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Hemifamity temperaments: Difference between revisions

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The hemifamity temperaments temper out the hemifamity comma, {{monzo| 10 -6 1 -1 }} = [[5120/5103]], dividing an exact or approximate septimal diesis, {{monzo| 2 2 -1 -1 }} = [[36/35]] into two equal steps.  
{{Technical data page}}
This is a collection of [[rank-2 temperament|rank-2]] [[regular temperament|temperaments]] [[tempering out]] the [[hemifamity comma]] ({{monzo|legend=1| 10 -6 1 -1 }}, [[ratio]]: 5120/5103). These temperaments divide an exact or approximate septimal quartertone, [[36/35]] into two equal steps, each representing [[81/80]][[~]][[64/63]], the syntonic comma or the septimal comma. Therefore, classical and septimal intervals are found by the same [[chain of fifths]] inflected by the syntonic~septimal comma to the opposite sides. In addition we may identify [[10/7]] by the augmented fourth and [[50/49]] by the [[Pythagorean comma]].  


Belonging to it and considered below are buzzard, undecental, leapday, mystery, quanic and ketchup. Other hemifamity temperaments are:
Temperaments belonging to this category and generated by the fifth are dominant, garibaldi, kwai, undecental, and leapday. Dominant has 5/4 mapped to M3. Garibaldi has 5/4 mapped to d4. Kwai has 5/4 mapped to 4A7. Undecental has 5/4 mapped to 5d7. Leapday has 5/4 mapped to 3A1.
* ''[[Dominant]]'', {36/35, 64/63} → [[Meantone family #Dominant|Meantone family]]
* [[Garibaldi]], {225/224, 3125/3087} → [[Schismatic family #Garibaldi|Schismatic family]]
* [[Rodan]], {245/243, 1029/1024} → [[Gamelismic clan #Rodan|Gamelismic clan]]
* [[Hemififths]], {2401/2400, 5120/5103} → [[Breedsmic temperaments #Hemififths|Breedsmic temperaments]]
* [[Misty]], {3136/3125, 5120/5103} → [[Misty family #Misty|Misty family]]
* [[Amity]], {4375/4374, 5120/5103} → [[Ragismic microtemperaments #Amity|Ragismic microtemperaments]] and [[Amity family #Septimal amity|Amity family]]
* ''[[Countercata]]'', {5120/5103, 15625/15552} → [[Kleismic family #Countercata|Kleismic family]]
* ''[[Kwai]]'', {5120/5103, 16875/16807} → [[Mirkwai clan #Kwai|Mirkwai clan]]
* ''[[Supers]]'', {5120/5103, 118098/117649} → [[Stearnsmic clan #Supers|Stearnsmic clan]]


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


Subgroup: 2.3.5.7
Temperaments discussed elsewhere are:
* [[Dominant (temperament)|Dominant]] (+36/35) → [[Meantone family #Dominant|Meantone family]]
* [[Garibaldi]] (+225/224) → [[Schismatic family #Garibaldi|Schismatic family]]
* [[Diaschismic]] (+126/125) → [[Diaschismic family #Septimal diaschismic|Diaschismic family]]
* [[Hemififths]] (+2401/2400) → [[Breedsmic temperaments #Hemififths|Breedsmic temperaments]]
* [[Rodan]] (+245/243) → [[Gamelismic clan #Rodan|Gamelismic clan]]
* ''[[Alphatrimot]]'' (+2430/2401) → [[Alphatricot family #Alphatrimot|Alphatricot family]]
* [[Misty]] (+3136/3125) → [[Misty family #Misty|Misty family]]
* [[Monkey]] (+875/864) → [[Tetracot family #Monkey|Tetracot family]]
* [[Buzzard]] (+1728/1715) → [[Buzzardsmic clan #Buzzard|Buzzardsmic clan]]
* ''[[Undim]]'' (+390625/388962) → [[Undim family #Septimal undim|Undim family]]
* ''[[Quinticosiennic]]'' (+395136/390625) → [[Quintaleap family #Quinticosiennic|Quintaleap family]]
* ''[[Quintakwai]]'' (+9765625/9680832) → [[Quindromeda family #Quintakwai|Quindromeda family]]
* [[Amity]] (+4375/4374) → [[Amity family #Septimal amity|Amity family]]
* ''[[Countercata]]'' (+15625/15552) → [[Kleismic family #Countercata|Kleismic family]]
* ''[[Abergravity]]'' (+177147/175000) → [[Gravity family #Abergravity|Gravity family]]
* ''[[Supers]]'' (+118098/117649) → [[Stearnsmic clan #Supers|Stearnsmic clan]]
* ''[[Warrior]]'' (+78732/78125) → [[Sensipent family #Warrior|Sensipent family]]
* ''[[Alphaquarter]]'' (+29360128/29296875) → [[Escapade family #Alphaquarter|Escapade family]]


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


[[Mapping]]: [{{val| 1 0 -6 4 }}, {{val| 0 4 21 -3 }}]
== Septiquarter ==
Septiquarter tempers out [[420175/419904]] and may be described as the {{nowrap| 94 & 99 }} temperament. Its [[ploidacot]] is epsilon-heptacot. [[99edo]] makes for an excellent tuning, and [[292edo]] an even better one. [[94edo]] and [[104edo]] in the 104c val are also among the possibilities.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 5120/5103, 420175/419904


{{Multival|legend=1| 4 21 -3 24 -16 -66 }}
{{Mapping|legend=1| 1 -4 -28 6 | 0 7 38 -4 }}
: mapping generators: ~2, ~243/140


[[POTE generator]]: ~21/16 = 475.636
[[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 }}


{{Val list|legend=1| 5, 43c, 48, 53, 111, 164d, 275d }}
{{Optimal ET sequence|legend=1| 94, 99, 292, 391, 881bd, 1272bcd }}


[[Badness]]: 0.047963
[[Badness]] (Sintel): 1.36


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


Comma list: 176/175, 540/539, 5120/5103
Comma list: 5120/5103, 9801/9800, 14641/14580


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


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


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


Badness: 0.034484
Badness (Sintel): 2.12


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


Comma list: 176/175, 351/350, 540/539, 676/675
Comma list: 352/351, 847/845, 1716/1715, 14641/14580


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


POTE generator: ~21/16 = 475.697
Optimal tunings:  
* WE: ~99/70 = 599.8565{{c}}, ~210/121 = 957.3261{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~210/121 = 957.5508{{c}}


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


Badness: 0.018842
Badness (Sintel): 1.44


==== 17-limit ====
== Kwai ==
Subgroup: 2.3.5.7.11.13.17
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Kwai]].''


Comma list: 176/175, 256/255, 351/350, 442/441, 540/539
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 }}.


Mapping: [{{val| 1 0 -6 4 -12 -7 14 }}, {{val| 0 4 21 -3 39 27 -25 }}]
[[Subgroup]]: 2.3.5.7


POTE generator: ~21/16 = 475.692
[[Comma list]]: 5120/5103, 16875/16807


Vals: {{Val list| 53, 58, 111, 321cdfg }}
{{Mapping|legend=1| 1 0 -50 -40 | 0 1 33 27 }}
: mapping generators: ~2, ~3


Badness: 0.018403
[[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 }}


==== 19-limit ====
{{Optimal ET sequence|legend=1| 41, 111, 152, 345, 497d }}
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 176/175, 256/255, 286/285, 324/323, 351/350, 540/539
[[Badness]] (Sintel): 1.38


Mapping: [{{val| 1 0 -6 4 -12 -7 14 -12 }}, {{val| 0 4 21 -3 39 27 -25 41 }}]
=== 11-limit ===
 
POTE generator: ~21/16 = 475.679
 
Vals: {{Val list| 53, 58h, 111 }}
 
Badness: 0.015649
 
=== Buteo ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 99/98, 385/384, 2200/2187
Comma list: 540/539, 1375/1372, 5120/5103


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


POTE generator: ~21/16 = 475.436
Optimal tunings:  
* WE: ~2 = 1199.6672{{c}}, ~3/2 = 702.4282{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6189{{c}}


Vals: {{Val list| 5, 48, 53 }}
{{Optimal ET sequence|legend=0| 41, 111, 152, 497de, 649dde }}


Badness: 0.060238
Badness (Sintel): 0.867


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


Comma list: 99/98, 275/273, 385/384, 572/567
Comma list: 352/351, 540/539, 729/728, 1375/1372


Mapping: [{{val| 1 0 -6 4 9 -7 }}, {{val| 0 4 21 -3 -14 27 }}]
Mapping: {{mapping| 1 0 -50 -40 32 27 | 0 1 33 27 -18 -21 }}


POTE generator: ~21/16 = 475.464
Optimal tunings:  
* WE: ~2 = 1199.4772{{c}}, ~3/2 = 702.3379{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6409{{c}}


Vals: {{Val list| 5, 48f, 53 }}
{{Optimal ET sequence|legend=0| 41, 111, 152f, 415dff }}


Badness: 0.039854
Badness (Sintel): 1.01


== Undecental ==
===== 17-limit =====
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7.11.13.17


[[Comma list]]: 5120/5103, 235298/234375
Comma list: 256/255, 352/351, 540/539, 715/714, 1089/1088


[[Mapping]]: [{{val| 1 0 61 71 }}, {{val| 0 1 -37 -43 }}]
Mapping: {{mapping| 1 0 -50 -40 32 27 58 | 0 1 33 27 -18 -21 -34 }}


{{Multival|legend=1| 1 -37 -43 -61 -71 4 }}
Optimal tunings:
* WE: ~2 = 1199.3537{{c}}, ~3/2 = 702.2850{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6589{{c}}


[[POTE generator]]: ~3/2 = 703.039
{{Optimal ET sequence|legend=0| 41, 70, 111, 152fg, 263dfg }}


{{Val list|legend=1| 29, 70, 99, 722bc, 821bc, 920bc, 1019bc, 1118bbcc, 1217bbcc, 1316bbccd }}
Badness (Sintel): 1.12


[[Badness]]: 0.094603
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


== Leapday ==
Comma list: 256/255, 352/351, 400/399, 456/455, 715/714, 847/845
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Leapday]].''
: ''Not to be confused with calendar-based modes such as those in [[293edo]] or [[Irvic scale|Irvian mode]].''


[[Leapday]] tempers out {{monzo|31 -21 1}} (trisayo) in the 5-limit. This temperament can be described as 29&amp;46 temperament, which tempers out the hemifamity and 686/675 (senga). Alternative extension [[Porwell temperaments #Polypyth|polypyth]] (46&amp;121) tempers out the same 5-limit comma as the leapday, but with the porwell (6144/6125) rather than the hemifamity tempered out.
Mapping: {{mapping| 1 0 -50 -40 32 27 58 -56 | 0 1 33 27 -18 -21 -34 38 }}


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


Subgroup: 2.3.5.7
{{Optimal ET sequence|legend=0| 41, 70h, 111, 152fg, 263dfgh }}


[[Comma list]]: 686/675, 5120/5103
Badness (Sintel): 1.03


[[Mapping]]: [{{val| 1 0 -31 -21 }}, {{val| 0 1 21 15 }}]
==== Hemikwai ====
 
{{Multival|legend=1| 1 21 15 31 21 -24 }}
 
[[POTE generator]]: ~3/2 = 704.263
 
{{Val list|legend=1| 17c, 29, 46, 167d, 213d, 259cdd, 305bcdd }}
 
[[Badness]]: 0.096123
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 121/120, 441/440, 686/675
 
Mapping: [{{val| 1 0 -31 -21 -14 }}, {{val| 0 1 21 15 11 }}]
 
POTE generator: ~3/2 = 704.250
 
Vals: {{Val list| 17c, 29, 46, 167de, 213de, 259cdde }}
 
Badness: 0.038624
 
=== 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: 540/539, 676/675, 1375/1372, 5120/5103


Mapping: [{{val| 1 0 -31 -21 -14 -9 }}, {{val| 0 1 21 15 11 8 }}]
Mapping: {{mapping| 1 0 -50 -40 32 -51 | 0 2 66 54 -36 69 }}
: mapping generators: ~2, ~26/15


POTE generator: ~3/2 = 704.214
Optimal tunings:
* WE: ~2 = 1199.6968{{c}}, ~26/15 = 951.0740{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.3123{{c}}


Vals: {{Val list| 17c, 29, 46, 121def, 167def, 213deff }}
{{Optimal ET sequence|legend=0| 82, 111, 193, 304d }}


Badness: 0.024732
Badness (Sintel): 1.82


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


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


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


Vals: {{Val list| 17cg, 29g, 46, 121defg, 167defg, 213deffg }}
{{Optimal ET sequence|legend=0| 82, 111, 193, 304d }}


Badness: 0.017863
Badness (Sintel): 1.31


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


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 -51 -30 -56 | 0 2 66 54 -36 69 43 76 }}


POTE generator: ~3/2 = 704.135
Optimal tunings:
* WE: ~2 = 1199.6718{{c}}, ~26/15 = 951.0526{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.3103{{c}}


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


Badness: 0.017356
Badness (Sintel): 1.16


==== Leapling ====
== Ketchup ==
Subgroup: 2.3.5.7.11.13.17.19
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.  


Comma list: 77/76, 91/90, 121/120, 136/135, 153/152, 169/168
[[Subgroup]]: 2.3.5.7


Mapping: [{{val| 1 0 -31 -21 -14 -9 -34 -37 }}, {{val| 0 1 21 15 11 8 24 26 }}]
[[Comma list]]: 5120/5103, 1071875/1062882
 
POTE generator: ~3/2 = 704.123
 
Vals: {{Val list| 17cgh, 29g, 46h, 75dfg, 121defghh }}
 
Badness: 0.019065
 
== Mystery ==
{{main| Mystery }}
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Mystery]].''


Subgroup: 2.3.5.7
{{Mapping|legend=1| 2 3 4 6 | 0 4 15 -9 }}
: mapping generators: ~1225/864, ~64/63


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


[[Mapping]]: [{{val| 29 46 0 14 }}, {{val| 0 0 1 1 }}]
{{Optimal ET sequence|legend=1| 46, 94, 140 }}


{{Multival|legend=1| 0 29 29 46 46 -14 }}
[[Badness]] (Sintel): 2.14
 
[[POTE generator]]: ~5/4 = 388.646
 
{{Val list|legend=1| 29, 58, 87, 145 }}
 
[[Badness]]: 0.103734


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


Comma list: 441/440, 896/891, 3388/3375
Comma list: 385/384, 1331/1323, 2200/2187


Mapping: [{{val| 29 46 0 14 33 }}, {{val| 0 0 1 1 1 }}]
Mapping: {{mapping| 2 3 4 6 7 | 0 4 15 -9 -2 }}


POTE generator: ~5/4 = 388.460
Optimal tunings:  
* WE: ~99/70 = 600.0678{{c}}, ~64/63 = 25.6963{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~64/63 = 25.6956{{c}}


Vals: {{Val list| 29, 58, 87, 145 }}
{{Optimal ET sequence|legend=0| 46, 94, 140 }}


Badness: 0.034291
Badness (Sintel): 1.31


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


Comma list: 196/195, 352/351, 364/363, 676/675
Comma list: 325/324, 352/351, 385/384, 1331/1323


Mapping: [{{val| 29 46 0 14 33 40 }}, {{val| 0 0 1 1 1 1 }}]
Mapping: {{mapping| 2 3 4 6 7 8 | 0 4 15 -9 -2 -14 }}


POTE generator: ~5/4 = 388.354
Optimal tunings:  
* WE: ~99/70 = 600.0612{{c}}, ~66/65 = 25.7000{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~66/65 = 25.6978{{c}}


Vals: {{Val list| 29, 58, 87, 145, 232, 377cef }}
{{Optimal ET sequence|legend=0| 46, 94, 140 }}


Badness: 0.018591
Badness (Sintel): 1.03


== Quanic ==
=== 17-limit ===
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7.11.13.17


[[Comma list]]: 5120/5103, 5832000/5764801
Comma list: 289/288, 325/324, 352/351, 385/384, 442/441


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


[[POTE generator]]: ~160/147 = 140.493
Optimal tunings:  
* WE: ~17/12 = 600.0896{{c}}, ~66/65 = 25.7048{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~66/65 = 25.7017{{c}}


{{Val list|legend=1| 94, 111, 205 }}
{{Optimal ET sequence|legend=0| 46, 94, 140 }}


[[Badness]]: 0.179475
Badness (Sintel): 0.845


=== 11-limit ===
=== 2.3.5.7.11.13.17.23 subgroup ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11.13.17.23


Comma list: 540/539, 1331/1323, 5120/5103
Comma list: 253/252, 289/288, 325/324, 352/351, 385/384, 391/390


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


POTE generator: ~88/81 = 140.489
Optimal tunings:  
* WE: ~17/12 = 600.1139{{c}}, ~66/65 = 25.7053{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~66/65 = 25.7013{{c}}


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


Badness: 0.058678
Badness (Sintel): 0.772


=== 13-limit ===
== Undecental ==
Subgroup: 2.3.5.7.11.13
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> }}.  


Comma list: 352/351, 540/539, 729/728, 1331/1323
[[Subgroup]]: 2.3.5.7


Mapping: [{{val| 1 1 -4 0 1 3 }}, {{val| 0 5 54 24 21 6 }}]
[[Comma list]]: 5120/5103, 235298/234375


POTE generator: ~13/12 = 140.496
{{Mapping|legend=1| 1 0 61 71 | 0 1 -37 -43 }}
: mapping generators: ~2, ~3


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


Badness: 0.032481
{{Optimal ET sequence|legend=1| 29, 70, 99, 722bc, 821bc, 920bc, 1019bc }}


=== 17-limit ===
[[Badness]] (Sintel): 2.39
Subgroup: 2.3.5.7.11.13.17


Comma list: 352/351, 442/441, 540/539, 715/714, 847/845
== Leapday ==
{{Main| Leapday }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Leapday]].''


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


POTE generator: ~13/12 = 140.497
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.  


Vals: {{Val list| 94, 111, 205 }}
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.


Badness: 0.021112
[[Subgroup]]: 2.3.5.7


=== 19-limit ===
[[Comma list]]: 686/675, 5120/5103
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 352/351, 400/399, 442/441, 456/455, 495/494, 715/714
{{Mapping|legend=1| 1 0 -31 -21 | 0 1 21 15 }}
: mapping generators: ~2, ~3


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


POTE generator: ~13/12 = 140.496
{{Optimal ET sequence|legend=1| 17c, 29, 46 }}


Vals: {{Val list| 94, 111, 205 }}
[[Badness]] (Sintel): 2.43
 
Badness: 0.017273
 
== Alphaquarter ==
{{see also|Escapade family}}
 
Subgroup: 2.3.5.7
 
[[Comma list]]: 5120/5103, 29360128/29296875
 
[[Mapping]]: [{{val| 1 2 2 0 }}, {{val| 0 -9 7 61 }}]
 
{{Multival|legend=1| 9 -7 -61 -32 -122 -122 }}
 
[[POTE generator]]: ~16128/15625 = 55.243
 
{{Val list|legend=1| 87, 152, 239, 391 }}
 
[[Badness]]: 0.116594


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


Comma list: 3025/3024, 4000/3993, 5120/5103
Comma list: 121/120, 441/440, 686/675


Mapping: [{{val| 1 2 2 0 3 }}, {{val| 0 -9 7 61 10 }}]
Mapping: {{mapping| 1 0 -31 -21 -14 | 0 1 21 15 11 }}


POTE generator: ~33/32 = 55.243
Optimal tunings:  
* WE: ~2 = 1200.0731{{c}}, ~3/2 = 704.2933{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2538{{c}}


Vals: {{Val list| 87, 152, 239, 391 }}
{{Optimal ET sequence|legend=0| 17c, 29, 46 }}


Badness: 0.029638
Badness (Sintel): 1.28


== Septiquarter ==
=== 13-limit ===
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7.11.13


[[Comma list]]: 5120/5103, 420175/419904
Comma list: 91/90, 121/120, 169/168, 352/351


[[Mapping]]: [{{val| 1 3 10 2 }}, {{val| 0 -7 -38 4 }}]
Mapping: {{mapping| 1 0 -31 -21 -14 -9 | 0 1 21 15 11 8 }}


{{Multival|legend=1| 7 38 -4 44 -26 -116 }}
Optimal tunings:
* WE: ~2 = 1200.4758{{c}}, ~3/2 = 704.4930{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2346{{c}}


[[POTE generator]]: ~147/128 = 242.453
{{Optimal ET sequence|legend=0| 17c, 29, 46, 121def }}


{{Val list|legend=1| 94, 99, 292, 391, 881bd, 1272bcd }}
Badness (Sintel): 1.02


[[Badness]]: 0.053760
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


=== Semiseptiquarter ===
Comma list: 91/90, 121/120, 136/135, 154/153, 169/168
Subgroup: 2.3.5.7.11
 
Comma list: 5120/5103, 9801/9800, 14641/14580
 
Mapping: [{{val| 2 6 20 4 15 }}, {{val| 0 -7 -38 4 -20 }}]
 
POTE generators: ~121/105 = 242.4511


Vals: {{Val list| 94, 198, 292, 490 }}
Mapping: {{mapping| 1 0 -31 -21 -14 -9 -34 | 0 1 21 15 11 8 24 }}


Badness: 0.064160
Optimal tunings:  
* WE: ~2 = 1200.4818{{c}}, ~3/2 = 704.5121{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2507{{c}}


==== 13-limit ====
{{Optimal ET sequence|legend=0| 17cg, 29g, 46, 121defg }}
Subgroup: 2.3.5.7.11.13


Comma list: 352/351, 847/845, 1716/1715, 14641/14580
Badness (Sintel): 0.910


Mapping: [{{val| 2 6 20 4 15 7 }}, {{val| 0 -7 -38 4 -20 1 }}]
=== 2.3.5.7.11.13.17.23 subgroup ===
Subgroup: 2.3.5.7.11.13.17.23


POTE generators: ~121/105 = 242.4448
Comma list: 91/90, 121/120, 136/135, 154/153, 161/160, 169/168


Vals: {{Val list| 94, 198, 490f }}
Mapping: {{mapping| 1 0 -31 -21 -14 -9 -34 -5 | 0 1 21 15 11 8 24 6 }}


Badness: 0.034834
Optimal tunings:  
* WE: ~2 = 1200.5169{{c}}, ~3/2 = 704.5279{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2450{{c}}


== Tricot ==
{{Optimal ET sequence|legend=0| 17cg, 29g, 46, 121defg }}
{{see also|Tricot family}}


The generator for tricot is the real cube root of third harmonic, 3<sup>1/3</sup>, tuned between 63/44 and 13/9. Tricot can be described as 53&amp;70 temperament (also known as ''trimot''), tempering out the [[tricot comma]], {{monzo| 39 -29 3 }} in the 5-limit, 2430/2401 (nuwell comma) and 5120/5103 in the 7-limit, 99/98 and 121/120 in the 11-limit, 169/168, 352/351, 640/637, and 729/728 in the 13-limit.
Badness (Sintel): 0.872


Subgroup: 2.3.5
== Mystery ==
{{Main| Mystery }}
: ''For the 5-limit version, see [[29th-octave temperaments #Mystery]].''


[[Comma]]: {{monzo | 39 -29 3 }} = 68719476736000/68630377364883
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.


[[Mapping]]: [{{val| 1 0 -13 }}, {{val| 0 3 29 }}]
[[Subgroup]]: 2.3.5.7


[[POTE generator]]: ~59049/40960 = 634.012
[[Comma list]]: 5120/5103, 50421/50000


{{Val list|legend=1| 53, 229, 282, 335, 388, 441, 1376, 1817, 2258 }}
{{Mapping|legend=1| 29 46 0 14 | 0 0 1 1 }}
: mapping generators: ~50/49, ~5


[[Badness]]: 0.046093
[[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 }}


=== 7-limit (Trimot) ===
{{Optimal ET sequence|legend=1| 29, 58, 87, 145 }}
Subgroup: 2.3.5.7


[[Comma list]]: 2430/2401, 5120/5103
[[Badness]] (Sintel): 2.63
 
[[Mapping]]: [{{val| 1 0 -13 -3 }}, {{val| 0 3 29 11 }}]
 
{{Multival|legend=1| 3 29 11 39 9 -56 }}
 
[[POTE generator]]: ~81/56 = 634.026
 
{{Val list|legend=1| 17c, 36c, 53, 229dd, 282dd }}
 
[[Badness]]: 0.100127


=== 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: 441/440, 896/891, 3388/3375


Mapping: [{{val| 1 0 -13 -3 -5 }}, {{val| 0 3 29 11 16 }}]
Mapping: {{mapping| 29 46 0 14 33 | 0 0 1 1 1 }}


POTE generator: ~63/44 = 634.027
Optimal tunings:
* WE: ~45/44 = 41.3637{{c}}, ~5/4 = 388.3136{{c}}
* CWE: ~45/44 = 41.3793{{c}}, ~5/4 = 388.0598{{c}}


Vals: {{Val list| 17c, 36ce, 53, 70, 123de }}
{{Optimal ET sequence|legend=0| 29, 58, 87, 145 }}


Badness: 0.056134
Badness (Sintel): 1.13


=== 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: 196/195, 352/351, 364/363, 676/675


Mapping: [{{val| 1 0 -13 -3 -5 0 }}, {{val| 0 3 29 11 16 7 }}]
Mapping: {{mapping| 29 46 0 14 33 40 | 0 0 1 1 1 1 }}


POTE generator: ~13/9 = 634.012
Optimal tunings:  
* WE: ~45/44 = 41.3623{{c}}, ~5/4 = 388.1942{{c}}
* CWE: ~40/39 = 41.3793{{c}}, ~5/4 = 387.9017{{c}}


Vals: {{Val list| 17c, 36ce, 53, 70, 123de }}
{{Optimal ET sequence|legend=0| 29, 58, 87, 145, 232 }}


Badness: 0.032102
Badness (Sintel): 0.768


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


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


[[Mapping]]: [{{val| 2 3 4 6 }}, {{val| 0 4 15 -9 }}]
[[Comma list]]: 5120/5103, 52734375/52706752


{{Multival|legend=1| 8 30 -18 29 -51 -126 }}
{{Mapping|legend=1| 1 0 38 48 | 0 2 -45 -57 }}
: mapping generator: ~2, ~12500/7203


[[POTE generator]]: ~64/63 = ~81/80 = 25.719
[[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| 46, 94, 140 }}
{{Optimal ET sequence|legend=1| 29, 82cd, 111, 140, 251, 391, 1424bbcdd }}


[[Badness]]: 0.084538
[[Badness]] (Sintel): 2.93


=== 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: 1331/1323, 1375/1372, 5120/5103


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


POTE generator: ~55/54 = ~64/63 = ~81/80 = 25.693
Optimal tunings:  
* WE: ~2 = 1199.8767{{c}}, ~400/231 = 951.3065{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~400/231 = 951.4063{{c}}


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


Badness: 0.039555
Badness (Sintel): 2.01


=== 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, 676/675, 847/845, 1331/1323


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


POTE generator: ~55/54 = ~64/63 = ~81/80 = 25.697
Optimal tunings:  
* WE: ~2 = 1199.8753{{c}}, ~26/15 = 951.3054{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.4064{{c}}


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


Badness: 0.024824
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: 289/288, 325/324, 352/351, 385/384, 561/560
Comma list: 352/351, 442/441, 561/560, 676/675, 715/714


Mapping: [{{val| 2 3 4 6 7 8 8 }}, {{val| 0 4 15 -9 -2 -14 4 }}]
Mapping: {{mapping| 1 0 38 48 32 37 58 | 0 2 -45 -57 -36 -42 -68 }}


POTE generator: ~55/54 = ~64/63 = ~81/80 = 25.701
Optimal tunings:  
* WE: ~2 = 1199.8770{{c}}, ~26/15 = 951.3039{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.4035{{c}}


Vals: {{Val list| 46, 94, 140 }}
{{Optimal ET sequence|legend=0| 29g, 82cdfg, 111, 140, 251, 391e }}


Badness: 0.016591
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: 190/189, 209/208, 289/288, 352/351, 385/384, 561/560
Comma list: 286/285, 352/351, 363/361, 442/441, 476/475, 561/560


Mapping: [{{val| 2 3 4 6 7 8 8 9 }}, {{val| 0 4 15 -9 -2 -14 4 -12 }}]
Mapping: {{mapping| 1 0 38 48 32 37 58 32 | 0 2 -45 -57 -36 -42 -68 -35 }}


POTE generator: ~55/54 = ~64/63 = ~81/80 = 25.660
Optimal tunings:  
* WE: ~2 = 1199.7534{{c}}, ~26/15 = 951.2024{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.4020{{c}}


Vals: {{Val list| 46, 94, 140h, 234eh }}
{{Optimal ET sequence|legend=0| 29g, 82cdfgh, 111, 140 }}


Badness: 0.018170
Badness (Sintel): 1.01


=== 23-limit ===
=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23
Subgroup: 2.3.5.7.11.13.17.19.23


Comma list: 190/189, 209/208, 253/252, 289/288, 323/322, 352/351, 385/384
Comma list: 253/252, 286/285, 352/351, 363/361, 391/390, 442/441, 460/459


Mapping: [{{val| 2 3 4 6 7 8 8 9 9 }}, {{val| 0 4 15 -9 -2 -14 4 -12 1 }}]
Mapping: {{mapping| 1 0 38 48 32 37 58 32 18 | 0 2 -45 -57 -36 -42 -68 -35 -17 }}


POTE generator: ~55/54 = ~64/63 = ~81/80 = 25.661
Optimal tunings:  
* WE: ~2 = 1199.9128{{c}}, ~26/15 = 951.3371{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.4076{{c}}


Vals: {{Val list| 46, 94, 140h, 234ehi }}
{{Optimal ET sequence|legend=0| 29g, 82cdfgh, 111, 140 }}


Badness: 0.014033
Badness (Sintel): 1.10


== Undim ==
== Countriton ==
Subgroup: 2.3.5
: ''For the 5-limit version, see [[Schismic–Mercator equivalence continuum #Countritonic]].''


[[Comma list]]: {{monzo| 41 -20 -4 }} = 2199023255552/2179240250625
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]]: [{{val| 4 0 41 }}, {{val| 0 1 -5 }}]
Countriton was named by [[Xenllium]] in 2022 as a counterpart of [[untriton]].


Mapping generators: ~1215/1024, ~3
[[Subgroup]]: 2.3.5.7


[[POTE generator]]: ~3/2 = 702.736
[[Comma list]]: 5120/5103, 7558272/7503125


{{Val list|legend=1| 12, 104, 116, 128, 140, 152, 620, 772, 924c, 1076bc, 1228bc }}
{{Mapping|legend=1| 1 -3 -15 13 | 0 9 34 -20 }}
: mapping generators: ~2, ~1225/864


[[Badness]]: 0.241703
[[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 }}


=== 7-limit ===
{{Optimal ET sequence|legend=1| 51c, 53, 157, 210, 473cdd }}
Subgroup: 2.3.5.7


[[Comma list]]: 5120/5103, 390625/388962
[[Badness]] (Sintel): 3.32


[[Mapping]]: [{{val| 4 0 41 81 }}, {{val| 0 1 -5 -11 }}]
=== 11-limit ===
Subgroup: 2.3.5.7.11


{{Multival|legend=1| 4 -20 -44 -41 -81 -46 }}
Comma list: 176/175, 5120/5103, 41503/41472


[[POTE generator]]: ~3/2 = 702.736
Mapping: {{mapping| 1 -3 -15 13 -21 | 0 9 34 -20 48 }}


{{Val list|legend=1| 12, 128, 140, 152, 292 }}
Optimal tunings:
* WE: ~2 = 1199.5178{{c}}, ~77/54 = 611.2097{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~77/54 = 611.4495{{c}}


[[Badness]]: 0.062754
{{Optimal ET sequence|legend=0| 51ce, 53, 104c, 157 }}


=== 11-limit ===
Badness (Sintel): 2.80
Subgroup: 2.3.5.7.11


Comma list: 1375/1372, 5120/5103, 5632/5625
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Mapping: [{{val| 4 0 41 81 128 }}, {{val| 0 1 -5 -11 -18 }}]
Comma list: 176/175, 351/350, 847/845, 2197/2187


POTE generator: ~3/2 = 702.689
Mapping: {{mapping| 1 -3 -15 13 -21 -7 | 0 9 34 -20 48 21 }}


Vals: {{Val list| 12, 128e, 140, 152, 292, 444d, 596d }}
Optimal tunings:  
* WE: ~2 = 1199.5944{{c}}, ~77/54 = 611.2491{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~77/54 = 611.4506{{c}}


Badness: 0.034837
{{Optimal ET sequence|legend=0| 51ce, 53, 104c, 157 }}


== Quintakwai ==
Badness (Sintel): 1.75
{{See also| 28ed5 #Regular temperaments }}


The ''quintakwai'' temperament (12&amp;193) tempers out the hemifamity comma (5120/5103) and the compass comma (9765625/9680832, quinruyoyo) in the 7-limit; 1375/1372 and 4375/4356 in the 11-limit. In the 2.3.5.7.17.19 subgroup, 225/224 (the difference between 15/14 and 16/15), 256/255 (between 16/15 and 17/16), 289/288 (between 17/16 and 18/17), 324/323 (between 18/17 and 19/18), and 361/360 (between 19/18 and 20/19) are equated together, and 400/399 (between 20/19 and 21/20) is tempered out. The name ''quintakwai'' is so named because the generator is 1/5 of the [[Mirkwai clan #Kwai|kwai]] fourth (~4/3, about 497.4 cents).
== Artoneutral ==
Artoneutral can be described as the {{nowrap| 87 & 94 }} temperament. It is generated by an artoneutral third of ~11/9 (or a tendoneutral sixth of ~18/11), nine of which make the [[12/1|12th harmonic]]; its ploidacot is thus beta-enneacot. [[181edo]] may be recommended as a tuning.  


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]]: 5120/5103, 9765625/9680832
[[Comma list]]: 5120/5103, 3828125/3779136


[[Mapping]]: [{{val|1 2 0 -2}}, {{val|0 -5 28 58}}]
{{Mapping|legend=1| 1 -1 -4 12 | 0 9 22 -32 }}
: mapping generators: ~2, ~128/105


{{Multival|legend=1|5 -28 -58 -56 -106 -56}}
[[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 }}


[[POTE generator]]: ~625/588 = 99.483
{{Optimal ET sequence|legend=1| 87, 94, 181 }}


{{Val list|legend=1| 12, 169, 181, 193 }}
[[Badness]] (Sintel): 3.98
 
[[Badness]]: 0.155536


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


Comma list: 1375/1372, 4375/4356, 5120/5103
Comma list: 385/384, 2200/2187, 4000/3993


Mapping: [{{val|1 2 0 -2 -4}}, {{val|0 -5 28 58 90}}]
Mapping: {{mapping| 1 -1 -4 12 -2 | 0 9 22 -32 19 }}


POTE generator: ~35/33 = 99.472
Optimal tunings:  
* WE: ~2 = 1200.1668{{c}}, ~11/9 = 344.8027{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 344.7557{{c}}


Vals: {{Val list| 12, 181, 193, 374, 567ce }}
{{Optimal ET sequence|legend=0| 87, 181 }}


Badness: 0.073158
Badness (Sintel): 1.52


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


Comma list: 325/324, 1375/1372, 1575/1573, 4096/4095
Comma list: 325/324, 352/351, 385/384, 1575/1573


Mapping: [{{val|1 2 0 -2 -4 10}}, {{val|0 -5 28 58 90 -76}}]
Mapping: {{mapping| 1 -1 -4 12 -2 6 | 0 9 22 -32 19 -8 }}


POTE generator: ~35/33 = 99.468
Optimal tunings:  
* WE: ~2 = 1200.0662{{c}}, ~11/9 = 344.7804{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 344.7617{{c}}


Vals: {{Val list| 12, 181, 193, 374, 567ce, 941bce }}
{{Optimal ET sequence|legend=0| 87, 181 }}


Badness: 0.062737
Badness (Sintel): 1.08


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


Comma list: 325/324, 375/374, 595/594, 1275/1274, 4096/4095
Comma list: 325/324, 352/351, 375/374, 385/384, 595/594


Mapping: [{{val|1 2 0 -2 -4 10 5}}, {{val|0 -5 28 58 90 -76 -11}}]
Mapping: {{mapping| 1 -1 -4 12 -2 6 -12 | 0 9 22 -32 19 -8 56 }}


POTE generator: ~18/17 = 99.469
Optimal tunings:  
* WE: ~2 = 1200.0346{{c}}, ~11/9 = 344.7589{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 344.7492{{c}}


Vals: {{Val list| 12, 181, 193, 374, 567ce, 941bceg }}
{{Optimal ET sequence|legend=0| 87, 94, 181 }}


Badness: 0.037855
Badness (Sintel): 1.16


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


Comma list: 325/324, 375/374, 400/399, 595/594, 1216/1215, 1275/1274
Comma list: 325/324, 352/351, 375/374, 385/384, 400/399, 595/594


Mapping: [{{val|1 2 0 -2 -4 10 5 4}}, {{val|0 -5 28 58 90 -76 -11 3}}]
Mapping: {{mapping| 1 -1 -4 12 -2 6 -12 -15 | 0 9 22 -32 19 -8 56 67 }}


POTE generator: ~18/17 = 99.469
Optimal tunings:  
* WE: ~2 = 1200.0282{{c}}, ~11/9 = 344.7532{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 344.7453{{c}}


Vals: {{Val list| 12, 181, 193, 374, 567ce, 941bcegh, 1508bccdeegghh }}
{{Optimal ET sequence|legend=0| 87, 94, 181 }}


Badness: 0.025861
Badness (Sintel): 1.19
 
=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23


=== Quinkwai ===
Comma list: 300/299, 325/324, 352/351, 375/374, 385/384, 400/399, 484/483
Subgroup: 2.3.5.7.11.13


Comma list: 352/351, 847/845, 1375/1372, 4375/4356
Mapping: {{mapping| 1 -1 -4 12 -2 6 -12 -15 -13 | 0 9 22 -32 19 -8 56 67 61 }}


Mapping: [{{val|1 2 0 -2 -4 -5}}, {{val|0 -5 28 58 90 105}}]
Optimal tunings:  
* WE: ~2 = 1200.0163{{c}}, ~11/9 = 344.7461{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 344.7416{{c}}


POTE generator: ~35/33 = 99.456
{{Optimal ET sequence|legend=0| 87, 94, 181 }}


Vals: {{Val list| 12f, 169e, 181 }}
Badness (Sintel): 1.17


Badness: 0.061873
== 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.  


==== 17-limit ====
[[Subgroup]]: 2.3.5.7
Subgroup: 2.3.5.7.11.13.17


Comma list: 352/351, 375/374, 595/594, 833/832, 1375/1372
[[Comma list]]: 5120/5103, 5832000/5764801


Mapping: [{{val|1 2 0 -2 -4 -5 5}}, {{val|0 -5 28 58 90 105 -11}}]
{{Mapping|legend=1| 1 1 -4 0 | 0 5 54 24 }}
: mapping generators: ~2, ~160/147


POTE generator: ~18/17 = 99.458
[[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 }}


Vals: {{Val list| 12f, 169e, 181, 374ff, 555cff }}
{{Optimal ET sequence|legend=1| 94, 111, 205 }}


Badness: 0.043506
[[Badness]] (Sintel): 4.54


==== 19-limit ====
=== 11-limit ===
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11


Comma list: 352/351, 375/374, 400/399, 495/494, 595/594, 1375/1372
Comma list: 540/539, 1331/1323, 5120/5103


Mapping: [{{val|1 2 0 -2 -4 -5 5 4}}, {{val|0 -5 28 58 90 105 -11 3}}]
Mapping: {{mapping| 1 1 -4 0 1 | 0 5 54 24 21 }}


POTE generator: ~18/17 = 99.459
Optimal tunings:  
* WE: ~2 = 1199.7834{{c}}, ~88/81 = 140.4635{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~88/81 = 140.4850{{c}}


Vals: {{Val list| 12f, 169e, 181, 374ff, 555cff }}
{{Optimal ET sequence|legend=0| 94, 111, 205 }}


Badness: 0.031313
Badness (Sintel): 1.94


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


Comma list: 364/363, 625/624, 1375/1372, 5120/5103
Comma list: 352/351, 540/539, 729/728, 1331/1323


Mapping: [{{val|1 2 0 -2 -4 -6}}, {{val|0 -5 28 58 90 117}}]
Mapping: {{mapping| 1 1 -4 0 1 3 | 0 5 54 24 21 6 }}


POTE generator: ~35/33 = 99.484
Optimal tunings:  
* WE: ~2 = 1199.6639{{c}}, ~13/12 = 140.4562{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.4904{{c}}


Vals: {{Val list| 12f, 181f, 193 }}
{{Optimal ET sequence|legend=0| 94, 111, 205 }}


Badness: 0.057357
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: 364/363, 375/374, 442/441, 595/594, 5120/5103
Comma list: 352/351, 442/441, 540/539, 715/714, 847/845


Mapping: [{{val|1 2 0 -2 -4 -6 5}}, {{val|0 -5 28 58 90 117 -11}}]
Mapping: {{mapping| 1 1 -4 0 1 3 -2 | 0 5 54 24 21 6 52 }}


POTE generator: ~18/17 = 99.484
Optimal tunings:  
* WE: ~2 = 1199.6699{{c}}, ~13/12 = 140.4586{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.4920{{c}}


Vals: {{Val list| 12f, 181f, 193 }}
{{Optimal ET sequence|legend=0| 94, 111, 205 }}


Badness: 0.039289
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: 364/363, 375/374, 400/399, 442/441, 595/594, 1216/1215
Comma list: 352/351, 400/399, 442/441, 456/455, 495/494, 715/714
 
Mapping: [{{val|1 2 0 -2 -4 -6 5 4}}, {{val|0 -5 28 58 90 117 -11 3}}]
 
POTE generator: ~18/17 = 99.485
 
Vals: {{Val list| 12f, 181f, 193 }}
 
Badness: 0.028101
 
== Quinticosiennic ==
{{See also| 16ed5/2 #Regular temperaments }}
 
The ''quinticosiennic'' temperament (12&amp;145) tempers out the hemifamity comma (5120/5103) and 395136/390625 (trizo-aquadbigu) in the 7-limit; 441/440 (werckisma), 896/891 (pentacircle), and 78408/78125 (lolosepgu) in the 11-limit. The word "quinticosiennic" means 5 (quintuple) × 29 (είκοσι εννέα) = 145, and so named because 1/5 of [[29edo|29EDO]] fourth, i.e. 12\145, is a possible generator.
 
 
Subgroup: 2.3.5.7
 
[[Comma list]]: 5120/5103, 395136/390625
 
[[Mapping]]: [{{val|1 2 1 -1}}, {{val|0 -5 16 46}}]
 
{{Multival|legend=1|5 -16 -46 -37 -87 -62}}
 
[[POTE generator]]: ~135/128 = 99.345


{{Val list|legend=1| 12, 133, 145, 157, 302c, 459bcc }}
Mapping: {{mapping| 1 1 -4 0 1 3 -2 -5 | 0 5 54 24 21 6 52 79 }}


[[Badness]]: 0.158041
Optimal tunings:  
 
* WE: ~2 = 1199.6745{{c}}, ~13/12 = 140.4574{{c}}
=== 11-limit ===
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.4908{{c}}
Subgroup: 2.3.5.7.11


Comma list: 441/440, 896/891, 78408/78125
{{Optimal ET sequence|legend=0| 94, 111, 205 }}


Mapping: [{{val|1 2 1 -1 -2}}, {{val|0 -5 16 46 66}}]
Badness (Sintel): 1.05


POTE generator: ~35/33 = 99.318
== Jorgensen ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Jorgensen]].''


Vals: {{Val list| 12, 133, 145 }}
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.


Badness: 0.080674
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>.  


=== 13-limit ===
[[Subgroup]]: 2.3.5.7
Subgroup: 2.3.5.7.11.13


Comma list: 196/195, 352/351, 364/363, 78408/78125
[[Comma list]]: 5120/5103, 578509309952/576650390625
 
Mapping: [{{val|1 2 1 -1 -2 -3}}, {{val|0 -5 16 46 66 81}}]
 
POTE generator: ~35/33 = 99.307
 
Vals: {{Val list| 12f, 133, 145 }}
 
Badness: 0.052464
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 196/195, 256/255, 352/351, 364/363, 3757/3750
 
Mapping: [{{val|1 2 1 -1 -2 -3 5}}, {{val|0 -5 16 46 66 81 -11}}]
 
POTE generator: ~18/17 = 99.308
 
Vals: {{Val list| 12f, 133, 145 }}
 
Badness: 0.037108
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 196/195, 256/255, 352/351, 361/360, 364/363, 476/475
{{Mapping|legend=1| 70 111 0 34 | 0 0 1 1 }}
: mapping generators: ~50421/50000, ~5


Mapping: [{{val|1 2 1 -1 -2 -3 5 4}}, {{val|0 -5 16 46 66 81 -11 3}}]
[[Optimal tuning]]s:  
* [[WE]]: ~50421/50000 = 17.1387{{c}}, ~5/4 = 386.8071{{c}}
: [[error map]]: {{val| -0.288 +0.445 -0.084 +0.121 }}
* [[CWE]]: ~50421/50000 = 17.1429{{c}}, ~5/4 = 386.6593{{c}}
: error map: {{val| 0.000 +0.902 +0.346 +0.690 }}


POTE generator: ~18/17 = 99.303
{{Optimal ET sequence|legend=1| 70, 140, 350, 490 }}


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


Badness: 0.028440
== References ==


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