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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 [[Meantone family #Dominant|dominant]], [[Schismatic family #Garibaldi|garibaldi]], [[Breedsmic temperaments #Hemififths|hemififths]], [[Ragismic microtemperaments #Amity|amity]], [[Hemimean clan #Misty|misty]], [[Gamelismic clan #Rodan|rodan]], [[Kleismic family #Countercata|countercata]] and [[Mirkwai clan #Kwai|kwai]].
Temperaments belonging to this category and generated by the fifth are dominant, garibaldi, kwai, undecental, and leapday. Dominant has 5/4 mapped to M3. Garibaldi has 5/4 mapped to d4. Kwai has 5/4 mapped to 4A7. Undecental has 5/4 mapped to 5d7. Leapday has 5/4 mapped to 3A1.  


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


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


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


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


{{Multival|legend=1| 4 21 -3 24 -16 -66 }}
[[Subgroup]]: 2.3.5.7


[[POTE generator]]: ~21/16 = 475.636
[[Comma list]]: 5120/5103, 420175/419904


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


[[Badness]]: 0.047963
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.7212{{c}}, ~243/140 = 957.3250{{c}}
: [[error map]]: {{val| -0.279 +0.435 -0.158 +0.201 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~243/140 = 957.5424{{c}}
: error map: {{val| 0.000 +0.842 +0.298 +1.004 }}


== 11-limit ==
{{Optimal ET sequence|legend=1| 94, 99, 292, 391, 881bd, 1272bcd }}


[[Badness]] (Sintel): 1.36
=== Semiseptiquarter ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 176/175, 540/539, 5120/5103
Comma list: 5120/5103, 9801/9800, 14641/14580
 
Mapping: {{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 tunings:
* WE: ~99/70 = 599.8565{{c}}, ~210/121 = 957.3261{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~210/121 = 957.5508{{c}}
 
{{Optimal ET sequence|legend=0| 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"<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
 
[[Comma list]]: 5120/5103, 16875/16807
 
{{Mapping|legend=1| 1 0 -50 -40 | 0 1 33 27 }}
: mapping generators: ~2, ~3
 
[[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 }}
 
{{Optimal ET sequence|legend=1| 41, 111, 152, 345, 497d }}
 
[[Badness]] (Sintel): 1.38
 
=== 11-limit ===
Subgroup: 2.3.5.7.11


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


POTE generator: ~21/16 = 475.700
Mapping: {{mapping| 1 0 -50 -40 32 | 0 1 33 27 -18 }}


Vals: {{Val list| 53, 58, 111, 280cd, 391cd }}
Optimal tunings:  
* WE: ~2 = 1199.6672{{c}}, ~3/2 = 702.4282{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6189{{c}}


Badness: 0.034484
{{Optimal ET sequence|legend=0| 41, 111, 152, 497de, 649dde }}


=== 13-limit ===
Badness (Sintel): 0.867


==== 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, 540/539, 729/728, 1375/1372
 
Mapping: {{mapping| 1 0 -50 -40 32 27 | 0 1 33 27 -18 -21 }}
 
Optimal tunings:
* WE: ~2 = 1199.4772{{c}}, ~3/2 = 702.3379{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6409{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 0 -50 -40 32 27 58 | 0 1 33 27 -18 -21 -34 }}
 
Optimal tunings:
* WE: ~2 = 1199.3537{{c}}, ~3/2 = 702.2850{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6589{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{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}}
 
{{Optimal ET sequence|legend=0| 41, 70h, 111, 152fg, 263dfgh }}
 
Badness (Sintel): 1.03
 
==== Hemikwai ====
Subgroup: 2.3.5.7.11.13


Mapping: [{{val| 1 0 -6 4 -12 -7 }}, {{val| 0 4 21 -3 39 27 }}]
Comma list: 540/539, 676/675, 1375/1372, 5120/5103


POTE generator: ~21/16 = 475.697
Mapping: {{mapping| 1 0 -50 -40 32 -51 | 0 2 66 54 -36 69 }}
: mapping generators: ~2, ~26/15


Vals: {{Val list| 53, 58, 111, 280cdf, 391cdf }}
Optimal tunings:  
* WE: ~2 = 1199.6968{{c}}, ~26/15 = 951.0740{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.3123{{c}}


Badness: 0.018842
{{Optimal ET sequence|legend=0| 82, 111, 193, 304d }}


=== 17-limit ===
Badness (Sintel): 1.82


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


Comma list: 176/175, 256/255, 351/350, 442/441, 540/539
Comma list: 442/441, 540/539, 676/675, 715/714, 5120/5103
 
Mapping: {{mapping| 1 0 -50 -40 32 -51 -30 | 0 2 66 54 -36 69 43 }}
 
Optimal tunings:
* WE: ~2 = 1199.6861{{c}}, ~26/15 = 951.0654{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.3120{{c}}
 
{{Optimal ET sequence|legend=0| 82, 111, 193, 304d }}
 
Badness (Sintel): 1.31
 
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


Mapping: [{{val| 1 0 -6 4 -12 -7 14 }}, {{val| 0 4 21 -3 39 27 -25 }}]
Comma list: 400/399, 442/441, 540/539, 676/675, 715/714, 1445/1444


POTE generator: ~21/16 = 475.692
Mapping: {{mapping| 1 0 -50 -40 32 -51 -30 -56 | 0 2 66 54 -36 69 43 76 }}


Vals: {{Val list| 53, 58, 111, 321cdfg }}
Optimal tunings:  
* WE: ~2 = 1199.6718{{c}}, ~26/15 = 951.0526{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.3103{{c}}


Badness: 0.018403
{{Optimal ET sequence|legend=0| 82, 111, 193, 304dh }}


=== 19-limit ===
Badness (Sintel): 1.16


Subgroup: 2.3.5.7.11.13.17.19
== Ketchup ==
Ketchup may be described as the {{nowrap| 46 & 94 }} temperament. It has a semi-octave period and a generator for a syntonic~septimal comma, four of which plus a period gives the perfect fifth; its [[ploidacot]] is diploid gamma-tetracot. [[140edo]] is an obvious tuning for this temperament.  


Comma list: 176/175, 256/255, 286/285, 324/323, 351/350, 540/539
[[Subgroup]]: 2.3.5.7


Mapping: [{{val| 1 0 -6 4 -12 -7 14 -12 }}, {{val| 0 4 21 -3 39 27 -25 41 }}]
[[Comma list]]: 5120/5103, 1071875/1062882


POTE generator: ~21/16 = 475.679
{{Mapping|legend=1| 2 3 4 6 | 0 4 15 -9 }}
: mapping generators: ~1225/864, ~64/63


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


Badness: 0.015649
{{Optimal ET sequence|legend=1| 46, 94, 140 }}


== Buteo ==
[[Badness]] (Sintel): 2.14


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


Comma list: 99/98, 385/384, 2200/2187
Comma list: 385/384, 1331/1323, 2200/2187


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


POTE generator: ~21/16 = 475.436
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| 5, 48, 53 }}
{{Optimal ET sequence|legend=0| 46, 94, 140 }}


Badness: 0.060238
Badness (Sintel): 1.31


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 325/324, 352/351, 385/384, 1331/1323
Mapping: {{mapping| 2 3 4 6 7 8 | 0 4 15 -9 -2 -14 }}


Subgroup: 2.3.5.7.11.13
Optimal tunings:  
* WE: ~99/70 = 600.0612{{c}}, ~66/65 = 25.7000{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~66/65 = 25.6978{{c}}


Comma list: 99/98, 275/273, 385/384, 572/567
{{Optimal ET sequence|legend=0| 46, 94, 140 }}


Mapping: [{{val| 1 0 -6 4 9 -7 }}, {{val| 0 4 21 -3 -14 27 }}]
Badness (Sintel): 1.03


POTE generator: ~21/16 = 475.464
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Vals: {{Val list| 5, 48f, 53 }}
Comma list: 289/288, 325/324, 352/351, 385/384, 442/441


Badness: 0.039854
Mapping: {{mapping| 2 3 4 6 7 8 8 | 0 4 15 -9 -2 -14 4 }}


= Undecental =
Optimal tunings:
* WE: ~17/12 = 600.0896{{c}}, ~66/65 = 25.7048{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~66/65 = 25.7017{{c}}


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


[[Comma list]]: 5120/5103, 235298/234375
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]]: [{{val| 1 0 61 71 }}, {{val| 0 1 -37 -43 }}]
Mapping: {{mapping| 2 3 4 6 7 8 8 9 | 0 4 15 -9 -2 -14 4 1 }}


{{Multival|legend=1| 1 -37 -43 -61 -71 4 }}
Optimal tunings:
* WE: ~17/12 = 600.1139{{c}}, ~66/65 = 25.7053{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~66/65 = 25.7013{{c}}


[[POTE generator]]: ~3/2 = 703.039
{{Optimal ET sequence|legend=0| 46, 94, 140 }}


{{Val list|legend=1| 29, 70, 99, 722bc, 821bc, 920bc, 1019bc, 1118bbcc, 1217bbcc, 1316bbccd }}
Badness (Sintel): 0.772


[[Badness]]: 0.094603
== 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> }}.


= Leapday =
[[Subgroup]]: 2.3.5.7


Subgroup: 2.3.5
[[Comma list]]: 5120/5103, 235298/234375


[[Comma list]]: 10737418240/10460353203
{{Mapping|legend=1| 1 0 61 71 | 0 1 -37 -43 }}
: mapping generators: ~2, ~3


[[Mapping]]: [{{val| 1 0 -31 }}, {{val| 0 1 21 }}]
[[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 }}


[[POTE generator]]: ~3/2 = 704.179
{{Optimal ET sequence|legend=1| 29, 70, 99, 722bc, 821bc, 920bc, 1019bc }}


{{Val list|legend=1| 29, 46, 121, 167, 455bc, 622bbc }}
[[Badness]] (Sintel): 2.39


[[Badness]]: 0.523182
== Leapday ==
{{Main| Leapday }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Leapday]].''


== 7-limit ==
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.


Subgroup: 2.3.5.7
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.  


[[Comma list]]: 686/675, 5120/5103
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.


[[Mapping]]: [{{val| 1 0 -31 -21 }}, {{val| 0 1 21 15 }}]
[[Subgroup]]: 2.3.5.7


{{Multival|legend=1| 1 21 15 31 21 -24 }}
[[Comma list]]: 686/675, 5120/5103


[[POTE generator]]: ~3/2 = 704.263
{{Mapping|legend=1| 1 0 -31 -21 | 0 1 21 15 }}
: mapping generators: ~2, ~3


{{Val list|legend=1| 17c, 29, 46, 167d, 213d, 259cdd, 305bcdd }}
[[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 }}


[[Badness]]: 0.096123
{{Optimal ET sequence|legend=1| 17c, 29, 46 }}


== 11-limit ==
[[Badness]] (Sintel): 2.43


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


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


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


POTE generator: ~3/2 = 704.250
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| 17c, 29, 46, 167de, 213de, 259cdde }}
{{Optimal ET sequence|legend=0| 17c, 29, 46 }}


Badness: 0.038624
Badness (Sintel): 1.28
 
== 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: 91/90, 121/120, 169/168, 352/351


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


POTE generator: ~3/2 = 704.214
Optimal tunings:  
* WE: ~2 = 1200.4758{{c}}, ~3/2 = 704.4930{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2346{{c}}


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


Badness: 0.024732
Badness (Sintel): 1.02
 
== 17-limit ==


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


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


Mapping: [{{val| 1 0 -31 -21 -14 -9 -34 }}, {{val| 0 1 21 15 11 8 24 }}]
Mapping: {{mapping| 1 0 -31 -21 -14 -9 -34 | 0 1 21 15 11 8 24 }}


POTE generator: ~3/2 = 704.229
Optimal tunings:  
* WE: ~2 = 1200.4818{{c}}, ~3/2 = 704.5121{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2507{{c}}


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


Badness: 0.017863
Badness (Sintel): 0.910


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


Subgroup: 2.3.5.7.11.13.17.19
Comma list: 91/90, 121/120, 136/135, 154/153, 161/160, 169/168


Comma list: 91/90, 121/120, 133/132, 136/135, 154/153, 169/168
Mapping: {{mapping| 1 0 -31 -21 -14 -9 -34 -5 | 0 1 21 15 11 8 24 6 }}


Mapping: [{{val| 1 0 -31 -21 -14 -9 -34 9 }}, {{val| 0 1 21 15 11 8 24 -3 }}]
Optimal tunings:  
* WE: ~2 = 1200.5169{{c}}, ~3/2 = 704.5279{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2450{{c}}


POTE generator: ~3/2 = 704.135
{{Optimal ET sequence|legend=0| 17cg, 29g, 46, 121defg }}


Vals: {{Val list| 17cg, 29g, 46, 75dfgh, 121defgh }}
Badness (Sintel): 0.872


Badness: 0.017356
== Mystery ==
{{Main| Mystery }}
: ''For the 5-limit version, see [[29th-octave temperaments #Mystery]].''


=== Leapling ===
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: 2.3.5.7.11.13.17.19
[[Subgroup]]: 2.3.5.7


Comma list: 77/76, 91/90, 121/120, 136/135, 153/152, 169/168
[[Comma list]]: 5120/5103, 50421/50000


Mapping: [&lt;1 0 -31 -21 -14 -9 -34 -37|, &lt;0 1 21 15 11 8 24 26|]
{{Mapping|legend=1| 29 46 0 14 | 0 0 1 1 }}
: mapping generators: ~50/49, ~5


POTE generator: ~3/2 = 704.123
[[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 }}


Vals: {{Val list| 17cgh, 29g, 46h, 75dfg, 121defghh }}
{{Optimal ET sequence|legend=1| 29, 58, 87, 145 }}


Badness: 0.019065
[[Badness]] (Sintel): 2.63


=Mystery=
=== 11-limit ===
Commas: 5120/5103, 50421/50000
Subgroup: 2.3.5.7.11


[[POTE_tuning|POTE generator]]: ~5/4 = 388.646
Comma list: 441/440, 896/891, 3388/3375


Map: [&lt;29 46 0 14|, &lt;0 0 1 1|]
Mapping: {{mapping| 29 46 0 14 33 | 0 0 1 1 1 }}


Wedgie: &lt;&lt;0 29 29 46 46 -14||
Optimal tunings:  
* WE: ~45/44 = 41.3637{{c}}, ~5/4 = 388.3136{{c}}
* CWE: ~45/44 = 41.3793{{c}}, ~5/4 = 388.0598{{c}}


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


Badness: 0.1037
Badness (Sintel): 1.13


==11-limit==
=== 13-limit ===
Commas: 441/440, 896/891, 3388/3375
Subgroup: 2.3.5.7.11.13


[[POTE_tuning|POTE generator]]: ~5/4 = 388.460
Comma list: 196/195, 352/351, 364/363, 676/675


Map: [&lt;29 46 0 14 33|, &lt;0 0 1 1 1|]
Mapping: {{mapping| 29 46 0 14 33 40 | 0 0 1 1 1 1 }}


EDOs: 29, 58, 87, 145
Optimal tunings:  
* WE: ~45/44 = 41.3623{{c}}, ~5/4 = 388.1942{{c}}
* CWE: ~40/39 = 41.3793{{c}}, ~5/4 = 387.9017{{c}}


Badness: 0.0343
{{Optimal ET sequence|legend=0| 29, 58, 87, 145, 232 }}


==13-limit==
Badness (Sintel): 0.768
Commas: 196/195, 352/351, 364/363, 676/675


[[POTE_tuning|POTE generator]]: ~5/4 = 388.354
== Hemidromeda ==
Hemidromeda may be described as the {{nowrap| 29 & 111 }} temperament. Named by [[Xenllium]] in 2023, ''hemidromeda'' comes from ''hemi-'' (Ancient Greek for "one half") and ''[[andromeda]]'', because the generator is 1/2 of andromeda's perfect twelfth (~3/1, about 1902.4 cents); the ploidacot for this temperament is alpha-dicot.  


Map: [&lt;29 46 0 14 33 40|, &lt;0 0 1 1 1 1|]
[[Subgroup]]: 2.3.5.7


EDOs: 29, 58, 87, 145, 232, 377
[[Comma list]]: 5120/5103, 52734375/52706752


Badness: 0.0186
{{Mapping|legend=1| 1 0 38 48 | 0 2 -45 -57 }}
: mapping generator: ~2, ~12500/7203


=Quanic=
[[Optimal tuning]]s:
Commas: 5120/5103, 5832000/5764801
* [[WE]]: ~2 = 1199.7236{{c}}, ~12500/7203 = 951.1864{{c}}
: [[error map]]: {{val| -0.276 +0.418 -0.205 +0.282 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~12500/7203 = 951.4098{{c}}
: error map: {{val| 0.000 +0.865 +0.243 +0.813 }}


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


Map: [&lt;1 1 -4 0|, &lt;0 5 54 24|]
[[Badness]] (Sintel): 2.93


EDOs: 94, 111, 205
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.1795
Comma list: 1331/1323, 1375/1372, 5120/5103


==11-limit==
Mapping: {{mapping| 1 0 38 48 32 | 0 2 -45 -57 -36 }}
Commas: 540/539, 1331/1323, 5120/5103


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


Map: [&lt;1 1 -4 0 1|, &lt;0 5 54 24 21|]
{{Optimal ET sequence|legend=0| 29, 82cd, 111, 140, 251, 391e }}


EDOs: 94, 111, 205
Badness (Sintel): 2.01


Badness: 0.0587
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


==13-limit==
Comma list: 352/351, 676/675, 847/845, 1331/1323
Commas: 352/351, 540/539, 729/728, 1331/1323


POTE generator: ~13/12 = 140.496
Mapping: {{mapping| 1 0 38 48 32 37 | 0 2 -45 -57 -36 -42 }}


Map: [&lt;1 1 -4 0 1 3|, &lt;0 5 54 24 21 6|]
Optimal tunings:  
* WE: ~2 = 1199.8753{{c}}, ~26/15 = 951.3054{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.4064{{c}}


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


Badness: 0.0325
Badness (Sintel): 1.18


==17-limit==
=== 17-limit ===
Commas: 352/351, 442/441, 540/539, 715/714, 847/845
Subgroup: 2.3.5.7.11.13.17


POTE generator: ~13/12 = 140.497
Comma list: 352/351, 442/441, 561/560, 676/675, 715/714


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


EDOs: 94, 111, 205
Optimal tunings:  
* WE: ~2 = 1199.8770{{c}}, ~26/15 = 951.3039{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.4035{{c}}


Badness: 0.0211
{{Optimal ET sequence|legend=0| 29g, 82cdfg, 111, 140, 251, 391e }}


==19-limit==
Badness (Sintel): 0.971
Commas: 352/351, 400/399, 442/441, 456/455, 495/494, 715/714


POTE generator: ~13/12 = 140.496
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


Map: [&lt;1 1 -4 0 1 3 -2 -5|, &lt;0 5 54 24 21 6 52 79|]
Comma list: 286/285, 352/351, 363/361, 442/441, 476/475, 561/560


EDOs: 94, 111, 205
Mapping: {{mapping| 1 0 38 48 32 37 58 32 | 0 2 -45 -57 -36 -42 -68 -35 }}


Badness: 0.0173
Optimal tunings:  
* WE: ~2 = 1199.7534{{c}}, ~26/15 = 951.2024{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.4020{{c}}


=Supers=
{{Optimal ET sequence|legend=0| 29g, 82cdfgh, 111, 140 }}
Commas: 5120/5103, 118098/117649


POTE generator: ~9/7 = 434.218
Badness (Sintel): 1.01


Map: [&lt;2 1 -12 2|, &lt;0 3 23 5|]
=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23


Wedgie: &lt;&lt;6 46 10 59 -1 -106||
Comma list: 253/252, 286/285, 352/351, 363/361, 391/390, 442/441, 460/459


EDOs: {{EDOs|58, 94, 152}}
Mapping: {{mapping| 1 0 38 48 32 37 58 32 18 | 0 2 -45 -57 -36 -42 -68 -35 -17 }}


Badness: 0.092748
Optimal tunings:  
* WE: ~2 = 1199.9128{{c}}, ~26/15 = 951.3371{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.4076{{c}}


==11-limit==
{{Optimal ET sequence|legend=0| 29g, 82cdfgh, 111, 140 }}
Commas: 540/539, 4000/3993, 5120/5103


POTE generator: ~9/7 = 434.217
Badness (Sintel): 1.10


Map: [&lt;2 1 -12 2 -9|, &lt;0 3 23 5 22|]
== Countriton ==
: ''For the 5-limit version, see [[Schismic–Mercator equivalence continuum #Countritonic]].''


EDOs: {{EDOs|58, 94, 152}}
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.


Badness: 0.028240
Countriton was named by [[Xenllium]] in 2022 as a counterpart of [[untriton]].  


==13-limit==
[[Subgroup]]: 2.3.5.7
Commas: 352/351, 540/539, 729/728, 1575/1573


POTE generator: ~9/7 = 434.221
[[Comma list]]: 5120/5103, 7558272/7503125


Map: [&lt;2 1 -12 2 -9 -2|, &lt;0 3 23 5 22 13|]
{{Mapping|legend=1| 1 -3 -15 13 | 0 9 34 -20 }}
: mapping generators: ~2, ~1225/864


EDOs: {{EDOs|58, 94, 152f}}
[[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 }}


Badness: 0.021645
{{Optimal ET sequence|legend=1| 51c, 53, 157, 210, 473cdd }}


==17-limit==
[[Badness]] (Sintel): 3.32
Commas: 170/169, 289/288, 352/351, 442/441, 561/560


POTE generator: ~9/7 = 434.181
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;2 1 -12 2 -9 -2 6|, &lt;0 3 23 5 22 13 3|]
Comma list: 176/175, 5120/5103, 41503/41472


EDOs: {{EDOs|58, 94, 152f}}
Mapping: {{mapping| 1 -3 -15 13 -21 | 0 9 34 -20 48 }}


Badness: 0.021316
Optimal tunings:  
* WE: ~2 = 1199.5178{{c}}, ~77/54 = 611.2097{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~77/54 = 611.4495{{c}}


=Alphaquarter=
{{Optimal ET sequence|legend=0| 51ce, 53, 104c, 157 }}
Commas: 5120/5103, 29360128/29296875


POTE generator: ~16128/15625 = 55.243
Badness (Sintel): 2.80


Map: [&lt;1 2 2 0|, &lt;0 -9 7 61|]
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Wedgie: &lt;&lt;9 -7 -61 -32 -122 -122||
Comma list: 176/175, 351/350, 847/845, 2197/2187


EDOs: 87, 152, 239, 391
Mapping: {{mapping| 1 -3 -15 13 -21 -7 | 0 9 34 -20 48 21 }}


Badness: 0.1166
Optimal tunings:  
* WE: ~2 = 1199.5944{{c}}, ~77/54 = 611.2491{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~77/54 = 611.4506{{c}}


==11-limit==
{{Optimal ET sequence|legend=0| 51ce, 53, 104c, 157 }}
Commas: 3025/3024, 4000/3993, 5120/5103


POTE generator: ~33/32 = 55.243
Badness (Sintel): 1.75


Map: [&lt;1 2 2 0 3|, &lt;0 -9 7 61 10|]
== 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.


EDOs: 87, 152, 239, 391
Artoneutral was named by [[Flora Canou]] in 2023 for its generator's quality.


Badness: 0.0296
[[Subgroup]]: 2.3.5.7


= Septiquarter =
[[Comma list]]: 5120/5103, 3828125/3779136
Commas: 5120/5103, 420175/419904


POTE generator: ~147/128 = 242.453
{{Mapping|legend=1| 1 -1 -4 12 | 0 9 22 -32 }}
: mapping generators: ~2, ~128/105


Map: [&lt;1 3 10 2|, &lt;0 -7 -38 4|]
[[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 }}


Wedgie: &lt;&lt;7 38 -4 44 -26 -116||
{{Optimal ET sequence|legend=1| 87, 94, 181 }}


EDOs: {{EDOs|94, 99, 292, 391, 881bd, 1272bcd}}
[[Badness]] (Sintel): 3.98


Badness: 0.0538
=== 11-limit ===
Subgroup: 2.3.5.7.11


== Semiseptiquarter ==
Comma list: 385/384, 2200/2187, 4000/3993
Commas: 5120/5103, 9801/9800, 14641/14580


POTE generators: ~121/105 = 242.4511
Mapping: {{mapping| 1 -1 -4 12 -2 | 0 9 22 -32 19 }}


Map: [<2 6 20 4 15|, <0 -7 -38 4 -20|]
Optimal tunings:  
* WE: ~2 = 1200.1668{{c}}, ~11/9 = 344.8027{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 344.7557{{c}}


EDOs: {{EDOs|94, 198, 292, 490}}
{{Optimal ET sequence|legend=0| 87, 181 }}


Badness: 0.0642
Badness (Sintel): 1.52


=== 13-limit ===
=== 13-limit ===
Commas: 352/351, 847/845, 1716/1715, 14641/14580
Subgroup: 2.3.5.7.11.13
 
Comma list: 325/324, 352/351, 385/384, 1575/1573
 
Mapping: {{mapping| 1 -1 -4 12 -2 6 | 0 9 22 -32 19 -8 }}
 
Optimal tunings:
* WE: ~2 = 1200.0662{{c}}, ~11/9 = 344.7804{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 344.7617{{c}}
 
{{Optimal ET sequence|legend=0| 87, 181 }}
 
Badness (Sintel): 1.08
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 325/324, 352/351, 375/374, 385/384, 595/594
 
Mapping: {{mapping| 1 -1 -4 12 -2 6 -12 | 0 9 22 -32 19 -8 56 }}
 
Optimal tunings:
* WE: ~2 = 1200.0346{{c}}, ~11/9 = 344.7589{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 344.7492{{c}}


POTE generators: ~121/105 = 242.4448
{{Optimal ET sequence|legend=0| 87, 94, 181 }}


Map: [<2 6 20 4 15 7|, <0 -7 -38 4 -20 1|]
Badness (Sintel): 1.16


EDOs: {{EDOs|94, 198, 490f}}
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


Badness: 0.0348
Comma list: 325/324, 352/351, 375/374, 385/384, 400/399, 595/594


= Tricot (aka Trimot) =
Mapping: {{mapping| 1 -1 -4 12 -2 6 -12 -15 | 0 9 22 -32 19 -8 56 67 }}
{{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.
Optimal tunings:
* WE: ~2 = 1200.0282{{c}}, ~11/9 = 344.7532{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 344.7453{{c}}


[[Comma list]]: 2430/2401, 5120/5103
{{Optimal ET sequence|legend=0| 87, 94, 181 }}


[[Mapping]]: [&lt;1 0 -13 -3|, &lt;0 3 29 11|]
Badness (Sintel): 1.19


[[Wedgie]]: &lt;&lt;3 29 11 39 9 -56||
=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23


[[POTE tuning|POTE generator]]: ~81/56 = 634.026
Comma list: 300/299, 325/324, 352/351, 375/374, 385/384, 400/399, 484/483


{{Val list|legend=1| 17c, 36c, 53, 229dd, 282dd }}
Mapping: {{mapping| 1 -1 -4 12 -2 6 -12 -15 -13 | 0 9 22 -32 19 -8 56 67 61 }}


[[Badness]]: 0.100127
Optimal tunings:  
* WE: ~2 = 1200.0163{{c}}, ~11/9 = 344.7461{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 344.7416{{c}}


== 11-limit ==
{{Optimal ET sequence|legend=0| 87, 94, 181 }}
Comma list: 99/98, 121/120, 5120/5103


Mapping: [&lt;1 0 -13 -3 -5|, &lt;0 3 29 11 16|]
Badness (Sintel): 1.17


POTE generator: ~63/44 = 634.027
== 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.  


Vals: {{Val list| 17c, 36ce, 53, 70, 123de }}
[[Subgroup]]: 2.3.5.7


Badness: 0.056134
[[Comma list]]: 5120/5103, 5832000/5764801


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


Mapping: [&lt;1 0 -13 -3 -5 0|, &lt;0 3 29 11 16 7|]
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.6159{{c}}, ~160/147 = 140.4483{{c}}
: [[error map]]: {{val| -0.384 -0.098 -0.570 +1.933 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~160/147 = 140.4862{{c}}
: error map: {{val| 0.000 +0.476 -0.061 +2.842 }}


POTE generator: ~13/9 = 634.012
{{Optimal ET sequence|legend=1| 94, 111, 205 }}


Vals: {{Val list| 17c, 36ce, 53, 70, 123de }}
[[Badness]] (Sintel): 4.54


Badness: 0.032102
=== 11-limit ===
Subgroup: 2.3.5.7.11


= Ketchup =
Comma list: 540/539, 1331/1323, 5120/5103
[[Comma list]]: 5120/5103, 1071875/1062882
 
Mapping: {{mapping| 1 1 -4 0 1 | 0 5 54 24 21 }}
 
Optimal tunings:
* WE: ~2 = 1199.7834{{c}}, ~88/81 = 140.4635{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~88/81 = 140.4850{{c}}


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


[[Wedgie]]: &lt;&lt;8 30 -18 29 -51 -126||
Badness (Sintel): 1.94


[[POTE tuning|POTE generator]]: ~64/63 = ~81/80 = 25.719
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


{{Val list|legend=1| 46, 94, 140 }}
Comma list: 352/351, 540/539, 729/728, 1331/1323


[[Badness]]: 0.084538
Mapping: {{mapping| 1 1 -4 0 1 3 | 0 5 54 24 21 6 }}


== 11-limit ==
Optimal tunings:
Comma list: 385/384, 1331/1323, 2200/2187
* WE: ~2 = 1199.6639{{c}}, ~13/12 = 140.4562{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.4904{{c}}


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


POTE generator: ~55/54 = ~64/63 = ~81/80 = 25.693
Badness (Sintel): 1.34


Vals: {{Val list| 46, 94, 140 }}
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Badness: 0.039555
Comma list: 352/351, 442/441, 540/539, 715/714, 847/845


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


Mapping: [&lt;2 3 4 6 7 8|, &lt;0 4 15 -9 -2 -14|]
Optimal tunings:  
* WE: ~2 = 1199.6699{{c}}, ~13/12 = 140.4586{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.4920{{c}}


POTE generator: ~55/54 = ~64/63 = ~81/80 = 25.697
{{Optimal ET sequence|legend=0| 94, 111, 205 }}


Vals: {{Val list| 46, 94, 140 }}
Badness (Sintel): 1.08


Badness: 0.024824
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


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


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


POTE generator: ~55/54 = ~64/63 = ~81/80 = 25.701
Optimal tunings:  
* WE: ~2 = 1199.6745{{c}}, ~13/12 = 140.4574{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.4908{{c}}


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


Badness: 0.016591
Badness (Sintel): 1.05


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


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


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


Vals: {{Val list| 46, 94, 140h, 234eh }}
[[Subgroup]]: 2.3.5.7


Badness: 0.018170
[[Comma list]]: 5120/5103, 578509309952/576650390625


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


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


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


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


Badness: 0.014033
== References ==


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