@@ Line 1: / Line 1: @@
-This is a collection of [[Rank-2 temperament|rank-2]] [[temperament]]s [[tempering out]] the hemifamity comma, {{monzo| 10 -6 1 -1 }} = [[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]].
+{{Technical data page}}
+This is a collection of [[rank-2 temperament|rank-2]] '''aberschismic temperaments''', which [[tempering out|temper out]] the [[aberschisma]] ({{monzo|legend=1| 10 -6 1 -1 }}, [[ratio]]: 5120/5103). These temperaments divide an exact or approximate septimal quartertone, [[36/35]] into two equal steps, each representing [[81/80]][[~]][[64/63]], the syntonic comma or the septimal comma. Therefore, classical and septimal intervals are found by the same [[chain of fifths]] inflected by the syntonic~septimal comma to the opposite sides. In addition we may identify [[10/7]] by the augmented fourth and [[50/49]] by the [[Pythagorean comma]].
 Temperaments belonging to this category and generated by the fifth are dominant, garibaldi, kwai, undecental, and leapday. Dominant has 5/4 mapped to M3. Garibaldi has 5/4 mapped to d4. Kwai has 5/4 mapped to 4A7. Undecental has 5/4 mapped to 5d7. Leapday has 5/4 mapped to 3A1.
-Diaschismic is generated by the fifth with a semi-octave period. Hemififths has the fifth sliced into two and 5/4 mapped to the hemififth + Pyth. comma. Rodan has the fifth sliced into three as does slendric. Trimot has the twelfth sliced into three as does tricot. Monkey has the fifth sliced into four as does tetracot. Buzzard has the twelfth sliced into four as does vulture. Misty is generated by the fifth with a 1/3-octave period. Supers has the fifth sliced into three with a semi-octave period. Undim is generated by the fifth with a 1/4-octave period. Quinticosiennic and quintakwai have the fourth sliced into five. Amity has the eleventh sliced into five. Countercata has the twelfth sliced into six as does hanson. Warrior has the 6th harmonic sliced into seven as does sensi. Finally, alphaquarter has the fourth sliced into nine as does escapade.
+Diaschismic is generated by the fifth with a semi-octave period. Hemififths has the fifth sliced into two and 5/4 mapped to the hemififth + Pyth. comma. Hemidromeda has the fourth sliced into two and 5/4 mapped to the hemifourth + 3d4. Rodan has the fifth sliced into three as does slendric. Alphatrimot has the twelfth sliced into three as does alphatricot. Monkey has the fifth sliced into four as does tetracot. Buzzard has the twelfth sliced into four as does vulture. Misty is generated by the fifth with a 1/3-octave period. Supers has the fifth sliced into three with a semi-octave period. Undim is generated by the fifth with a 1/4-octave period. Quinticosiennic and quintakwai have the fourth sliced into five. Amity has the eleventh sliced into five. Countercata has the twelfth sliced into six as does hanson. Warrior has the 6th harmonic sliced into seven as does sensi. Finally, alphaquarter has the fourth sliced into nine as does escapade.
-Temperaments considered below are undecental, leapday, mystery, quanic, septiquarter, countriton, ketchup, and artoneutral. Discussed elsewhere are:
+Temperaments discussed elsewhere are:
-* ''[[Dominant]]'' (+36/35) → [[Meantone family #Dominant|Meantone family]]
+* [[Dominant (temperament)|Dominant]] (+36/35) → [[Meantone family #Dominant|Meantone family]]
 * [[Garibaldi]] (+225/224) → [[Schismatic family #Garibaldi|Schismatic family]]
-* ''[[Kwai]]'' (+16875/16807) → [[Mirkwai clan #Kwai|Mirkwai clan]]
+* [[Diaschismic]] (+126/125) → [[Diaschismic family #Septimal diaschismic|Diaschismic family]]
-* ''[[Diaschismic]]'' (+126/125) → [[Diaschismic family #Diaschismic|Diaschismic family]]
 * [[Hemififths]] (+2401/2400) → [[Breedsmic temperaments #Hemififths|Breedsmic temperaments]]
 * [[Rodan]] (+245/243) → [[Gamelismic clan #Rodan|Gamelismic clan]]
-* ''[[Trimot]]'' (+2430/2401) → [[Tricot family #Trimot|Tricot family]]
+* ''[[Alphatrimot]]'' (+2430/2401) → [[Alphatricot family #Alphatrimot|Alphatricot family]]
-* ''[[Monkey]]'' (+875/864) → [[Tetracot family #Monkey|Tetracot family]]
-* [[Buzzard]] (+1728/1715) → [[Vulture family #Buzzard|Vulture family]]
 * [[Misty]] (+3136/3125) → [[Misty family #Misty|Misty family]]
-* ''[[Supers]]'' (+118098/117649) → [[Stearnsmic clan #Supers|Stearnsmic clan]]
+* [[Monkey]] (+875/864) → [[Tetracot family #Monkey|Tetracot family]]
+* [[Buzzard]] (+1728/1715) → [[Buzzardsmic clan #Buzzard|Buzzardsmic clan]]
 * ''[[Undim]]'' (+390625/388962) → [[Undim family #Septimal undim|Undim family]]
 * ''[[Quinticosiennic]]'' (+395136/390625) → [[Quintaleap family #Quinticosiennic|Quintaleap family]]
@@ Line 22: / Line 21: @@
 * [[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]]
-== Undecental ==
+Considered below are septiquarter, kwai, ketchup, undecental, leapday, mystery, hemidromeda, countriton, artoneutral, quanic and jorgensen, in the order of increasing [[TE logflat badness]].
-Undecental adds the triwellisma to the comma list and may be described as the 29 & 70 temperament. 5/4 is mapped to the quintuple diminished seventh (5d7) or equivalently the perfect fourth (P4) - 3 Pyth. commas. [[99edo|58\99]] is an almost perfect generator, just as the name suggests. Another interesting choice is the argent fifth, 2<sup>(2 - sqrt (2))</sup>.
+== Septiquarter ==
+Septiquarter tempers out [[420175/419904]] and may be described as the {{nowrap| 94 & 99 }} temperament. Its [[ploidacot]] is epsilon-heptacot. [[99edo]] makes for an excellent tuning, and [[292edo]] an even better one. [[94edo]] and [[104edo]] in the 104c val are also among the possibilities.
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 5120/5103, 235298/234375
+[[Comma list]]: 5120/5103, 420175/419904
+{{Mapping|legend=1| 1 -4 -28 6 | 0 7 38 -4 }}
+: mapping generators: ~2, ~243/140
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.7212{{c}}, ~243/140 = 957.3250{{c}}
+: [[error map]]: {{val| -0.279 +0.435 -0.158 +0.201 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~243/140 = 957.5424{{c}}
+: error map: {{val| 0.000 +0.842 +0.298 +1.004 }}
+{{Optimal ET sequence|legend=1| 94, 99, 292, 391, 881bd, 1272bcd }}
+[[Badness]] (Sintel): 1.36
+=== Semiseptiquarter ===
+Subgroup: 2.3.5.7.11
+Comma list: 5120/5103, 9801/9800, 14641/14580
+Mapping: {{mapping| 2 -8 -56 12 -25 | 0 7 38 -4 20 }}
+Optimal tunings:
+* WE: ~99/70 = 599.8953{{c}}, ~210/121 = 957.3819{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~210/121 = 957.5449{{c}}
+{{Optimal ET sequence|legend=0| 94, 198, 292, 490 }}
+Badness (Sintel): 2.12
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
-{{Mapping|legend=1| 1 0 61 71 | 0 1 -37 -43 }}
+Comma list: 352/351, 847/845, 1716/1715, 14641/14580
-{{Multival|legend=1| 1 -37 -43 -61 -71 4 }}
+Mapping: {{mapping| 2 -8 -56 12 -25 9 | 0 7 38 -4 20 -1 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 703.039
+Optimal tunings:
+* WE: ~99/70 = 599.8565{{c}}, ~210/121 = 957.3261{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~210/121 = 957.5508{{c}}
-{{Optimal ET sequence|legend=1| 29, 70, 99, 722bc, 821bc, 920bc, 1019bc, 1118bbcc, 1217bbcc, 1316bbccd }}
+{{Optimal ET sequence|legend=0| 94, 198, 490f }}
-[[Badness]]: 0.094603
+Badness (Sintel): 1.44
-== Leapday ==
+== Kwai ==
-{{Main| Leapday }}
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Kwai]].''
-: ''For the 5-limit version of this temperament, see [[High badness temperaments #Leapday]].''
-Leapday tempers out the leapday comma, {{monzo| 31 -21 1 }}, in the 5-limit, mapping 5/4 to the triple augmented unison (3A1) or equivalently the minor third (m3) + 2 Pyth. commas. This temperament can be described as the 29 &amp; 46 temperament, which tempers out the hemifamity and [[686/675]] (senga). The 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.
+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]]: 686/675, 5120/5103
+[[Comma list]]: 5120/5103, 16875/16807
-{{Mapping|legend=1| 1 0 -31 -21 | 0 1 21 15 }}
-{{Multival|legend=1| 1 21 15 31 21 -24 }}
+{{Mapping|legend=1| 1 0 -50 -40 | 0 1 33 27 }}
+: mapping generators: ~2, ~3
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 704.263
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.7337{{c}}, ~3/2 = 702.4600{{c}}
+: [[error map]]: {{val| -0.266 +0.239 -0.607 +1.055 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.6085{{c}}
+: error map: {{val| 0.000 +0.653 -0.234 +1.603 }}
-{{Optimal ET sequence|legend=1| 17c, 29, 46, 167d, 213d, 259cdd, 305bcdd }}
+{{Optimal ET sequence|legend=1| 41, 111, 152, 345, 497d }}
-[[Badness]]: 0.096123
+[[Badness]] (Sintel): 1.38
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 121/120, 441/440, 686/675
+Comma list: 540/539, 1375/1372, 5120/5103
-Mapping: {{mapping| 1 0 -31 -21 -14 | 0 1 21 15 11 }}
+Mapping: {{mapping| 1 0 -50 -40 32 | 0 1 33 27 -18 }}
-Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.250
+Optimal tunings:
+* WE: ~2 = 1199.6672{{c}}, ~3/2 = 702.4282{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6189{{c}}
-{{Optimal ET sequence|legend=1| 17c, 29, 46, 167de, 213de, 259cdde }}
+{{Optimal ET sequence|legend=0| 41, 111, 152, 497de, 649dde }}
-Badness: 0.038624
+Badness (Sintel): 0.867
-=== 13-limit ===
+==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 91/90, 121/120, 169/168, 352/351
+Comma list: 352/351, 540/539, 729/728, 1375/1372
-Mapping: {{mapping| 1 0 -31 -21 -14 -9 | 0 1 21 15 11 8 }}
+Mapping: {{mapping| 1 0 -50 -40 32 27 | 0 1 33 27 -18 -21 }}
-Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.214
+Optimal tunings:
+* WE: ~2 = 1199.4772{{c}}, ~3/2 = 702.3379{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6409{{c}}
-{{Optimal ET sequence|legend=1| 17c, 29, 46, 121def, 167def, 213deff }}
+{{Optimal ET sequence|legend=0| 41, 111, 152f, 415dff }}
-Badness: 0.024732
+Badness (Sintel): 1.01
-=== 17-limit ===
+===== 17-limit =====
 Subgroup: 2.3.5.7.11.13.17
-Comma list: 91/90, 121/120, 136/135, 154/153, 169/168
+Comma list: 256/255, 352/351, 540/539, 715/714, 1089/1088
-Mapping: {{mapping| 1 0 -31 -21 -14 -9 -34 | 0 1 21 15 11 8 24 }}
+Mapping: {{mapping| 1 0 -50 -40 32 27 58 | 0 1 33 27 -18 -21 -34 }}
-Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.229
+Optimal tunings:
+* WE: ~2 = 1199.3537{{c}}, ~3/2 = 702.2850{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6589{{c}}
-{{Optimal ET sequence|legend=1| 17cg, 29g, 46, 121defg, 167defg, 213deffg }}
+{{Optimal ET sequence|legend=0| 41, 70, 111, 152fg, 263dfg }}
-Badness: 0.017863
+Badness (Sintel): 1.12
-==== 19-limit ====
+===== 19-limit =====
 Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 91/90, 121/120, 133/132, 136/135, 154/153, 169/168
+Comma list: 256/255, 352/351, 400/399, 456/455, 715/714, 847/845
-Mapping: {{mapping| 1 0 -31 -21 -14 -9 -34 9 | 0 1 21 15 11 8 24 -3 }}
+Mapping: {{mapping| 1 0 -50 -40 32 27 58 -56 | 0 1 33 27 -18 -21 -34 38 }}
-Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.135
+Optimal tunings:
+* WE: ~2 = 1199.3401{{c}}, ~3/2 = 702.2705{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6548{{c}}
-{{Optimal ET sequence|legend=1| 17cg, 29g, 46, 75dfgh, 121defgh }}
+{{Optimal ET sequence|legend=0| 41, 70h, 111, 152fg, 263dfgh }}
-Badness: 0.017356
+Badness (Sintel): 1.03
-===== 23-limit =====
+==== Hemikwai ====
-Subgroup: 2.3.5.7.11.13.17.19.23
+Subgroup: 2.3.5.7.11.13
-Comma list: 91/90, 121/120, 133/132, 136/135, 154/153, 161/160, 169/168
+Comma list: 540/539, 676/675, 1375/1372, 5120/5103
-Mapping: {{mapping| 1 0 -31 -21 -14 -9 -34 9 -5 | 0 1 21 15 11 8 24 -3 6 }}
+Mapping: {{mapping| 1 0 -50 -40 32 -51 | 0 2 66 54 -36 69 }}
+: mapping generators: ~2, ~26/15
-Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.141
+Optimal tunings:
+* WE: ~2 = 1199.6968{{c}}, ~26/15 = 951.0740{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.3123{{c}}
-{{Optimal ET sequence|legend=1| 17cg, 29g, 46, 75dfgh, 121defgh }}
+{{Optimal ET sequence|legend=0| 82, 111, 193, 304d }}
-Badness: 0.014065
+Badness (Sintel): 1.82
-==== Leapling ====
+===== 17-limit =====
-Subgroup: 2.3.5.7.11.13.17.19
+Subgroup: 2.3.5.7.11.13.17
-Comma list: 77/76, 91/90, 121/120, 136/135, 153/152, 169/168
+Comma list: 442/441, 540/539, 676/675, 715/714, 5120/5103
-Mapping: {{mapping| 1 0 -31 -21 -14 -9 -34 -37 | 0 1 21 15 11 8 24 26 }}
+Mapping: {{mapping| 1 0 -50 -40 32 -51 -30 | 0 2 66 54 -36 69 43 }}
-Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.123
+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=1| 17cgh, 29g, 46h, 75dfg, 121defghh }}
+{{Optimal ET sequence|legend=0| 82, 111, 193, 304d }}
-Badness: 0.019065
+Badness (Sintel): 1.31
-===== 23-limit =====
+===== 19-limit =====
-Subgroup: 2.3.5.7.11.13.17.19.23
+Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 77/76, 91/90, 115/114, 121/120, 136/135, 153/152, 161/160
+Comma list: 400/399, 442/441, 540/539, 676/675, 715/714, 1445/1444
-Mapping: {{mapping| 1 0 -31 -21 -14 -9 -34 -37 -5 | 0 1 21 15 11 8 24 26 6 }}
+Mapping: {{mapping| 1 0 -50 -40 32 -51 -30 -56 | 0 2 66 54 -36 69 43 76 }}
-Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.114
+Optimal tunings:
+* WE: ~2 = 1199.6718{{c}}, ~26/15 = 951.0526{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.3103{{c}}
-{{Optimal ET sequence|legend=1| 17cgh, 29g, 46h, 75dfg, 121defghh }}
+{{Optimal ET sequence|legend=0| 82, 111, 193, 304dh }}
-Badness: 0.016067
+Badness (Sintel): 1.16
-== Mystery ==
+== Ketchup ==
-{{Main| Mystery }}
+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.
-: ''For the 5-limit version of this temperament, see [[29th-octave temperaments #Mystery]].''
-Mystery has a 1\29 period and primes 5, 7, 11 and 13 are all reached by one generator step. [[145edo]] or [[232edo]] are good candidates for tunings.
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 5120/5103, 50421/50000
+[[Comma list]]: 5120/5103, 1071875/1062882
-{{Mapping|legend=1| 29 46 0 14 | 0 0 1 1 }}
+{{Mapping|legend=1| 2 3 4 6 | 0 4 15 -9 }}
+: mapping generators: ~1225/864, ~64/63
-{{Multival|legend=1| 0 29 29 46 46 -14 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~1225/864 = 599.9685{{c}}, ~64/63 = 25.7181{{c}}
+: [[error map]]: {{val| -0.063 +0.823 -0.668 -0.478 }}
+* [[CWE]]: ~1225/864 = 600.0000{{c}}, ~64/63 = 25.7181{{c}}
+: error map: {{val| 0.000 +0.917 -0.543 -0.288 }}
-[[Optimal tuning]] ([[POTE]]): ~50/49 = 1\29, ~5/4 = 388.646
+{{Optimal ET sequence|legend=1| 46, 94, 140 }}
-{{Optimal ET sequence|legend=1| 29, 58, 87, 145 }}
+[[Badness]] (Sintel): 2.14
-[[Badness]]: 0.103734
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 441/440, 896/891, 3388/3375
+Comma list: 385/384, 1331/1323, 2200/2187
-Mapping: {{mapping| 29 46 0 14 33 | 0 0 1 1 1 }}
+Mapping: {{mapping| 2 3 4 6 7 | 0 4 15 -9 -2 }}
-Optimal tuning (POTE): ~45/44 = 1\29, ~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}}
-{{Optimal ET sequence|legend=1| 29, 58, 87, 145 }}
+{{Optimal ET sequence|legend=0| 46, 94, 140 }}
-Badness: 0.034291
+Badness (Sintel): 1.31
 === 13-limit ===
 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: {{mapping| 2 3 4 6 7 8 | 0 4 15 -9 -2 -14 }}
+Optimal tunings:
+* WE: ~99/70 = 600.0612{{c}}, ~66/65 = 25.7000{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~66/65 = 25.6978{{c}}
+{{Optimal ET sequence|legend=0| 46, 94, 140 }}
+Badness (Sintel): 1.03
+=== 17-limit ===
+Subgroup: 2.3.5.7.11.13.17
+Comma list: 289/288, 325/324, 352/351, 385/384, 442/441
+Mapping: {{mapping| 2 3 4 6 7 8 8 | 0 4 15 -9 -2 -14 4 }}
+Optimal tunings:
+* WE: ~17/12 = 600.0896{{c}}, ~66/65 = 25.7048{{c}}
+* CWE: ~17/12 = 600.0000{{c}}, ~66/65 = 25.7017{{c}}
+{{Optimal ET sequence|legend=0| 46, 94, 140 }}
+Badness (Sintel): 0.845
+=== 2.3.5.7.11.13.17.23 subgroup ===
+Subgroup: 2.3.5.7.11.13.17.23
+Comma list: 253/252, 289/288, 325/324, 352/351, 385/384, 391/390
+Mapping: {{mapping| 2 3 4 6 7 8 8 9 | 0 4 15 -9 -2 -14 4 1 }}
+Optimal tunings:
+* WE: ~17/12 = 600.1139{{c}}, ~66/65 = 25.7053{{c}}
+* CWE: ~17/12 = 600.0000{{c}}, ~66/65 = 25.7013{{c}}
+{{Optimal ET sequence|legend=0| 46, 94, 140 }}
+Badness (Sintel): 0.772
+== Undecental ==
+Undecental adds the triwellisma to the comma list and may be described as the {{nowrap| 29 & 70 }} temperament. 5/4 is mapped to the quintuple-diminished seventh or equivalently the perfect fourth minus three [[diesis (scale theory)|dieses]]. [[99edo|58\99]] is an almost perfect generator, just as the name suggests. Another interesting tuning choice is the argent fifth, {{nowrap| 2<sup>(2 - sqrt (2))</sup> }}.
+[[Subgroup]]: 2.3.5.7
+[[Comma list]]: 5120/5103, 235298/234375
+{{Mapping|legend=1| 1 0 61 71 | 0 1 -37 -43 }}
+: mapping generators: ~2, ~3
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.6543{{c}}, ~3/2 = 702.8370{{c}}
+: [[error map]]: {{val| -0.346 +0.536 +0.423 -0.494 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.0465{{c}}
+: error map: {{val| 0.000 +1.092 +0.966 +0.175 }}
+{{Optimal ET sequence|legend=1| 29, 70, 99, 722bc, 821bc, 920bc, 1019bc }}
+[[Badness]] (Sintel): 2.39
-Mapping: {{mapping| 29 46 0 14 33 40 | 0 0 1 1 1 1 }}
+== Leapday ==
+{{Main| Leapday }}
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Leapday]].''
-Optimal tuning (POTE): ~45/44 = 1\29, ~5/4 = 388.354
+Leapday tempers out [[686/675]], the senga, in addition to the aberschisma, 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.
-{{Optimal ET sequence|legend=1| 29, 58, 87, 145, 232, 377cef }}
+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: 0.018591
+Leapday has an alternative extension called [[porwell temperaments #Polypyth|polypyth]], which tempers out the same 5-limit comma as leapday, but with the porwell comma ([[6144/6125]]) rather than the aberschisma tempered out.
-== Quanic ==
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 5120/5103, 5832000/5764801
+[[Comma list]]: 686/675, 5120/5103
-{{Mapping|legend=1| 1 1 -4 0 | 0 5 54 24 }}
+{{Mapping|legend=1| 1 0 -31 -21 | 0 1 21 15 }}
+: mapping generators: ~2, ~3
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~160/147 = 140.493
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.7167{{c}}, ~3/2 = 704.0971{{c}}
+: [[error map]]: {{val| -0.283 +1.859 +2.559 -5.669 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.2504{{c}}
+: error map: {{val| 0.000 +2.295 +2.945 -5.070 }}
-{{Optimal ET sequence|legend=1| 94, 111, 205 }}
+{{Optimal ET sequence|legend=1| 17c, 29, 46 }}
-[[Badness]]: 0.179475
+[[Badness]] (Sintel): 2.43
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 540/539, 1331/1323, 5120/5103
+Comma list: 121/120, 441/440, 686/675
-Mapping: {{mapping| 1 1 -4 0 1 | 0 5 54 24 21 }}
+Mapping: {{mapping| 1 0 -31 -21 -14 | 0 1 21 15 11 }}
-Optimal tuning (POTE): ~2 = 1\1, ~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}}
-{{Optimal ET sequence|legend=1| 94, 111, 205 }}
+{{Optimal ET sequence|legend=0| 17c, 29, 46 }}
-Badness: 0.058678
+Badness (Sintel): 1.28
 === 13-limit ===
 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: {{mapping| 1 1 -4 0 1 3 | 0 5 54 24 21 6 }}
+Mapping: {{mapping| 1 0 -31 -21 -14 -9 | 0 1 21 15 11 8 }}
-Optimal tuning (POTE): ~2 = 1\1, ~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}}
-{{Optimal ET sequence|legend=1| 94, 111, 205 }}
+{{Optimal ET sequence|legend=0| 17c, 29, 46, 121def }}
-Badness: 0.032481
+Badness (Sintel): 1.02
 === 17-limit ===
 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 }}
+Optimal tunings:
+* WE: ~2 = 1200.4818{{c}}, ~3/2 = 704.5121{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2507{{c}}
-Mapping: {{mapping| 1 1 -4 0 1 3 -2 | 0 5 54 24 21 6 52 }}
+{{Optimal ET sequence|legend=0| 17cg, 29g, 46, 121defg }}
-Optimal tuning (POTE): ~2 = 1\1, ~13/12 = 140.497
+Badness (Sintel): 0.910
-{{Optimal ET sequence|legend=1| 94, 111, 205 }}
+=== 2.3.5.7.11.13.17.23 subgroup ===
+Subgroup: 2.3.5.7.11.13.17.23
-Badness: 0.021112
+Comma list: 91/90, 121/120, 136/135, 154/153, 161/160, 169/168
-=== 19-limit ===
+Mapping: {{mapping| 1 0 -31 -21 -14 -9 -34 -5 | 0 1 21 15 11 8 24 6 }}
-Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 352/351, 400/399, 442/441, 456/455, 495/494, 715/714
+Optimal tunings:
+* WE: ~2 = 1200.5169{{c}}, ~3/2 = 704.5279{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2450{{c}}
-Mapping: {{mapping| 1 1 -4 0 1 3 -2 -5 | 0 5 54 24 21 6 52 79 }}
+{{Optimal ET sequence|legend=0| 17cg, 29g, 46, 121defg }}
-Optimal tuning (POTE): ~2 = 1\1, ~13/12 = 140.496
+Badness (Sintel): 0.872
-{{Optimal ET sequence|legend=1| 94, 111, 205 }}
+== Mystery ==
+{{Main| Mystery }}
+: ''For the 5-limit version, see [[29th-octave temperaments #Mystery]].''
-Badness: 0.017273
+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.
-== Septiquarter ==
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 5120/5103, 420175/419904
+[[Comma list]]: 5120/5103, 50421/50000
+{{Mapping|legend=1| 29 46 0 14 | 0 0 1 1 }}
+: mapping generators: ~50/49, ~5
+[[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 }}
+{{Optimal ET sequence|legend=1| 29, 58, 87, 145 }}
+[[Badness]] (Sintel): 2.63
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
+Comma list: 441/440, 896/891, 3388/3375
+Mapping: {{mapping| 29 46 0 14 33 | 0 0 1 1 1 }}
+Optimal tunings:
+* WE: ~45/44 = 41.3637{{c}}, ~5/4 = 388.3136{{c}}
+* CWE: ~45/44 = 41.3793{{c}}, ~5/4 = 388.0598{{c}}
+{{Optimal ET sequence|legend=0| 29, 58, 87, 145 }}
+Badness (Sintel): 1.13
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
+Comma list: 196/195, 352/351, 364/363, 676/675
+Mapping: {{mapping| 29 46 0 14 33 40 | 0 0 1 1 1 1 }}
+Optimal tunings:
+* WE: ~45/44 = 41.3623{{c}}, ~5/4 = 388.1942{{c}}
+* CWE: ~40/39 = 41.3793{{c}}, ~5/4 = 387.9017{{c}}
+{{Optimal ET sequence|legend=0| 29, 58, 87, 145, 232 }}
+Badness (Sintel): 0.768
+== Hemidromeda ==
+Hemidromeda may be described as the {{nowrap| 29 & 111 }} temperament. Named by [[Xenllium]] in 2023, ''hemidromeda'' comes from ''hemi-'' (Ancient Greek for "one half") and ''[[andromeda]]'', because the generator is 1/2 of andromeda's perfect twelfth (~3/1, about 1902.4 cents); the ploidacot for this temperament is alpha-dicot.
+[[Subgroup]]: 2.3.5.7
-{{Mapping|legend=1| 1 3 10 2 | 0 -7 -38 4 }}
+[[Comma list]]: 5120/5103, 52734375/52706752
-{{Multival|legend=1| 7 38 -4 44 -26 -116 }}
+{{Mapping|legend=1| 1 0 38 48 | 0 2 -45 -57 }}
+: mapping generator: ~2, ~12500/7203
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~147/128 = 242.453
+[[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 }}
-{{Optimal ET sequence|legend=1| 94, 99, 292, 391, 881bd, 1272bcd }}
+{{Optimal ET sequence|legend=1| 29, 82cd, 111, 140, 251, 391, 1424bbcdd }}
-[[Badness]]: 0.053760
+[[Badness]] (Sintel): 2.93
-=== Semiseptiquarter ===
+=== 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 5120/5103, 9801/9800, 14641/14580
+Comma list: 1331/1323, 1375/1372, 5120/5103
-Mapping: {{mapping| 2 6 20 4 15 | 0 -7 -38 4 -20 }}
+Mapping: {{mapping| 1 0 38 48 32 | 0 2 -45 -57 -36 }}
-Optimal tuning (POTE): ~2 = 1\1, ~121/105 = 242.4511
+Optimal tunings:
+* WE: ~2 = 1199.8767{{c}}, ~400/231 = 951.3065{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~400/231 = 951.4063{{c}}
-{{Optimal ET sequence|legend=1| 94, 198, 292, 490 }}
+{{Optimal ET sequence|legend=0| 29, 82cd, 111, 140, 251, 391e }}
-Badness: 0.064160
+Badness (Sintel): 2.01
-==== 13-limit ====
+=== 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 352/351, 847/845, 1716/1715, 14641/14580
+Comma list: 352/351, 676/675, 847/845, 1331/1323
+Mapping: {{mapping| 1 0 38 48 32 37 | 0 2 -45 -57 -36 -42 }}
+Optimal tunings:
+* WE: ~2 = 1199.8753{{c}}, ~26/15 = 951.3054{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.4064{{c}}
+{{Optimal ET sequence|legend=0| 29, 82cdf, 111, 140, 251, 391e }}
+Badness (Sintel): 1.18
+=== 17-limit ===
+Subgroup: 2.3.5.7.11.13.17
+Comma list: 352/351, 442/441, 561/560, 676/675, 715/714
+Mapping: {{mapping| 1 0 38 48 32 37 58 | 0 2 -45 -57 -36 -42 -68 }}
+Optimal tunings:
+* WE: ~2 = 1199.8770{{c}}, ~26/15 = 951.3039{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.4035{{c}}
+{{Optimal ET sequence|legend=0| 29g, 82cdfg, 111, 140, 251, 391e }}
+Badness (Sintel): 0.971
+=== 19-limit ===
+Subgroup: 2.3.5.7.11.13.17.19
+Comma list: 286/285, 352/351, 363/361, 442/441, 476/475, 561/560
+Mapping: {{mapping| 1 0 38 48 32 37 58 32 | 0 2 -45 -57 -36 -42 -68 -35 }}
+Optimal tunings:
+* WE: ~2 = 1199.7534{{c}}, ~26/15 = 951.2024{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.4020{{c}}
+{{Optimal ET sequence|legend=0| 29g, 82cdfgh, 111, 140 }}
+Badness (Sintel): 1.01
+=== 23-limit ===
+Subgroup: 2.3.5.7.11.13.17.19.23
+Comma list: 253/252, 286/285, 352/351, 363/361, 391/390, 442/441, 460/459
-Mapping: {{mapping| 2 6 20 4 15 7 | 0 -7 -38 4 -20 1 }}
+Mapping: {{mapping| 1 0 38 48 32 37 58 32 18 | 0 2 -45 -57 -36 -42 -68 -35 -17 }}
-Optimal tuning (POTE): ~2 = 1\1, ~121/105 = 242.4448
+Optimal tunings:
+* WE: ~2 = 1199.9128{{c}}, ~26/15 = 951.3371{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.4076{{c}}
-{{Optimal ET sequence|legend=1| 94, 198, 490f }}
+{{Optimal ET sequence|legend=0| 29g, 82cdfgh, 111, 140 }}
-Badness: 0.034834
+Badness (Sintel): 1.10
 == Countriton ==
-: ''For the 5-limit version of this temperament, see [[Schismic-Mercator equivalence continuum #Countritonic]] and [[High badness temperaments #Countritonic]].''
+: ''For the 5-limit version, see [[Schismic–Mercator equivalence continuum #Countritonic]].''
+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.
+Countriton was named by [[Xenllium]] in 2022 as a counterpart of [[untriton]].
 [[Subgroup]]: 2.3.5.7
@@ Line 312: / Line 552: @@
 [[Comma list]]: 5120/5103, 7558272/7503125
-{{Mapping|legend=1| 1 6 19 -7 | 0 -9 -34 20 }}
+{{Mapping|legend=1| 1 -3 -15 13 | 0 9 34 -20 }}
+: mapping generators: ~2, ~1225/864
-{{Multival|legend=1| 9 34 -20 33 -57 -142 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1728/1225 = 588.582
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.4179{{c}}, ~1225/864 = 611.1213{{c}}
+: [[error map]]: {{val| -0.582 -0.117 +0.541 +1.181 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~1225/864 = 611.4120{{c}}
+: error map: {{val| 0.000 +0.753 +1.695 +2.934 }}
-{{Optimal ET sequence|legend=1| 53, 157, 210 }}
+{{Optimal ET sequence|legend=1| 51c, 53, 157, 210, 473cdd }}
-[[Badness]]: 0.131191
+[[Badness]] (Sintel): 3.32
 === 11-limit ===
@@ Line 327: / Line 570: @@
 Comma list: 176/175, 5120/5103, 41503/41472
-Mapping: {{mapping| 1 6 19 -7 27 | 0 -9 -34 20 -48 }}
+Mapping: {{mapping| 1 -3 -15 13 -21 | 0 9 34 -20 48 }}
-Optimal tuning (POTE): ~2 = 1\1, ~108/77 = 588.545
+Optimal tunings:
+* WE: ~2 = 1199.5178{{c}}, ~77/54 = 611.2097{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~77/54 = 611.4495{{c}}
-{{Optimal ET sequence|legend=1| 53, 104c, 157 }}
+{{Optimal ET sequence|legend=0| 51ce, 53, 104c, 157 }}
-Badness: 0.084782
+Badness (Sintel): 2.80
 === 13-limit ===
@@ Line 340: / Line 585: @@
 Comma list: 176/175, 351/350, 847/845, 2197/2187
-Mapping: {{mapping| 1 6 19 -7 27 14 | 0 -9 -34 20 -48 -21 }}
+Mapping: {{mapping| 1 -3 -15 13 -21 -7 | 0 9 34 -20 48 21 }}
+Optimal tunings:
+* WE: ~2 = 1199.5944{{c}}, ~77/54 = 611.2491{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~77/54 = 611.4506{{c}}
+{{Optimal ET sequence|legend=0| 51ce, 53, 104c, 157 }}
-Optimal tuning (POTE): ~2 = 1\1, ~108/77 = 588.544
+Badness (Sintel): 1.75
-{{Optimal ET sequence|legend=1| 53, 104c, 157 }}
+== 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.
-Badness: 0.042321
+Artoneutral was named by [[Flora Canou]] in 2023 for its generator's quality.
-== Ketchup ==
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 5120/5103, 1071875/1062882
+[[Comma list]]: 5120/5103, 3828125/3779136
-{{Mapping|legend=1| 2 3 4 6 | 0 4 15 -9 }}
+{{Mapping|legend=1| 1 -1 -4 12 | 0 9 22 -32 }}
+: mapping generators: ~2, ~128/105
-{{Multival|legend=1| 8 30 -18 29 -51 -126 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.1400{{c}}, ~128/105 = 344.7929{{c}}
+: [[error map]]: {{val| +0.140 +1.041 -1.430 -0.518 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~128/105 = 344.7531{{c}}
+: error map: {{val| 0.000 +0.823 -1.746 -0.925 }}
-[[Optimal tuning]] ([[POTE]]): ~1225/864 = 1\2, ~64/63 = 25.719
+{{Optimal ET sequence|legend=1| 87, 94, 181 }}
-{{Optimal ET sequence|legend=1| 46, 94, 140 }}
+[[Badness]] (Sintel): 3.98
-[[Badness]]: 0.084538
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 385/384, 1331/1323, 2200/2187
+Comma list: 385/384, 2200/2187, 4000/3993
-Mapping: {{mapping| 2 3 4 6 7 | 0 4 15 -9 -2 }}
+Mapping: {{mapping| 1 -1 -4 12 -2 | 0 9 22 -32 19 }}
-Optimal tuning (POTE): ~99/70 = 1\2, ~64/63 = 25.693
+Optimal tunings:
+* WE: ~2 = 1200.1668{{c}}, ~11/9 = 344.8027{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 344.7557{{c}}
-{{Optimal ET sequence|legend=1| 46, 94, 140 }}
+{{Optimal ET sequence|legend=0| 87, 181 }}
-Badness: 0.039555
+Badness (Sintel): 1.52
 === 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 325/324, 352/351, 385/384, 1331/1323
+Comma list: 325/324, 352/351, 385/384, 1575/1573
-Mapping: {{mapping| 2 3 4 6 7 8 | 0 4 15 -9 -2 -14 }}
+Mapping: {{mapping| 1 -1 -4 12 -2 6 | 0 9 22 -32 19 -8 }}
-Optimal tuning (POTE): ~99/70 = 1\2, ~66/65 = 25.697
+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=1| 46, 94, 140 }}
+{{Optimal ET sequence|legend=0| 87, 181 }}
-Badness: 0.024824
+Badness (Sintel): 1.08
 === 17-limit ===
 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: {{mapping| 2 3 4 6 7 8 8 | 0 4 15 -9 -2 -14 4 }}
+Mapping: {{mapping| 1 -1 -4 12 -2 6 -12 | 0 9 22 -32 19 -8 56 }}
-Optimal tuning (POTE): ~17/12 = 1\2, ~66/65 = 25.701
+Optimal tunings:
+* WE: ~2 = 1200.0346{{c}}, ~11/9 = 344.7589{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 344.7492{{c}}
-{{Optimal ET sequence|legend=1| 46, 94, 140 }}
+{{Optimal ET sequence|legend=0| 87, 94, 181 }}
-Badness: 0.016591
+Badness (Sintel): 1.16
 === 19-limit ===
 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: {{mapping| 2 3 4 6 7 8 8 9 | 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 }}
-Optimal tuning (POTE): ~17/12 = 1\2, ~66/65 = 25.660
+Optimal tunings:
+* WE: ~2 = 1200.0282{{c}}, ~11/9 = 344.7532{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 344.7453{{c}}
-{{Optimal ET sequence|legend=1| 46, 94, 140h, 234eh }}
+{{Optimal ET sequence|legend=0| 87, 94, 181 }}
-Badness: 0.018170
+Badness (Sintel): 1.19
 === 23-limit ===
 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: {{mapping| 2 3 4 6 7 8 8 9 9 | 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 }}
-Optimal tuning (POTE): ~17/12 = 1\2, ~66/65 = 25.661
+Optimal tunings:
+* WE: ~2 = 1200.0163{{c}}, ~11/9 = 344.7461{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 344.7416{{c}}
-{{Optimal ET sequence|legend=1| 46, 94, 140h, 234ehi }}
+{{Optimal ET sequence|legend=0| 87, 94, 181 }}
-Badness: 0.014033
+Badness (Sintel): 1.17
-== Artoneutral ==
+== Quanic ==
-Artoneutral is generated by an artoneutral third of ~11/9 (or a tendoneutral sixth of ~18/11) and can be described as the 87 & 94 temperament. [[181edo]] is a recommendable tuning.
+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.
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 5120/5103, 3828125/3779136
+[[Comma list]]: 5120/5103, 5832000/5764801
-{{Mapping|legend=1| 1 8 18 -20 | 0 -9 -22 32 }}
+{{Mapping|legend=1| 1 1 -4 0 | 0 5 54 24 }}
+: mapping generators: ~2, ~160/147
-: mapping generators: ~2, ~105/64
+[[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 }}
-[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~105/64 = 855.2452
+{{Optimal ET sequence|legend=1| 94, 111, 205 }}
-{{Optimal ET sequence|legend=1| 87, 94, 181 }}
-[[Badness]]: 0.157
+[[Badness]] (Sintel): 4.54
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 385/384, 2200/2187, 4000/3993
+Comma list: 540/539, 1331/1323, 5120/5103
-Mapping: {{mapping| 1 8 18 -20 17 | 0 -9 -22 32 -19 }}
+Mapping: {{mapping| 1 1 -4 0 1 | 0 5 54 24 21 }}
-Optimal tuning (CTE): ~2 = 1\1, ~18/11 = 855.2397
+Optimal tunings:
+* WE: ~2 = 1199.7834{{c}}, ~88/81 = 140.4635{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~88/81 = 140.4850{{c}}
-{{Optimal ET sequence|legend=1| 87, 181 }}
+{{Optimal ET sequence|legend=0| 94, 111, 205 }}
-Badness: 0.0459
+Badness (Sintel): 1.94
 === 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 325/324, 352/351, 385/384, 1575/1573
+Comma list: 352/351, 540/539, 729/728, 1331/1323
-Mapping: {{mapping| 1 8 18 -20 17 -2 | 0 -9 -22 32 -19 8 }}
+Mapping: {{mapping| 1 1 -4 0 1 3 | 0 5 54 24 21 6 }}
-Optimal tuning (CTE): ~2 = 1\1, ~18/11 = 855.2369
+Optimal tunings:
+* WE: ~2 = 1199.6639{{c}}, ~13/12 = 140.4562{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.4904{{c}}
-{{Optimal ET sequence|legend=1| 87, 181 }}
+{{Optimal ET sequence|legend=0| 94, 111, 205 }}
-Badness: 0.0263
+Badness (Sintel): 1.34
 === 17-limit ===
 Subgroup: 2.3.5.7.11.13.17
-Comma list: 325/324, 352/351, 375/374, 385/384, 595/594
+Comma list: 352/351, 442/441, 540/539, 715/714, 847/845
-Mapping: {{mapping| 1 8 18 -20 17 -2 44 | 0 -9 -22 32 -19 8 -56 }}
+Mapping: {{mapping| 1 1 -4 0 1 3 -2 | 0 5 54 24 21 6 52 }}
-Optimal tuning (CTE): ~2 = 1\1, ~18/11 = 855.2495
+Optimal tunings:
+* WE: ~2 = 1199.6699{{c}}, ~13/12 = 140.4586{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.4920{{c}}
-{{Optimal ET sequence|legend=1| 87, 94, 181 }}
+{{Optimal ET sequence|legend=0| 94, 111, 205 }}
-Badness: 0.0227
+Badness (Sintel): 1.08
 === 19-limit ===
 Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 325/324, 352/351, 375/374, 385/384, 400/399, 595/594
+Comma list: 352/351, 400/399, 442/441, 456/455, 495/494, 715/714
+Mapping: {{mapping| 1 1 -4 0 1 3 -2 -5 | 0 5 54 24 21 6 52 79 }}
+Optimal tunings:
+* WE: ~2 = 1199.6745{{c}}, ~13/12 = 140.4574{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.4908{{c}}
+{{Optimal ET sequence|legend=0| 94, 111, 205 }}
+Badness (Sintel): 1.05
-Mapping: {{mapping| 1 8 18 -20 17 -2 44 52 | 0 -9 -22 32 -19 8 -56 -67 }}
+== Jorgensen ==
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Jorgensen]].''
-Optimal tuning (CTE): ~2 = 1\1, ~18/11 = 855.2534
+Jorgensen tempers out the [[linus comma]] in addition to the aberschisma, and may be described as the {{nowrap| 70 & 140 }} temperament, with a 70th-octave period. Its ploidacot is 70-ploid acot.
-{{Optimal ET sequence|legend=1| 87, 94, 181 }}
+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>.
-Badness: 0.0196
+[[Subgroup]]: 2.3.5.7
-=== 23-limit ===
+[[Comma list]]: 5120/5103, 578509309952/576650390625
-Subgroup: 2.3.5.7.11.13.17.19.23
-Comma list: 300/299, 325/324, 352/351, 375/374, 385/384, 400/399, 484/483
+{{Mapping|legend=1| 70 111 0 34 | 0 0 1 1 }}
+: mapping generators: ~50421/50000, ~5
-Mapping: {{mapping| 1 8 18 -20 17 -2 44 52 48 | 0 -9 -22 32 -19 8 -56 -67 -61 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~50421/50000 = 17.1387{{c}}, ~5/4 = 386.8071{{c}}
+: [[error map]]: {{val| -0.288 +0.445 -0.084 +0.121 }}
+* [[CWE]]: ~50421/50000 = 17.1429{{c}}, ~5/4 = 386.6593{{c}}
+: error map: {{val| 0.000 +0.902 +0.346 +0.690 }}
-Optimal tuning (CTE): ~2 = 1\1, ~18/11 = 855.2576
+{{Optimal ET sequence|legend=1| 70, 140, 350, 490 }}
-{{Optimal ET sequence|legend=1| 87, 94, 181 }}
+[[Badness]] (Sintel): 5.40
-Badness: 0.0163
+== References ==
 [[Category:Temperament collections]]
-[[Category:Hemifamity temperaments| ]] <!-- main article -->
+[[Category:Aberschismic temperaments| ]] <!-- main article -->
-[[Category:Hamifamity| ]] <!-- key article -->
 [[Category:Rank 2]]

Aberschismic temperaments: Difference between revisions