Aberschismic temperaments: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Technical data page}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
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]].  
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-02-05 10:18:25 UTC</tt>.<br>
: The original revision id was <tt>487486574</tt>.<br>
: The revision comment was: <tt></tt><br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
<h4>Original Wikitext content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">[[toc]]
The hemifamity temperaments temper out the hemifamity comma, |10 -6 1 -1&gt; = 5120/5103. Belonging to it and considered below are buzzard, undecental, leapday, mystery, quanic and ketchup. Other hemifamity temperaments are dominant, garibaldi, hemififths, amity, misty, rodan, countercata and kwai.


=Buzzard=
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.
Commas: 1728/1715, 5120/5103


[[POTE tuning|POTE generator]]: ~320/243 = 475.636
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.  


Map: [&lt;1 0 -6 4|, &lt;0 4 21 -3|]
Temperaments discussed elsewhere are:
Wedgie: &lt;&lt;4 21 -3 24 -16 -66||
* [[Dominant (temperament)|Dominant]] (+36/35) → [[Meantone family #Dominant|Meantone family]]
EDOs: 48, 53, 111, 164d, 275d
* [[Garibaldi]] (+225/224) → [[Schismatic family #Garibaldi|Schismatic family]]
Badness: 0.0480
* [[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]]


==11-limit==
Considered below are septiquarter, kwai, ketchup, undecental, leapday, mystery, hemidromeda, countriton, artoneutral, quanic and jorgensen, in the order of increasing [[TE logflat badness]].
Commas: 176/175, 540/539, 5120/5103


[[POTE tuning|POTE generator]]: ~320/243 = 475.700
== 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.  


Map: [&lt;1 0 -6 4 -12|, &lt;0 4 21 -3 39|]
[[Subgroup]]: 2.3.5.7
EDOs: 53, 58, 111, 280cd, 391cd
Badness: 0.0345


==13-limit==
[[Comma list]]: 5120/5103, 420175/419904
Commas: 176/175, 351/350, 540/539, 676/675


[[POTE tuning|POTE generator]]: ~320/243 = 475.697
{{Mapping|legend=1| 1 -4 -28 6 | 0 7 38 -4 }}
: mapping generators: ~2, ~243/140


Map: [&lt;1 0 -6 4 -12 -7|, &lt;0 4 21 -3 39 27|]
[[Optimal tuning]]s:
EDOs: 53, 58, 111, 280cdf, 391cdf
* [[WE]]: ~2 = 1199.7212{{c}}, ~243/140 = 957.3250{{c}}
Badness: 0.0188
: [[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 }}


==17-limit==
{{Optimal ET sequence|legend=1| 94, 99, 292, 391, 881bd, 1272bcd }}
Commas: 176/175, 256/255, 351/350, 442/441, 540/539


[[POTE tuning|POTE generator]]: ~320/243 = 475.692
[[Badness]] (Sintel): 1.36


Map: [&lt;1 0 -6 4 -12 -7 14|, &lt;0 4 21 -3 39 27 -25|]
=== Semiseptiquarter ===
EDOs: 53, 58, 111, 321cdfg
Subgroup: 2.3.5.7.11
Badness: 0.0184


==19-limit==
Comma list: 5120/5103, 9801/9800, 14641/14580
Commas: 176/175, 256/255, 286/285, 324/323, 351/350, 540/539


[[POTE tuning|POTE generator]]: ~320/243 = 475.679
Mapping: {{mapping| 2 -8 -56 12 -25 | 0 7 38 -4 20 }}


Map: [&lt;1 0 -6 4 -12 -7 14 -12|, &lt;0 4 21 -3 39 27 -25 41|]
Optimal tunings:  
EDOs: 53, 58h, 111
* WE: ~99/70 = 599.8953{{c}}, ~210/121 = 957.3819{{c}}
Badness: 0.0156
* CWE: ~99/70 = 600.0000{{c}}, ~210/121 = 957.5449{{c}}


==Buteo==
{{Optimal ET sequence|legend=0| 94, 198, 292, 490 }}
Commas: 99/98, 385/384, 2200/2187


POTE generator: ~21/16 = 475.436
Badness (Sintel): 2.12


Map: [&lt;1 0 -6 4 9|, &lt;0 4 21 -3 -14|]
==== 13-limit ====
EDOs: 5, 48, 53
Subgroup: 2.3.5.7.11.13
Badness: 0.0602


===13-limit===
Comma list: 352/351, 847/845, 1716/1715, 14641/14580
Commas: 99/98, 275/273, 385/384, 572/567


POTE generator: ~21/16 = 475.464
Mapping: {{mapping| 2 -8 -56 12 -25 9 | 0 7 38 -4 20 -1 }}


Map: [&lt;1 0 -6 4 9 -7|, &lt;0 4 21 -3 -14 27|]
Optimal tunings:  
EDOs: 5, 53
* WE: ~99/70 = 599.8565{{c}}, ~210/121 = 957.3261{{c}}
Badness: 0.0390
* CWE: ~99/70 = 600.0000{{c}}, ~210/121 = 957.5508{{c}}


=Undecental=
{{Optimal ET sequence|legend=0| 94, 198, 490f }}
Commas: 5120/5103, 235298/234375


[[POTE tuning|POTE generator]]: ~3/2 = 703.039
Badness (Sintel): 1.44


Map: [&lt;1 0 61 71|, &lt;0 1 -37 -43|]
== Kwai ==
Wedgie: &lt;&lt;1 -37 -43 -61 -71 4||
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Kwai]].''
EDOs: 12, 29, 70, 99


=Leapday=
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 }}.
Comma: 10737418240/10460353203


POTE generator: ~3/2 = 704.179
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 0 -31|, &lt;0 1 21|]
[[Comma list]]: 5120/5103, 16875/16807
EDOs: 29, 46, 121, 167, 455bc, 622bc
Badness: 0.5232


==7-limit==
{{Mapping|legend=1| 1 0 -50 -40 | 0 1 33 27 }}
Commas: 686/675, 5120/5103
: mapping generators: ~2, ~3


[[POTE tuning|POTE generator]]: ~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 }}


Map: [&lt;1 0 -31 -21|, &lt;0 1 21 15|]
{{Optimal ET sequence|legend=1| 41, 111, 152, 345, 497d }}
Wedgie: &lt;&lt;1 21 15 31 21 -24||
EDOs: 29, 46, 305
Badness: 0.0961


==11-limit==
[[Badness]] (Sintel): 1.38
Commas: 121/120, 441/440, 686/675


[[POTE tuning|POTE generator]]: ~3/2 = 704.250
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 0 -31 -21 -14|, &lt;0 1 21 15 11|]
Comma list: 540/539, 1375/1372, 5120/5103
EDOs: 29, 46, 259
Badness: 0.0386


==13-limit==
Mapping: {{mapping| 1 0 -50 -40 32 | 0 1 33 27 -18 }}
Commas: 91/90, 121/120, 169/168, 441/440


[[POTE tuning|POTE generator]]: ~3/2 = 704.214
Optimal tunings:  
* WE: ~2 = 1199.6672{{c}}, ~3/2 = 702.4282{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6189{{c}}


Map: [&lt;1 0 -31 -21 -14 -9|, &lt;0 1 21 15 11 8|]
{{Optimal ET sequence|legend=0| 41, 111, 152, 497de, 649dde }}
EDOs: 29, 46, 167, 213, 380
Badness: 0.0247


==17-limit==
Badness (Sintel): 0.867
Commas: 91/90, 121/120, 136/135, 154/153, 169/168


[[POTE tuning|POTE generator]]: ~3/2 = 704.229
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


[&lt;1 0 -31 -21 -14 -9 -34|, &lt;0 1 21 15 11 8 24|]
Comma list: 352/351, 540/539, 729/728, 1375/1372
EDOs: 29g, 46, 121defg, 167defg, 213defg
Badness: 0.0179


==19-limit==
Mapping: {{mapping| 1 0 -50 -40 32 27 | 0 1 33 27 -18 -21 }}
Commas: 91/90, 121/120, 133/132, 136/135, 154/153, 169/168


[[POTE tuning|POTE generator]]: ~3/2 = 704.135
Optimal tunings:  
* WE: ~2 = 1199.4772{{c}}, ~3/2 = 702.3379{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6409{{c}}


Map: [&lt;1 0 -31 -21 -14 -9 -34 9|, &lt;0 1 21 15 11 8 24 -3|]
{{Optimal ET sequence|legend=0| 41, 111, 152f, 415dff }}
EDOs: 29g, 46, 75dfgh, 121defgh
Badness: 0.0174


===Leapling===
Badness (Sintel): 1.01
Commas: 77/76, 91/90, 121/120, 136/135, 153/152, 169/168


[[POTE tuning|POTE generator]]: ~3/2 = 704.123
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


Map: [&lt;1 0 -31 -21 -14 -9 -34 -37|, &lt;0 1 21 15 11 8 24 26|]
Comma list: 256/255, 352/351, 540/539, 715/714, 1089/1088
EDOs: 29g, 46h, 75dfg
Badness: 0.0191


=Mystery=
Mapping: {{mapping| 1 0 -50 -40 32 27 58 | 0 1 33 27 -18 -21 -34 }}
Commas: 5120/5103, 50421/50000


[[POTE tuning|POTE generator]]: ~5/4 = 388.646
Optimal tunings:  
* WE: ~2 = 1199.3537{{c}}, ~3/2 = 702.2850{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6589{{c}}


Map: [&lt;29 46 0 14|, &lt;0 0 1 1|]
{{Optimal ET sequence|legend=0| 41, 70, 111, 152fg, 263dfg }}
Wedgie: &lt;&lt;0 29 29 46 46 -14||
EDOs: 29, 58, 87, 145
Badness: 0.1037


==11-limit==
Badness (Sintel): 1.12
Commas: 441/440, 896/891, 3388/3375


[[POTE tuning|POTE generator]]: ~5/4 = 388.460
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


Map: [&lt;29 46 0 14 33|, &lt;0 0 1 1 1|]
Comma list: 256/255, 352/351, 400/399, 456/455, 715/714, 847/845
EDOs: 29, 58, 87, 145
Badness: 0.0343


==13-limit==
Mapping: {{mapping| 1 0 -50 -40 32 27 58 -56 | 0 1 33 27 -18 -21 -34 38 }}
Commas: 196/195, 352/351, 364/363, 676/675


[[POTE tuning|POTE generator]]: ~5/4 = 388.354
Optimal tunings:  
* WE: ~2 = 1199.3401{{c}}, ~3/2 = 702.2705{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6548{{c}}


Map: [&lt;29 46 0 14 33 40|, &lt;0 0 1 1 1 1|]
{{Optimal ET sequence|legend=0| 41, 70h, 111, 152fg, 263dfgh }}
EDOs: 29, 58, 87, 145, 232, 377
Badness: 0.0186


=Quanic=
Badness (Sintel): 1.03
Commas: 5120/5103, 5832000/5764801


POTE generator: ~160/147 = 140.493
==== Hemikwai ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 1 -4 0|, &lt;0 5 54 24|]
Comma list: 540/539, 676/675, 1375/1372, 5120/5103
EDOs: 94, 111, 205
Badness: 0.1795


==11-limit==
Mapping: {{mapping| 1 0 -50 -40 32 -51 | 0 2 66 54 -36 69 }}
Commas: 540/539, 1331/1323, 5120/5103
: mapping generators: ~2, ~26/15


POTE generator: ~88/81 = 140.489
Optimal tunings:  
* WE: ~2 = 1199.6968{{c}}, ~26/15 = 951.0740{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.3123{{c}}


Map: [&lt;1 1 -4 0 1|, &lt;0 5 54 24 21|]
{{Optimal ET sequence|legend=0| 82, 111, 193, 304d }}
EDOs: 94, 111, 205
Badness: 0.0587


==13-limit==
Badness (Sintel): 1.82
Commas: 352/351, 540/539, 729/728, 1331/1323


POTE generator: ~13/12 = 140.496
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


Map: [&lt;1 1 -4 0 1 3|, &lt;0 5 54 24 21 6|]
Comma list: 442/441, 540/539, 676/675, 715/714, 5120/5103
EDOs: 94, 111, 205
Badness: 0.0325


==17-limit==
Mapping: {{mapping| 1 0 -50 -40 32 -51 -30 | 0 2 66 54 -36 69 43 }}
Commas: 352/351, 442/441, 540/539, 715/714, 847/845


POTE generator: ~13/12 = 140.497
Optimal tunings:  
* WE: ~2 = 1199.6861{{c}}, ~26/15 = 951.0654{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.3120{{c}}


Map: [&lt;1 1 -4 0 1 3 -2|, &lt;0 5 54 24 21 6 52|]
{{Optimal ET sequence|legend=0| 82, 111, 193, 304d }}
EDOs: 94, 111, 205
Badness: 0.0211


==19-limit==
Badness (Sintel): 1.31
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: 400/399, 442/441, 540/539, 676/675, 715/714, 1445/1444
EDOs: 94, 111, 205
Badness: 0.0173


=Supers=
Mapping: {{mapping| 1 0 -50 -40 32 -51 -30 -56 | 0 2 66 54 -36 69 43 76 }}
Commas: 5120/5103, 118098/117649


POTE generator: ~9/7 = 434.218
Optimal tunings:  
* WE: ~2 = 1199.6718{{c}}, ~26/15 = 951.0526{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.3103{{c}}


Map: [&lt;2 1 -12 2|, &lt;0 3 23 5|]
{{Optimal ET sequence|legend=0| 82, 111, 193, 304dh }}
Wedgie: &lt;&lt;6 46 10 59 -1 -106||
EDOs: 58, 94, 152
Badness: 92.748


==11-limit==
Badness (Sintel): 1.16
Commas: 540/539, 4000/3993, 5120/5103


POTE generator: ~9/7 = 434.217
== 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.  


Map: [&lt;2 1 -12 2 -9|, &lt;0 3 23 5 22|]
[[Subgroup]]: 2.3.5.7
EDOs: 58, 94, 152
Badness: 0.0282


==13-limit==
[[Comma list]]: 5120/5103, 1071875/1062882
Commas: 352/351, 540/539, 729/728, 1575/1573


POTE generator: ~9/7 = 434.221
{{Mapping|legend=1| 2 3 4 6 | 0 4 15 -9 }}
: mapping generators: ~1225/864, ~64/63


Map: [&lt;2 1 -12 2 -9 -2|, &lt;0 3 23 5 22 13|]
[[Optimal tuning]]s:
EDOs: 58, 94, 152f
* [[WE]]: ~1225/864 = 599.9685{{c}}, ~64/63 = 25.7181{{c}}
Badness: 0.0216
: [[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 }}


=Alphaquarter=
{{Optimal ET sequence|legend=1| 46, 94, 140 }}
Commas: 5120/5103, 29360128/29296875


POTE generator: ~16128/15625 = 55.243
[[Badness]] (Sintel): 2.14


Map: [&lt;1 2 2 0|, &lt;0 -9 7 61|]
=== 11-limit ===
Wedgie: &lt;&lt;9 -7 -61 -32 -122 -122||
Subgroup: 2.3.5.7.11
EDOs: 87, 152, 239, 391
Badness: 0.1166


==11-limit==
Comma list: 385/384, 1331/1323, 2200/2187
Commas: 3025/3024, 4000/3993, 5120/5103


POTE generator: ~33/32 = 55.243
Mapping: {{mapping| 2 3 4 6 7 | 0 4 15 -9 -2 }}


EDOs: 87, 152, 239, 391
Optimal tunings:  
Badness: 0.0296
* WE: ~99/70 = 600.0678{{c}}, ~64/63 = 25.6963{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~64/63 = 25.6956{{c}}


=Septiquarter=
{{Optimal ET sequence|legend=0| 46, 94, 140 }}
Commas: 5120/5103, 420175/419904


POTE generator: ~147/128 = 242.453
Badness (Sintel): 1.31


Map: [&lt;1 3 10 2|, &lt;0 -7 -38 4|]
=== 13-limit ===
Wedgie: &lt;&lt;7 38 -4 44 -26 -116||
Subgroup: 2.3.5.7.11.13
EDOs: 94, 99, 292, 391, 881bd, 1272bcd
Badness: 0.0538


=Tricot=
Comma list: 325/324, 352/351, 385/384, 1331/1323
Commas: 2430/2401, 5120/5103


POTE generator: ~81/56 = 634.026
Mapping: {{mapping| 2 3 4 6 7 8 | 0 4 15 -9 -2 -14 }}


Map: [[&lt; 3 16 8|, &lt;0 -3 -29 -11|]
Optimal tunings:  
Wedgie: &lt;&lt;3 29 11 39 9 -56||
* WE: ~99/70 = 600.0612{{c}}, ~66/65 = 25.7000{{c}}
EDOs: 53, 229d, 282d
* CWE: ~99/70 = 600.0000{{c}}, ~66/65 = 25.6978{{c}}
Badness: 0.1001


==11-limit==
{{Optimal ET sequence|legend=0| 46, 94, 140 }}
Commas: 99/98, 121/120, 5120/5103


POTE generator: ~63/44 = 634.027
Badness (Sintel): 1.03


Map: [[&lt; 3 16 8 11|, &lt;0 -3 -29 -11 -16|]
=== 17-limit ===
EDOs: 53, 176de, 229
Subgroup: 2.3.5.7.11.13.17
Badness: 0.0561


==13-limit==
Comma list: 289/288, 325/324, 352/351, 385/384, 442/441
Commas: 99/98, 121/120, 169/168, 352/351


POTE generator: ~13/9 = 634.012
Mapping: {{mapping| 2 3 4 6 7 8 8 | 0 4 15 -9 -2 -14 4 }}


Map: [[&lt; 3 16 8 11 7|, &lt;0 -3 -29 -11 -16 -7|]
Optimal tunings:  
EDOs: 53
* WE: ~17/12 = 600.0896{{c}}, ~66/65 = 25.7048{{c}}
Badness: 0.0321
* CWE: ~17/12 = 600.0000{{c}}, ~66/65 = 25.7017{{c}}


=Ketchup=
{{Optimal ET sequence|legend=0| 46, 94, 140 }}
Commas: 5120/5103, 1071875/1062882


POTE generator: ~64/63 = ~81/80 = 25.719
Badness (Sintel): 0.845


Map: [&lt;2 3 4 6|, &lt;0 4 15 -9|]
=== 2.3.5.7.11.13.17.23 subgroup ===
EDOS: 46, 94, 140
Subgroup: 2.3.5.7.11.13.17.23
Badness: 0.0845


==11-limit==
Comma list: 253/252, 289/288, 325/324, 352/351, 385/384, 391/390
Commas: 385/384, 1331/1323, 2200/2187


POTE generator: ~55/54 = ~64/63 = ~81/80 = 25.693
Mapping: {{mapping| 2 3 4 6 7 8 8 9 | 0 4 15 -9 -2 -14 4 1 }}


Map: [&lt;2 3 4 6 7|, &lt;0 4 15 -9 -2|]
Optimal tunings:  
EDOs: 46, 94, 140
* WE: ~17/12 = 600.1139{{c}}, ~66/65 = 25.7053{{c}}
Badness: 0.0396
* CWE: ~17/12 = 600.0000{{c}}, ~66/65 = 25.7013{{c}}


==13-limit==
{{Optimal ET sequence|legend=0| 46, 94, 140 }}
Commas: 325/324, 352/351, 847/845, 1331/1323


POTE generator: ~55/54 = ~64/63 = ~81/80 = 25.697
Badness (Sintel): 0.772


Map: [&lt;2 3 4 6 7 8|, &lt;0 4 15 -9 -2 -14|]
== Undecental ==
EDOs: 46, 94, 140
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> }}.  
Badness: 0.0248


==17-limit==
[[Subgroup]]: 2.3.5.7
Commas: 289/288, 325/324, 352/351, 385/384, 561/560


POTE generator: ~55/54 = ~64/63 = ~81/80 = 25.701
[[Comma list]]: 5120/5103, 235298/234375


Map: [&lt;2 3 4 6 7 8 8|, &lt;0 4 15 -9 -2 -14 4|]
{{Mapping|legend=1| 1 0 61 71 | 0 1 -37 -43 }}
EDOs: 46, 94, 140
: mapping generators: ~2, ~3
Badness: 0.0166


==19-limit==
[[Optimal tuning]]s:
Commas: 190/189, 209/208, 289/288, 352/351, 385/384, 561/560
* [[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: ~55/54 = ~64/63 = ~81/80 = 25.660
{{Optimal ET sequence|legend=1| 29, 70, 99, 722bc, 821bc, 920bc, 1019bc }}


Map: [&lt;2 3 4 6 7 8 8 9|, &lt;0 4 15 -9 -2 -14 4 -12|]
[[Badness]] (Sintel): 2.39
EDOs: 46, 94, 140h, 234eh
Badness: 0.0182


==23-limit==
== Leapday ==
Commas: 190/189, 209/208, 253/252, 289/288, 323/322, 352/351, 385/384
{{Main| Leapday }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Leapday]].''


POTE generator: ~55/54 = ~64/63 = ~81/80 = 25.661
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.  


Map: [&lt;2 3 4 6 7 8 8 9 9|, &lt;0 4 15 -9 -2 -14 4 -12 1|]
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.
EDOs: 46, 94, 140h, 234ehi
 
Badness: 0.0140
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.
</pre></div>
 
<h4>Original HTML content:</h4>
[[Subgroup]]: 2.3.5.7
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Hemifamity temperaments&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:76:&amp;lt;img id=&amp;quot;wikitext@@toc@@normal&amp;quot; class=&amp;quot;WikiMedia WikiMediaToc&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/normal?w=225&amp;amp;h=100&amp;quot;/&amp;gt; --&gt;&lt;div id="toc"&gt;&lt;h1 class="nopad"&gt;Table of Contents&lt;/h1&gt;&lt;!-- ws:end:WikiTextTocRule:76 --&gt;&lt;!-- ws:start:WikiTextTocRule:77: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Buzzard"&gt;Buzzard&lt;/a&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:77 --&gt;&lt;!-- ws:start:WikiTextTocRule:78: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Buzzard-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
[[Comma list]]: 686/675, 5120/5103
&lt;!-- ws:end:WikiTextTocRule:78 --&gt;&lt;!-- ws:start:WikiTextTocRule:79: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Buzzard-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:79 --&gt;&lt;!-- ws:start:WikiTextTocRule:80: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Buzzard-17-limit"&gt;17-limit&lt;/a&gt;&lt;/div&gt;
{{Mapping|legend=1| 1 0 -31 -21 | 0 1 21 15 }}
&lt;!-- ws:end:WikiTextTocRule:80 --&gt;&lt;!-- ws:start:WikiTextTocRule:81: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Buzzard-19-limit"&gt;19-limit&lt;/a&gt;&lt;/div&gt;
: mapping generators: ~2, ~3
&lt;!-- ws:end:WikiTextTocRule:81 --&gt;&lt;!-- ws:start:WikiTextTocRule:82: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Buzzard-Buteo"&gt;Buteo&lt;/a&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:82 --&gt;&lt;!-- ws:start:WikiTextTocRule:83: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Buzzard-Buteo-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
[[Optimal tuning]]s:  
&lt;!-- ws:end:WikiTextTocRule:83 --&gt;&lt;!-- ws:start:WikiTextTocRule:84: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Undecental"&gt;Undecental&lt;/a&gt;&lt;/div&gt;
* [[WE]]: ~2 = 1199.7167{{c}}, ~3/2 = 704.0971{{c}}
&lt;!-- ws:end:WikiTextTocRule:84 --&gt;&lt;!-- ws:start:WikiTextTocRule:85: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Leapday"&gt;Leapday&lt;/a&gt;&lt;/div&gt;
: [[error map]]: {{val| -0.283 +1.859 +2.559 -5.669 }}
&lt;!-- ws:end:WikiTextTocRule:85 --&gt;&lt;!-- ws:start:WikiTextTocRule:86: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Leapday-7-limit"&gt;7-limit&lt;/a&gt;&lt;/div&gt;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.2504{{c}}
&lt;!-- ws:end:WikiTextTocRule:86 --&gt;&lt;!-- ws:start:WikiTextTocRule:87: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Leapday-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
: error map: {{val| 0.000 +2.295 +2.945 -5.070 }}
&lt;!-- ws:end:WikiTextTocRule:87 --&gt;&lt;!-- ws:start:WikiTextTocRule:88: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Leapday-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:88 --&gt;&lt;!-- ws:start:WikiTextTocRule:89: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Leapday-17-limit"&gt;17-limit&lt;/a&gt;&lt;/div&gt;
{{Optimal ET sequence|legend=1| 17c, 29, 46 }}
&lt;!-- ws:end:WikiTextTocRule:89 --&gt;&lt;!-- ws:start:WikiTextTocRule:90: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Leapday-19-limit"&gt;19-limit&lt;/a&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:90 --&gt;&lt;!-- ws:start:WikiTextTocRule:91: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Leapday-19-limit-Leapling"&gt;Leapling&lt;/a&gt;&lt;/div&gt;
[[Badness]] (Sintel): 2.43
&lt;!-- ws:end:WikiTextTocRule:91 --&gt;&lt;!-- ws:start:WikiTextTocRule:92: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Mystery"&gt;Mystery&lt;/a&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:92 --&gt;&lt;!-- ws:start:WikiTextTocRule:93: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Mystery-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
=== 11-limit ===
&lt;!-- ws:end:WikiTextTocRule:93 --&gt;&lt;!-- ws:start:WikiTextTocRule:94: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Mystery-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
Subgroup: 2.3.5.7.11
&lt;!-- ws:end:WikiTextTocRule:94 --&gt;&lt;!-- ws:start:WikiTextTocRule:95: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Quanic"&gt;Quanic&lt;/a&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:95 --&gt;&lt;!-- ws:start:WikiTextTocRule:96: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Quanic-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
Comma list: 121/120, 441/440, 686/675
&lt;!-- ws:end:WikiTextTocRule:96 --&gt;&lt;!-- ws:start:WikiTextTocRule:97: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Quanic-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:97 --&gt;&lt;!-- ws:start:WikiTextTocRule:98: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Quanic-17-limit"&gt;17-limit&lt;/a&gt;&lt;/div&gt;
Mapping: {{mapping| 1 0 -31 -21 -14 | 0 1 21 15 11 }}
&lt;!-- ws:end:WikiTextTocRule:98 --&gt;&lt;!-- ws:start:WikiTextTocRule:99: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Quanic-19-limit"&gt;19-limit&lt;/a&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:99 --&gt;&lt;!-- ws:start:WikiTextTocRule:100: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Supers"&gt;Supers&lt;/a&gt;&lt;/div&gt;
Optimal tunings:  
&lt;!-- ws:end:WikiTextTocRule:100 --&gt;&lt;!-- ws:start:WikiTextTocRule:101: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Supers-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
* WE: ~2 = 1200.0731{{c}}, ~3/2 = 704.2933{{c}}
&lt;!-- ws:end:WikiTextTocRule:101 --&gt;&lt;!-- ws:start:WikiTextTocRule:102: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Supers-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2538{{c}}
&lt;!-- ws:end:WikiTextTocRule:102 --&gt;&lt;!-- ws:start:WikiTextTocRule:103: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Alphaquarter"&gt;Alphaquarter&lt;/a&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:103 --&gt;&lt;!-- ws:start:WikiTextTocRule:104: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Alphaquarter-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
{{Optimal ET sequence|legend=0| 17c, 29, 46 }}
&lt;!-- ws:end:WikiTextTocRule:104 --&gt;&lt;!-- ws:start:WikiTextTocRule:105: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Septiquarter"&gt;Septiquarter&lt;/a&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:105 --&gt;&lt;!-- ws:start:WikiTextTocRule:106: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Tricot"&gt;Tricot&lt;/a&gt;&lt;/div&gt;
Badness (Sintel): 1.28
&lt;!-- ws:end:WikiTextTocRule:106 --&gt;&lt;!-- ws:start:WikiTextTocRule:107: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Tricot-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:107 --&gt;&lt;!-- ws:start:WikiTextTocRule:108: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Tricot-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
=== 13-limit ===
&lt;!-- ws:end:WikiTextTocRule:108 --&gt;&lt;!-- ws:start:WikiTextTocRule:109: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Ketchup"&gt;Ketchup&lt;/a&gt;&lt;/div&gt;
Subgroup: 2.3.5.7.11.13
&lt;!-- ws:end:WikiTextTocRule:109 --&gt;&lt;!-- ws:start:WikiTextTocRule:110: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Ketchup-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:110 --&gt;&lt;!-- ws:start:WikiTextTocRule:111: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Ketchup-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
Comma list: 91/90, 121/120, 169/168, 352/351
&lt;!-- ws:end:WikiTextTocRule:111 --&gt;&lt;!-- ws:start:WikiTextTocRule:112: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Ketchup-17-limit"&gt;17-limit&lt;/a&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:112 --&gt;&lt;!-- ws:start:WikiTextTocRule:113: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Ketchup-19-limit"&gt;19-limit&lt;/a&gt;&lt;/div&gt;
Mapping: {{mapping| 1 0 -31 -21 -14 -9 | 0 1 21 15 11 8 }}
&lt;!-- ws:end:WikiTextTocRule:113 --&gt;&lt;!-- ws:start:WikiTextTocRule:114: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Ketchup-23-limit"&gt;23-limit&lt;/a&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:114 --&gt;&lt;!-- ws:start:WikiTextTocRule:115: --&gt;&lt;/div&gt;
Optimal tunings:  
&lt;!-- ws:end:WikiTextTocRule:115 --&gt;The hemifamity temperaments temper out the hemifamity comma, |10 -6 1 -1&amp;gt; = 5120/5103. Belonging to it and considered below are buzzard, undecental, leapday, mystery, quanic and ketchup. Other hemifamity temperaments are dominant, garibaldi, hemififths, amity, misty, rodan, countercata and kwai.&lt;br /&gt;
* WE: ~2 = 1200.4758{{c}}, ~3/2 = 704.4930{{c}}
&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2346{{c}}
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Buzzard"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Buzzard&lt;/h1&gt;
 
Commas: 1728/1715, 5120/5103&lt;br /&gt;
{{Optimal ET sequence|legend=0| 17c, 29, 46, 121def }}
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~320/243 = 475.636&lt;br /&gt;
Badness (Sintel): 1.02
&lt;br /&gt;
 
Map: [&amp;lt;1 0 -6 4|, &amp;lt;0 4 21 -3|]&lt;br /&gt;
=== 17-limit ===
Wedgie: &amp;lt;&amp;lt;4 21 -3 24 -16 -66||&lt;br /&gt;
Subgroup: 2.3.5.7.11.13.17
EDOs: 48, 53, 111, 164d, 275d&lt;br /&gt;
 
Badness: 0.0480&lt;br /&gt;
Comma list: 91/90, 121/120, 136/135, 154/153, 169/168
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="Buzzard-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;11-limit&lt;/h2&gt;
Mapping: {{mapping| 1 0 -31 -21 -14 -9 -34 | 0 1 21 15 11 8 24 }}
Commas: 176/175, 540/539, 5120/5103&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~320/243 = 475.700&lt;br /&gt;
* WE: ~2 = 1200.4818{{c}}, ~3/2 = 704.5121{{c}}
&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2507{{c}}
Map: [&amp;lt;1 0 -6 4 -12|, &amp;lt;0 4 21 -3 39|]&lt;br /&gt;
 
EDOs: 53, 58, 111, 280cd, 391cd&lt;br /&gt;
{{Optimal ET sequence|legend=0| 17cg, 29g, 46, 121defg }}
Badness: 0.0345&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 0.910
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Buzzard-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;13-limit&lt;/h2&gt;
 
Commas: 176/175, 351/350, 540/539, 676/675&lt;br /&gt;
=== 2.3.5.7.11.13.17.23 subgroup ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11.13.17.23
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~320/243 = 475.697&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 91/90, 121/120, 136/135, 154/153, 161/160, 169/168
Map: [&amp;lt;1 0 -6 4 -12 -7|, &amp;lt;0 4 21 -3 39 27|]&lt;br /&gt;
 
EDOs: 53, 58, 111, 280cdf, 391cdf&lt;br /&gt;
Mapping: {{mapping| 1 0 -31 -21 -14 -9 -34 -5 | 0 1 21 15 11 8 24 6 }}
Badness: 0.0188&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:  
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="Buzzard-17-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;17-limit&lt;/h2&gt;
* WE: ~2 = 1200.5169{{c}}, ~3/2 = 704.5279{{c}}
Commas: 176/175, 256/255, 351/350, 442/441, 540/539&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2450{{c}}
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~320/243 = 475.692&lt;br /&gt;
{{Optimal ET sequence|legend=0| 17cg, 29g, 46, 121defg }}
&lt;br /&gt;
 
Map: [&amp;lt;1 0 -6 4 -12 -7 14|, &amp;lt;0 4 21 -3 39 27 -25|]&lt;br /&gt;
Badness (Sintel): 0.872
EDOs: 53, 58, 111, 321cdfg&lt;br /&gt;
 
Badness: 0.0184&lt;br /&gt;
== Mystery ==
&lt;br /&gt;
{{Main| Mystery }}
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="Buzzard-19-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;19-limit&lt;/h2&gt;
: ''For the 5-limit version, see [[29th-octave temperaments #Mystery]].''
Commas: 176/175, 256/255, 286/285, 324/323, 351/350, 540/539&lt;br /&gt;
 
&lt;br /&gt;
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.
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~320/243 = 475.679&lt;br /&gt;
 
&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
Map: [&amp;lt;1 0 -6 4 -12 -7 14 -12|, &amp;lt;0 4 21 -3 39 27 -25 41|]&lt;br /&gt;
 
EDOs: 53, 58h, 111&lt;br /&gt;
[[Comma list]]: 5120/5103, 50421/50000
Badness: 0.0156&lt;br /&gt;
 
&lt;br /&gt;
{{Mapping|legend=1| 29 46 0 14 | 0 0 1 1 }}
&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="Buzzard-Buteo"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Buteo&lt;/h2&gt;
: mapping generators: ~50/49, ~5
Commas: 99/98, 385/384, 2200/2187&lt;br /&gt;
 
&lt;br /&gt;
[[Optimal tuning]]s:  
POTE generator: ~21/16 = 475.436&lt;br /&gt;
* [[WE]]: ~50/49 = 41.3652{{c}}, ~5/4 = 388.5128{{c}}
&lt;br /&gt;
: [[error map]]: {{val| -0.410 +0.842 +1.378 -2.022 }}
Map: [&amp;lt;1 0 -6 4 9|, &amp;lt;0 4 21 -3 -14|]&lt;br /&gt;
* [[CWE]]: ~50/49 = 41.3793{{c}}, ~5/4 = 388.3030{{c}}
EDOs: 5, 48, 53&lt;br /&gt;
: error map: {{val| 0.000 +1.493 +1.989 -1.213 }}
Badness: 0.0602&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=1| 29, 58, 87, 145 }}
&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc6"&gt;&lt;a name="Buzzard-Buteo-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;13-limit&lt;/h3&gt;
 
Commas: 99/98, 275/273, 385/384, 572/567&lt;br /&gt;
[[Badness]] (Sintel): 2.63
&lt;br /&gt;
 
POTE generator: ~21/16 = 475.464&lt;br /&gt;
=== 11-limit ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11
Map: [&amp;lt;1 0 -6 4 9 -7|, &amp;lt;0 4 21 -3 -14 27|]&lt;br /&gt;
 
EDOs: 5, 53&lt;br /&gt;
Comma list: 441/440, 896/891, 3388/3375
Badness: 0.0390&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 29 46 0 14 33 | 0 0 1 1 1 }}
&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc7"&gt;&lt;a name="Undecental"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;Undecental&lt;/h1&gt;
 
Commas: 5120/5103, 235298/234375&lt;br /&gt;
Optimal tunings:  
&lt;br /&gt;
* WE: ~45/44 = 41.3637{{c}}, ~5/4 = 388.3136{{c}}
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~3/2 = 703.039&lt;br /&gt;
* CWE: ~45/44 = 41.3793{{c}}, ~5/4 = 388.0598{{c}}
&lt;br /&gt;
 
Map: [&amp;lt;1 0 61 71|, &amp;lt;0 1 -37 -43|]&lt;br /&gt;
{{Optimal ET sequence|legend=0| 29, 58, 87, 145 }}
Wedgie: &amp;lt;&amp;lt;1 -37 -43 -61 -71 4||&lt;br /&gt;
 
EDOs: 12, 29, 70, 99&lt;br /&gt;
Badness (Sintel): 1.13
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc8"&gt;&lt;a name="Leapday"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;Leapday&lt;/h1&gt;
=== 13-limit ===
Comma: 10737418240/10460353203&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
&lt;br /&gt;
 
POTE generator: ~3/2 = 704.179&lt;br /&gt;
Comma list: 196/195, 352/351, 364/363, 676/675
&lt;br /&gt;
 
Map: [&amp;lt;1 0 -31|, &amp;lt;0 1 21|]&lt;br /&gt;
Mapping: {{mapping| 29 46 0 14 33 40 | 0 0 1 1 1 1 }}
EDOs: 29, 46, 121, 167, 455bc, 622bc&lt;br /&gt;
 
Badness: 0.5232&lt;br /&gt;
Optimal tunings:  
&lt;br /&gt;
* WE: ~45/44 = 41.3623{{c}}, ~5/4 = 388.1942{{c}}
&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc9"&gt;&lt;a name="Leapday-7-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;7-limit&lt;/h2&gt;
* CWE: ~40/39 = 41.3793{{c}}, ~5/4 = 387.9017{{c}}
Commas: 686/675, 5120/5103&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 29, 58, 87, 145, 232 }}
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~3/2 = 704.263&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 0.768
Map: [&amp;lt;1 0 -31 -21|, &amp;lt;0 1 21 15|]&lt;br /&gt;
 
Wedgie: &amp;lt;&amp;lt;1 21 15 31 21 -24||&lt;br /&gt;
== Hemidromeda ==
EDOs: 29, 46, 305&lt;br /&gt;
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.
Badness: 0.0961&lt;br /&gt;
 
&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc10"&gt;&lt;a name="Leapday-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;11-limit&lt;/h2&gt;
 
Commas: 121/120, 441/440, 686/675&lt;br /&gt;
[[Comma list]]: 5120/5103, 52734375/52706752
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~3/2 = 704.250&lt;br /&gt;
{{Mapping|legend=1| 1 0 38 48 | 0 2 -45 -57 }}
&lt;br /&gt;
: mapping generator: ~2, ~12500/7203
Map: [&amp;lt;1 0 -31 -21 -14|, &amp;lt;0 1 21 15 11|]&lt;br /&gt;
 
EDOs: 29, 46, 259&lt;br /&gt;
[[Optimal tuning]]s:  
Badness: 0.0386&lt;br /&gt;
* [[WE]]: ~2 = 1199.7236{{c}}, ~12500/7203 = 951.1864{{c}}
&lt;br /&gt;
: [[error map]]: {{val| -0.276 +0.418 -0.205 +0.282 }}
&lt;!-- ws:start:WikiTextHeadingRule:22:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc11"&gt;&lt;a name="Leapday-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&gt;13-limit&lt;/h2&gt;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~12500/7203 = 951.4098{{c}}
Commas: 91/90, 121/120, 169/168, 441/440&lt;br /&gt;
: error map: {{val| 0.000 +0.865 +0.243 +0.813 }}
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~3/2 = 704.214&lt;br /&gt;
{{Optimal ET sequence|legend=1| 29, 82cd, 111, 140, 251, 391, 1424bbcdd }}
&lt;br /&gt;
 
Map: [&amp;lt;1 0 -31 -21 -14 -9|, &amp;lt;0 1 21 15 11 8|]&lt;br /&gt;
[[Badness]] (Sintel): 2.93
EDOs: 29, 46, 167, 213, 380&lt;br /&gt;
 
Badness: 0.0247&lt;br /&gt;
=== 11-limit ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11
&lt;!-- ws:start:WikiTextHeadingRule:24:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc12"&gt;&lt;a name="Leapday-17-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:24 --&gt;17-limit&lt;/h2&gt;
 
Commas: 91/90, 121/120, 136/135, 154/153, 169/168&lt;br /&gt;
Comma list: 1331/1323, 1375/1372, 5120/5103
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~3/2 = 704.229&lt;br /&gt;
Mapping: {{mapping| 1 0 38 48 32 | 0 2 -45 -57 -36 }}
&lt;br /&gt;
 
[&amp;lt;1 0 -31 -21 -14 -9 -34|, &amp;lt;0 1 21 15 11 8 24|]&lt;br /&gt;
Optimal tunings:  
EDOs: 29g, 46, 121defg, 167defg, 213defg&lt;br /&gt;
* WE: ~2 = 1199.8767{{c}}, ~400/231 = 951.3065{{c}}
Badness: 0.0179&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~400/231 = 951.4063{{c}}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:26:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc13"&gt;&lt;a name="Leapday-19-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:26 --&gt;19-limit&lt;/h2&gt;
{{Optimal ET sequence|legend=0| 29, 82cd, 111, 140, 251, 391e }}
Commas: 91/90, 121/120, 133/132, 136/135, 154/153, 169/168&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 2.01
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~3/2 = 704.135&lt;br /&gt;
 
&lt;br /&gt;
=== 13-limit ===
Map: [&amp;lt;1 0 -31 -21 -14 -9 -34 9|, &amp;lt;0 1 21 15 11 8 24 -3|]&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
EDOs: 29g, 46, 75dfgh, 121defgh&lt;br /&gt;
 
Badness: 0.0174&lt;br /&gt;
Comma list: 352/351, 676/675, 847/845, 1331/1323
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:28:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc14"&gt;&lt;a name="Leapday-19-limit-Leapling"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:28 --&gt;Leapling&lt;/h3&gt;
Mapping: {{mapping| 1 0 38 48 32 37 | 0 2 -45 -57 -36 -42 }}
Commas: 77/76, 91/90, 121/120, 136/135, 153/152, 169/168&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:  
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~3/2 = 704.123&lt;br /&gt;
* WE: ~2 = 1199.8753{{c}}, ~26/15 = 951.3054{{c}}
&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.4064{{c}}
Map: [&amp;lt;1 0 -31 -21 -14 -9 -34 -37|, &amp;lt;0 1 21 15 11 8 24 26|]&lt;br /&gt;
 
EDOs: 29g, 46h, 75dfg&lt;br /&gt;
{{Optimal ET sequence|legend=0| 29, 82cdf, 111, 140, 251, 391e }}
Badness: 0.0191&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 1.18
&lt;!-- ws:start:WikiTextHeadingRule:30:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc15"&gt;&lt;a name="Mystery"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:30 --&gt;Mystery&lt;/h1&gt;
 
Commas: 5120/5103, 50421/50000&lt;br /&gt;
=== 17-limit ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11.13.17
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~5/4 = 388.646&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 352/351, 442/441, 561/560, 676/675, 715/714
Map: [&amp;lt;29 46 0 14|, &amp;lt;0 0 1 1|]&lt;br /&gt;
 
Wedgie: &amp;lt;&amp;lt;0 29 29 46 46 -14||&lt;br /&gt;
Mapping: {{mapping| 1 0 38 48 32 37 58 | 0 2 -45 -57 -36 -42 -68 }}
EDOs: 29, 58, 87, 145&lt;br /&gt;
 
Badness: 0.1037&lt;br /&gt;
Optimal tunings:  
&lt;br /&gt;
* WE: ~2 = 1199.8770{{c}}, ~26/15 = 951.3039{{c}}
&lt;!-- ws:start:WikiTextHeadingRule:32:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc16"&gt;&lt;a name="Mystery-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:32 --&gt;11-limit&lt;/h2&gt;
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.4035{{c}}
Commas: 441/440, 896/891, 3388/3375&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 29g, 82cdfg, 111, 140, 251, 391e }}
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~5/4 = 388.460&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 0.971
Map: [&amp;lt;29 46 0 14 33|, &amp;lt;0 0 1 1 1|]&lt;br /&gt;
 
EDOs: 29, 58, 87, 145&lt;br /&gt;
=== 19-limit ===
Badness: 0.0343&lt;br /&gt;
Subgroup: 2.3.5.7.11.13.17.19
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:34:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc17"&gt;&lt;a name="Mystery-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:34 --&gt;13-limit&lt;/h2&gt;
Comma list: 286/285, 352/351, 363/361, 442/441, 476/475, 561/560
Commas: 196/195, 352/351, 364/363, 676/675&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 1 0 38 48 32 37 58 32 | 0 2 -45 -57 -36 -42 -68 -35 }}
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~5/4 = 388.354&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:  
Map: [&amp;lt;29 46 0 14 33 40|, &amp;lt;0 0 1 1 1 1|]&lt;br /&gt;
* WE: ~2 = 1199.7534{{c}}, ~26/15 = 951.2024{{c}}
EDOs: 29, 58, 87, 145, 232, 377&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.4020{{c}}
Badness: 0.0186&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 29g, 82cdfgh, 111, 140 }}
&lt;!-- ws:start:WikiTextHeadingRule:36:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc18"&gt;&lt;a name="Quanic"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:36 --&gt;Quanic&lt;/h1&gt;
 
Commas: 5120/5103, 5832000/5764801&lt;br /&gt;
Badness (Sintel): 1.01
&lt;br /&gt;
 
POTE generator: ~160/147 = 140.493&lt;br /&gt;
=== 23-limit ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11.13.17.19.23
Map: [&amp;lt;1 1 -4 0|, &amp;lt;0 5 54 24|]&lt;br /&gt;
 
EDOs: 94, 111, 205&lt;br /&gt;
Comma list: 253/252, 286/285, 352/351, 363/361, 391/390, 442/441, 460/459
Badness: 0.1795&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 1 0 38 48 32 37 58 32 18 | 0 2 -45 -57 -36 -42 -68 -35 -17 }}
&lt;!-- ws:start:WikiTextHeadingRule:38:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc19"&gt;&lt;a name="Quanic-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:38 --&gt;11-limit&lt;/h2&gt;
 
Commas: 540/539, 1331/1323, 5120/5103&lt;br /&gt;
Optimal tunings:  
&lt;br /&gt;
* WE: ~2 = 1199.9128{{c}}, ~26/15 = 951.3371{{c}}
POTE generator: ~88/81 = 140.489&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.4076{{c}}
&lt;br /&gt;
 
Map: [&amp;lt;1 1 -4 0 1|, &amp;lt;0 5 54 24 21|]&lt;br /&gt;
{{Optimal ET sequence|legend=0| 29g, 82cdfgh, 111, 140 }}
EDOs: 94, 111, 205&lt;br /&gt;
 
Badness: 0.0587&lt;br /&gt;
Badness (Sintel): 1.10
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:40:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc20"&gt;&lt;a name="Quanic-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:40 --&gt;13-limit&lt;/h2&gt;
== Countriton ==
Commas: 352/351, 540/539, 729/728, 1331/1323&lt;br /&gt;
: ''For the 5-limit version, see [[Schismic–Mercator equivalence continuum #Countritonic]].''
&lt;br /&gt;
 
POTE generator: ~13/12 = 140.496&lt;br /&gt;
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.
&lt;br /&gt;
 
Map: [&amp;lt;1 1 -4 0 1 3|, &amp;lt;0 5 54 24 21 6|]&lt;br /&gt;
Countriton was named by [[Xenllium]] in 2022 as a counterpart of [[untriton]].
EDOs: 94, 111, 205&lt;br /&gt;
 
Badness: 0.0325&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:42:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc21"&gt;&lt;a name="Quanic-17-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:42 --&gt;17-limit&lt;/h2&gt;
[[Comma list]]: 5120/5103, 7558272/7503125
Commas: 352/351, 442/441, 540/539, 715/714, 847/845&lt;br /&gt;
 
&lt;br /&gt;
{{Mapping|legend=1| 1 -3 -15 13 | 0 9 34 -20 }}
POTE generator: ~13/12 = 140.497&lt;br /&gt;
: mapping generators: ~2, ~1225/864
&lt;br /&gt;
 
Map: [&amp;lt;1 1 -4 0 1 3 -2|, &amp;lt;0 5 54 24 21 6 52|]&lt;br /&gt;
[[Optimal tuning]]s:  
EDOs: 94, 111, 205&lt;br /&gt;
* [[WE]]: ~2 = 1199.4179{{c}}, ~1225/864 = 611.1213{{c}}
Badness: 0.0211&lt;br /&gt;
: [[error map]]: {{val| -0.582 -0.117 +0.541 +1.181 }}
&lt;br /&gt;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~1225/864 = 611.4120{{c}}
&lt;!-- ws:start:WikiTextHeadingRule:44:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc22"&gt;&lt;a name="Quanic-19-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:44 --&gt;19-limit&lt;/h2&gt;
: error map: {{val| 0.000 +0.753 +1.695 +2.934 }}
Commas: 352/351, 400/399, 442/441, 456/455, 495/494, 715/714&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=1| 51c, 53, 157, 210, 473cdd }}
POTE generator: ~13/12 = 140.496&lt;br /&gt;
 
&lt;br /&gt;
[[Badness]] (Sintel): 3.32
Map: [&amp;lt;1 1 -4 0 1 3 -2 -5|, &amp;lt;0 5 54 24 21 6 52 79|]&lt;br /&gt;
 
EDOs: 94, 111, 205&lt;br /&gt;
=== 11-limit ===
Badness: 0.0173&lt;br /&gt;
Subgroup: 2.3.5.7.11
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:46:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc23"&gt;&lt;a name="Supers"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:46 --&gt;Supers&lt;/h1&gt;
Comma list: 176/175, 5120/5103, 41503/41472
Commas: 5120/5103, 118098/117649&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 1 -3 -15 13 -21 | 0 9 34 -20 48 }}
POTE generator: ~9/7 = 434.218&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:  
Map: [&amp;lt;2 1 -12 2|, &amp;lt;0 3 23 5|]&lt;br /&gt;
* WE: ~2 = 1199.5178{{c}}, ~77/54 = 611.2097{{c}}
Wedgie: &amp;lt;&amp;lt;6 46 10 59 -1 -106||&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~77/54 = 611.4495{{c}}
EDOs: 58, 94, 152&lt;br /&gt;
 
Badness: 92.748&lt;br /&gt;
{{Optimal ET sequence|legend=0| 51ce, 53, 104c, 157 }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:48:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc24"&gt;&lt;a name="Supers-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:48 --&gt;11-limit&lt;/h2&gt;
Badness (Sintel): 2.80
Commas: 540/539, 4000/3993, 5120/5103&lt;br /&gt;
 
&lt;br /&gt;
=== 13-limit ===
POTE generator: ~9/7 = 434.217&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
&lt;br /&gt;
 
Map: [&amp;lt;2 1 -12 2 -9|, &amp;lt;0 3 23 5 22|]&lt;br /&gt;
Comma list: 176/175, 351/350, 847/845, 2197/2187
EDOs: 58, 94, 152&lt;br /&gt;
 
Badness: 0.0282&lt;br /&gt;
Mapping: {{mapping| 1 -3 -15 13 -21 -7 | 0 9 34 -20 48 21 }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:50:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc25"&gt;&lt;a name="Supers-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:50 --&gt;13-limit&lt;/h2&gt;
Optimal tunings:  
Commas: 352/351, 540/539, 729/728, 1575/1573&lt;br /&gt;
* WE: ~2 = 1199.5944{{c}}, ~77/54 = 611.2491{{c}}
&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~77/54 = 611.4506{{c}}
POTE generator: ~9/7 = 434.221&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 51ce, 53, 104c, 157 }}
Map: [&amp;lt;2 1 -12 2 -9 -2|, &amp;lt;0 3 23 5 22 13|]&lt;br /&gt;
 
EDOs: 58, 94, 152f&lt;br /&gt;
Badness (Sintel): 1.75
Badness: 0.0216&lt;br /&gt;
 
&lt;br /&gt;
== Artoneutral ==
&lt;!-- ws:start:WikiTextHeadingRule:52:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc26"&gt;&lt;a name="Alphaquarter"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:52 --&gt;Alphaquarter&lt;/h1&gt;
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.
Commas: 5120/5103, 29360128/29296875&lt;br /&gt;
 
&lt;br /&gt;
Artoneutral was named by [[Flora Canou]] in 2023 for its generator's quality.
POTE generator: ~16128/15625 = 55.243&lt;br /&gt;
 
&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
Map: [&amp;lt;1 2 2 0|, &amp;lt;0 -9 7 61|]&lt;br /&gt;
 
Wedgie: &amp;lt;&amp;lt;9 -7 -61 -32 -122 -122||&lt;br /&gt;
[[Comma list]]: 5120/5103, 3828125/3779136
EDOs: 87, 152, 239, 391&lt;br /&gt;
 
Badness: 0.1166&lt;br /&gt;
{{Mapping|legend=1| 1 -1 -4 12 | 0 9 22 -32 }}
&lt;br /&gt;
: mapping generators: ~2, ~128/105
&lt;!-- ws:start:WikiTextHeadingRule:54:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc27"&gt;&lt;a name="Alphaquarter-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:54 --&gt;11-limit&lt;/h2&gt;
 
Commas: 3025/3024, 4000/3993, 5120/5103&lt;br /&gt;
[[Optimal tuning]]s:  
&lt;br /&gt;
* [[WE]]: ~2 = 1200.1400{{c}}, ~128/105 = 344.7929{{c}}
POTE generator: ~33/32 = 55.243&lt;br /&gt;
: [[error map]]: {{val| +0.140 +1.041 -1.430 -0.518 }}
&lt;br /&gt;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~128/105 = 344.7531{{c}}
EDOs: 87, 152, 239, 391&lt;br /&gt;
: error map: {{val| 0.000 +0.823 -1.746 -0.925 }}
Badness: 0.0296&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=1| 87, 94, 181 }}
&lt;!-- ws:start:WikiTextHeadingRule:56:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc28"&gt;&lt;a name="Septiquarter"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:56 --&gt;Septiquarter&lt;/h1&gt;
 
Commas: 5120/5103, 420175/419904&lt;br /&gt;
[[Badness]] (Sintel): 3.98
&lt;br /&gt;
 
POTE generator: ~147/128 = 242.453&lt;br /&gt;
=== 11-limit ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11
Map: [&amp;lt;1 3 10 2|, &amp;lt;0 -7 -38 4|]&lt;br /&gt;
 
Wedgie: &amp;lt;&amp;lt;7 38 -4 44 -26 -116||&lt;br /&gt;
Comma list: 385/384, 2200/2187, 4000/3993
EDOs: 94, 99, 292, 391, 881bd, 1272bcd&lt;br /&gt;
 
Badness: 0.0538&lt;br /&gt;
Mapping: {{mapping| 1 -1 -4 12 -2 | 0 9 22 -32 19 }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:58:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc29"&gt;&lt;a name="Tricot"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:58 --&gt;Tricot&lt;/h1&gt;
Optimal tunings:  
Commas: 2430/2401, 5120/5103&lt;br /&gt;
* WE: ~2 = 1200.1668{{c}}, ~11/9 = 344.8027{{c}}
&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 344.7557{{c}}
POTE generator: ~81/56 = 634.026&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 87, 181 }}
Map: [[&amp;lt; 3 16 8|, &amp;lt;0 -3 -29 -11|]&lt;br /&gt;
 
Wedgie: &amp;lt;&amp;lt;3 29 11 39 9 -56||&lt;br /&gt;
Badness (Sintel): 1.52
EDOs: 53, 229d, 282d&lt;br /&gt;
 
Badness: 0.1001&lt;br /&gt;
=== 13-limit ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
&lt;!-- ws:start:WikiTextHeadingRule:60:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc30"&gt;&lt;a name="Tricot-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:60 --&gt;11-limit&lt;/h2&gt;
 
Commas: 99/98, 121/120, 5120/5103&lt;br /&gt;
Comma list: 325/324, 352/351, 385/384, 1575/1573
&lt;br /&gt;
 
POTE generator: ~63/44 = 634.027&lt;br /&gt;
Mapping: {{mapping| 1 -1 -4 12 -2 6 | 0 9 22 -32 19 -8 }}
&lt;br /&gt;
 
Map: [[&amp;lt; 3 16 8 11|, &amp;lt;0 -3 -29 -11 -16|]&lt;br /&gt;
Optimal tunings:  
EDOs: 53, 176de, 229&lt;br /&gt;
* WE: ~2 = 1200.0662{{c}}, ~11/9 = 344.7804{{c}}
Badness: 0.0561&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 344.7617{{c}}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:62:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc31"&gt;&lt;a name="Tricot-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:62 --&gt;13-limit&lt;/h2&gt;
{{Optimal ET sequence|legend=0| 87, 181 }}
Commas: 99/98, 121/120, 169/168, 352/351&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 1.08
POTE generator: ~13/9 = 634.012&lt;br /&gt;
 
&lt;br /&gt;
=== 17-limit ===
Map: [[&amp;lt; 3 16 8 11 7|, &amp;lt;0 -3 -29 -11 -16 -7|]&lt;br /&gt;
Subgroup: 2.3.5.7.11.13.17
EDOs: 53&lt;br /&gt;
 
Badness: 0.0321&lt;br /&gt;
Comma list: 325/324, 352/351, 375/374, 385/384, 595/594
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:64:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc32"&gt;&lt;a name="Ketchup"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:64 --&gt;Ketchup&lt;/h1&gt;
Mapping: {{mapping| 1 -1 -4 12 -2 6 -12 | 0 9 22 -32 19 -8 56 }}
Commas: 5120/5103, 1071875/1062882&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:  
POTE generator: ~64/63 = ~81/80 = 25.719&lt;br /&gt;
* WE: ~2 = 1200.0346{{c}}, ~11/9 = 344.7589{{c}}
&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 344.7492{{c}}
Map: [&amp;lt;2 3 4 6|, &amp;lt;0 4 15 -9|]&lt;br /&gt;
 
EDOS: 46, 94, 140&lt;br /&gt;
{{Optimal ET sequence|legend=0| 87, 94, 181 }}
Badness: 0.0845&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 1.16
&lt;!-- ws:start:WikiTextHeadingRule:66:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc33"&gt;&lt;a name="Ketchup-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:66 --&gt;11-limit&lt;/h2&gt;
 
Commas: 385/384, 1331/1323, 2200/2187&lt;br /&gt;
=== 19-limit ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11.13.17.19
POTE generator: ~55/54 = ~64/63 = ~81/80 = 25.693&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 325/324, 352/351, 375/374, 385/384, 400/399, 595/594
Map: [&amp;lt;2 3 4 6 7|, &amp;lt;0 4 15 -9 -2|]&lt;br /&gt;
 
EDOs: 46, 94, 140&lt;br /&gt;
Mapping: {{mapping| 1 -1 -4 12 -2 6 -12 -15 | 0 9 22 -32 19 -8 56 67 }}
Badness: 0.0396&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:  
&lt;!-- ws:start:WikiTextHeadingRule:68:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc34"&gt;&lt;a name="Ketchup-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:68 --&gt;13-limit&lt;/h2&gt;
* WE: ~2 = 1200.0282{{c}}, ~11/9 = 344.7532{{c}}
Commas: 325/324, 352/351, 847/845, 1331/1323&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 344.7453{{c}}
&lt;br /&gt;
 
POTE generator: ~55/54 = ~64/63 = ~81/80 = 25.697&lt;br /&gt;
{{Optimal ET sequence|legend=0| 87, 94, 181 }}
&lt;br /&gt;
 
Map: [&amp;lt;2 3 4 6 7 8|, &amp;lt;0 4 15 -9 -2 -14|]&lt;br /&gt;
Badness (Sintel): 1.19
EDOs: 46, 94, 140&lt;br /&gt;
 
Badness: 0.0248&lt;br /&gt;
=== 23-limit ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11.13.17.19.23
&lt;!-- ws:start:WikiTextHeadingRule:70:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc35"&gt;&lt;a name="Ketchup-17-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:70 --&gt;17-limit&lt;/h2&gt;
 
Commas: 289/288, 325/324, 352/351, 385/384, 561/560&lt;br /&gt;
Comma list: 300/299, 325/324, 352/351, 375/374, 385/384, 400/399, 484/483
&lt;br /&gt;
 
POTE generator: ~55/54 = ~64/63 = ~81/80 = 25.701&lt;br /&gt;
Mapping: {{mapping| 1 -1 -4 12 -2 6 -12 -15 -13 | 0 9 22 -32 19 -8 56 67 61 }}
&lt;br /&gt;
 
Map: [&amp;lt;2 3 4 6 7 8 8|, &amp;lt;0 4 15 -9 -2 -14 4|]&lt;br /&gt;
Optimal tunings:
EDOs: 46, 94, 140&lt;br /&gt;
* WE: ~2 = 1200.0163{{c}}, ~11/9 = 344.7461{{c}}
Badness: 0.0166&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 344.7416{{c}}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:72:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc36"&gt;&lt;a name="Ketchup-19-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:72 --&gt;19-limit&lt;/h2&gt;
{{Optimal ET sequence|legend=0| 87, 94, 181 }}
Commas: 190/189, 209/208, 289/288, 352/351, 385/384, 561/560&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 1.17
POTE generator: ~55/54 = ~64/63 = ~81/80 = 25.660&lt;br /&gt;
 
&lt;br /&gt;
== Quanic ==
Map: [&amp;lt;2 3 4 6 7 8 8 9|, &amp;lt;0 4 15 -9 -2 -14 4 -12|]&lt;br /&gt;
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.
EDOs: 46, 94, 140h, 234eh&lt;br /&gt;
 
Badness: 0.0182&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:74:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc37"&gt;&lt;a name="Ketchup-23-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:74 --&gt;23-limit&lt;/h2&gt;
[[Comma list]]: 5120/5103, 5832000/5764801
Commas: 190/189, 209/208, 253/252, 289/288, 323/322, 352/351, 385/384&lt;br /&gt;
 
&lt;br /&gt;
{{Mapping|legend=1| 1 1 -4 0 | 0 5 54 24 }}
POTE generator: ~55/54 = ~64/63 = ~81/80 = 25.661&lt;br /&gt;
: mapping generators: ~2, ~160/147
&lt;br /&gt;
 
Map: [&amp;lt;2 3 4 6 7 8 8 9 9|, &amp;lt;0 4 15 -9 -2 -14 4 -12 1|]&lt;br /&gt;
[[Optimal tuning]]s:
EDOs: 46, 94, 140h, 234ehi&lt;br /&gt;
* [[WE]]: ~2 = 1199.6159{{c}}, ~160/147 = 140.4483{{c}}
Badness: 0.0140&lt;/body&gt;&lt;/html&gt;</pre></div>
: [[error map]]: {{val| -0.384 -0.098 -0.570 +1.933 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~160/147 = 140.4862{{c}}
: error map: {{val| 0.000 +0.476 -0.061 +2.842 }}
 
{{Optimal ET sequence|legend=1| 94, 111, 205 }}
 
[[Badness]] (Sintel): 4.54
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 1331/1323, 5120/5103
 
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}}
 
{{Optimal ET sequence|legend=0| 94, 111, 205 }}
 
Badness (Sintel): 1.94
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 352/351, 540/539, 729/728, 1331/1323
 
Mapping: {{mapping| 1 1 -4 0 1 3 | 0 5 54 24 21 6 }}
 
Optimal tunings:
* WE: ~2 = 1199.6639{{c}}, ~13/12 = 140.4562{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.4904{{c}}
 
{{Optimal ET sequence|legend=0| 94, 111, 205 }}
 
Badness (Sintel): 1.34
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 352/351, 442/441, 540/539, 715/714, 847/845
 
Mapping: {{mapping| 1 1 -4 0 1 3 -2 | 0 5 54 24 21 6 52 }}
 
Optimal tunings:  
* WE: ~2 = 1199.6699{{c}}, ~13/12 = 140.4586{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.4920{{c}}
 
{{Optimal ET sequence|legend=0| 94, 111, 205 }}
 
Badness (Sintel): 1.08
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 352/351, 400/399, 442/441, 456/455, 495/494, 715/714
 
Mapping: {{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
 
== Jorgensen ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Jorgensen]].''
 
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.
 
It is the natural 7-limit extension of the 5-limit temperament tempering out the 70-comma, named by [[Mike Battaglia]] in 2012 for historical interests<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_103982.html Yahoo! Tuning Group | ''Jorgensen Temperament'']</ref>.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 5120/5103, 578509309952/576650390625
 
{{Mapping|legend=1| 70 111 0 34 | 0 0 1 1 }}
: mapping generators: ~50421/50000, ~5
 
[[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 ET sequence|legend=1| 70, 140, 350, 490 }}
 
[[Badness]] (Sintel): 5.40
 
== References ==
 
[[Category:Temperament collections]]
[[Category:Aberschismic temperaments| ]] <!-- main article -->
[[Category:Rank 2]]

Latest revision as of 12:45, 6 June 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

This is a collection of rank-2 aberschismic temperaments, which temper out the aberschisma (monzo: [10 -6 1 -1⟩, ratio: 5120/5103). These temperaments divide an exact or approximate septimal quartertone, 36/35 into two equal steps, each representing 81/80~64/63, the syntonic comma or the septimal comma. Therefore, classical and septimal intervals are found by the same chain of fifths inflected by the syntonic~septimal comma to the opposite sides. In addition we may identify 10/7 by the augmented fourth and 50/49 by the Pythagorean comma.

Temperaments belonging to this category and generated by the fifth are dominant, garibaldi, kwai, undecental, and leapday. Dominant has 5/4 mapped to M3. Garibaldi has 5/4 mapped to d4. Kwai has 5/4 mapped to 4A7. Undecental has 5/4 mapped to 5d7. Leapday has 5/4 mapped to 3A1.

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

Temperaments discussed elsewhere are:

Considered below are septiquarter, kwai, ketchup, undecental, leapday, mystery, hemidromeda, countriton, artoneutral, quanic and jorgensen, in the order of increasing TE logflat badness.

Septiquarter

Septiquarter tempers out 420175/419904 and may be described as the 94 & 99 temperament. Its ploidacot is epsilon-heptacot. 99edo makes for an excellent tuning, and 292edo an even better one. 94edo and 104edo in the 104c val are also among the possibilities.

Subgroup: 2.3.5.7

Comma list: 5120/5103, 420175/419904

Mapping: [⟨1 -4 -28 6], ⟨0 7 38 -4]]

mapping generators: ~2, ~243/140

Optimal tunings:

  • WE: ~2 = 1199.7212 ¢, ~243/140 = 957.3250 ¢
error map: ⟨-0.279 +0.435 -0.158 +0.201]
  • CWE: ~2 = 1200.0000 ¢, ~243/140 = 957.5424 ¢
error map: ⟨0.000 +0.842 +0.298 +1.004]

Optimal ET sequence: 94, 99, 292, 391, 881bd, 1272bcd

Badness (Sintel): 1.36

Semiseptiquarter

Subgroup: 2.3.5.7.11

Comma list: 5120/5103, 9801/9800, 14641/14580

Mapping: [⟨2 -8 -56 12 -25], ⟨0 7 38 -4 20]]

Optimal tunings:

  • WE: ~99/70 = 599.8953 ¢, ~210/121 = 957.3819 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~210/121 = 957.5449 ¢

Optimal ET sequence: 94, 198, 292, 490

Badness (Sintel): 2.12

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨2 -8 -56 12 -25 9], ⟨0 7 38 -4 20 -1]]

Optimal tunings:

  • WE: ~99/70 = 599.8565 ¢, ~210/121 = 957.3261 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~210/121 = 957.5508 ¢

Optimal ET sequence: 94, 198, 490f

Badness (Sintel): 1.44

Kwai

For the 5-limit version, see Miscellaneous 5-limit temperaments #Kwai.

Named by Gene Ward Smith in 2004 for its "bridgeability"[1], kwai is generated by a perfect fifth, and can be described as 41 & 70.

Subgroup: 2.3.5.7

Comma list: 5120/5103, 16875/16807

Mapping: [⟨1 0 -50 -40], ⟨0 1 33 27]]

mapping generators: ~2, ~3

Optimal tunings:

  • WE: ~2 = 1199.7337 ¢, ~3/2 = 702.4600 ¢
error map: ⟨-0.266 +0.239 -0.607 +1.055]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.6085 ¢
error map: ⟨0.000 +0.653 -0.234 +1.603]

Optimal ET sequence: 41, 111, 152, 345, 497d

Badness (Sintel): 1.38

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 0 -50 -40 32], ⟨0 1 33 27 -18]]

Optimal tunings:

  • WE: ~2 = 1199.6672 ¢, ~3/2 = 702.4282 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.6189 ¢

Optimal ET sequence: 41, 111, 152, 497de, 649dde

Badness (Sintel): 0.867

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 540/539, 729/728, 1375/1372

Mapping: [⟨1 0 -50 -40 32 27], ⟨0 1 33 27 -18 -21]]

Optimal tunings:

  • WE: ~2 = 1199.4772 ¢, ~3/2 = 702.3379 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.6409 ¢

Optimal ET sequence: 41, 111, 152f, 415dff

Badness (Sintel): 1.01

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 256/255, 352/351, 540/539, 715/714, 1089/1088

Mapping: [⟨1 0 -50 -40 32 27 58], ⟨0 1 33 27 -18 -21 -34]]

Optimal tunings:

  • WE: ~2 = 1199.3537 ¢, ~3/2 = 702.2850 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.6589 ¢

Optimal ET sequence: 41, 70, 111, 152fg, 263dfg

Badness (Sintel): 1.12

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 256/255, 352/351, 400/399, 456/455, 715/714, 847/845

Mapping: [⟨1 0 -50 -40 32 27 58 -56], ⟨0 1 33 27 -18 -21 -34 38]]

Optimal tunings:

  • WE: ~2 = 1199.3401 ¢, ~3/2 = 702.2705 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.6548 ¢

Optimal ET sequence: 41, 70h, 111, 152fg, 263dfgh

Badness (Sintel): 1.03

Hemikwai

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 676/675, 1375/1372, 5120/5103

Mapping: [⟨1 0 -50 -40 32 -51], ⟨0 2 66 54 -36 69]]

mapping generators: ~2, ~26/15

Optimal tunings:

  • WE: ~2 = 1199.6968 ¢, ~26/15 = 951.0740 ¢
  • CWE: ~2 = 1200.0000 ¢, ~26/15 = 951.3123 ¢

Optimal ET sequence: 82, 111, 193, 304d

Badness (Sintel): 1.82

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 0 -50 -40 32 -51 -30], ⟨0 2 66 54 -36 69 43]]

Optimal tunings:

  • WE: ~2 = 1199.6861 ¢, ~26/15 = 951.0654 ¢
  • CWE: ~2 = 1200.0000 ¢, ~26/15 = 951.3120 ¢

Optimal ET sequence: 82, 111, 193, 304d

Badness (Sintel): 1.31

19-limit

Subgroup: 2.3.5.7.11.13.17.19

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

Mapping: [⟨1 0 -50 -40 32 -51 -30 -56], ⟨0 2 66 54 -36 69 43 76]]

Optimal tunings:

  • WE: ~2 = 1199.6718 ¢, ~26/15 = 951.0526 ¢
  • CWE: ~2 = 1200.0000 ¢, ~26/15 = 951.3103 ¢

Optimal ET sequence: 82, 111, 193, 304dh

Badness (Sintel): 1.16

Ketchup

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

Subgroup: 2.3.5.7

Comma list: 5120/5103, 1071875/1062882

Mapping: [⟨2 3 4 6], ⟨0 4 15 -9]]

mapping generators: ~1225/864, ~64/63

Optimal tunings:

  • WE: ~1225/864 = 599.9685 ¢, ~64/63 = 25.7181 ¢
error map: ⟨-0.063 +0.823 -0.668 -0.478]
  • CWE: ~1225/864 = 600.0000 ¢, ~64/63 = 25.7181 ¢
error map: ⟨0.000 +0.917 -0.543 -0.288]

Optimal ET sequence: 46, 94, 140

Badness (Sintel): 2.14

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨2 3 4 6 7], ⟨0 4 15 -9 -2]]

Optimal tunings:

  • WE: ~99/70 = 600.0678 ¢, ~64/63 = 25.6963 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~64/63 = 25.6956 ¢

Optimal ET sequence: 46, 94, 140

Badness (Sintel): 1.31

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨2 3 4 6 7 8], ⟨0 4 15 -9 -2 -14]]

Optimal tunings:

  • WE: ~99/70 = 600.0612 ¢, ~66/65 = 25.7000 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~66/65 = 25.6978 ¢

Optimal ET sequence: 46, 94, 140

Badness (Sintel): 1.03

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨2 3 4 6 7 8 8], ⟨0 4 15 -9 -2 -14 4]]

Optimal tunings:

  • WE: ~17/12 = 600.0896 ¢, ~66/65 = 25.7048 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~66/65 = 25.7017 ¢

Optimal ET sequence: 46, 94, 140

Badness (Sintel): 0.845

2.3.5.7.11.13.17.23 subgroup

Subgroup: 2.3.5.7.11.13.17.23

Comma list: 253/252, 289/288, 325/324, 352/351, 385/384, 391/390

Mapping: [⟨2 3 4 6 7 8 8 9], ⟨0 4 15 -9 -2 -14 4 1]]

Optimal tunings:

  • WE: ~17/12 = 600.1139 ¢, ~66/65 = 25.7053 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~66/65 = 25.7013 ¢

Optimal ET sequence: 46, 94, 140

Badness (Sintel): 0.772

Undecental

Undecental adds the triwellisma to the comma list and may be described as the 29 & 70 temperament. 5/4 is mapped to the quintuple-diminished seventh or equivalently the perfect fourth minus three dieses. 58\99 is an almost perfect generator, just as the name suggests. Another interesting tuning choice is the argent fifth, 2(2 - sqrt (2)).

Subgroup: 2.3.5.7

Comma list: 5120/5103, 235298/234375

Mapping: [⟨1 0 61 71], ⟨0 1 -37 -43]]

mapping generators: ~2, ~3

Optimal tunings:

  • WE: ~2 = 1199.6543 ¢, ~3/2 = 702.8370 ¢
error map: ⟨-0.346 +0.536 +0.423 -0.494]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 703.0465 ¢
error map: ⟨0.000 +1.092 +0.966 +0.175]

Optimal ET sequence: 29, 70, 99, 722bc, 821bc, 920bc, 1019bc

Badness (Sintel): 2.39

Leapday

Main article: Leapday
For the 5-limit version, see Miscellaneous 5-limit temperaments #Leapday.

Leapday tempers out 686/675, the senga, in addition to the aberschisma, and may be described as the 29 & 46 temperament. It extends leapfrog, such that 7/4 is found by 15 generators up, as a double-augmented fifth (a major sixth and a diesis). 5/4 is found by a tritone above that, as a triple-augmented unison (a minor third and two dieses). 46edo itself is an excellent tuning for this.

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

Leapday has an alternative extension called polypyth, which tempers out the same 5-limit comma as leapday, but with the porwell comma (6144/6125) rather than the aberschisma tempered out.

Subgroup: 2.3.5.7

Comma list: 686/675, 5120/5103

Mapping: [⟨1 0 -31 -21], ⟨0 1 21 15]]

mapping generators: ~2, ~3

Optimal tunings:

  • WE: ~2 = 1199.7167 ¢, ~3/2 = 704.0971 ¢
error map: ⟨-0.283 +1.859 +2.559 -5.669]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.2504 ¢
error map: ⟨0.000 +2.295 +2.945 -5.070]

Optimal ET sequence: 17c, 29, 46

Badness (Sintel): 2.43

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 0 -31 -21 -14], ⟨0 1 21 15 11]]

Optimal tunings:

  • WE: ~2 = 1200.0731 ¢, ~3/2 = 704.2933 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.2538 ¢

Optimal ET sequence: 17c, 29, 46

Badness (Sintel): 1.28

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 121/120, 169/168, 352/351

Mapping: [⟨1 0 -31 -21 -14 -9], ⟨0 1 21 15 11 8]]

Optimal tunings:

  • WE: ~2 = 1200.4758 ¢, ~3/2 = 704.4930 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.2346 ¢

Optimal ET sequence: 17c, 29, 46, 121def

Badness (Sintel): 1.02

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 0 -31 -21 -14 -9 -34], ⟨0 1 21 15 11 8 24]]

Optimal tunings:

  • WE: ~2 = 1200.4818 ¢, ~3/2 = 704.5121 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.2507 ¢

Optimal ET sequence: 17cg, 29g, 46, 121defg

Badness (Sintel): 0.910

2.3.5.7.11.13.17.23 subgroup

Subgroup: 2.3.5.7.11.13.17.23

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

Mapping: [⟨1 0 -31 -21 -14 -9 -34 -5], ⟨0 1 21 15 11 8 24 6]]

Optimal tunings:

  • WE: ~2 = 1200.5169 ¢, ~3/2 = 704.5279 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.2450 ¢

Optimal ET sequence: 17cg, 29g, 46, 121defg

Badness (Sintel): 0.872

Mystery

Main article: Mystery
For the 5-limit version, see 29th-octave temperaments #Mystery.

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

Subgroup: 2.3.5.7

Comma list: 5120/5103, 50421/50000

Mapping: [⟨29 46 0 14], ⟨0 0 1 1]]

mapping generators: ~50/49, ~5

Optimal tunings:

  • WE: ~50/49 = 41.3652 ¢, ~5/4 = 388.5128 ¢
error map: ⟨-0.410 +0.842 +1.378 -2.022]
  • CWE: ~50/49 = 41.3793 ¢, ~5/4 = 388.3030 ¢
error map: ⟨0.000 +1.493 +1.989 -1.213]

Optimal ET sequence: 29, 58, 87, 145

Badness (Sintel): 2.63

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 896/891, 3388/3375

Mapping: [⟨29 46 0 14 33], ⟨0 0 1 1 1]]

Optimal tunings:

  • WE: ~45/44 = 41.3637 ¢, ~5/4 = 388.3136 ¢
  • CWE: ~45/44 = 41.3793 ¢, ~5/4 = 388.0598 ¢

Optimal ET sequence: 29, 58, 87, 145

Badness (Sintel): 1.13

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 352/351, 364/363, 676/675

Mapping: [⟨29 46 0 14 33 40], ⟨0 0 1 1 1 1]]

Optimal tunings:

  • WE: ~45/44 = 41.3623 ¢, ~5/4 = 388.1942 ¢
  • CWE: ~40/39 = 41.3793 ¢, ~5/4 = 387.9017 ¢

Optimal ET sequence: 29, 58, 87, 145, 232

Badness (Sintel): 0.768

Hemidromeda

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

Subgroup: 2.3.5.7

Comma list: 5120/5103, 52734375/52706752

Mapping: [⟨1 0 38 48], ⟨0 2 -45 -57]]

mapping generator: ~2, ~12500/7203

Optimal tunings:

  • WE: ~2 = 1199.7236 ¢, ~12500/7203 = 951.1864 ¢
error map: ⟨-0.276 +0.418 -0.205 +0.282]
  • CWE: ~2 = 1200.0000 ¢, ~12500/7203 = 951.4098 ¢
error map: ⟨0.000 +0.865 +0.243 +0.813]

Optimal ET sequence: 29, 82cd, 111, 140, 251, 391, 1424bbcdd

Badness (Sintel): 2.93

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 0 38 48 32], ⟨0 2 -45 -57 -36]]

Optimal tunings:

  • WE: ~2 = 1199.8767 ¢, ~400/231 = 951.3065 ¢
  • CWE: ~2 = 1200.0000 ¢, ~400/231 = 951.4063 ¢

Optimal ET sequence: 29, 82cd, 111, 140, 251, 391e

Badness (Sintel): 2.01

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 676/675, 847/845, 1331/1323

Mapping: [⟨1 0 38 48 32 37], ⟨0 2 -45 -57 -36 -42]]

Optimal tunings:

  • WE: ~2 = 1199.8753 ¢, ~26/15 = 951.3054 ¢
  • CWE: ~2 = 1200.0000 ¢, ~26/15 = 951.4064 ¢

Optimal ET sequence: 29, 82cdf, 111, 140, 251, 391e

Badness (Sintel): 1.18

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 352/351, 442/441, 561/560, 676/675, 715/714

Mapping: [⟨1 0 38 48 32 37 58], ⟨0 2 -45 -57 -36 -42 -68]]

Optimal tunings:

  • WE: ~2 = 1199.8770 ¢, ~26/15 = 951.3039 ¢
  • CWE: ~2 = 1200.0000 ¢, ~26/15 = 951.4035 ¢

Optimal ET sequence: 29g, 82cdfg, 111, 140, 251, 391e

Badness (Sintel): 0.971

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 286/285, 352/351, 363/361, 442/441, 476/475, 561/560

Mapping: [⟨1 0 38 48 32 37 58 32], ⟨0 2 -45 -57 -36 -42 -68 -35]]

Optimal tunings:

  • WE: ~2 = 1199.7534 ¢, ~26/15 = 951.2024 ¢
  • CWE: ~2 = 1200.0000 ¢, ~26/15 = 951.4020 ¢

Optimal ET sequence: 29g, 82cdfgh, 111, 140

Badness (Sintel): 1.01

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 253/252, 286/285, 352/351, 363/361, 391/390, 442/441, 460/459

Mapping: [⟨1 0 38 48 32 37 58 32 18], ⟨0 2 -45 -57 -36 -42 -68 -35 -17]]

Optimal tunings:

  • WE: ~2 = 1199.9128 ¢, ~26/15 = 951.3371 ¢
  • CWE: ~2 = 1200.0000 ¢, ~26/15 = 951.4076 ¢

Optimal ET sequence: 29g, 82cdfgh, 111, 140

Badness (Sintel): 1.10

Countriton

For the 5-limit version, see Schismic–Mercator equivalence continuum #Countritonic.

Countriton may be described as the 51c & 53 temperament. It splits the 24th harmonic into nine tritone generators; its ploidacot is thus delta-enneacot. Among the possible tunings are 157edo and 210edo, as well as 104edo in the 104c val.

Countriton was named by Xenllium in 2022 as a counterpart of untriton.

Subgroup: 2.3.5.7

Comma list: 5120/5103, 7558272/7503125

Mapping: [⟨1 -3 -15 13], ⟨0 9 34 -20]]

mapping generators: ~2, ~1225/864

Optimal tunings:

  • WE: ~2 = 1199.4179 ¢, ~1225/864 = 611.1213 ¢
error map: ⟨-0.582 -0.117 +0.541 +1.181]
  • CWE: ~2 = 1200.0000 ¢, ~1225/864 = 611.4120 ¢
error map: ⟨0.000 +0.753 +1.695 +2.934]

Optimal ET sequence: 51c, 53, 157, 210, 473cdd

Badness (Sintel): 3.32

11-limit

Subgroup: 2.3.5.7.11

Comma list: 176/175, 5120/5103, 41503/41472

Mapping: [⟨1 -3 -15 13 -21], ⟨0 9 34 -20 48]]

Optimal tunings:

  • WE: ~2 = 1199.5178 ¢, ~77/54 = 611.2097 ¢
  • CWE: ~2 = 1200.0000 ¢, ~77/54 = 611.4495 ¢

Optimal ET sequence: 51ce, 53, 104c, 157

Badness (Sintel): 2.80

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 176/175, 351/350, 847/845, 2197/2187

Mapping: [⟨1 -3 -15 13 -21 -7], ⟨0 9 34 -20 48 21]]

Optimal tunings:

  • WE: ~2 = 1199.5944 ¢, ~77/54 = 611.2491 ¢
  • CWE: ~2 = 1200.0000 ¢, ~77/54 = 611.4506 ¢

Optimal ET sequence: 51ce, 53, 104c, 157

Badness (Sintel): 1.75

Artoneutral

Artoneutral can be described as the 87 & 94 temperament. It is generated by an artoneutral third of ~11/9 (or a tendoneutral sixth of ~18/11), nine of which make the 12th harmonic; its ploidacot is thus beta-enneacot. 181edo may be recommended as a tuning.

Artoneutral was named by Flora Canou in 2023 for its generator's quality.

Subgroup: 2.3.5.7

Comma list: 5120/5103, 3828125/3779136

Mapping: [⟨1 -1 -4 12], ⟨0 9 22 -32]]

mapping generators: ~2, ~128/105

Optimal tunings:

  • WE: ~2 = 1200.1400 ¢, ~128/105 = 344.7929 ¢
error map: ⟨+0.140 +1.041 -1.430 -0.518]
  • CWE: ~2 = 1200.0000 ¢, ~128/105 = 344.7531 ¢
error map: ⟨0.000 +0.823 -1.746 -0.925]

Optimal ET sequence: 87, 94, 181

Badness (Sintel): 3.98

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 2200/2187, 4000/3993

Mapping: [⟨1 -1 -4 12 -2], ⟨0 9 22 -32 19]]

Optimal tunings:

  • WE: ~2 = 1200.1668 ¢, ~11/9 = 344.8027 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 344.7557 ¢

Optimal ET sequence: 87, 181

Badness (Sintel): 1.52

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 352/351, 385/384, 1575/1573

Mapping: [⟨1 -1 -4 12 -2 6], ⟨0 9 22 -32 19 -8]]

Optimal tunings:

  • WE: ~2 = 1200.0662 ¢, ~11/9 = 344.7804 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 344.7617 ¢

Optimal ET sequence: 87, 181

Badness (Sintel): 1.08

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 325/324, 352/351, 375/374, 385/384, 595/594

Mapping: [⟨1 -1 -4 12 -2 6 -12], ⟨0 9 22 -32 19 -8 56]]

Optimal tunings:

  • WE: ~2 = 1200.0346 ¢, ~11/9 = 344.7589 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 344.7492 ¢

Optimal ET sequence: 87, 94, 181

Badness (Sintel): 1.16

19-limit

Subgroup: 2.3.5.7.11.13.17.19

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

Mapping: [⟨1 -1 -4 12 -2 6 -12 -15], ⟨0 9 22 -32 19 -8 56 67]]

Optimal tunings:

  • WE: ~2 = 1200.0282 ¢, ~11/9 = 344.7532 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 344.7453 ¢

Optimal ET sequence: 87, 94, 181

Badness (Sintel): 1.19

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

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

Mapping: [⟨1 -1 -4 12 -2 6 -12 -15 -13], ⟨0 9 22 -32 19 -8 56 67 61]]

Optimal tunings:

  • WE: ~2 = 1200.0163 ¢, ~11/9 = 344.7461 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 344.7416 ¢

Optimal ET sequence: 87, 94, 181

Badness (Sintel): 1.17

Quanic

Quanic may be described as the 94 & 111 temperament. It splits the perfect fifth into five generators which in the 13-limit extension may be taken as ~13/12; its ploidacot is thus pentacot. 205edo may be recommended as a tuning.

Subgroup: 2.3.5.7

Comma list: 5120/5103, 5832000/5764801

Mapping: [⟨1 1 -4 0], ⟨0 5 54 24]]

mapping generators: ~2, ~160/147

Optimal tunings:

  • WE: ~2 = 1199.6159 ¢, ~160/147 = 140.4483 ¢
error map: ⟨-0.384 -0.098 -0.570 +1.933]
  • CWE: ~2 = 1200.0000 ¢, ~160/147 = 140.4862 ¢
error map: ⟨0.000 +0.476 -0.061 +2.842]

Optimal ET sequence: 94, 111, 205

Badness (Sintel): 4.54

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 1 -4 0 1], ⟨0 5 54 24 21]]

Optimal tunings:

  • WE: ~2 = 1199.7834 ¢, ~88/81 = 140.4635 ¢
  • CWE: ~2 = 1200.0000 ¢, ~88/81 = 140.4850 ¢

Optimal ET sequence: 94, 111, 205

Badness (Sintel): 1.94

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 540/539, 729/728, 1331/1323

Mapping: [⟨1 1 -4 0 1 3], ⟨0 5 54 24 21 6]]

Optimal tunings:

  • WE: ~2 = 1199.6639 ¢, ~13/12 = 140.4562 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/12 = 140.4904 ¢

Optimal ET sequence: 94, 111, 205

Badness (Sintel): 1.34

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 1 -4 0 1 3 -2], ⟨0 5 54 24 21 6 52]]

Optimal tunings:

  • WE: ~2 = 1199.6699 ¢, ~13/12 = 140.4586 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/12 = 140.4920 ¢

Optimal ET sequence: 94, 111, 205

Badness (Sintel): 1.08

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 352/351, 400/399, 442/441, 456/455, 495/494, 715/714

Mapping: [⟨1 1 -4 0 1 3 -2 -5], ⟨0 5 54 24 21 6 52 79]]

Optimal tunings:

  • WE: ~2 = 1199.6745 ¢, ~13/12 = 140.4574 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/12 = 140.4908 ¢

Optimal ET sequence: 94, 111, 205

Badness (Sintel): 1.05

Jorgensen

For the 5-limit version, see Miscellaneous 5-limit temperaments #Jorgensen.

Jorgensen tempers out the linus comma in addition to the aberschisma, and may be described as the 70 & 140 temperament, with a 70th-octave period. Its ploidacot is 70-ploid acot.

It is the natural 7-limit extension of the 5-limit temperament tempering out the 70-comma, named by Mike Battaglia in 2012 for historical interests[2].

Subgroup: 2.3.5.7

Comma list: 5120/5103, 578509309952/576650390625

Mapping: [⟨70 111 0 34], ⟨0 0 1 1]]

mapping generators: ~50421/50000, ~5

Optimal tunings:

  • WE: ~50421/50000 = 17.1387 ¢, ~5/4 = 386.8071 ¢
error map: ⟨-0.288 +0.445 -0.084 +0.121]
  • CWE: ~50421/50000 = 17.1429 ¢, ~5/4 = 386.6593 ¢
error map: ⟨0.000 +0.902 +0.346 +0.690]

Optimal ET sequence: 70, 140, 350, 490

Badness (Sintel): 5.40

References

  1. ↑ Yahoo! Tuning Group | Kwai
  2. ↑ Yahoo! Tuning Group | Jorgensen Temperament
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