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 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 circle 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]].  
{{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 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]]'' → [[Meantone family #Dominant|Meantone family]]
: +36/35, the 12 & 17c temperament, generated by the fifth with 5/4 mapped to the M3.  
* [[Garibaldi]] → [[Schismatic family #Garibaldi|Schismatic family]]
: +225/224, the 41 & 53 temperament, generated by the fifth with 5/4 mapped to the d4.  
* ''[[Kwai]]'' → [[Mirkwai clan #Kwai|Mirkwai clan]]
: +16875/16807, the 41 & 70 temperament, generated by the fifth with 5/4 mapped to the 4A7 aka m3 + 3 Pyth. commas.
* ''[[Diaschismic]]'' → [[Diaschismic family #Diaschismic|Diaschismic family]]
: +126/125, the 46 & 58 temperament, generated by the fifth and using a semioctave period.  
* [[Hemififths]] → [[Breedsmic temperaments #Hemififths|Breedsmic temperaments]]
: +2401/2400, the 41 & 58 temperament, fifth sliced into two and 5/4 mapped to the hemififth + Pyth. comma.
* [[Rodan]] → [[Gamelismic clan #Rodan|Gamelismic clan]]
: +245/243, the 41 & 46 temperament, fifth sliced into three as does slendric.
* ''[[Monkey]]'' → [[Tetracot family #Monkey|Tetracot family]]
: +875/864, the 34 & 41 temperament, fifth sliced into four as does tetracot.
* [[Buzzard]] → [[Vulture family #Buzzard|Vulture family]]
: +1728/1715, the 53 & 58 temperament, twelfth sliced into four as does vulture.
* [[Misty]] → [[Misty family #Misty|Misty family]]
: +3136/3125, the 12 & 99 temperament, generated by the fifth and using a 1/3-octave period.
* ''[[Supers]]'' → [[Stearnsmic clan #Supers|Stearnsmic clan]]
: +118098/117649, the 36c & 58 temperament, fifth sliced into three and using a semioctave period.
* ''[[Undim]]'' → [[Undim family #Septimal undim|Undim family]]
: +390625/388962, the 12 & 140 temperament, generated by the fifth and using a 1/4-octave period.
* ''[[Quinticosiennic]]'' → [[Quintaleap family #Quinticosiennic|Quintaleap family]]
: +395136/390625, the 12 & 145 temperament, fourth sliced into five.
* ''[[Quintakwai]]'' → [[Quindromeda family #Quintakwai|Quindromeda family]]
: +9765625/9680832, the 12 & 181 temperament, fourth sliced into five.
* [[Amity]] → [[Amity family #Septimal amity|Amity family]]
: +4375/4374, the 46 & 53 temperament, eleventh sliced into five.
* ''[[Countercata]]'' → [[Kleismic family #Countercata|Kleismic family]]
: +15625/15552, the 34 & 53 temperament, twelfth sliced into six as does hanson.
* ''[[Warrior]]'' → [[Sensipent family #Warrior|Sensipent family]]
: +78732/78125, the 46 & 65d temperament, 6th harmonic sliced into seven as does sensi.
* ''[[Alphaquarter]]'' → [[Escapade family #Alphaquarter|Escapade family]]
: +29360128/29296875, the 65d & 87 temperament, fourth sliced into nine as does escapade.  


== Undecental ==
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.
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 (5d7) or equivalently the perfect fourth (P4) - 3 Pyth. commas.  
 
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]]
 
Considered below are septiquarter, kwai, ketchup, undecental, 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 {{nowrap| 94 & 99 }} temperament. Its [[ploidacot]] is epsilon-heptacot. [[99edo]] makes for an excellent tuning, and [[292edo]] an even better one. [[94edo]] and [[104edo]] in the 104c val are also among the possibilities.  


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


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


[[Mapping]]: [{{val| 1 0 61 71 }}, {{val| 0 1 -37 -43 }}]
Comma list: 352/351, 847/845, 1716/1715, 14641/14580


{{Multival|legend=1| 1 -37 -43 -61 -71 4 }}
Mapping: {{mapping| 2 -8 -56 12 -25 9 | 0 7 38 -4 20 -1 }}


[[Optimal tuning]] ([[POTE]]): ~3/2 = 703.039
Optimal tunings:  
* WE: ~99/70 = 599.8565{{c}}, ~210/121 = 957.3261{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~210/121 = 957.5508{{c}}


{{Val list|legend=1| 29, 70, 99, 722bc, 821bc, 920bc, 1019bc, 1118bbcc, 1217bbcc, 1316bbccd }}
{{Optimal ET sequence|legend=0| 94, 198, 490f }}


[[Badness]]: 0.094603
Badness (Sintel): 1.44


== Leapday ==
== Kwai ==
{{Main| Leapday }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Kwai]].''
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Leapday]].''


Leapday tempers out {{monzo|31 -21 1}} (trisayo) in the 5-limit, mapping 5/4 to the triple augmented unison (3A1) or equivalently the minor third (m3) + 2 Pyth. commas. This temperament can be described as the 29 & 46 temperament, which tempers out the hemifamity and [[686/675]] (senga). The alternative extension [[Porwell temperaments #Polypyth|polypyth]] (46 & 121) tempers out the same 5-limit comma as the leapday, but with the porwell ([[6144/6125]]) rather than the hemifamity tempered out.
Named by [[Gene Ward Smith]] in 2004 for its "bridgeability"<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_10766.html Yahoo! Tuning Group | ''Kwai'']</ref>, kwai is generated by a [[3/2|perfect fifth]], and can be described as {{nowrap| 41 & 70 }}.  


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


[[Comma list]]: 686/675, 5120/5103
[[Comma list]]: 5120/5103, 16875/16807
 
[[Mapping]]: [{{val| 1 0 -31 -21 }}, {{val| 0 1 21 15 }}]


{{Multival|legend=1| 1 21 15 31 21 -24 }}
{{Mapping|legend=1| 1 0 -50 -40 | 0 1 33 27 }}
: mapping generators: ~2, ~3


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 704.263
[[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 }}


{{Val list|legend=1| 17c, 29, 46, 167d, 213d, 259cdd, 305bcdd }}
{{Optimal ET sequence|legend=1| 41, 111, 152, 345, 497d }}


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


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


Comma list: 121/120, 441/440, 686/675
Comma list: 540/539, 1375/1372, 5120/5103


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


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.250
Optimal tunings:
* WE: ~2 = 1199.6672{{c}}, ~3/2 = 702.4282{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6189{{c}}


Optimal GPV sequence: {{Val list| 17c, 29, 46, 167de, 213de, 259cdde }}
{{Optimal ET sequence|legend=0| 41, 111, 152, 497de, 649dde }}


Badness: 0.038624
Badness (Sintel): 0.867


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


Comma list: 91/90, 121/120, 169/168, 352/351
Comma list: 352/351, 540/539, 729/728, 1375/1372


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


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.214
Optimal tunings:
* WE: ~2 = 1199.4772{{c}}, ~3/2 = 702.3379{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6409{{c}}


Optimal GPV sequence: {{Val list| 17c, 29, 46, 121def, 167def, 213deff }}
{{Optimal ET sequence|legend=0| 41, 111, 152f, 415dff }}


Badness: 0.024732
Badness (Sintel): 1.01


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


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


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


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.229
Optimal tunings:
* WE: ~2 = 1199.3537{{c}}, ~3/2 = 702.2850{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6589{{c}}


Optimal GPV sequence: {{Val list| 17cg, 29g, 46, 121defg, 167defg, 213deffg }}
{{Optimal ET sequence|legend=0| 41, 70, 111, 152fg, 263dfg }}


Badness: 0.017863
Badness (Sintel): 1.12


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


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


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


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.135
Optimal tunings:
* WE: ~2 = 1199.3401{{c}}, ~3/2 = 702.2705{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6548{{c}}


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


Badness: 0.017356
Badness (Sintel): 1.03


===== 23-limit =====
==== Hemikwai ====
Subgroup: 2.3.5.7.11.13.17.19.23
Subgroup: 2.3.5.7.11.13
 
Comma list: 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
 
Optimal tunings:
* WE: ~2 = 1199.6968{{c}}, ~26/15 = 951.0740{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.3123{{c}}
 
{{Optimal ET sequence|legend=0| 82, 111, 193, 304d }}
 
Badness (Sintel): 1.82
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


Comma list: 91/90, 121/120, 133/132, 136/135, 154/153, 161/160, 169/168
Comma list: 442/441, 540/539, 676/675, 715/714, 5120/5103


Mapping: [{{val| 1 0 -31 -21 -14 -9 -34 9 -5 }}, {{val| 0 1 21 15 11 8 24 -3 6 }}]
Mapping: {{mapping| 1 0 -50 -40 32 -51 -30 | 0 2 66 54 -36 69 43 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.141
Optimal tunings:
* WE: ~2 = 1199.6861{{c}}, ~26/15 = 951.0654{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.3120{{c}}


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


Badness: 0.014065
Badness (Sintel): 1.31


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


Comma list: 77/76, 91/90, 121/120, 136/135, 153/152, 169/168
Comma list: 400/399, 442/441, 540/539, 676/675, 715/714, 1445/1444
 
Mapping: {{mapping| 1 0 -50 -40 32 -51 -30 -56 | 0 2 66 54 -36 69 43 76 }}
 
Optimal tunings:
* WE: ~2 = 1199.6718{{c}}, ~26/15 = 951.0526{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.3103{{c}}
 
{{Optimal ET sequence|legend=0| 82, 111, 193, 304dh }}
 
Badness (Sintel): 1.16
 
== Ketchup ==
Ketchup may be described as the {{nowrap| 46 & 94 }} temperament. It has a semi-octave period and a generator for a syntonic~septimal comma, four of which plus a period gives the perfect fifth; its [[ploidacot]] is diploid gamma-tetracot. [[140edo]] is an obvious tuning for this temperament.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 5120/5103, 1071875/1062882
 
{{Mapping|legend=1| 2 3 4 6 | 0 4 15 -9 }}
: mapping generators: ~1225/864, ~64/63
 
[[Optimal tuning]]s:
* [[WE]]: ~1225/864 = 599.9685{{c}}, ~64/63 = 25.7181{{c}}
: [[error map]]: {{val| -0.063 +0.823 -0.668 -0.478 }}
* [[CWE]]: ~1225/864 = 600.0000{{c}}, ~64/63 = 25.7181{{c}}
: error map: {{val| 0.000 +0.917 -0.543 -0.288 }}
 
{{Optimal ET sequence|legend=1| 46, 94, 140 }}
 
[[Badness]] (Sintel): 2.14
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 385/384, 1331/1323, 2200/2187
 
Mapping: {{mapping| 2 3 4 6 7 | 0 4 15 -9 -2 }}
 
Optimal tunings:
* WE: ~99/70 = 600.0678{{c}}, ~64/63 = 25.6963{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~64/63 = 25.6956{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{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
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 289/288, 325/324, 352/351, 385/384, 442/441
 
Mapping: {{mapping| 2 3 4 6 7 8 8 | 0 4 15 -9 -2 -14 4 }}
 
Optimal tunings:
* WE: ~17/12 = 600.0896{{c}}, ~66/65 = 25.7048{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~66/65 = 25.7017{{c}}
 
{{Optimal ET sequence|legend=0| 46, 94, 140 }}
 
Badness (Sintel): 0.845
 
=== 2.3.5.7.11.13.17.23 subgroup ===
Subgroup: 2.3.5.7.11.13.17.23


Mapping: [{{val| 1 0 -31 -21 -14 -9 -34 -37 }}, {{val| 0 1 21 15 11 8 24 26 }}]
Comma list: 253/252, 289/288, 325/324, 352/351, 385/384, 391/390


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.123
Mapping: {{mapping| 2 3 4 6 7 8 8 9 | 0 4 15 -9 -2 -14 4 1 }}


Optimal GPV sequence: {{Val list| 17cgh, 29g, 46h, 75dfg, 121defghh }}
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: 0.019065
{{Optimal ET sequence|legend=0| 46, 94, 140 }}


===== 23-limit =====
Badness (Sintel): 0.772
Subgroup: 2.3.5.7.11.13.17.19.23
 
== Undecental ==
Undecental adds the triwellisma to the comma list and may be described as the {{nowrap| 29 & 70 }} temperament. 5/4 is mapped to the quintuple-diminished seventh or equivalently the perfect fourth minus three [[diesis (scale theory)|dieses]]. [[99edo|58\99]] is an almost perfect generator, just as the name suggests. Another interesting tuning choice is the argent fifth, {{nowrap| 2<sup>(2 - sqrt (2))</sup> }}.
 
[[Subgroup]]: 2.3.5.7


Comma list: 77/76, 91/90, 115/114, 121/120, 136/135, 153/152, 161/160
[[Comma list]]: 5120/5103, 235298/234375


Mapping: [{{val| 1 0 -31 -21 -14 -9 -34 -37 -5 }}, {{val| 0 1 21 15 11 8 24 26 6 }}]
{{Mapping|legend=1| 1 0 61 71 | 0 1 -37 -43 }}
: mapping generators: ~2, ~3


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.114
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.6543{{c}}, ~3/2 = 702.8370{{c}}
: [[error map]]: {{val| -0.346 +0.536 +0.423 -0.494 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.0465{{c}}
: error map: {{val| 0.000 +1.092 +0.966 +0.175 }}


Optimal GPV sequence: {{Val list| 17cgh, 29g, 46h, 75dfg, 121defghh }}
{{Optimal ET sequence|legend=1| 29, 70, 99, 722bc, 821bc, 920bc, 1019bc }}


Badness: 0.016067
[[Badness]] (Sintel): 2.39


== Mystery ==
== Mystery ==
{{main| Mystery }}
{{Main| Mystery }}
: ''For the 5-limit version of this temperament, see [[High badness temperaments #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.
 
[[Subgroup]]: 2.3.5.7


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


[[Mapping]]: [{{val| 29 46 0 14 }}, {{val| 0 0 1 1 }}]
{{Mapping|legend=1| 29 46 0 14 | 0 0 1 1 }}
: mapping generators: ~50/49, ~5


{{Multival|legend=1| 0 29 29 46 46 -14 }}
[[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 }}


[[POTE generator]]: ~5/4 = 388.646
{{Optimal ET sequence|legend=1| 29, 58, 87, 145 }}


{{Val list|legend=1| 29, 58, 87, 145 }}
[[Badness]] (Sintel): 2.63
 
[[Badness]]: 0.103734


=== 11-limit ===
=== 11-limit ===
Line 188: Line 335:
Comma list: 441/440, 896/891, 3388/3375
Comma list: 441/440, 896/891, 3388/3375


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


POTE generator: ~5/4 = 388.460
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 GPV sequence: {{Val list| 29, 58, 87, 145 }}
{{Optimal ET sequence|legend=0| 29, 58, 87, 145 }}


Badness: 0.034291
Badness (Sintel): 1.13


=== 13-limit ===
=== 13-limit ===
Line 201: Line 350:
Comma list: 196/195, 352/351, 364/363, 676/675
Comma list: 196/195, 352/351, 364/363, 676/675


Mapping: [{{val| 29 46 0 14 33 40 }}, {{val| 0 0 1 1 1 1 }}]
Mapping: {{mapping| 29 46 0 14 33 40 | 0 0 1 1 1 1 }}


POTE generator: ~5/4 = 388.354
Optimal tunings:  
* WE: ~45/44 = 41.3623{{c}}, ~5/4 = 388.1942{{c}}
* CWE: ~40/39 = 41.3793{{c}}, ~5/4 = 387.9017{{c}}


Optimal GPV sequence: {{Val list| 29, 58, 87, 145, 232, 377cef }}
{{Optimal ET sequence|legend=0| 29, 58, 87, 145, 232 }}


Badness: 0.018591
Badness (Sintel): 0.768


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


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


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


[[POTE generator]]: ~160/147 = 140.493
[[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, 111, 205 }}
{{Optimal ET sequence|legend=1| 29, 82cd, 111, 140, 251, 391, 1424bbcdd }}


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


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


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


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


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


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


Badness: 0.058678
Badness (Sintel): 2.01


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


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


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


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


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


Badness: 0.032481
Badness (Sintel): 1.18


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


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


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


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


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


Badness: 0.021112
Badness (Sintel): 0.971


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


Comma list: 352/351, 400/399, 442/441, 456/455, 495/494, 715/714
Comma list: 286/285, 352/351, 363/361, 442/441, 476/475, 561/560
 
Mapping: {{mapping| 1 0 38 48 32 37 58 32 | 0 2 -45 -57 -36 -42 -68 -35 }}
 
Optimal tunings:
* WE: ~2 = 1199.7534{{c}}, ~26/15 = 951.2024{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.4020{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{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}}


Mapping: [{{val| 1 1 -4 0 1 3 -2 -5 }}, {{val| 0 5 54 24 21 6 52 79 }}]
{{Optimal ET sequence|legend=0| 29g, 82cdfgh, 111, 140 }}


POTE generator: ~13/12 = 140.496
Badness (Sintel): 1.10


Optimal GPV sequence: {{Val list| 94, 111, 205 }}
== Countriton ==
: ''For the 5-limit version, see [[Schismic–Mercator equivalence continuum #Countritonic]].''


Badness: 0.017273
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.  


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


[[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, 7558272/7503125


{{Multival|legend=1| 7 38 -4 44 -26 -116 }}
{{Mapping|legend=1| 1 -3 -15 13 | 0 9 34 -20 }}
: mapping generators: ~2, ~1225/864


[[POTE generator]]: ~147/128 = 242.453
[[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| 94, 99, 292, 391, 881bd, 1272bcd }}
{{Optimal ET sequence|legend=1| 51c, 53, 157, 210, 473cdd }}


[[Badness]]: 0.053760
[[Badness]] (Sintel): 3.32


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


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


POTE generators: ~121/105 = 242.4511
Optimal tunings:  
* WE: ~2 = 1199.5178{{c}}, ~77/54 = 611.2097{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~77/54 = 611.4495{{c}}


Optimal GPV sequence: {{Val list| 94, 198, 292, 490 }}
{{Optimal ET sequence|legend=0| 51ce, 53, 104c, 157 }}


Badness: 0.064160
Badness (Sintel): 2.80


==== 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: 176/175, 351/350, 847/845, 2197/2187


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


POTE generators: ~121/105 = 242.4448
Optimal tunings:  
* WE: ~2 = 1199.5944{{c}}, ~77/54 = 611.2491{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~77/54 = 611.4506{{c}}


Optimal GPV sequence: {{Val list| 94, 198, 490f }}
{{Optimal ET sequence|legend=0| 51ce, 53, 104c, 157 }}


Badness: 0.034834
Badness (Sintel): 1.75


== Tricot ==
== Artoneutral ==
{{see also| Tricot family }}
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.


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 called as "trimot", as confirmed by the data from [http://x31eq.com/cgi-bin/rt.cgi?ets=53_17c&limit=7 x31eq]), 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.
Artoneutral was named by [[Flora Canou]] in 2023 for its generator's quality.  


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2430/2401, 5120/5103


[[Mapping]]: [{{val| 1 0 -13 -3 }}, {{val| 0 3 29 11 }}]
[[Comma list]]: 5120/5103, 3828125/3779136


{{Multival|legend=1| 3 29 11 39 9 -56 }}
{{Mapping|legend=1| 1 -1 -4 12 | 0 9 22 -32 }}
: mapping generators: ~2, ~128/105


[[POTE generator]]: ~81/56 = 634.026
[[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 }}


{{Val list|legend=1| 17c, 36c, 53, 229dd, 282dd }}
{{Optimal ET sequence|legend=1| 87, 94, 181 }}


[[Badness]]: 0.100127
[[Badness]] (Sintel): 3.98


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


Comma list: 99/98, 121/120, 5120/5103
Comma list: 385/384, 2200/2187, 4000/3993


Mapping: [{{val| 1 0 -13 -3 -5 }}, {{val| 0 3 29 11 16 }}]
Mapping: {{mapping| 1 -1 -4 12 -2 | 0 9 22 -32 19 }}


POTE generator: ~63/44 = 634.027
Optimal tunings:  
* WE: ~2 = 1200.1668{{c}}, ~11/9 = 344.8027{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 344.7557{{c}}


Optimal GPV sequence: {{Val list| 17c, 36ce, 53, 70, 123de }}
{{Optimal ET sequence|legend=0| 87, 181 }}


Badness: 0.056134
Badness (Sintel): 1.52


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


Comma list: 99/98, 121/120, 169/168, 352/351
Comma list: 325/324, 352/351, 385/384, 1575/1573


Mapping: [{{val| 1 0 -13 -3 -5 0 }}, {{val| 0 3 29 11 16 7 }}]
Mapping: {{mapping| 1 -1 -4 12 -2 6 | 0 9 22 -32 19 -8 }}


POTE generator: ~13/9 = 634.012
Optimal tunings:  
* WE: ~2 = 1200.0662{{c}}, ~11/9 = 344.7804{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 344.7617{{c}}


Optimal GPV sequence: {{Val list| 17c, 36ce, 53, 70, 123de }}
{{Optimal ET sequence|legend=0| 87, 181 }}


Badness: 0.032102
Badness (Sintel): 1.08


== Countriton ==
=== 17-limit ===
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Countriton]].''
Subgroup: 2.3.5.7.11.13.17


Subgroup: 2.3.5.7
Comma list: 325/324, 352/351, 375/374, 385/384, 595/594


[[Comma list]]: 5120/5103, 7558272/7503125
Mapping: {{mapping| 1 -1 -4 12 -2 6 -12 | 0 9 22 -32 19 -8 56 }}


[[Mapping]]: [{{val|1 6 19 -7}}, {{val|0 -9 -34 20}}]
Optimal tunings:  
* WE: ~2 = 1200.0346{{c}}, ~11/9 = 344.7589{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 344.7492{{c}}


{{Multival|legend=1| 9 34 -20 33 -57 -142 }}
{{Optimal ET sequence|legend=0| 87, 94, 181 }}


[[POTE generator]]: ~1728/1225 = 588.582
Badness (Sintel): 1.16


{{Val list|legend=1| 53, 157, 210 }}
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


[[Badness]]: 0.131191
Comma list: 325/324, 352/351, 375/374, 385/384, 400/399, 595/594


=== 11-limit ===
Mapping: {{mapping| 1 -1 -4 12 -2 6 -12 -15 | 0 9 22 -32 19 -8 56 67 }}
Subgroup: 2.3.5.7.11


Comma list: 176/175, 5120/5103, 41503/41472
Optimal tunings:  
* WE: ~2 = 1200.0282{{c}}, ~11/9 = 344.7532{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 344.7453{{c}}


Mapping: [{{val|1 6 19 -7 27}}, {{val|0 -9 -34 20 -48}}]
{{Optimal ET sequence|legend=0| 87, 94, 181 }}


POTE generator: ~108/77 = 588.545
Badness (Sintel): 1.19


Optimal GPV sequence: {{Val list| 53, 104c, 157 }}
=== 23-limit ===
 
Subgroup: 2.3.5.7.11.13.17.19.23
Badness: 0.084782
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Comma list: 176/175, 351/350, 847/845, 2197/2187
Comma list: 300/299, 325/324, 352/351, 375/374, 385/384, 400/399, 484/483


Mapping: [{{val|1 6 19 -7 27 14}}, {{val|0 -9 -34 20 -48 -21}}]
Mapping: {{mapping| 1 -1 -4 12 -2 6 -12 -15 -13 | 0 9 22 -32 19 -8 56 67 61 }}


POTE generator: ~108/77 = 588.544
Optimal tunings:  
* WE: ~2 = 1200.0163{{c}}, ~11/9 = 344.7461{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 344.7416{{c}}


Optimal GPV sequence: {{Val list| 53, 104c, 157 }}
{{Optimal ET sequence|legend=0| 87, 94, 181 }}


Badness: 0.042321
Badness (Sintel): 1.17


== Ketchup ==
== Quanic ==
Subgroup: 2.3.5.7
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.  


[[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, 5832000/5764801


{{Multival|legend=1| 8 30 -18 29 -51 -126 }}
{{Mapping|legend=1| 1 1 -4 0 | 0 5 54 24 }}
: mapping generators: ~2, ~160/147


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


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


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


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


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


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


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


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


Badness: 0.039555
Badness (Sintel): 1.94


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


Comma list: 325/324, 352/351, 847/845, 1331/1323
Comma list: 352/351, 540/539, 729/728, 1331/1323


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


POTE generator: ~55/54 = ~64/63 = ~81/80 = 25.697
Optimal tunings:  
* WE: ~2 = 1199.6639{{c}}, ~13/12 = 140.4562{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.4904{{c}}


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


Badness: 0.024824
Badness (Sintel): 1.34


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


Comma list: 289/288, 325/324, 352/351, 385/384, 561/560
Comma list: 352/351, 442/441, 540/539, 715/714, 847/845


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


POTE generator: ~55/54 = ~64/63 = ~81/80 = 25.701
Optimal tunings:  
* WE: ~2 = 1199.6699{{c}}, ~13/12 = 140.4586{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.4920{{c}}


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


Badness: 0.016591
Badness (Sintel): 1.08


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


Comma list: 190/189, 209/208, 289/288, 352/351, 385/384, 561/560
Comma list: 352/351, 400/399, 442/441, 456/455, 495/494, 715/714


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


POTE generator: ~55/54 = ~64/63 = ~81/80 = 25.660
Optimal tunings:  
* WE: ~2 = 1199.6745{{c}}, ~13/12 = 140.4574{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.4908{{c}}


Optimal GPV sequence: {{Val list| 46, 94, 140h, 234eh }}
{{Optimal ET sequence|legend=0| 94, 111, 205 }}


Badness: 0.018170
Badness (Sintel): 1.05


=== 23-limit ===
== Jorgensen ==
Subgroup: 2.3.5.7.11.13.17.19.23
: ''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


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


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


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


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


Badness: 0.014033
== References ==


[[Category:Aberschismic temperaments| ]] <!-- main article -->
[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Hemifamity temperaments| ]] <!-- main article -->
[[Category:Catalogs of rank-2 temperaments]]
[[Category:Rank 2]]

Latest revision as of 07:50, 15 August 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, 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

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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