Hemifamity temperaments: Difference between revisions

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Diaschismic is generated by the fifth with a semi-octave period. Hemififths has the fifth sliced into two and 5/4 mapped to the hemififth + Pyth. comma. Hemidromeda has the fourth sliced into two and 5/4 mapped to the hemifourth + 3d4. Rodan has the fifth sliced into three as does slendric. Alphatrimot has the twelfth sliced into three as does alphatricot. Monkey has the fifth sliced into four as does tetracot. Buzzard has the twelfth sliced into four as does vulture. Misty is generated by the fifth with a 1/3-octave period. Supers has the fifth sliced into three with a semi-octave period. Undim is generated by the fifth with a 1/4-octave period. Quinticosiennic and quintakwai have the fourth sliced into five. Amity has the eleventh sliced into five. Countercata has the twelfth sliced into six as does hanson. Warrior has the 6th harmonic sliced into seven as does sensi. Finally, alphaquarter has the fourth sliced into nine as does escapade.  
Diaschismic is generated by the fifth with a semi-octave period. Hemififths has the fifth sliced into two and 5/4 mapped to the hemififth + Pyth. comma. Hemidromeda has the fourth sliced into two and 5/4 mapped to the hemifourth + 3d4. Rodan has the fifth sliced into three as does slendric. Alphatrimot has the twelfth sliced into three as does alphatricot. Monkey has the fifth sliced into four as does tetracot. Buzzard has the twelfth sliced into four as does vulture. Misty is generated by the fifth with a 1/3-octave period. Supers has the fifth sliced into three with a semi-octave period. Undim is generated by the fifth with a 1/4-octave period. Quinticosiennic and quintakwai have the fourth sliced into five. Amity has the eleventh sliced into five. Countercata has the twelfth sliced into six as does hanson. Warrior has the 6th harmonic sliced into seven as does sensi. Finally, alphaquarter has the fourth sliced into nine as does escapade.  


Temperaments considered below are undecental, leapday, hemidromeda, mystery, quanic, septiquarter, countriton, artoneutral and ketchup. Discussed elsewhere are:
Temperaments discussed elsewhere are:
* [[Dominant (temperament)|Dominant]] (+36/35) → [[Meantone family #Dominant|Meantone family]]
* [[Dominant (temperament)|Dominant]] (+36/35) → [[Meantone family #Dominant|Meantone family]]
* [[Garibaldi]] (+225/224) → [[Schismatic family #Garibaldi|Schismatic family]]
* [[Garibaldi]] (+225/224) → [[Schismatic family #Garibaldi|Schismatic family]]
* ''[[Kwai]]'' (+16875/16807) → [[Mirkwai clan #Kwai|Mirkwai clan]]
* [[Diaschismic]] (+126/125) → [[Diaschismic family #Septimal diaschismic|Diaschismic family]]
* [[Diaschismic]] (+126/125) → [[Diaschismic family #Septimal diaschismic|Diaschismic family]]
* [[Hemififths]] (+2401/2400) → [[Breedsmic temperaments #Hemififths|Breedsmic temperaments]]
* [[Hemififths]] (+2401/2400) → [[Breedsmic temperaments #Hemififths|Breedsmic temperaments]]
* [[Rodan]] (+245/243) → [[Gamelismic clan #Rodan|Gamelismic clan]]
* [[Rodan]] (+245/243) → [[Gamelismic clan #Rodan|Gamelismic clan]]
* ''[[Alphatrimot]]'' (+2430/2401) → [[Alphatricot family #Alphatrimot|Alphatricot family]]
* ''[[Alphatrimot]]'' (+2430/2401) → [[Alphatricot family #Alphatrimot|Alphatricot family]]
* ''[[Monkey]]'' (+875/864) → [[Tetracot family #Monkey|Tetracot family]]
* [[Buzzard]] (+1728/1715) → [[Vulture family #Buzzard|Vulture family]]
* [[Misty]] (+3136/3125) → [[Misty family #Misty|Misty family]]
* [[Misty]] (+3136/3125) → [[Misty family #Misty|Misty family]]
* ''[[Supers]]'' (+118098/117649) → [[Stearnsmic clan #Supers|Stearnsmic clan]]
* [[Monkey]] (+875/864) → [[Tetracot family #Monkey|Tetracot family]]
* [[Buzzard]] (+1728/1715) → [[Buzzardsmic clan #Buzzard|Buzzardsmic clan]]
* ''[[Undim]]'' (+390625/388962) → [[Undim family #Septimal undim|Undim family]]
* ''[[Undim]]'' (+390625/388962) → [[Undim family #Septimal undim|Undim family]]
* ''[[Quinticosiennic]]'' (+395136/390625) → [[Quintaleap family #Quinticosiennic|Quintaleap family]]
* ''[[Quinticosiennic]]'' (+395136/390625) → [[Quintaleap family #Quinticosiennic|Quintaleap family]]
Line 23: Line 21:
* [[Amity]] (+4375/4374) → [[Amity family #Septimal amity|Amity family]]
* [[Amity]] (+4375/4374) → [[Amity family #Septimal amity|Amity family]]
* ''[[Countercata]]'' (+15625/15552) → [[Kleismic family #Countercata|Kleismic family]]
* ''[[Countercata]]'' (+15625/15552) → [[Kleismic family #Countercata|Kleismic family]]
* ''[[Abergravity]]'' (+177147/175000) → [[Gravity family #Abergravity|Gravity family]]
* ''[[Supers]]'' (+118098/117649) → [[Stearnsmic clan #Supers|Stearnsmic clan]]
* ''[[Warrior]]'' (+78732/78125) → [[Sensipent family #Warrior|Sensipent family]]
* ''[[Warrior]]'' (+78732/78125) → [[Sensipent family #Warrior|Sensipent family]]
* ''[[Alphaquarter]]'' (+29360128/29296875) → [[Escapade family #Alphaquarter|Escapade family]]
* ''[[Alphaquarter]]'' (+29360128/29296875) → [[Escapade family #Alphaquarter|Escapade family]]


== Undecental ==
Considered below are septiquarter, kwai, ketchup, undecental, leapday, mystery, hemidromeda, countriton, artoneutral, quanic and jorgensen, in the order of increasing [[TE logflat badness]].
Undecental adds the triwellisma to the comma list and may be described as the {{nowrap| 29 & 70 }} temperament. 5/4 is mapped to the quintuple-diminished seventh or equivalently the perfect fourth minus three [[diesis (scale theory)|dieses]]. [[99edo|58\99]] is an almost perfect generator, just as the name suggests. Another interesting tuning choice is the argent fifth, {{nowrap| 2<sup>(2 - sqrt (2))</sup> }}.  
 
== Septiquarter ==
Septiquarter tempers out [[420175/419904]] and may be described as the {{nowrap| 94 & 99 }} temperament. Its [[ploidacot]] is epsilon-heptacot. [[99edo]] makes for an excellent tuning, and [[292edo]] an even better one. [[94edo]] and [[104edo]] in the 104c val are also among the possibilities.  


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


[[Comma list]]: 5120/5103, 235298/234375
[[Comma list]]: 5120/5103, 420175/419904


{{Mapping|legend=1| 1 0 61 71 | 0 1 -37 -43 }}
{{Mapping|legend=1| 1 -4 -28 6 | 0 7 38 -4 }}
: mapping generators: ~2, ~3
: mapping generators: ~2, ~243/140


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.6543{{c}}, ~3/2 = 702.8370{{c}}
* [[WE]]: ~2 = 1199.7212{{c}}, ~243/140 = 957.3250{{c}}
: [[error map]]: {{val| -0.346 +0.536 +0.423 -0.494 }}
: [[error map]]: {{val| -0.279 +0.435 -0.158 +0.201 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.0465{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~243/140 = 957.5424{{c}}
: error map: {{val| 0.000 +1.092 +0.966 +0.175 }}
: error map: {{val| 0.000 +0.842 +0.298 +1.004 }}
<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~3/2 = 703.039{{c}} -->
 
{{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
 
Comma list: 352/351, 847/845, 1716/1715, 14641/14580
 
Mapping: {{mapping| 2 -8 -56 12 -25 9 | 0 7 38 -4 20 -1 }}


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


[[Badness]] (Sintel): 2.39
{{Optimal ET sequence|legend=0| 94, 198, 490f }}


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


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


It has an alternative extension called [[porwell temperaments #Polypyth|polypyth]], which tempers out the same 5-limit comma as leapday, but with the porwell ([[6144/6125]]) rather than the hemifamity comma tempered out.
Named by [[Gene Ward Smith]] in 2004 for its "bridgeability"<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_10766.html Yahoo! Tuning Group | ''Kwai'']</ref>, kwai is generated by a [[3/2|perfect fifth]], and can be described as {{nowrap| 41 & 70 }}.  


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


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


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


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.7167{{c}}, ~3/2 = 704.0971{{c}}
* [[WE]]: ~2 = 1199.7337{{c}}, ~3/2 = 702.4600{{c}}
: [[error map]]: {{val| -0.283 +1.859 +2.559 -5.669 }}
: [[error map]]: {{val| -0.266 +0.239 -0.607 +1.055 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.2504{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.6085{{c}}
: error map: {{val| 0.000 +2.295 +2.945 -5.070 }}
: error map: {{val| 0.000 +0.653 -0.234 +1.603 }}
<!-- * [[CTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.2257{{c}}
* [[POTE]]: ~2 = 1200.000{{c}}, ~3/2 = 704.263{{c}} -->


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


[[Badness]] (Sintel): 2.43
[[Badness]] (Sintel): 1.38


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


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


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


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.0731{{c}}, ~3/2 = 704.2933{{c}}
* WE: ~2 = 1199.6672{{c}}, ~3/2 = 702.4282{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2538{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6189{{c}}
<!-- * CTE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2625{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 704.250{{c}} -->


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


Badness (Sintel): 1.28
Badness (Sintel): 0.867


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


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


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


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.4758{{c}}, ~3/2 = 704.4930{{c}}
* WE: ~2 = 1199.4772{{c}}, ~3/2 = 702.3379{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2346{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6409{{c}}
<!-- * CTE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2924{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 704.214{{c}} -->


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


Badness (Sintel): 1.02
Badness (Sintel): 1.01


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


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


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


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.4818{{c}}, ~3/2 = 704.5121{{c}}
* WE: ~2 = 1199.3537{{c}}, ~3/2 = 702.2850{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2507{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6589{{c}}
<!-- * CTE: ~2 = 1200.0000{{c}}, ~3/2 = 704.3098{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 704.229{{c}} -->


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


Badness (Sintel): 0.910
Badness (Sintel): 1.12


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


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


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


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1201.0192{{c}}, ~3/2 = 704.7333{{c}}
* WE: ~2 = 1199.3401{{c}}, ~3/2 = 702.2705{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.1680{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6548{{c}}
<!-- * CTE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2990{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 704.135{{c}} -->


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


Badness (Sintel): 1.06
Badness (Sintel): 1.03


===== 23-limit =====
==== Hemikwai ====
Subgroup: 2.3.5.7.11.13.17.19.23
Subgroup: 2.3.5.7.11.13


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


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


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.9738{{c}}, ~3/2 = 704.7120{{c}}
* WE: ~2 = 1199.6968{{c}}, ~26/15 = 951.0740{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.1695{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.3123{{c}}
<!-- * CTE: ~2 = 1200.0000{{c}}, ~3/2 = 704.3035{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 704.141{{c}} -->


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


Badness (Sintel): 1.01
Badness (Sintel): 1.82


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


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


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


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.4745{{c}}, ~3/2 = 704.4016{{c}}
* WE: ~2 = 1199.6861{{c}}, ~26/15 = 951.0654{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.1442{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.3120{{c}}
<!-- * CTE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2037{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 704.123{{c}} -->


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


Badness (Sintel): 1.16
Badness (Sintel): 1.31


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


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


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


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.5425{{c}}, ~3/2 = 704.4319{{c}}
* WE: ~2 = 1199.6718{{c}}, ~26/15 = 951.0526{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.1349{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.3103{{c}}
<!-- * CTE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2082{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 704.114{{c}} -->


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


Badness (Sintel): 1.15
Badness (Sintel): 1.16


== Hemidromeda ==
== Ketchup ==
Hemidromeda may be described as the {{nowrap| 29 & 111 }} temperament. The name ''hemidromeda'' comes from "hemi-" (Ancient Greek for "one half") and ''[[andromeda]]'', because the generator is 1/2 of andromeda's perfect twelfth (~3/1, about 1902.4 cents).
Ketchup may be described as the {{nowrap| 46 & 94 }} temperament. It has a semi-octave period and a generator for a syntonic~septimal comma, four of which plus a period gives the perfect fifth; its [[ploidacot]] is diploid gamma-tetracot. [[140edo]] is an obvious tuning for this temperament.  


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


[[Comma list]]: 5120/5103, 52734375/52706752
[[Comma list]]: 5120/5103, 1071875/1062882


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


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.7236{{c}}, ~12500/7203 = 951.1864{{c}}
* [[WE]]: ~1225/864 = 599.9685{{c}}, ~64/63 = 25.7181{{c}}
: [[error map]]: {{val| -0.276 +0.418 -0.205 +0.282 }}
: [[error map]]: {{val| -0.063 +0.823 -0.668 -0.478 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~12500/7203 = 951.4098{{c}}
* [[CWE]]: ~1225/864 = 600.0000{{c}}, ~64/63 = 25.7181{{c}}
: error map: {{val| 0.000 +0.865 +0.243 +0.813 }}
: error map: {{val| 0.000 +0.917 -0.543 -0.288 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~12500/7203 = 951.419{{c}} -->


{{Optimal ET sequence|legend=1| 29, 82cd, 111, 140, 251, 391, 1424bbcdd }}
{{Optimal ET sequence|legend=1| 46, 94, 140 }}


[[Badness]] (Sintel): 2.93
[[Badness]] (Sintel): 2.14


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


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


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


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.8767{{c}}, ~400/231 = 951.3065{{c}}
* WE: ~99/70 = 600.0678{{c}}, ~64/63 = 25.6963{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~400/231 = 951.4063{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~64/63 = 25.6956{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~400/231 = 951.411{{c}} -->


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


Badness (Sintel): 2.01
Badness (Sintel): 1.31


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


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


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


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.8753{{c}}, ~26/15 = 951.3054{{c}}
* WE: ~99/70 = 600.0612{{c}}, ~66/65 = 25.7000{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.4064{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~66/65 = 25.6978{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~26/15 = 951.412{{c}} -->


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


Badness (Sintel): 1.18
Badness (Sintel): 1.03


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


Comma list: 352/351, 442/441, 561/560, 676/675, 715/714
Comma list: 289/288, 325/324, 352/351, 385/384, 442/441
 
Mapping: {{mapping| 2 3 4 6 7 8 8 | 0 4 15 -9 -2 -14 4 }}
 
Optimal tunings:
* WE: ~17/12 = 600.0896{{c}}, ~66/65 = 25.7048{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~66/65 = 25.7017{{c}}
 
{{Optimal ET sequence|legend=0| 46, 94, 140 }}
 
Badness (Sintel): 0.845
 
=== 2.3.5.7.11.13.17.23 subgroup ===
Subgroup: 2.3.5.7.11.13.17.23
 
Comma list: 253/252, 289/288, 325/324, 352/351, 385/384, 391/390
 
Mapping: {{mapping| 2 3 4 6 7 8 8 9 | 0 4 15 -9 -2 -14 4 1 }}
 
Optimal tunings:
* WE: ~17/12 = 600.1139{{c}}, ~66/65 = 25.7053{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~66/65 = 25.7013{{c}}
 
{{Optimal ET sequence|legend=0| 46, 94, 140 }}
 
Badness (Sintel): 0.772
 
== Undecental ==
Undecental adds the triwellisma to the comma list and may be described as the {{nowrap| 29 & 70 }} temperament. 5/4 is mapped to the quintuple-diminished seventh or equivalently the perfect fourth minus three [[diesis (scale theory)|dieses]]. [[99edo|58\99]] is an almost perfect generator, just as the name suggests. Another interesting tuning choice is the argent fifth, {{nowrap| 2<sup>(2 - sqrt (2))</sup> }}.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 5120/5103, 235298/234375
 
{{Mapping|legend=1| 1 0 61 71 | 0 1 -37 -43 }}
: mapping generators: ~2, ~3
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.6543{{c}}, ~3/2 = 702.8370{{c}}
: [[error map]]: {{val| -0.346 +0.536 +0.423 -0.494 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.0465{{c}}
: error map: {{val| 0.000 +1.092 +0.966 +0.175 }}
 
{{Optimal ET sequence|legend=1| 29, 70, 99, 722bc, 821bc, 920bc, 1019bc }}
 
[[Badness]] (Sintel): 2.39
 
== Leapday ==
{{Main| Leapday }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Leapday]].''
 
Leapday tempers out [[686/675]], the senga, in addition to the hemifamity comma, and may be described as the {{nowrap| 29 & 46 }} temperament. It extends [[leapfrog]], such that [[7/4]] is found by 15 generators up, as a double-augmented fifth (a major sixth and a diesis). 5/4 is found by a tritone above that, as a triple-augmented unison (a minor third and two dieses). [[46edo]] itself is an excellent tuning for this.
 
Leapday is more notable in the higher limits than the lower, as it nails the 13-limit pretty well from identifying [[14/11]] by a major third and [[13/11]] by a minor third, tempering out not only [[352/351]] and [[364/363]] but [[91/90]], [[121/120]], [[169/168]] and [[196/195]]. It can be further extended to include the [[17/1|17th]] and [[23/1|23rd]] [[harmonic]]s. Adding 17 would fix the valid diamond monotone tuning to 46edo, however.
 
Leapday has an alternative extension called [[porwell temperaments #Polypyth|polypyth]], which tempers out the same 5-limit comma as leapday, but with the porwell ([[6144/6125]]) rather than the hemifamity comma tempered out.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 686/675, 5120/5103
 
{{Mapping|legend=1| 1 0 -31 -21 | 0 1 21 15 }}
: mapping generators: ~2, ~3
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.7167{{c}}, ~3/2 = 704.0971{{c}}
: [[error map]]: {{val| -0.283 +1.859 +2.559 -5.669 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.2504{{c}}
: error map: {{val| 0.000 +2.295 +2.945 -5.070 }}
 
{{Optimal ET sequence|legend=1| 17c, 29, 46 }}
 
[[Badness]] (Sintel): 2.43
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 121/120, 441/440, 686/675
 
Mapping: {{mapping| 1 0 -31 -21 -14 | 0 1 21 15 11 }}
 
Optimal tunings:
* WE: ~2 = 1200.0731{{c}}, ~3/2 = 704.2933{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2538{{c}}
 
{{Optimal ET sequence|legend=0| 17c, 29, 46 }}
 
Badness (Sintel): 1.28
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 91/90, 121/120, 169/168, 352/351


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


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.8770{{c}}, ~26/15 = 951.3039{{c}}
* WE: ~2 = 1200.4758{{c}}, ~3/2 = 704.4930{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.4035{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2346{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~26/15 = 951.409{{c}} -->


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


Badness (Sintel): 0.971
Badness (Sintel): 1.02


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


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


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


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.7534{{c}}, ~26/15 = 951.2024{{c}}
* WE: ~2 = 1200.4818{{c}}, ~3/2 = 704.5121{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.4020{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2507{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~26/15 = 951.413{{c}} -->


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


Badness (Sintel): 1.01
Badness (Sintel): 0.910


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


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


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


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.9128{{c}}, ~26/15 = 951.3371{{c}}
* WE: ~2 = 1200.5169{{c}}, ~3/2 = 704.5279{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.4076{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2450{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~26/15 = 951.412{{c}} -->


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


Badness (Sintel): 1.10
Badness (Sintel): 0.872


== Mystery ==
== Mystery ==
Line 298: Line 397:
: ''For the 5-limit version, see [[29th-octave temperaments #Mystery]].''
: ''For the 5-limit version, see [[29th-octave temperaments #Mystery]].''


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


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 312: Line 411:
* [[CWE]]: ~50/49 = 41.3793{{c}}, ~5/4 = 388.3030{{c}}
* [[CWE]]: ~50/49 = 41.3793{{c}}, ~5/4 = 388.3030{{c}}
: error map: {{val| 0.000 +1.493 +1.989 -1.213 }}
: error map: {{val| 0.000 +1.493 +1.989 -1.213 }}
<!-- * [[POTE]]: ~50/49 = 41.379{{c}}, ~5/4 = 388.646{{c}} -->


{{Optimal ET sequence|legend=1| 29, 58, 87, 145 }}
{{Optimal ET sequence|legend=1| 29, 58, 87, 145 }}
Line 326: Line 424:


Optimal tunings:  
Optimal tunings:  
* WE: ~50/49 = 41.3637{{c}}, ~5/4 = 388.3136{{c}}
* WE: ~45/44 = 41.3637{{c}}, ~5/4 = 388.3136{{c}}
* CWE: ~50/49 = 41.3793{{c}}, ~5/4 = 388.0598{{c}}
* CWE: ~45/44 = 41.3793{{c}}, ~5/4 = 388.0598{{c}}
<!-- * POTE: ~45/44 = 41.379{{c}}, ~5/4 = 388.460{{c}} -->


{{Optimal ET sequence|legend=0| 29, 58, 87, 145 }}
{{Optimal ET sequence|legend=0| 29, 58, 87, 145 }}
Line 342: Line 439:


Optimal tunings:  
Optimal tunings:  
* WE: ~50/49 = 41.3623{{c}}, ~5/4 = 388.1942{{c}}
* WE: ~45/44 = 41.3623{{c}}, ~5/4 = 388.1942{{c}}
* CWE: ~50/49 = 41.3793{{c}}, ~5/4 = 387.9017{{c}}
* CWE: ~40/39 = 41.3793{{c}}, ~5/4 = 387.9017{{c}}
<!-- * POTE: ~45/44 = 41.379{{c}}, ~5/4 = 388.354{{c}} -->


{{Optimal ET sequence|legend=0| 29, 58, 87, 145, 232 }}
{{Optimal ET sequence|legend=0| 29, 58, 87, 145, 232 }}
Line 350: Line 446:
Badness (Sintel): 0.768
Badness (Sintel): 0.768


== Quanic ==
== Hemidromeda ==
Hemidromeda may be described as the {{nowrap| 29 & 111 }} temperament. Named by [[Xenllium]] in 2023, ''hemidromeda'' comes from ''hemi-'' (Ancient Greek for "one half") and ''[[andromeda]]'', because the generator is 1/2 of andromeda's perfect twelfth (~3/1, about 1902.4 cents); the ploidacot for this temperament is alpha-dicot.
 
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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


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


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.6159{{c}}, ~160/147 = 140.4483{{c}}
* [[WE]]: ~2 = 1199.7236{{c}}, ~12500/7203 = 951.1864{{c}}
: [[error map]]: {{val| -0.384 -0.098 -0.570 +1.933 }}
: [[error map]]: {{val| -0.276 +0.418 -0.205 +0.282 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~160/147 = 140.4862{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~12500/7203 = 951.4098{{c}}
: error map: {{val| 0.000 +0.476 -0.061 +2.842 }}
: error map: {{val| 0.000 +0.865 +0.243 +0.813 }}
<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~160/147 = 140.493{{c}} -->


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


[[Badness]] (Sintel): 4.54
[[Badness]] (Sintel): 2.93


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


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


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


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.7834{{c}}, ~88/81 = 140.4635{{c}}
* WE: ~2 = 1199.8767{{c}}, ~400/231 = 951.3065{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~88/81 = 140.4850{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~400/231 = 951.4063{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~88/81 = 140.489{{c}} -->


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


Badness (Sintel): 1.94
Badness (Sintel): 2.01


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


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


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


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.6639{{c}}, ~13/12 = 140.4562{{c}}
* WE: ~2 = 1199.8753{{c}}, ~26/15 = 951.3054{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.4904{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.4064{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~13/12 = 140.496{{c}} -->


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


Badness (Sintel): 1.34
Badness (Sintel): 1.18


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


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


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


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.6699{{c}}, ~13/12 = 140.4586{{c}}
* WE: ~2 = 1199.8770{{c}}, ~26/15 = 951.3039{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.4920{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.4035{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~13/12 = 140.497{{c}} -->


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


Badness (Sintel): 1.08
Badness (Sintel): 0.971


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


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


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


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.6745{{c}}, ~13/12 = 140.4574{{c}}
* WE: ~2 = 1199.7534{{c}}, ~26/15 = 951.2024{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.4908{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.4020{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~13/12 = 140.496{{c}} -->


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


Badness (Sintel): 1.05
Badness (Sintel): 1.01


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


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


{{Mapping|legend=1| 1 -4 -28 6 | 0 7 38 -4 }}
Mapping: {{mapping| 1 0 38 48 32 37 58 32 18 | 0 2 -45 -57 -36 -42 -68 -35 -17 }}
: mapping generators: ~2, ~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 }}
<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~243/140 = 957.547{{c}} -->
 
{{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:  
Optimal tunings:  
* WE: ~99/70 = 599.8953{{c}}, ~210/121 = 957.3819{{c}}
* WE: ~2 = 1199.9128{{c}}, ~26/15 = 951.3371{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~210/121 = 957.5449{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.4076{{c}}
<!-- * POTE: ~2 = 1200.0000{{c}}, ~210/121 = 957.5489{{c}} -->


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


Badness (Sintel): 2.12
Badness (Sintel): 1.10


==== 13-limit ====
== Countriton ==
Subgroup: 2.3.5.7.11.13
: ''For the 5-limit version, see [[Schismic–Mercator equivalence continuum #Countritonic]].''


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


Mapping: {{mapping| 2 -8 -56 12 -25 9 | 0 7 38 -4 20 -1 }}
Countriton was named by [[Xenllium]] in 2022 as a counterpart of [[untriton]].  
 
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: ~2 = 1200.0000{{c}}, ~210/121 = 957.5552{{c}} -->
 
{{Optimal ET sequence|legend=0| 94, 198, 490f }}
 
Badness (Sintel): 1.44
 
== Countriton ==
: ''For the 5-limit version, see [[Schismic–Mercator equivalence continuum #Countritonic]].''


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 491: Line 552:
[[Comma list]]: 5120/5103, 7558272/7503125
[[Comma list]]: 5120/5103, 7558272/7503125


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


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


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


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


=== 11-limit ===
=== 11-limit ===
Line 505: Line 570:
Comma list: 176/175, 5120/5103, 41503/41472
Comma list: 176/175, 5120/5103, 41503/41472


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


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


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


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


=== 13-limit ===
=== 13-limit ===
Line 518: Line 585:
Comma list: 176/175, 351/350, 847/845, 2197/2187
Comma list: 176/175, 351/350, 847/845, 2197/2187


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


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


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


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


== Artoneutral ==
== Artoneutral ==
Artoneutral is generated by an artoneutral third of ~11/9 (or a tendoneutral sixth of ~18/11) and can be described as the {{nowrap| 87 & 94 }} temperament. [[181edo]] may be recommended as a tuning.  
Artoneutral can be described as the {{nowrap| 87 & 94 }} temperament. It is generated by an artoneutral third of ~11/9 (or a tendoneutral sixth of ~18/11), nine of which make the [[12/1|12th harmonic]]; its ploidacot is thus beta-enneacot. [[181edo]] may be recommended as a tuning.
 
Artoneutral was named by [[Flora Canou]] in 2023 for its generator's quality.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 533: Line 604:
[[Comma list]]: 5120/5103, 3828125/3779136
[[Comma list]]: 5120/5103, 3828125/3779136


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


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


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


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


=== 11-limit ===
=== 11-limit ===
Line 547: Line 622:
Comma list: 385/384, 2200/2187, 4000/3993
Comma list: 385/384, 2200/2187, 4000/3993


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


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


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


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


=== 13-limit ===
=== 13-limit ===
Line 560: Line 637:
Comma list: 325/324, 352/351, 385/384, 1575/1573
Comma list: 325/324, 352/351, 385/384, 1575/1573


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


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


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


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


=== 17-limit ===
=== 17-limit ===
Line 573: Line 652:
Comma list: 325/324, 352/351, 375/374, 385/384, 595/594
Comma list: 325/324, 352/351, 375/374, 385/384, 595/594


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


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


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


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


=== 19-limit ===
=== 19-limit ===
Line 586: Line 667:
Comma list: 325/324, 352/351, 375/374, 385/384, 400/399, 595/594
Comma list: 325/324, 352/351, 375/374, 385/384, 400/399, 595/594


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


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


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


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


=== 23-limit ===
=== 23-limit ===
Line 599: Line 682:
Comma list: 300/299, 325/324, 352/351, 375/374, 385/384, 400/399, 484/483
Comma list: 300/299, 325/324, 352/351, 375/374, 385/384, 400/399, 484/483


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


Comma list: 325/324, 352/351, 385/384, 1331/1323
Comma list: 352/351, 540/539, 729/728, 1331/1323


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


Badness (Smith): 0.014033
== References ==


[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Pages with mostly numerical content]]
[[Category:Hemifamity temperaments| ]] <!-- main article -->
[[Category:Hemifamity temperaments| ]] <!-- main article -->
[[Category:Hemifamity| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]

Latest revision as of 07:57, 26 May 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 temperaments tempering out the hemifamity comma (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 hemifamity comma, 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 (6144/6125) rather than the hemifamity comma 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 hemifamity comma, 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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