! Tuning Group | Kwai
  2. ↑ Yahoo! Tuning Group | Jorgensen Temperament
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