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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: [{{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 }}]


POTE generator: ~21/16 = 475.697
Mapping: {{mapping| 2 -8 -56 12 -25 9 | 0 7 38 -4 20 -1 }}


Vals: {{Val list| 53, 58, 111, 280cdf, 391cdf }}
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: 0.018842
{{Optimal ET sequence|legend=0| 94, 198, 490f }}


==== 17-limit ====
Badness (Sintel): 1.44
Subgroup: 2.3.5.7.11.13.17


Comma list: 176/175, 256/255, 351/350, 442/441, 540/539
== Kwai ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Kwai]].''


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


POTE generator: ~21/16 = 475.692
[[Subgroup]]: 2.3.5.7


Vals: {{Val list| 53, 58, 111, 321cdfg }}
[[Comma list]]: 5120/5103, 16875/16807


Badness: 0.018403
{{Mapping|legend=1| 1 0 -50 -40 | 0 1 33 27 }}
: mapping generators: ~2, ~3


==== 19-limit ====
[[Optimal tuning]]s:
Subgroup: 2.3.5.7.11.13.17.19
* [[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 }}


Comma list: 176/175, 256/255, 286/285, 324/323, 351/350, 540/539
{{Optimal ET sequence|legend=1| 41, 111, 152, 345, 497d }}


Mapping: [{{val| 1 0 -6 4 -12 -7 14 -12 }}, {{val| 0 4 21 -3 39 27 -25 41 }}]
[[Badness]] (Sintel): 1.38


POTE generator: ~21/16 = 475.679
=== 11-limit ===
 
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
[[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.


Subgroup: 2.3.5
Mapping: {{mapping| 1 0 -50 -40 32 27 58 -56 | 0 1 33 27 -18 -21 -34 38 }}


[[Comma list]]: 10737418240/10460353203
Optimal tunings:  
* WE: ~2 = 1199.3401{{c}}, ~3/2 = 702.2705{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6548{{c}}


[[Mapping]]: [{{val| 1 0 -31 }}, {{val| 0 1 21 }}]
{{Optimal ET sequence|legend=0| 41, 70h, 111, 152fg, 263dfgh }}


[[POTE generator]]: ~3/2 = 704.179
Badness (Sintel): 1.03


{{Val list|legend=1| 29, 46, 121, 167, 455bc, 622bbc }}
==== Hemikwai ====
Subgroup: 2.3.5.7.11.13


[[Badness]]: 0.523182
Comma list: 540/539, 676/675, 1375/1372, 5120/5103


=== 7-limit ===
Mapping: {{mapping| 1 0 -50 -40 32 -51 | 0 2 66 54 -36 69 }}
Subgroup: 2.3.5.7
: mapping generators: ~2, ~26/15


[[Comma list]]: 686/675, 5120/5103
Optimal tunings:  
* WE: ~2 = 1199.6968{{c}}, ~26/15 = 951.0740{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.3123{{c}}


[[Mapping]]: [{{val| 1 0 -31 -21 }}, {{val| 0 1 21 15 }}]
{{Optimal ET sequence|legend=0| 82, 111, 193, 304d }}


{{Multival|legend=1| 1 21 15 31 21 -24 }}
Badness (Sintel): 1.82


[[POTE generator]]: ~3/2 = 704.263
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


{{Val list|legend=1| 17c, 29, 46, 167d, 213d, 259cdd, 305bcdd }}
Comma list: 442/441, 540/539, 676/675, 715/714, 5120/5103


[[Badness]]: 0.096123
Mapping: {{mapping| 1 0 -50 -40 32 -51 -30 | 0 2 66 54 -36 69 43 }}


=== 11-limit ===
Optimal tunings:
Subgroup: 2.3.5.7.11
* WE: ~2 = 1199.6861{{c}}, ~26/15 = 951.0654{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.3120{{c}}


Comma list: 121/120, 441/440, 686/675
{{Optimal ET sequence|legend=0| 82, 111, 193, 304d }}


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


POTE generator: ~3/2 = 704.250
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


Vals: {{Val list| 17c, 29, 46, 167de, 213de, 259cdde }}
Comma list: 400/399, 442/441, 540/539, 676/675, 715/714, 1445/1444


Badness: 0.038624
Mapping: {{mapping| 1 0 -50 -40 32 -51 -30 -56 | 0 2 66 54 -36 69 43 76 }}


=== 13-limit ===
Optimal tunings:
Subgroup: 2.3.5.7.11.13
* WE: ~2 = 1199.6718{{c}}, ~26/15 = 951.0526{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.3103{{c}}


Comma list: 91/90, 121/120, 169/168, 352/351
{{Optimal ET sequence|legend=0| 82, 111, 193, 304dh }}


Mapping: [{{val| 1 0 -31 -21 -14 -9 }}, {{val| 0 1 21 15 11 8 }}]
Badness (Sintel): 1.16


POTE generator: ~3/2 = 704.214
== 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.  


Vals: {{Val list| 17c, 29, 46, 121def, 167def, 213deff }}
[[Subgroup]]: 2.3.5.7


Badness: 0.024732
[[Comma list]]: 5120/5103, 1071875/1062882


=== 17-limit ===
{{Mapping|legend=1| 2 3 4 6 | 0 4 15 -9 }}
Subgroup: 2.3.5.7.11.13.17
: mapping generators: ~1225/864, ~64/63


Comma list: 91/90, 121/120, 136/135, 154/153, 169/168
[[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| 1 0 -31 -21 -14 -9 -34 }}, {{val| 0 1 21 15 11 8 24 }}]
{{Optimal ET sequence|legend=1| 46, 94, 140 }}


POTE generator: ~3/2 = 704.229
[[Badness]] (Sintel): 2.14


Vals: {{Val list| 17cg, 29g, 46, 121defg, 167defg, 213deffg }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.017863
Comma list: 385/384, 1331/1323, 2200/2187


=== 19-limit ===
Mapping: {{mapping| 2 3 4 6 7 | 0 4 15 -9 -2 }}
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 91/90, 121/120, 133/132, 136/135, 154/153, 169/168
Optimal tunings:  
* WE: ~99/70 = 600.0678{{c}}, ~64/63 = 25.6963{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~64/63 = 25.6956{{c}}


Mapping: [{{val| 1 0 -31 -21 -14 -9 -34 9 }}, {{val| 0 1 21 15 11 8 24 -3 }}]
{{Optimal ET sequence|legend=0| 46, 94, 140 }}


POTE generator: ~3/2 = 704.135
Badness (Sintel): 1.31


Vals: {{Val list| 17cg, 29g, 46, 75dfgh, 121defgh }}
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Badness: 0.017356
Comma list: 325/324, 352/351, 385/384, 1331/1323


=== Leapling ===
Mapping: {{mapping| 2 3 4 6 7 8 | 0 4 15 -9 -2 -14 }}
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 77/76, 91/90, 121/120, 136/135, 153/152, 169/168
Optimal tunings:  
* WE: ~99/70 = 600.0612{{c}}, ~66/65 = 25.7000{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~66/65 = 25.6978{{c}}


Mapping: [{{val| 1 0 -31 -21 -14 -9 -34 -37 }}, {{val| 0 1 21 15 11 8 24 26 }}]
{{Optimal ET sequence|legend=0| 46, 94, 140 }}


POTE generator: ~3/2 = 704.123
Badness (Sintel): 1.03


Vals: {{Val list| 17cgh, 29g, 46h, 75dfg, 121defghh }}
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Badness: 0.019065
Comma list: 289/288, 325/324, 352/351, 385/384, 442/441


== Mystery ==
Mapping: {{mapping| 2 3 4 6 7 8 8 | 0 4 15 -9 -2 -14 4 }}
{{main| Mystery }}


Subgroup: 2.3.5
Optimal tunings:  
* WE: ~17/12 = 600.0896{{c}}, ~66/65 = 25.7048{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~66/65 = 25.7017{{c}}


[[Comma list]]: {{monzo| 46 -29 }}
{{Optimal ET sequence|legend=0| 46, 94, 140 }}


[[Mapping]]: [{{val| 29 46 0 }}, {{val| 0 0 1 }}]
Badness (Sintel): 0.845


Mapping generators: ~531441/524288, ~5
=== 2.3.5.7.11.13.17.23 subgroup ===
Subgroup: 2.3.5.7.11.13.17.23


[[POTE generator]]: ~5/4 = 387.408
Comma list: 253/252, 289/288, 325/324, 352/351, 385/384, 391/390


{{Val list|legend=1| 29, 58, 87, 232, 319 }}
Mapping: {{mapping| 2 3 4 6 7 8 8 9 | 0 4 15 -9 -2 -14 4 1 }}


[[Badness]]: 1.020556
Optimal tunings:  
* WE: ~17/12 = 600.1139{{c}}, ~66/65 = 25.7053{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~66/65 = 25.7013{{c}}


=== 7-limit ===
{{Optimal ET sequence|legend=0| 46, 94, 140 }}
Subgroup: 2.3.5.7


[[Comma list]]: 5120/5103, 50421/50000
Badness (Sintel): 0.772


[[Mapping]]: [{{val| 29 46 0 14 }}, {{val| 0 0 1 1 }}]
== 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> }}.


{{Multival|legend=1| 0 29 29 46 46 -14 }}
[[Subgroup]]: 2.3.5.7


[[POTE generator]]: ~5/4 = 388.646
[[Comma list]]: 5120/5103, 235298/234375


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


[[Badness]]: 0.103734
[[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 }}


=== 11-limit ===
{{Optimal ET sequence|legend=1| 29, 70, 99, 722bc, 821bc, 920bc, 1019bc }}
Subgroup: 2.3.5.7.11


Comma list: 441/440, 896/891, 3388/3375
[[Badness]] (Sintel): 2.39


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


POTE generator: ~5/4 = 388.460
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.  


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


Badness: 0.034291
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.


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


Comma list: 196/195, 352/351, 364/363, 676/675
[[Comma list]]: 686/675, 5120/5103


Mapping: [{{val| 29 46 0 14 33 40 }}, {{val| 0 0 1 1 1 1 }}]
{{Mapping|legend=1| 1 0 -31 -21 | 0 1 21 15 }}
: mapping generators: ~2, ~3


POTE generator: ~5/4 = 388.354
[[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 }}


Vals: {{Val list| 29, 58, 87, 145, 232, 377cef }}
{{Optimal ET sequence|legend=1| 17c, 29, 46 }}


Badness: 0.018591
[[Badness]] (Sintel): 2.43
 
== Quanic ==
Subgroup: 2.3.5.7
 
[[Comma list]]: 5120/5103, 5832000/5764801
 
[[Mapping]]: [{{val| 1 1 -4 0 }}, {{val| 0 5 54 24 }}]
 
[[POTE generator]]: ~160/147 = 140.493
 
{{Val list|legend=1| 94, 111, 205 }}
 
[[Badness]]: 0.179475


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


Comma list: 540/539, 1331/1323, 5120/5103
Comma list: 121/120, 441/440, 686/675


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


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


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


Badness: 0.058678
Badness (Sintel): 1.28


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


Comma list: 352/351, 540/539, 729/728, 1331/1323
Comma list: 91/90, 121/120, 169/168, 352/351


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


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


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


Badness: 0.032481
Badness (Sintel): 1.02


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


Comma list: 352/351, 442/441, 540/539, 715/714, 847/845
Comma list: 91/90, 121/120, 136/135, 154/153, 169/168
 
Mapping: {{mapping| 1 0 -31 -21 -14 -9 -34 | 0 1 21 15 11 8 24 }}


Mapping: [{{val| 1 1 -4 0 1 3 -2 }}, {{val| 0 5 54 24 21 6 52 }}]
Optimal tunings:  
* WE: ~2 = 1200.4818{{c}}, ~3/2 = 704.5121{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2507{{c}}


POTE generator: ~13/12 = 140.497
{{Optimal ET sequence|legend=0| 17cg, 29g, 46, 121defg }}


Vals: {{Val list| 94, 111, 205 }}
Badness (Sintel): 0.910


Badness: 0.021112
=== 2.3.5.7.11.13.17.23 subgroup ===
Subgroup: 2.3.5.7.11.13.17.23


=== 19-limit ===
Comma list: 91/90, 121/120, 136/135, 154/153, 161/160, 169/168
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 352/351, 400/399, 442/441, 456/455, 495/494, 715/714
Mapping: {{mapping| 1 0 -31 -21 -14 -9 -34 -5 | 0 1 21 15 11 8 24 6 }}


Mapping: [{{val| 1 1 -4 0 1 3 -2 -5 }}, {{val| 0 5 54 24 21 6 52 79 }}]
Optimal tunings:  
* WE: ~2 = 1200.5169{{c}}, ~3/2 = 704.5279{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2450{{c}}


POTE generator: ~13/12 = 140.496
{{Optimal ET sequence|legend=0| 17cg, 29g, 46, 121defg }}


Vals: {{Val list| 94, 111, 205 }}
Badness (Sintel): 0.872


Badness: 0.017273
== Mystery ==
{{Main| Mystery }}
: ''For the 5-limit version, see [[29th-octave temperaments #Mystery]].''


== Supers ==
Mystery tempers out [[50421/50000]] and may be described as the {{nowrap| 29 & 58 }} temperament. It has a 1\29 period and primes 5, 7, 11 and 13 are all reached by one generator step; its ploidacot is 29-ploid acot. [[145edo]] or [[232edo]] are good candidates for tunings.  
Subgroup: 2.3.5.7


[[Comma list]]: 5120/5103, 118098/117649
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val| 2 1 -12 2 }}, {{val| 0 3 23 5 }}]
[[Comma list]]: 5120/5103, 50421/50000


{{Multival|legend=1| 6 46 10 59 -1 -106 }}
{{Mapping|legend=1| 29 46 0 14 | 0 0 1 1 }}
: mapping generators: ~50/49, ~5


[[POTE generator]]: ~9/7 = 434.218
[[Optimal tuning]]s:  
* [[WE]]: ~50/49 = 41.3652{{c}}, ~5/4 = 388.5128{{c}}
: [[error map]]: {{val| -0.410 +0.842 +1.378 -2.022 }}
* [[CWE]]: ~50/49 = 41.3793{{c}}, ~5/4 = 388.3030{{c}}
: error map: {{val| 0.000 +1.493 +1.989 -1.213 }}


{{Val list|legend=1| 58, 94, 152 }}
{{Optimal ET sequence|legend=1| 29, 58, 87, 145 }}


[[Badness]]: 0.092748
[[Badness]] (Sintel): 2.63


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


Comma list: 540/539, 4000/3993, 5120/5103
Comma list: 441/440, 896/891, 3388/3375


Mapping: [{{val| 2 1 -12 2 -9 }}, {{val| 0 3 23 5 22 }}]
Mapping: {{mapping| 29 46 0 14 33 | 0 0 1 1 1 }}


POTE generator: ~9/7 = 434.217
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| 58, 94, 152 }}
{{Optimal ET sequence|legend=0| 29, 58, 87, 145 }}


Badness: 0.028240
Badness (Sintel): 1.13


=== 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, 1575/1573
Comma list: 196/195, 352/351, 364/363, 676/675
 
Mapping: [{{val| 2 1 -12 2 -9 -2 }}, {{val| 0 3 23 5 22 13 }}]


POTE generator: ~9/7 = 434.221
Mapping: {{mapping| 29 46 0 14 33 40 | 0 0 1 1 1 1 }}


Vals: {{Val list| 58, 94, 152f }}
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: 0.021645
{{Optimal ET sequence|legend=0| 29, 58, 87, 145, 232 }}


=== 17-limit ===
Badness (Sintel): 0.768
Subgroup: 2.3.5.7.11.13.17


Comma list: 170/169, 289/288, 352/351, 442/441, 561/560
== 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.


Mapping: [{{val| 2 1 -12 2 -9 -2 6 }}, {{val| 0 3 23 5 22 13 3 }}]
[[Subgroup]]: 2.3.5.7


POTE generator: ~9/7 = 434.181
[[Comma list]]: 5120/5103, 52734375/52706752


Vals: {{Val list| 58, 94, 152f }}
{{Mapping|legend=1| 1 0 38 48 | 0 2 -45 -57 }}
: mapping generator: ~2, ~12500/7203


Badness: 0.021316
[[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 }}


== Alphaquarter ==
{{Optimal ET sequence|legend=1| 29, 82cd, 111, 140, 251, 391, 1424bbcdd }}
{{see also|Escapade family}}


Subgroup: 2.3.5.7
[[Badness]] (Sintel): 2.93
 
[[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: 1331/1323, 1375/1372, 5120/5103


Mapping: [{{val| 1 2 2 0 3 }}, {{val| 0 -9 7 61 10 }}]
Mapping: {{mapping| 1 0 38 48 32 | 0 2 -45 -57 -36 }}


POTE generator: ~33/32 = 55.243
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| 87, 152, 239, 391 }}
{{Optimal ET sequence|legend=0| 29, 82cd, 111, 140, 251, 391e }}


Badness: 0.029638
Badness (Sintel): 2.01


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


[[Comma list]]: 5120/5103, 420175/419904
Comma list: 352/351, 676/675, 847/845, 1331/1323


[[Mapping]]: [{{val| 1 3 10 2 }}, {{val| 0 -7 -38 4 }}]
Mapping: {{mapping| 1 0 38 48 32 37 | 0 2 -45 -57 -36 -42 }}


{{Multival|legend=1| 7 38 -4 44 -26 -116 }}
Optimal tunings:
* WE: ~2 = 1199.8753{{c}}, ~26/15 = 951.3054{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.4064{{c}}


[[POTE generator]]: ~147/128 = 242.453
{{Optimal ET sequence|legend=0| 29, 82cdf, 111, 140, 251, 391e }}


{{Val list|legend=1| 94, 99, 292, 391, 881bd, 1272bcd }}
Badness (Sintel): 1.18


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


=== Semiseptiquarter ===
Comma list: 352/351, 442/441, 561/560, 676/675, 715/714
Subgroup: 2.3.5.7.11


Comma list: 5120/5103, 9801/9800, 14641/14580
Mapping: {{mapping| 1 0 38 48 32 37 58 | 0 2 -45 -57 -36 -42 -68 }}


Mapping: [{{val| 2 6 20 4 15 }}, {{val| 0 -7 -38 4 -20 }}]
Optimal tunings:  
* WE: ~2 = 1199.8770{{c}}, ~26/15 = 951.3039{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.4035{{c}}


POTE generators: ~121/105 = 242.4511
{{Optimal ET sequence|legend=0| 29g, 82cdfg, 111, 140, 251, 391e }}


Vals: {{Val list| 94, 198, 292, 490 }}
Badness (Sintel): 0.971


Badness: 0.064160
=== 19-limit ===
 
Subgroup: 2.3.5.7.11.13.17.19
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Comma list: 352/351, 847/845, 1716/1715, 14641/14580
Comma list: 286/285, 352/351, 363/361, 442/441, 476/475, 561/560


Mapping: [{{val| 2 6 20 4 15 7 }}, {{val| 0 -7 -38 4 -20 1 }}]
Mapping: {{mapping| 1 0 38 48 32 37 58 32 | 0 2 -45 -57 -36 -42 -68 -35 }}


POTE generators: ~121/105 = 242.4448
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| 94, 198, 490f }}
{{Optimal ET sequence|legend=0| 29g, 82cdfgh, 111, 140 }}


Badness: 0.034834
Badness (Sintel): 1.01


== Tricot ==
=== 23-limit ===
{{see also|Tricot family}}
Subgroup: 2.3.5.7.11.13.17.19.23


The generator for tricot is the real cube root of third harmonic, 3<sup>1/3</sup>, tuned between 63/44 and 13/9. Tricot can be described as 53&amp;70 temperament (also known as ''trimot''), tempering out the [[tricot comma]], {{monzo| 39 -29 3 }} in the 5-limit, 2430/2401 (nuwell comma) and 5120/5103 in the 7-limit, 99/98 and 121/120 in the 11-limit, 169/168, 352/351, 640/637, and 729/728 in the 13-limit.
Comma list: 253/252, 286/285, 352/351, 363/361, 391/390, 442/441, 460/459


Subgroup: 2.3.5
Mapping: {{mapping| 1 0 38 48 32 37 58 32 18 | 0 2 -45 -57 -36 -42 -68 -35 -17 }}


[[Comma]]: {{monzo | 39 -29 3 }} = 68719476736000/68630377364883
Optimal tunings:  
* WE: ~2 = 1199.9128{{c}}, ~26/15 = 951.3371{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.4076{{c}}


[[Mapping]]: [&lt;1 0 -13|, &lt;0 3 29|]
{{Optimal ET sequence|legend=0| 29g, 82cdfgh, 111, 140 }}


[[POTE tuning|POTE generator]]: ~59049/40960 = 634.012
Badness (Sintel): 1.10


{{Val list|legend=1| 53, 229, 282, 335, 388, 441, 1376, 1817, 2258 }}
== Countriton ==
: ''For the 5-limit version, see [[Schismic–Mercator equivalence continuum #Countritonic]].''


[[Badness]]: 0.046093
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.  


=== 7-limit (aka Trimot) ===
Countriton was named by [[Xenllium]] in 2022 as a counterpart of [[untriton]].  
Subgroup: 2.3.5.7


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


[[Mapping]]: [{{val| 1 0 -13 -3 }}, {{val| 0 3 29 11 }}]
[[Comma list]]: 5120/5103, 7558272/7503125


{{Multival|legend=1| 3 29 11 39 9 -56 }}
{{Mapping|legend=1| 1 -3 -15 13 | 0 9 34 -20 }}
: mapping generators: ~2, ~1225/864


[[POTE generator]]: ~81/56 = 634.026
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.4179{{c}}, ~1225/864 = 611.1213{{c}}
: [[error map]]: {{val| -0.582 -0.117 +0.541 +1.181 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~1225/864 = 611.4120{{c}}
: error map: {{val| 0.000 +0.753 +1.695 +2.934 }}


{{Val list|legend=1| 17c, 36c, 53, 229dd, 282dd }}
{{Optimal ET sequence|legend=1| 51c, 53, 157, 210, 473cdd }}


[[Badness]]: 0.100127
[[Badness]] (Sintel): 3.32


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


Mapping: [{{val| 1 0 -13 -3 -5 }}, {{val| 0 3 29 11 16 }}]
Mapping: {{mapping| 1 -3 -15 13 -21 | 0 9 34 -20 48 }}


POTE generator: ~63/44 = 634.027
Optimal tunings:  
* WE: ~2 = 1199.5178{{c}}, ~77/54 = 611.2097{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~77/54 = 611.4495{{c}}


Vals: {{Val list| 17c, 36ce, 53, 70, 123de }}
{{Optimal ET sequence|legend=0| 51ce, 53, 104c, 157 }}


Badness: 0.056134
Badness (Sintel): 2.80


=== 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: 176/175, 351/350, 847/845, 2197/2187
 
Mapping: {{mapping| 1 -3 -15 13 -21 -7 | 0 9 34 -20 48 21 }}


Mapping: [{{val| 1 0 -13 -3 -5 0 }}, {{val| 0 3 29 11 16 7 }}]
Optimal tunings:  
* WE: ~2 = 1199.5944{{c}}, ~77/54 = 611.2491{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~77/54 = 611.4506{{c}}


POTE generator: ~13/9 = 634.012
{{Optimal ET sequence|legend=0| 51ce, 53, 104c, 157 }}


Vals: {{Val list| 17c, 36ce, 53, 70, 123de }}
Badness (Sintel): 1.75


Badness: 0.032102
== 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.  


== Ketchup ==
Artoneutral was named by [[Flora Canou]] in 2023 for its generator's quality.  
Subgroup: 2.3.5.7


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


[[Mapping]]: [{{val| 2 3 4 6 }}, {{val| 0 4 15 -9 }}]
[[Comma list]]: 5120/5103, 3828125/3779136


{{Multival|legend=1| 8 30 -18 29 -51 -126 }}
{{Mapping|legend=1| 1 -1 -4 12 | 0 9 22 -32 }}
: mapping generators: ~2, ~128/105


[[POTE generator]]: ~64/63 = ~81/80 = 25.719
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.1400{{c}}, ~128/105 = 344.7929{{c}}
: [[error map]]: {{val| +0.140 +1.041 -1.430 -0.518 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~128/105 = 344.7531{{c}}
: error map: {{val| 0.000 +0.823 -1.746 -0.925 }}


{{Val list|legend=1| 46, 94, 140 }}
{{Optimal ET sequence|legend=1| 87, 94, 181 }}


[[Badness]]: 0.084538
[[Badness]] (Sintel): 3.98


=== 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: 385/384, 2200/2187, 4000/3993


Mapping: [{{val| 2 3 4 6 7 }}, {{val| 0 4 15 -9 -2 }}]
Mapping: {{mapping| 1 -1 -4 12 -2 | 0 9 22 -32 19 }}


POTE generator: ~55/54 = ~64/63 = ~81/80 = 25.693
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| 46, 94, 140 }}
{{Optimal ET sequence|legend=0| 87, 181 }}


Badness: 0.039555
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, 352/351, 847/845, 1331/1323
Comma list: 325/324, 352/351, 385/384, 1575/1573


Mapping: [{{val| 2 3 4 6 7 8 }}, {{val| 0 4 15 -9 -2 -14 }}]
Mapping: {{mapping| 1 -1 -4 12 -2 6 | 0 9 22 -32 19 -8 }}


POTE generator: ~55/54 = ~64/63 = ~81/80 = 25.697
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| 46, 94, 140 }}
{{Optimal ET sequence|legend=0| 87, 181 }}


Badness: 0.024824
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: 289/288, 325/324, 352/351, 385/384, 561/560
Comma list: 325/324, 352/351, 375/374, 385/384, 595/594


Mapping: [{{val| 2 3 4 6 7 8 8 }}, {{val| 0 4 15 -9 -2 -14 4 }}]
Mapping: {{mapping| 1 -1 -4 12 -2 6 -12 | 0 9 22 -32 19 -8 56 }}


POTE generator: ~55/54 = ~64/63 = ~81/80 = 25.701
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| 46, 94, 140 }}
{{Optimal ET sequence|legend=0| 87, 94, 181 }}


Badness: 0.016591
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: 190/189, 209/208, 289/288, 352/351, 385/384, 561/560
Comma list: 325/324, 352/351, 375/374, 385/384, 400/399, 595/594


Mapping: [{{val| 2 3 4 6 7 8 8 9 }}, {{val| 0 4 15 -9 -2 -14 4 -12 }}]
Mapping: {{mapping| 1 -1 -4 12 -2 6 -12 -15 | 0 9 22 -32 19 -8 56 67 }}


POTE generator: ~55/54 = ~64/63 = ~81/80 = 25.660
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| 46, 94, 140h, 234eh }}
{{Optimal ET sequence|legend=0| 87, 94, 181 }}


Badness: 0.018170
Badness (Sintel): 1.19


=== 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: 300/299, 325/324, 352/351, 375/374, 385/384, 400/399, 484/483


Mapping: [{{val| 2 3 4 6 7 8 8 9 9 }}, {{val| 0 4 15 -9 -2 -14 4 -12 1 }}]
Mapping: {{mapping| 1 -1 -4 12 -2 6 -12 -15 -13 | 0 9 22 -32 19 -8 56 67 61 }}


POTE generator: ~55/54 = ~64/63 = ~81/80 = 25.661
Optimal tunings:  
* WE: ~2 = 1200.0163{{c}}, ~11/9 = 344.7461{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 344.7416{{c}}


Vals: {{Val list| 46, 94, 140h, 234ehi }}
{{Optimal ET sequence|legend=0| 87, 94, 181 }}


Badness: 0.014033
Badness (Sintel): 1.17


== Undim ==
== Quanic ==
Subgroup: 2.3.5
Quanic may be described as the {{nowrap| 94 & 111 }} temperament. It splits the perfect fifth into five generators which in the 13-limit extension may be taken as ~13/12; its ploidacot is thus pentacot. [[205edo]] may be recommended as a tuning.  


[[Comma list]]: {{monzo| 41 -20 -4 }} = 2199023255552/2179240250625
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val| 4 0 41 }}, {{val| 0 1 -5 }}]
[[Comma list]]: 5120/5103, 5832000/5764801
 
Mapping generators: ~1215/1024, ~3
 
[[POTE generator]]: ~3/2 = 702.736
 
{{Val list|legend=1| 12, 104, 116, 128, 140, 152, 620, 772, 924c, 1076bc, 1228bc }}
 
[[Badness]]: 0.241703
 
=== 7-limit ===
Subgroup: 2.3.5.7


[[Comma list]]: 5120/5103, 390625/388962
{{Mapping|legend=1| 1 1 -4 0 | 0 5 54 24 }}
: mapping generators: ~2, ~160/147


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


{{Multival|legend=1| 4 -20 -44 -41 -81 -46 }}
{{Optimal ET sequence|legend=1| 94, 111, 205 }}


[[POTE generator]]: ~3/2 = 702.736
[[Badness]] (Sintel): 4.54
 
{{Val list|legend=1| 12, 128, 140, 152, 292 }}
 
[[Badness]]: 0.062754


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


Comma list: 1375/1372, 5120/5103, 5632/5625
Comma list: 540/539, 1331/1323, 5120/5103


Mapping: [{{val| 4 0 41 81 128 }}, {{val| 0 1 -5 -11 -18 }}]
Mapping: {{mapping| 1 1 -4 0 1 | 0 5 54 24 21 }}


POTE generator: ~3/2 = 702.689
Optimal tunings:
* WE: ~2 = 1199.7834{{c}}, ~88/81 = 140.4635{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~88/81 = 140.4850{{c}}


Vals: {{Val list| 12, 128e, 140, 152, 292, 444d, 596d }}
{{Optimal ET sequence|legend=0| 94, 111, 205 }}


Badness: 0.034837
Badness (Sintel): 1.94


== Quintakwai ==
=== 13-limit ===
{{See also| 28ed5 #28ed5 as a generator }}
Subgroup: 2.3.5.7.11.13


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 (the difference between 16/15 and 17/16), 289/288 (the difference between 17/16 and 18/17), 324/323 (the difference between 18/17 and 19/18), and 361/360 (the difference between 19/18 and 20/19) are equated together, and 400/399 (the difference 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).
Comma list: 352/351, 540/539, 729/728, 1331/1323


Subgroup: 2.3.5.7
Mapping: {{mapping| 1 1 -4 0 1 3 | 0 5 54 24 21 6 }}


[[Comma list]]: 5120/5103, 9765625/9680832
Optimal tunings:  
* WE: ~2 = 1199.6639{{c}}, ~13/12 = 140.4562{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.4904{{c}}


[[Mapping]]: [{{val|1 2 0 -2}}, {{val|0 -5 28 58}}]
{{Optimal ET sequence|legend=0| 94, 111, 205 }}


{{Multival|legend=1|5 -28 -58 -56 -106 -56}}
Badness (Sintel): 1.34


[[POTE generator]]: ~625/588 = 99.483
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


{{Val list|legend=1| 12, 169, 181, 193 }}
Comma list: 352/351, 442/441, 540/539, 715/714, 847/845


[[Badness]]: 0.155536
Mapping: {{mapping| 1 1 -4 0 1 3 -2 | 0 5 54 24 21 6 52 }}


=== 11-limit ===
Optimal tunings:
Subgroup: 2.3.5.7.11
* WE: ~2 = 1199.6699{{c}}, ~13/12 = 140.4586{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.4920{{c}}


Comma list: 1375/1372, 4375/4356, 5120/5103
{{Optimal ET sequence|legend=0| 94, 111, 205 }}


Mapping: [{{val|1 2 0 -2 -4}}, {{val|0 -5 28 58 90}}]
Badness (Sintel): 1.08


POTE generator: ~35/33 = 99.472
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


Vals: {{Val list| 12, 181, 193, 374, 567ce }}
Comma list: 352/351, 400/399, 442/441, 456/455, 495/494, 715/714


Badness: 0.073158
Mapping: {{mapping| 1 1 -4 0 1 3 -2 -5 | 0 5 54 24 21 6 52 79 }}


=== 13-limit ===
Optimal tunings:
Subgroup: 2.3.5.7.11.13
* WE: ~2 = 1199.6745{{c}}, ~13/12 = 140.4574{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.4908{{c}}


Comma list: 325/324, 1375/1372, 1575/1573, 4096/4095
{{Optimal ET sequence|legend=0| 94, 111, 205 }}


Mapping: [{{val|1 2 0 -2 -4 10}}, {{val|0 -5 28 58 90 -76}}]
Badness (Sintel): 1.05


POTE generator: ~35/33 = 99.468
== Jorgensen ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Jorgensen]].''


Vals: {{Val list| 12, 181, 193, 374, 567ce, 941bce }}
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: 0.062737
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>.  


=== 17-limit ===
[[Subgroup]]: 2.3.5.7
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 325/324, 375/374, 595/594, 1275/1274, 4096/4095
 
Mapping: [{{val|1 2 0 -2 -4 10 5}}, {{val|0 -5 28 58 90 -76 -11}}]
 
POTE generator: ~18/17 = 99.469
 
Vals: {{Val list| 12, 181, 193, 374, 567ce, 941bceg }}
 
Badness: 0.037855


=== 19-limit ===
[[Comma list]]: 5120/5103, 578509309952/576650390625
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 325/324, 375/374, 400/399, 595/594, 1216/1215, 1275/1274
{{Mapping|legend=1| 70 111 0 34 | 0 0 1 1 }}
: mapping generators: ~50421/50000, ~5


Mapping: [{{val|1 2 0 -2 -4 10 5 4}}, {{val|0 -5 28 58 90 -76 -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 = 99.469
{{Optimal ET sequence|legend=1| 70, 140, 350, 490 }}


Vals: {{Val list| 12, 181, 193, 374, 567ce, 941bcegh, 1508bccdeegghh }}
[[Badness]] (Sintel): 5.40


Badness: 0.025861
== References ==


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