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=== Overview to extensions ===
=== Overview to extensions ===
==== Full 7-limit extensions ====
==== Full 7-limit extensions ====
To the gamelisma itself we need to add the comma which appears next on the modified [[Normal lists #Normal interval list|normal comma list]] for the full 7-limit. The second comma on the list for mothra is [[81/80]], for rodan [[245/243]], for guiron [[32805/32768]], for gorgo [[36/35]], and for gidorah [[256/245]]. These all use ~8/7 as a generator, though in the case of gidorah that is the same as ~6/5.  
To the gamelisma itself we need to add the comma which appears next on the modified [[Normal lists #Normal interval list|normal comma list]] for the full 7-limit. The second comma on the list for mothra is [[81/80]], for rodan [[245/243]], for guiron [[32805/32768]], for gorgo [[36/35]], and for gidorah [[256/245]]. These all use ~8/7 as a generator, though in the case of gidorah that is the same as ~[[6/5]].  


Miracle adds [[33075/32768]] and uses the [[secor]], half an ~8/7, as generator. Lemba adds [[525/512]] to the list, and has a half-octave [[period]]. Valentine adds [[6144/6125]] with a generator of ~21/20 and superkleismic adds [[875/864]] with a generator of ~6/5. Unidec adds [[4375/4374]], and has a generator of ~10/9 with a half-octave period. Hemithirds adds [[65625/65536]] with a generator half of a classical major third. Finally, tritikleismic adds [[15625/15552]] and has a generator of 6/5 with a 1/3-octave period.
Miracle adds [[33075/32768]] and uses the [[secor]], half an ~8/7, as generator. Lemba adds [[525/512]] to the list, and has a half-octave [[period]]. Valentine adds [[6144/6125]] with a generator of ~[[21/20]] and superkleismic adds [[875/864]] with a generator of ~6/5. Unidec adds [[4375/4374]], and has a generator of ~[[10/9]] with a half-octave period. Hemithirds adds [[65625/65536]] with a generator half of a classical major third. Finally, tritikleismic adds [[15625/15552]] and has a generator of 6/5 with a 1/3-octave period.


Full 7-limit temperaments discussed elsewhere are:
Full 7-limit temperaments discussed elsewhere are:
* [[Blackwood]] (+28/27) → [[Limmic temperaments #Blackwood|Limmic temperaments]]
* [[Blackwood]] (+28/27) → [[Blackwood family #Blackwood|Blackwood family]]
* [[Lemba]] (+50/49) → [[Jubilismic clan #Lemba|Jubilismic clan]]
* [[Lemba]] (+50/49) → [[Jubilismic clan #Lemba|Jubilismic clan]]
* [[Trisected]] (+128/125) → [[Augmented family #Trisected|Augmented family]]
* [[Trisected]] (+128/125) → [[Augmented family #Trisected|Augmented family]]
Line 46: Line 46:


==== Subgroup extensions ====
==== Subgroup extensions ====
No-five subgroup extensions of slendric include radon, a 2.3.7.11-subgroup extension that may be viewed as no-five rodan, considered below, euslendric, a 2.3.7.13-subgroup extension, and baladic, a weak 2.3.7.13.17-subgroup extension, considered in [[#Other subgroup extensions]]. Dicussed elsewhere is [[No-fives subgroup temperaments #Gigapyth|gigapyth]] in the 2.3.7.85 subgroup.  
No-five subgroup extensions of slendric include radon, a 2.3.7.11-subgroup extension that may be viewed as no-five rodan, considered below, euslendric, a 2.3.7.13-subgroup extension, baladic, a weak 2.3.7.13.17-subgroup extension, and gigapyth, a 2.3.7.85-subgroup extension, considered in [[#Other subgroup extensions]]. Dicussed elsewhere is [[Subgroup temperaments #Trisect|trisect]] in the 2.3.7.11/5 subgroup.


=== Radon ===
=== Radon ===
{{See also|Chromatic pairs #Radon}}
Radon is the no-fives version of [[rodan]], equating the diatonic major third to [[14/11]].
Radon is the no-fives version of [[rodan]], equating the diatonic major third to [[14/11]].


Line 70: Line 72:
{{Main| Mothra }}
{{Main| Mothra }}


Mothra tempers out [[81/80]] and finds the prime 5 at a stack of four fifths as does any temperament in the [[meantone family]]. It also tempers out [[1728/1715]], the orwellisma. It can be described as the {{nowrap| 26 & 31 }}. Using [[31edo]] with a generator of 6/31 is an excellent tuning choice. However, a pure mos mothra scale is often described as directionless and has limited chord-building potential<ref>[https://www.youtube.com/watch?v=uH3ahBzDSrs 31-EDO Music Theory: Supermajor Hexatonic Scale] by [[Zhea Erose]]</ref>, so something other than a mos may be used as a scale to get the most out of mothra. There are examples of non-mos mothra scales in 31edo [[Strictly proper 7-tone 31edo scales|in the article on strictly proper 7-tone 31edo scales]].  
Mothra tempers out [[81/80]] and finds the prime 5 at a stack of four fifths as does any temperament in the [[meantone family]]. It also tempers out [[1728/1715]], the orwellisma. It can be described as the {{nowrap| 26 & 31 }}. Using [[31edo]] with a generator of 6\31 is an excellent tuning choice. However, a pure mos mothra scale is often described as directionless and has limited chord-building potential<ref>[https://www.youtube.com/watch?v=uH3ahBzDSrs 31-EDO Music Theory: Supermajor Hexatonic Scale] by [[Zhea Erose]]</ref>, so something other than a mos may be used as a scale to get the most out of mothra. There are examples of non-mos mothra scales in 31edo [[Strictly proper 7-tone 31edo scales|in the article on strictly proper 7-tone 31edo scales]].  


Note that mothra is also called '''cynder''' in the 7-limit, which can be a little confusing sometimes.  
Note that mothra is also called '''cynder''' in the 7-limit, which can be a little confusing sometimes.  
Line 106: Line 108:
Comma list: 81/80, 99/98, 385/384
Comma list: 81/80, 99/98, 385/384


Mapping: {{mapping| 1 1 0 3 5 | 0 3 12 -1 -8 }}
{{Mapping|legend=0| 1 1 0 3 5 | 0 3 12 -1 -8 }}


Optimal tunings:  
Optimal tunings:  
Line 121: Line 123:
Comma list: 81/80, 99/98, 105/104, 144/143
Comma list: 81/80, 99/98, 105/104, 144/143


Mapping: {{mapping| 1 1 0 3 5 1 | 0 3 12 -1 -8 14 }}
{{Mapping|legend=0| 1 1 0 3 5 1 | 0 3 12 -1 -8 14 }}


Optimal tunings:  
Optimal tunings:  
Line 130: Line 132:


Badness (Sintel): 0.990
Badness (Sintel): 0.990
; Music
* ''Prelude for solo piano'' (2014) by [[Chris Vaisvil]] – [https://web.archive.org/web/20201127013310/http://micro.soonlabel.com/16-ET/mothra/20141028_mothra16br4.mp3 play] | [https://www.chrisvaisvil.com/prelude-for-solo-piano-in-mothra16-brat-4-tuning/ blog] – in Mothra[16], brat 4 tuning


==== 17-limit ====
==== 17-limit ====
Line 139: Line 138:
Comma list: 81/80, 99/98, 105/104, 120/119, 144/143
Comma list: 81/80, 99/98, 105/104, 120/119, 144/143


Mapping: {{mapping| 1 1 0 3 5 1 | 0 3 12 -1 -8 14 16 }}
{{Mapping|legend=0| 1 1 0 3 5 1 | 0 3 12 -1 -8 14 16 }}


Optimal tunings:  
Optimal tunings:  
Line 154: Line 153:
Comma list: 81/80, 99/98, 105/104, 120/119, 144/143, 153/152
Comma list: 81/80, 99/98, 105/104, 120/119, 144/143, 153/152


Mapping: {{mapping| 1 1 0 3 5 1 | 0 3 12 -1 -8 14 16 22 }}
{{Mapping|legend=0| 1 1 0 3 5 1 | 0 3 12 -1 -8 14 16 22 }}


Optimal tunings:  
Optimal tunings:  
Line 171: Line 170:
Comma list: 81/80, 176/175, 540/539
Comma list: 81/80, 176/175, 540/539


Mapping: {{mapping| 1 1 0 3 -1 | 0 3 12 -1 23 }}
{{Mapping|legend=0| 1 1 0 3 -1 | 0 3 12 -1 23 }}


Optimal tunings:  
Optimal tunings:  
Line 186: Line 185:
Comma list: 81/80, 144/143, 176/175, 196/195
Comma list: 81/80, 144/143, 176/175, 196/195


Mapping: {{mapping| 1 1 0 3 -1 7 | 0 3 12 -1 23 -17 }}
{{Mapping|legend=0| 1 1 0 3 -1 7 | 0 3 12 -1 23 -17 }}


Optimal tunings:  
Optimal tunings:  
Line 201: Line 200:
Comma list: 81/80, 144/143, 176/175, 189/187, 196/195
Comma list: 81/80, 144/143, 176/175, 189/187, 196/195


Mapping: {{mapping| 1 1 0 3 -1 7 | 0 3 12 -1 23 -17 -15 }}
{{Mapping|legend=0| 1 1 0 3 -1 7 | 0 3 12 -1 23 -17 -15 }}


Optimal tunings:  
Optimal tunings:  
Line 216: Line 215:
Comma list: 81/80, 96/95, 144/143, 153/152, 176/175, 196/195
Comma list: 81/80, 96/95, 144/143, 153/152, 176/175, 196/195


Mapping: {{mapping| 1 1 0 3 -1 7 | 0 3 12 -1 23 -17 -15 -9 }}
{{Mapping|legend=0| 1 1 0 3 -1 7 | 0 3 12 -1 23 -17 -15 -9 }}


Optimal tunings:  
Optimal tunings:  
Line 231: Line 230:
Comma list: 45/44, 81/80, 1029/1024
Comma list: 45/44, 81/80, 1029/1024


Mapping: {{mapping| 1 1 0 3 0 | 0 3 12 -1 18 }}
{{Mapping|legend=0| 1 1 0 3 0 | 0 3 12 -1 18 }}


Optimal tunings:  
Optimal tunings:  
Line 246: Line 245:
Comma list: 45/44, 78/77, 81/80, 640/637
Comma list: 45/44, 78/77, 81/80, 640/637


Mapping: {{mapping| 1 1 0 3 0 1 | 0 3 12 -1 18 14 }}
{{Mapping|legend=0| 1 1 0 3 0 1 | 0 3 12 -1 18 14 }}


Optimal tunings:  
Optimal tunings:  
Line 290: Line 289:
Comma list: 245/243, 385/384, 441/440
Comma list: 245/243, 385/384, 441/440


Mapping: {{mapping| 1 1 -1 3 6 | 0 3 17 -1 -13 }}
{{Mapping|legend=0| 1 1 -1 3 6 | 0 3 17 -1 -13 }}


Optimal tunings:  
Optimal tunings:  
Line 312: Line 311:
Comma list: 196/195, 245/243, 352/351, 364/363
Comma list: 196/195, 245/243, 352/351, 364/363


Mapping: {{mapping| 1 1 -1 3 6 8 | 0 3 17 -1 -13 -22 }}
{{Mapping|legend=0| 1 1 -1 3 6 8 | 0 3 17 -1 -13 -22 }}


Optimal tunings:  
Optimal tunings:  
Line 333: Line 332:
Comma list: 154/153, 196/195, 245/243, 256/255, 273/272
Comma list: 154/153, 196/195, 245/243, 256/255, 273/272


Mapping: {{mapping| 1 1 -1 3 6 8 8 | 0 3 17 -1 -13 -22 -20 }}
{{Mapping|legend=0| 1 1 -1 3 6 8 8 | 0 3 17 -1 -13 -22 -20 }}


Optimal tunings:  
Optimal tunings:  
Line 352: Line 351:
Comma list: 91/90, 245/243, 385/384, 441/440
Comma list: 91/90, 245/243, 385/384, 441/440


Mapping: {{mapping| 1 1 -1 3 6 -1 | 0 3 17 -1 -13 24 }}
{{Mapping|legend=0| 1 1 -1 3 6 -1 | 0 3 17 -1 -13 24 }}


Optimal tunings:  
Optimal tunings:  
Line 367: Line 366:
Comma list: 176/175, 245/243, 1029/1024
Comma list: 176/175, 245/243, 1029/1024


Mapping: {{mapping| 1 1 -1 3 -3 | 0 3 17 -1 33 }}
{{Mapping|legend=0| 1 1 -1 3 -3 | 0 3 17 -1 33 }}


Optimal tunings:  
Optimal tunings:  
Line 382: Line 381:
Comma list: 91/90, 176/175, 245/243, 847/845
Comma list: 91/90, 176/175, 245/243, 847/845


Mapping: {{mapping| 1 1 -1 3 -3 -1 | 0 3 17 -1 33 24 }}
{{Mapping|legend=0| 1 1 -1 3 -3 -1 | 0 3 17 -1 33 24 }}


Optimal tunings:  
Optimal tunings:  
Line 397: Line 396:
Comma list: 100/99, 245/243, 1029/1024
Comma list: 100/99, 245/243, 1029/1024


Mapping: {{mapping| 1 1 -1 3 -2 | 0 3 17 -1 28 }}
{{Mapping|legend=0| 1 1 -1 3 -2 | 0 3 17 -1 28 }}


Optimal tunings:  
Optimal tunings:  
Line 412: Line 411:
Comma list: 100/99, 105/104, 245/243, 352/351
Comma list: 100/99, 105/104, 245/243, 352/351


Mapping: {{mapping| 1 1 -1 3 -2 0 | 0 3 17 -1 28 19 }}
{{Mapping|legend=0| 1 1 -1 3 -2 0 | 0 3 17 -1 28 19 }}


Optimal tunings:  
Optimal tunings:  
Line 451: Line 450:
Comma list: 385/384, 441/440, 10976/10935
Comma list: 385/384, 441/440, 10976/10935


Mapping: {{mapping| 1 1 7 3 -2 | 0 3 -24 -1 28 }}
{{Mapping|legend=0| 1 1 7 3 -2 | 0 3 -24 -1 28 }}


Optimal tunings:  
Optimal tunings:  
Line 471: Line 470:
Comma list: 196/195, 352/351, 385/384, 729/728
Comma list: 196/195, 352/351, 385/384, 729/728


Mapping: {{mapping| 1 1 7 3 -2 0 | 0 3 -24 -1 28 19 }}
{{Mapping|legend=0| 1 1 7 3 -2 0 | 0 3 -24 -1 28 19 }}


Optimal tunings:  
Optimal tunings:  
Line 510: Line 509:
Comma list: 36/35, 45/44, 1029/1024
Comma list: 36/35, 45/44, 1029/1024


Mapping: {{mapping| 1 1 1 3 1 | 0 3 7 -1 13 }}
{{Mapping|legend=0| 1 1 1 3 1 | 0 3 7 -1 13 }}


Optimal tunings:  
Optimal tunings:  
Line 525: Line 524:
Comma list: 27/26, 36/35, 45/44, 507/500
Comma list: 27/26, 36/35, 45/44, 507/500


Mapping: {{mapping| 1 1 1 3 1 2 | 0 3 7 -1 13 9 }}
{{Mapping|legend=0| 1 1 1 3 1 2 | 0 3 7 -1 13 9 }}


Optimal tunings:  
Optimal tunings:  
Line 540: Line 539:
Comma list: 36/35, 56/55, 1029/1024
Comma list: 36/35, 56/55, 1029/1024


Mapping: {{mapping| 1 1 1 3 5 | 0 3 7 -1 -8 }}
{{Mapping|legend=0| 1 1 1 3 5 | 0 3 7 -1 -8 }}


Optimal tunings:  
Optimal tunings:  
Line 555: Line 554:
Comma list: 27/26, 36/35, 56/55, 507/500
Comma list: 27/26, 36/35, 56/55, 507/500


Mapping: {{mapping| 1 1 1 3 5 2 | 0 3 7 -1 -8 9 }}
{{Mapping|legend=0| 1 1 1 3 5 2 | 0 3 7 -1 -8 9 }}


Optimal tunings:  
Optimal tunings:  
Line 655: Line 654:
Comma list: 33/32, 45/44, 352/343
Comma list: 33/32, 45/44, 352/343


Mapping: {{mapping| 1 1 4 3 4 | 0 3 -9 -1 -3 }}
{{Mapping|legend=0| 1 1 4 3 4 | 0 3 -9 -1 -3 }}


Optimal tunings:  
Optimal tunings:  
Line 669: Line 668:
: ''For the 5-limit version, see [[Syntonic–31 equivalence continuum #Ampersand]].''
: ''For the 5-limit version, see [[Syntonic–31 equivalence continuum #Ampersand]].''


Miracle is one of the most important entries of this temperament clan. It tempers out [[225/224]], splitting the ~8/7 generator of slendric into 15/14~16/15, and can be described as the {{nowrap| 31 & 41 }} temperament. Its ploidacot is hexacot. It is then extremely natural to equate the neutral third, three generators up, to [[11/9]] and thereby extend miracle to the full [[11-limit]] with essentially no further damage. [[72edo]] makes for an excellent tuning.  
Miracle is one of the most important entries of this temperament clan. It tempers out [[225/224]], splitting the ~8/7 generator of slendric into [[15/14]]~[[16/15]], and can be described as the {{nowrap| 31 & 41 }} temperament. Its ploidacot is hexacot. It is then extremely natural to equate the neutral third, three generators up, to [[11/9]] and thereby extend miracle to the full [[11-limit]] with essentially no further damage. [[72edo]] makes for an excellent tuning.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 708: Line 707:
Comma list: 225/224, 243/242, 385/384
Comma list: 225/224, 243/242, 385/384


Mapping: {{mapping| 1 1 3 3 2 | 0 6 -7 -2 15 }}
{{Mapping|legend=0| 1 1 3 3 2 | 0 6 -7 -2 15 }}


Optimal tunings:  
Optimal tunings:  
Line 734: Line 733:
Comma list: 105/104, 144/143, 196/195, 243/242
Comma list: 105/104, 144/143, 196/195, 243/242


Mapping: {{mapping| 1 1 3 3 2 4 | 0 6 -7 -2 15 -3 }}
{{Mapping|legend=0| 1 1 3 3 2 4 | 0 6 -7 -2 15 -3 }}


Optimal tunings:  
Optimal tunings:  
Line 749: Line 748:
Comma list: 105/104, 120/119, 144/143, 154/153, 170/169
Comma list: 105/104, 120/119, 144/143, 154/153, 170/169


Mapping: {{mapping| 1 1 3 3 2 4 4 | 0 6 -7 -2 15 -3 1 }}
{{Mapping|legend=0| 1 1 3 3 2 4 4 | 0 6 -7 -2 15 -3 1 }}


Optimal tunings:  
Optimal tunings:  
Line 778: Line 777:
Comma list: 225/224, 243/242, 351/350, 385/384
Comma list: 225/224, 243/242, 351/350, 385/384


Mapping: {{mapping| 1 1 3 3 2 7 | 0 6 -7 -2 15 -34 }}
{{Mapping|legend=0| 1 1 3 3 2 7 | 0 6 -7 -2 15 -34 }}


Optimal tunings:  
Optimal tunings:  
Line 793: Line 792:
Comma list: 225/224, 243/242, 273/272, 351/350, 375/374
Comma list: 225/224, 243/242, 273/272, 351/350, 375/374


Mapping: {{mapping| 1 1 3 3 2 7 7 | 0 6 -7 -2 15 -34 -30 }}
{{Mapping|legend=0| 1 1 3 3 2 7 7 | 0 6 -7 -2 15 -34 -30 }}


Optimal tunings:  
Optimal tunings:  
Line 822: Line 821:
Comma list: 225/224, 243/242, 325/324, 385/384
Comma list: 225/224, 243/242, 325/324, 385/384


Mapping: {{mapping| 1 1 3 3 2 0 | 0 6 -7 -2 15 38 }}
{{Mapping|legend=0| 1 1 3 3 2 0 | 0 6 -7 -2 15 38 }}


Optimal tunings:  
Optimal tunings:  
Line 837: Line 836:
Comma list: 225/224, 243/242, 273/272, 325/324, 385/384
Comma list: 225/224, 243/242, 273/272, 325/324, 385/384


Mapping: {{mapping| 1 1 3 3 2 0 0 | 0 6 -7 -2 15 38 42 }}
{{Mapping|legend=0| 1 1 3 3 2 0 0 | 0 6 -7 -2 15 38 42 }}


Optimal tunings:  
Optimal tunings:  
Line 866: Line 865:
Comma list: 169/168, 225/224, 243/242, 385/384
Comma list: 169/168, 225/224, 243/242, 385/384


Mapping: {{mapping| 2 2 6 6 4 7 | 0 6 -7 -2 15 2 }}
{{Mapping|legend=0| 2 2 6 6 4 7 | 0 6 -7 -2 15 2 }}
: mapping generators: ~55/39, ~15/14
: mapping generators: ~55/39, ~15/14


Line 882: Line 881:
Comma list: 169/168, 221/220, 225/224, 243/242, 273/272
Comma list: 169/168, 221/220, 225/224, 243/242, 273/272


Mapping: {{mapping| 2 2 6 6 4 7 7 | 0 6 -7 -2 15 2 6 }}
{{Mapping|legend=0| 2 2 6 6 4 7 7 | 0 6 -7 -2 15 2 6 }}


Optimal tunings:  
Optimal tunings:  
Line 911: Line 910:
Comma list: 225/224, 243/242, 385/384, 847/845
Comma list: 225/224, 243/242, 385/384, 847/845


Mapping: {{mapping| 1 1 3 3 2 2 | 0 12 -14 -4 30 35 }}
{{Mapping|legend=0| 1 1 3 3 2 2 | 0 12 -14 -4 30 35 }}
: mapping generators: ~2, ~27/26
: mapping generators: ~2, ~27/26


Line 927: Line 926:
Comma list: 225/224, 243/242, 273/272, 385/384, 847/845
Comma list: 225/224, 243/242, 273/272, 385/384, 847/845


Mapping: {{mapping| 1 1 3 3 2 2 2 | 0 12 -14 -4 30 35 43 }}
{{Mapping|legend=0| 1 1 3 3 2 2 2 | 0 12 -14 -4 30 35 43 }}


Optimal tunings:  
Optimal tunings:  
Line 956: Line 955:
Comma list: 225/224, 243/242, 289/288, 385/384, 847/845
Comma list: 225/224, 243/242, 289/288, 385/384, 847/845


Mapping: {{mapping| 2 2 6 6 4 4 7 | 0 12 -14 -4 30 35 12 }}
{{Mapping|legend=0| 2 2 6 6 4 4 7 | 0 12 -14 -4 30 35 12 }}
: mapping generators: ~17/12, ~27/26
: mapping generators: ~17/12, ~27/26


Line 972: Line 971:
Comma list: 209/208, 225/224, 243/242, 289/288, 361/360, 385/384
Comma list: 209/208, 225/224, 243/242, 289/288, 361/360, 385/384


Mapping: {{mapping| 2 2 6 6 4 4 7 8 | 0 12 -14 -4 30 35 12 5 }}
{{Mapping|legend=0| 2 2 6 6 4 4 7 8 | 0 12 -14 -4 30 35 12 5 }}


Optimal tunings:  
Optimal tunings:  
Line 987: Line 986:
Comma list: 209/208, 225/224, 243/242, 289/288, 323/322, 361/360, 385/384
Comma list: 209/208, 225/224, 243/242, 289/288, 323/322, 361/360, 385/384


Mapping: {{mapping| 2 2 6 6 4 4 7 8 7 | 0 12 -14 -4 30 35 12 5 21 }}
{{Mapping|legend=0| 2 2 6 6 4 4 7 8 7 | 0 12 -14 -4 30 35 12 5 21 }}


Optimal tunings:  
Optimal tunings:  
Line 1,002: Line 1,001:
Comma list: 225/224, 243/242, 385/384, 2200/2197
Comma list: 225/224, 243/242, 385/384, 2200/2197


Mapping: {{mapping| 1 -11 17 7 -28 3 | 0 18 -21 -6 45 1 }}
{{Mapping|legend=0| 1 -11 17 7 -28 3 | 0 18 -21 -6 45 1 }}
: mapping generators: ~2, ~13/8
: mapping generators: ~2, ~13/8


Line 1,018: Line 1,017:
Comma list: 225/224, 243/242, 273/272, 385/384, 2200/2197
Comma list: 225/224, 243/242, 273/272, 385/384, 2200/2197


Mapping: {{mapping| 1 -11 17 7 -28 3 -5 | 0 18 -21 -6 45 1 13 }}
{{Mapping|legend=0| 1 -11 17 7 -28 3 -5 | 0 18 -21 -6 45 1 13 }}


Optimal tunings:  
Optimal tunings:  
Line 1,047: Line 1,046:
Comma list: 99/98, 176/175, 1029/1024
Comma list: 99/98, 176/175, 1029/1024


Mapping: {{mapping| 1 1 3 3 5 | 0 6 -7 -2 -16 }}
{{Mapping|legend=0| 1 1 3 3 5 | 0 6 -7 -2 -16 }}


Optimal tunings:  
Optimal tunings:  
Line 1,062: Line 1,061:
Comma list: 66/65, 99/98, 105/104, 512/507
Comma list: 66/65, 99/98, 105/104, 512/507


Mapping: {{mapping| 1 1 3 3 5 4 | 0 6 -7 -2 -16 -3 }}
{{Mapping|legend=0| 1 1 3 3 5 4 | 0 6 -7 -2 -16 -3 }}


Optimal tunings:  
Optimal tunings:  
Line 1,077: Line 1,076:
Comma list: 225/224, 245/242, 1029/1024
Comma list: 225/224, 245/242, 1029/1024


Mapping: {{mapping| 1 1 3 3 4 | 0 12 -14 -4 -11 }}
{{Mapping|legend=0| 1 1 3 3 4 | 0 12 -14 -4 -11 }}
: mapping generators: ~2, ~33/32
: mapping generators: ~2, ~33/32


Line 1,093: Line 1,092:
Comma list: 105/104, 196/195, 245/242, 512/507
Comma list: 105/104, 196/195, 245/242, 512/507


Mapping: {{mapping| 1 1 3 3 4 4 | 0 12 -14 -4 -11 -6 }}
{{Mapping|legend=0| 1 1 3 3 4 4 | 0 12 -14 -4 -11 -6 }}


Optimal tunings:  
Optimal tunings:  
Line 1,104: Line 1,103:


=== Oracle ===
=== Oracle ===
The name is a portmanteau of [[orwell]] and [[miracle]]: Oracle is a weak extension of 7-limit miracle, splitting its ~[[16/15]] generator and an octave into two ~[[16/11]] generators. Additionally, when [[restriction|restricted]] to the 2.15.7.11 subgroup, oracle's generator corresponds to 2 stacked orwell generators.
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 121/120, 225/224, 1029/1024
Comma list: 121/120, 225/224, 1029/1024


Mapping: {{mapping| 1 -5 10 5 4 | 0 12 -14 -4 -1 }}
{{Mapping|legend=0| 1 -5 10 5 4 | 0 12 -14 -4 -1 }}
: mapping generators: ~2, ~16/11
: mapping generators: ~2, ~16/11


Line 1,144: Line 1,145:
Comma list: 385/384, 441/440, 19683/19600
Comma list: 385/384, 441/440, 19683/19600


Mapping: {{mapping| 1 -2 -15 4 16 | 0 6 29 -2 -21 }}
{{Mapping|legend=0| 1 -2 -15 4 16 | 0 6 29 -2 -21 }}


Optimal tunings:  
Optimal tunings:  
Line 1,159: Line 1,160:
Comma list: 351/350, 385/384, 441/440, 676/675
Comma list: 351/350, 385/384, 441/440, 676/675


Mapping: {{mapping| 1 -2 -15 4 16 -19 | 0 6 29 -2 -21 38 }}
{{Mapping|legend=0| 1 -2 -15 4 16 -19 | 0 6 29 -2 -21 38 }}


Optimal tunings:  
Optimal tunings:  
Line 1,174: Line 1,175:
Comma list: 273/272, 351/350, 385/384, 441/440, 676/675
Comma list: 273/272, 351/350, 385/384, 441/440, 676/675


Mapping: {{mapping| 1 -2 -15 4 16 -19 -21 | 0 6 29 -2 -21 38 42 }}
{{Mapping|legend=0| 1 -2 -15 4 16 -19 -21 | 0 6 29 -2 -21 38 42 }}


Optimal tunings:  
Optimal tunings:  
Line 1,188: Line 1,189:
: ''For the 5-limit version, see [[Syntonic–31 equivalence continuum #Valentine (5-limit)]].''
: ''For the 5-limit version, see [[Syntonic–31 equivalence continuum #Valentine (5-limit)]].''


Valentine tempers out [[126/125]] and [[6144/6125]] as well as 1029/1024. It has a generator of [[~]][[21/20]], three of which make the slendric generator ~8/7. 21/20 can be stripped of its 2 and taken as 3 × 7/5. In this respect it resembles miracle, with a generator of 3 × 5/7, and casablanca, with a generator of 5 × 7/3. These three generators are the simplest in terms of the relationship of tetrads in the [[7-limit symmetrical lattices|lattice of 7-limit tetrads]]. Valentine can be described as the {{nowrap| 31 & 46 }} temperament; its ploidacot is enneacot. [[77edo]], [[108edo]], or [[185edo]] make for excellent tunings, which also happen to be excellent tunings for [[starling]], the rank-3 temperament tempering out 126/125. Hence 7-limit valentine can be used whenever starling is wanted, with the extra tempering out of 1029/1024 having no discernible effect on tuning accuracy. Another tuning for valentine uses (3/2)<sup>1/9</sup> as a generator, giving pure 3/2 fifths. Valentine extends naturally to the 11-limit, tempering out 121/120 and 441/440; 46edo has a valentine generator 3\46 which is only 0.0117 cents sharp of the minimax generator, (11/7)<sup>1/10</sup>.
Valentine tempers out [[126/125]] and [[6144/6125]] as well as 1029/1024. It has a generator of [[~]][[21/20]], three of which make the slendric generator ~8/7. 21/20 can be stripped of its 2 and taken as 3 × 7/5. In this respect it resembles miracle, with a generator of 3 × 5/7, and casablanca, with a generator of 5 × 7/3. These three generators are the simplest in terms of the relationship of tetrads in the [[7-limit symmetrical lattices|lattice of 7-limit tetrads]]. Valentine can be described as the {{nowrap| 31 & 46 }} temperament; its ploidacot is enneacot. [[77edo]], [[108edo]], or [[185edo]] make for excellent tunings, which also happen to be excellent tunings for [[starling]], the rank-3 temperament tempering out 126/125. Hence 7-limit valentine can be used whenever starling is wanted, with the extra tempering out of 1029/1024 having no discernible effect on tuning accuracy. Another tuning for valentine uses (3/2)<sup>1/9</sup> as a generator, giving pure 3/2 fifths. Valentine extends naturally to the 11-limit, tempering out 121/120 and 441/440; 46edo has a valentine generator 3\46 which is only 0.0117 cents sharp of the minimax generator, ([[11/7]])<sup>1/10</sup>.


Valentine has a very straighforward [[S-expression]]-based comma list in the [[11-limit]] add-23 (i.e. the 2.3.5.7.11.23 subgroup) of {([[176/175|S8/S10 = S22 × S23 × S24]], [[121/120|S11]]), [[441/440|S21]], [[484/483|S22]], [[529/528|S23]], [[576/575|S24]]}, so it is the temperament that equalizes the 20::25 segment of the harmonic series.
Valentine has a very straighforward [[S-expression]]-based comma list in the [[11-limit]] add-23 (i.e. the 2.3.5.7.11.23 subgroup) of {([[176/175|S8/S10 = S22 × S23 × S24]], [[121/120|S11]]), [[441/440|S21]], [[484/483|S22]], [[529/528|S23]], [[576/575|S24]]}, so it is the temperament that equalizes the 20::25 segment of the harmonic series.
Line 1,224: Line 1,225:
Comma list: 121/120, 126/125, 176/175
Comma list: 121/120, 126/125, 176/175


Mapping: {{mapping| 1 1 2 3 3 | 0 9 5 -3 7 }}
{{Mapping|legend=0| 1 1 2 3 3 | 0 9 5 -3 7 }}


Optimal tunings:  
Optimal tunings:  
Line 1,246: Line 1,247:
Comma list: 121/120, 126/125, 176/175, 196/195
Comma list: 121/120, 126/125, 176/175, 196/195


Mapping: {{mapping| 1 1 2 3 3 5 | 0 9 5 -3 7 -20 }}
{{Mapping|legend=0| 1 1 2 3 3 5 | 0 9 5 -3 7 -20 }}


Optimal tunings:  
Optimal tunings:  
Line 1,261: Line 1,262:
Comma list: 121/120, 126/125, 154/153, 176/175, 196/195
Comma list: 121/120, 126/125, 154/153, 176/175, 196/195


Mapping: {{mapping| 1 1 2 3 3 5 5 | 0 9 5 -3 7 -20 -14 }}
{{Mapping|legend=0| 1 1 2 3 3 5 5 | 0 9 5 -3 7 -20 -14 }}


Optimal tunings:  
Optimal tunings:  
Line 1,276: Line 1,277:
Comma list: 66/65, 105/104, 121/120, 126/125
Comma list: 66/65, 105/104, 121/120, 126/125


Mapping: {{mapping| 1 1 2 3 3 3 | 0 9 5 -3 7 11 }}
{{Mapping|legend=0| 1 1 2 3 3 3 | 0 9 5 -3 7 11 }}


Optimal tunings:  
Optimal tunings:  
Line 1,291: Line 1,292:
Comma list: 91/90, 121/120, 126/125, 176/175
Comma list: 91/90, 121/120, 126/125, 176/175


Mapping: {{mapping| 1 1 2 3 3 2 | 0 9 5 -3 7 26 }}
{{Mapping|legend=0| 1 1 2 3 3 2 | 0 9 5 -3 7 26 }}


Optimal tunings:  
Optimal tunings:  
Line 1,306: Line 1,307:
Comma list: 121/120, 126/125, 169/168, 176/175
Comma list: 121/120, 126/125, 169/168, 176/175


Mapping: {{mapping| 2 2 4 6 6 7 | 0 9 5 -3 7 3 }}
{{Mapping|legend=0| 2 2 4 6 6 7 | 0 9 5 -3 7 3 }}
: mapping generators: ~55/39, ~22/21
: mapping generators: ~55/39, ~22/21


Line 1,322: Line 1,323:
Comma list: 121/120, 126/125, 176/175, 343/338
Comma list: 121/120, 126/125, 176/175, 343/338


Mapping: {{mapping| 1 1 2 3 3 4 | 0 18 10 -6 14 -9 }}
{{Mapping|legend=0| 1 1 2 3 3 4 | 0 18 10 -6 14 -9 }}
: mapping generators: ~2, ~40/39
: mapping generators: ~2, ~40/39


Line 1,338: Line 1,339:
Comma list: 121/120, 126/125, 176/175, 676/675
Comma list: 121/120, 126/125, 176/175, 676/675


Mapping: {{mapping| 1 -8 -3 6 -4 -16 | 0 18 10 -6 14 37 }}
{{Mapping|legend=0| 1 -8 -3 6 -4 -16 | 0 18 10 -6 14 37 }}
: mapping generators: ~2, ~13/9
: mapping generators: ~2, ~13/9


Line 1,354: Line 1,355:
Comma list: 126/125, 243/242, 1029/1024
Comma list: 126/125, 243/242, 1029/1024


Mapping: {{mapping| 1 1 2 3 2 | 0 18 10 -6 45 }}
{{Mapping|legend=0| 1 1 2 3 2 | 0 18 10 -6 45 }}


Optimal tunings:  
Optimal tunings:  
Line 1,369: Line 1,370:
Comma list: 126/125, 196/195, 243/242, 1029/1024
Comma list: 126/125, 196/195, 243/242, 1029/1024


Mapping: {{mapping| 1 1 2 3 2 5 | 0 18 10 -6 45 -40 }}
{{Mapping|legend=0| 1 1 2 3 2 5 | 0 18 10 -6 45 -40 }}


Optimal tunings:  
Optimal tunings:  
Line 1,384: Line 1,385:
Comma list: 126/125, 144/143, 243/242, 343/338
Comma list: 126/125, 144/143, 243/242, 343/338


Mapping: {{mapping| 1 1 2 3 2 4 | 0 18 10 -6 45 -9 }}
{{Mapping|legend=0| 1 1 2 3 2 4 | 0 18 10 -6 45 -9 }}


Optimal tunings:  
Optimal tunings:  
Line 1,398: Line 1,399:
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Shibboleth]].''
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Shibboleth]].''


Superkleismic tempers out the keema, [[875/864]], and can be described as the {{nowrap| 15 & 26 }} temperament. It splits the ~7/4 into three ~6/5 generators of around 322 cents. This is noticeably sharper than the [[kleismic]] generator, hence the name.  
Superkleismic tempers out the keema, [[875/864]], and can be described as the {{nowrap| 15 & 26 }} temperament. It splits the ~[[7/4]] into three ~6/5 generators of around 322 cents. This is noticeably sharper than the [[kleismic]] generator, hence the name.  


In the 11-limit, two generator steps can be identified with ~16/11, and in the 13-limit, the same step can be treated as ~13/9. The [[S-expression]]-based comma list of 13-limit superkleismic is {[[875/864|S5/S6]], [[1029/1024|S7/S8]], [[100/99|S10]], [[144/143|S12]], ([[441/440|S21]])}. Through careful observation of the equivalences therein one can derive the mapping of the full 13-limit.  
In the 11-limit, two generator steps can be identified with ~16/11, and in the 13-limit, the same step can be treated as ~13/9. The [[S-expression]]-based comma list of 13-limit superkleismic is {[[875/864|S5/S6]], [[1029/1024|S7/S8]], [[100/99|S10]], [[144/143|S12]], ([[441/440|S21]])}. Through careful observation of the equivalences therein one can derive the mapping of the full 13-limit.  
Line 1,430: Line 1,431:
Comma list: 100/99, 245/242, 385/384
Comma list: 100/99, 245/242, 385/384


Mapping: {{mapping| 1 -5 -5 5 2 | 0 9 10 -3 2 }}
{{Mapping|legend=0| 1 -5 -5 5 2 | 0 9 10 -3 2 }}


Optimal tunings:  
Optimal tunings:  
Line 1,445: Line 1,446:
Comma list: 100/99, 133/132, 190/189, 385/384
Comma list: 100/99, 133/132, 190/189, 385/384


Mapping: {{mapping| 1 -5 -5 5 2 -6 | 0 9 10 -3 2 14 }}
{{Mapping|legend=0| 1 -5 -5 5 2 -6 | 0 9 10 -3 2 14 }}


Optimal tunings:  
Optimal tunings:  
Line 1,462: Line 1,463:
Comma list: 100/99, 105/104, 144/143, 245/242
Comma list: 100/99, 105/104, 144/143, 245/242


Mapping: {{mapping| 1 -5 -5 5 2 -8 | 0 9 10 -3 2 16 }}
{{Mapping|legend=0| 1 -5 -5 5 2 -8 | 0 9 10 -3 2 16 }}


Optimal tunings:  
Optimal tunings:  
Line 1,477: Line 1,478:
Comma list: 100/99, 105/104, 120/119, 144/143, 245/242
Comma list: 100/99, 105/104, 120/119, 144/143, 245/242


Mapping: {{mapping| 1 -5 -5 5 2 -8 -12 | 0 9 10 -3 2 16 22 }}
{{Mapping|legend=0| 1 -5 -5 5 2 -8 -12 | 0 9 10 -3 2 16 22 }}


Optimal tunings:  
Optimal tunings:  
Line 1,492: Line 1,493:
Comma list: 100/99, 105/104, 120/119, 144/143, 133/132, 190/189
Comma list: 100/99, 105/104, 120/119, 144/143, 133/132, 190/189


Mapping: {{mapping| 1 -5 -5 5 2 -8 -12 -6 | 0 9 10 -3 2 16 22 14 }}
{{Mapping|legend=0| 1 -5 -5 5 2 -8 -12 -6 | 0 9 10 -3 2 16 22 14 }}


Optimal tunings:  
Optimal tunings:  
Line 1,509: Line 1,510:
Comma list: 100/99, 196/195, 245/242, 385/384
Comma list: 100/99, 196/195, 245/242, 385/384


Mapping: {{mapping| 1 -5 -5 5 2 22 | 0 9 10 -3 2 -25 }}
{{Mapping|legend=0| 1 -5 -5 5 2 22 | 0 9 10 -3 2 -25 }}


Optimal tunings:  
Optimal tunings:  
Line 1,524: Line 1,525:
Comma list: 100/99, 154/153, 196/195, 245/242, 256/255
Comma list: 100/99, 154/153, 196/195, 245/242, 256/255


Mapping: {{mapping| 1 -5 -5 5 2 22 18 | 0 9 10 -3 2 -25 -19 }}
{{Mapping|legend=0| 1 -5 -5 5 2 22 18 | 0 9 10 -3 2 -25 -19 }}


Optimal tunings:  
Optimal tunings:  
Line 1,539: Line 1,540:
Comma list: 100/99, 133/132, 154/153, 190/189, 196/195, 256/255
Comma list: 100/99, 133/132, 154/153, 190/189, 196/195, 256/255


Mapping: {{mapping| 1 -5 -5 5 2 22 18 -6 | 0 9 10 -3 2 -25 -19 14 }}
{{Mapping|legend=0| 1 -5 -5 5 2 22 18 -6 | 0 9 10 -3 2 -25 -19 14 }}


Optimal tunings:  
Optimal tunings:  
Line 1,573: Line 1,574:
Comma list: 385/384, 441/440, 2460375/2458624
Comma list: 385/384, 441/440, 2460375/2458624


Mapping: {{mapping| 1 -5 25 5 -28 | 0 9 -31 -3 43 }}
{{Mapping|legend=0| 1 -5 25 5 -28 | 0 9 -31 -3 43 }}


Optimal tunings:  
Optimal tunings:  
Line 1,586: Line 1,587:
{{Main| Unidec }}
{{Main| Unidec }}


Unidec tempers out the ragisma, [[4375/4374]], and may be described as the {{nowrap| 26 & 46 }} temperament. It has a [[semi-octave]] [[period]] and a generator of ~80/63, two of which minus a period make slendric's generator; its [[ploidacot]] is therefore diploid gamma-hexacot. In the 11-limit, the generator represents [[14/11]]. [[190edo]] makes for an excellent tuning in both the 7-limit and 11-limit.  
Unidec tempers out the ragisma, [[4375/4374]], and may be described as the {{nowrap| 26 & 46 }} temperament. It has a [[semi-octave]] [[period]] and a generator of ~[[80/63]], two of which minus a period make slendric's generator; its [[ploidacot]] is therefore diploid gamma-hexacot. In the 11-limit, the generator represents [[14/11]]. [[190edo]] makes for an excellent tuning in both the 7-limit and 11-limit.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 1,617: Line 1,618:
Comma list: 385/384, 441/440, 4375/4374
Comma list: 385/384, 441/440, 4375/4374


Mapping: {{mapping| 2 -1 -3 7 9 | 0 6 11 -2 -3 }}
{{Mapping|legend=0| 2 -1 -3 7 9 | 0 6 11 -2 -3 }}


Optimal tunings:  
Optimal tunings:  
Line 1,637: Line 1,638:
Comma list: 385/384, 441/440, 625/624, 729/728
Comma list: 385/384, 441/440, 625/624, 729/728


Mapping: {{mapping| 2 -1 -3 7 9 -19 | 0 6 11 -2 -3 38 }}
{{Mapping|legend=0| 2 -1 -3 7 9 -19 | 0 6 11 -2 -3 38 }}


Optimal tunings:  
Optimal tunings:  
Line 1,652: Line 1,653:
Comma list: 169/168, 325/324, 364/363, 385/384
Comma list: 169/168, 325/324, 364/363, 385/384


Mapping: {{mapping| 2 -1 -3 7 9 6 | 0 6 11 -2 -3 2 }}
{{Mapping|legend=0| 2 -1 -3 7 9 6 | 0 6 11 -2 -3 2 }}


Optimal tunings:  
Optimal tunings:  
Line 1,667: Line 1,668:
Comma list: 169/168, 221/220, 273/272, 325/324, 364/363
Comma list: 169/168, 221/220, 273/272, 325/324, 364/363


Mapping: {{mapping| 2 -1 -3 7 9 6 4 | 0 6 11 -2 -3 2 6 }}
{{Mapping|legend=0| 2 -1 -3 7 9 6 4 | 0 6 11 -2 -3 2 6 }}


Optimal tunings:  
Optimal tunings:  
Line 1,677: Line 1,678:
Badness (Sintel): 0.595
Badness (Sintel): 0.595


== Necromanteion ==
== Restles ==
Necromanteion, named by [[Johannes Werpup]] in 2014<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_106371.html Yahoo! Tuning Group | ''Temperament ideas: A cuckoo, and two oracles'']</ref> may be described as the {{nowrap| 31 & 51c }} temperament. The generator is a subfifth representing 35/24, four of which minus two octaves make slendric's generator, so its [[ploidacot]] is beta-dodecacot.  
{{See also| Lesser tendoneutralic }}
 
Restles may be described as the {{nowrap| 77 & 87 }} temperament, and has a [[ploidacot]] signature of gamma-dodecacot. It was named by [[Petr Pařízek]] in 2011 for it is some sort of opposite to [[beatles]]<ref name="petr's long post">[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group | ''Suggested names for the unclasified temperaments'']</ref>.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 1029/1024, 5103/5000
[[Comma list]]: 1029/1024, 153664/151875


{{Mapping|legend=1| 1 -5 -7 5 | 0 12 17 -4 }}
{{Mapping|legend=1| 1 -2 8 4 | 0 12 -19 -4 }}
: mapping generators: ~2, ~35/24
: mapping generators: ~2. ~315/256


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.2959{{c}}, ~35/24 = 658.3833{{c}}
* [[WE]]: ~2 = 1200.0322{{c}}, ~315/256 = 358.5581{{c}}
: [[error map]]: {{val| +0.296 -2.835 +4.130 -0.879 }}
: [[error map]]: {{val| +0.032 +0.678 +1.340 -2.930 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~35/24 = 658.2313{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~315/256 = 358.5484{{c}}
: error map: {{val| 0.000 -3.179 +3.619 -1.751 }}
: error map: {{val| 0.000 +0.626 +1.267 -3.019 }}


{{Optimal ET sequence|legend=1| 11c, 20c, 31, 144c, 175c }}
{{Optimal ET sequence|legend=1| 77, 87, 164 }}


[[Badness]] (Sintel): 2.98
[[Badness]] (Sintel): 2.73


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


Comma list: 176/175, 243/242, 1029/1024
Comma list: 385/384, 441/440, 153664/151875


Mapping: {{mapping| 1 -5 -7 5 -13 | 0 12 17 -4 30 }}
{{Mapping|legend=0| 1 -2 8 4 -7 | 0 12 -19 -4 35 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.2862{{c}}, ~22/15 = 658.4276{{c}}
* WE: ~2 = 1200.1110{{c}}, ~27/22 = 358.6045{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/15 = 658.2805{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~27/22 = 358.5720{{c}}


{{Optimal ET sequence|legend=0| 20ce, 31, 113c, 144c }}
{{Optimal ET sequence|legend=0| 77, 87, 164, 251d }}


Badness (Sintel): 1.77
Badness (Sintel): 1.81


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


Comma list: 144/143, 176/175, 243/242, 343/338
Comma list: 196/195, 352/351, 385/384, 676/675


Mapping: {{mapping| 1 -5 -7 5 -13 7 | 0 12 17 -4 30 -6 }}
{{Mapping|legend=0| 1 -2 8 4 -7 4 | 0 12 -19 -4 35 -1 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.3663{{c}}, ~22/15 = 658.0465{{c}}
* WE: ~2 = 1200.0482{{c}}, ~~16/13 = 358.5883{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/15 = 658.3800{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/13 = 358.5741{{c}}


{{Optimal ET sequence|legend=0| 20ce, 31, 82cf, 113cf }}
{{Optimal ET sequence|legend=0| 77, 87, 164, 251d }}


Badness (Sintel): 1.94
Badness (Sintel): 1.16


== Restles ==
== Necromanteion ==
{{See also| Lesser tendoneutralic }}
Necromanteion, named by [[Johannes Werpup]] in 2014<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_106371.html Yahoo! Tuning Group | ''Temperament ideas: A cuckoo, and two oracles'']</ref> may be described as the {{nowrap| 31 & 51c }} temperament. The generator is a subfifth representing ~[[35/24]], four of which minus two octaves make slendric's generator. Therefore, its [[ploidacot]] is wau-dodecacot.
 
Restles may be described as the {{nowrap| 77 & 87 }} temperament, and has a [[ploidacot]] signature of wau-dodecacot. It was named by [[Petr Pařízek]] in 2011 for it is some sort of opposite to [[beatles]]<ref name="petr's long post">[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group | ''Suggested names for the unclasified temperaments'']</ref>.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 1029/1024, 153664/151875
[[Comma list]]: 1029/1024, 5103/5000


{{Mapping|legend=1| 1 -2 8 4 | 0 12 -19 -4 }}
{{Mapping|legend=1| 1 -5 -7 5 | 0 12 17 -4 }}
: mapping generators: ~2. ~315/256
: mapping generators: ~2, ~35/24


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.0322{{c}}, ~315/256 = 358.5581{{c}}
* [[WE]]: ~2 = 1200.2959{{c}}, ~35/24 = 658.3833{{c}}
: [[error map]]: {{val| +0.032 +0.678 +1.340 -2.930 }}
: [[error map]]: {{val| +0.296 -2.835 +4.130 -0.879 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~315/256 = 358.5484{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~35/24 = 658.2313{{c}}
: error map: {{val| 0.000 +0.626 +1.267 -3.019 }}
: error map: {{val| 0.000 -3.179 +3.619 -1.751 }}


{{Optimal ET sequence|legend=1| 77, 87, 164 }}
{{Optimal ET sequence|legend=1| 11c, 20c, 31, 144c, 175c }}


[[Badness]] (Sintel): 2.73
[[Badness]] (Sintel): 2.98


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


Comma list: 385/384, 441/440, 153664/151875
Comma list: 176/175, 243/242, 1029/1024


Mapping: {{mapping| 1 -2 8 4 -7 | 0 12 -19 -4 35 }}
{{Mapping|legend=0| 1 -5 -7 5 -13 | 0 12 17 -4 30 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.1110{{c}}, ~27/22 = 358.6045{{c}}
* WE: ~2 = 1200.2862{{c}}, ~22/15 = 658.4276{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~27/22 = 358.5720{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/15 = 658.2805{{c}}


{{Optimal ET sequence|legend=0| 77, 87, 164, 251d }}
{{Optimal ET sequence|legend=0| 20ce, 31, 113c, 144c }}


Badness (Sintel): 1.81
Badness (Sintel): 1.77


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


Comma list: 196/195, 352/351, 385/384, 676/675
Comma list: 144/143, 176/175, 243/242, 343/338


Mapping: {{mapping| 1 -2 8 4 -7 4 | 0 12 -19 -4 35 -1 }}
{{Mapping|legend=0| 1 -5 -7 5 -13 7 | 0 12 17 -4 30 -6 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.0482{{c}}, ~~16/13 = 358.5883{{c}}
* WE: ~2 = 1199.3663{{c}}, ~22/15 = 658.0465{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/13 = 358.5741{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/15 = 658.3800{{c}}


{{Optimal ET sequence|legend=0| 77, 87, 164, 251d }}
{{Optimal ET sequence|legend=0| 20ce, 31, 82cf, 113cf }}


Badness (Sintel): 1.16
Badness (Sintel): 1.94


== Lagaca ==
== Lagaca ==
Cryptically named by [[Petr Pařízek]] in 2011<ref name="petr's long post"/>, lagaca may be described as the {{nowrap| 10 & 118 }} temperament with a [[ploidacot]] signature of diploid wau-enneacot. The name actually refers to the fact that 12 generator steps in this temperament make ~7/3, where "l", "g", "c" are integers alphabetically converted to letters.  
Cryptically named by [[Petr Pařízek]] in 2011<ref name="petr's long post"/>, lagaca may be described as the {{nowrap| 10 & 118 }} temperament with a [[ploidacot]] signature of diploid wau-enneacot. The name actually refers to the fact that 12 generator steps in this temperament make ~[[7/3]], where "l", "g", "c" are integers alphabetically converted to letters.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 1,826: Line 1,827:
Comma list: 385/384, 441/440, 800000/793881
Comma list: 385/384, 441/440, 800000/793881


Mapping: {{mapping| 1 -17 -26 9 7 | 0 21 32 -7 -4 }}
{{Mapping|legend=0| 1 -17 -26 9 7 | 0 21 32 -7 -4 }}


Optimal tunings:  
Optimal tunings:  
Line 1,841: Line 1,842:
Comma list: 325/324, 364/363, 385/384, 2200/2197
Comma list: 325/324, 364/363, 385/384, 2200/2197


Mapping: {{mapping| 1 -17 -26 9 7 -14 | 0 21 32 -7 -4 20 }}
{{Mapping|legend=0| 1 -17 -26 9 7 -14 | 0 21 32 -7 -4 20 }}


Optimal tunings:  
Optimal tunings:  
Line 1,878: Line 1,879:
Comma list: 385/384, 441/440, 43923/43750
Comma list: 385/384, 441/440, 43923/43750


Mapping: {{mapping| 1 -11 -7 7 -4 | 0 27 20 -9 16 }}
{{Mapping|legend=0| 1 -11 -7 7 -4 | 0 27 20 -9 16 }}


Optimal tunings:  
Optimal tunings:  
Line 1,910: Line 1,911:
Comma list: 385/384, 441/440, 234375/234256
Comma list: 385/384, 441/440, 234375/234256


Mapping: {{mapping| 1 16 8 -2 17 | 0 -33 -13 11 -31 }}
{{Mapping|legend=0| 1 16 8 -2 17 | 0 -33 -13 11 -31 }}


Optimal tunings:  
Optimal tunings:  
Line 1,925: Line 1,926:
Comma list: 385/384, 441/440, 625/624, 847/845
Comma list: 385/384, 441/440, 625/624, 847/845


Mapping: {{mapping| 1 16 8 -2 17 12 | 0 -33 -13 11 -31 -19 }}
{{Mapping|legend=0| 1 16 8 -2 17 12 | 0 -33 -13 11 -31 -19 }}


Optimal tunings:  
Optimal tunings:  
Line 1,936: Line 1,937:


== Other subgroup extensions ==
== Other subgroup extensions ==
=== Euslendric ===
=== Euslendric (2.3.7.13) ===
Forms of slendric in the most optimal range for the 2.3.7 temperament ({{nowrap| 36 & 77 }}) lack an obvious strong mapping of prime 5 or prime 11. However, slendric can extend well to the no-fives no-elevens [[29-limit]] by tempering out [[273/272]], [[343/342]], [[378/377]], [[392/391]], [[513/512]], and [[729/728]], or a comma basis defined in terms of [[S-expression]]s as {S7/S8, S14/S16, S15/S20, S24/S26, S27, S28}. [[113edo]] is an obvious tuning.
Forms of slendric in the most optimal range for the 2.3.7 temperament ({{nowrap| 36 & 77 }}) lack an obvious strong mapping of prime 5 or prime 11. However, slendric can extend well to the no-fives no-elevens [[29-limit]] by tempering out [[273/272]], [[343/342]], [[378/377]], [[392/391]], [[513/512]], and [[729/728]], or a comma basis defined in terms of [[S-expression]]s as {S7/S8, S14/S16, S15/S20, S24/S26, S27, S28}. [[113edo]] is an obvious tuning.


Line 2,023: Line 2,024:
Badness (Sintel): 0.473
Badness (Sintel): 0.473


=== Baladic ===
=== Baladic (2.3.7.13) ===
Baladic is a 2.3.7.13.17-subgroup temperament that attempts to approximate the Maqam Sikah Baladi scale. It tempers out [[169/168]] ({{S|13}}), which splits [[7/6]] in half ([[13/12]]~[[14/13]]) and one finds that the octave is therefore split in half via the interval [[91/64]], which is then equated to [[17/12]]. 36edo is an excellent baladic tuning.
Baladic is a 2.3.7.13.17-subgroup temperament that attempts to approximate the Maqam Sikah Baladi scale. It tempers out [[169/168]] ({{S|13}}), which splits [[7/6]] in half ([[13/12]]~[[14/13]]) and one finds that the octave is therefore split in half via the interval [[91/64]], which is then equated to [[17/12]]. 36edo is an excellent baladic tuning.


Line 2,059: Line 2,060:


Badness (Sintel): 0.253
Badness (Sintel): 0.253
=== Gigapyth (2.3.7.85) ===
Subgroup: 2.3.7.85
Comma list: 1029/1024, 7225/7203
Subgroup-val mapping: {{mapping| 1 -2 4 7 | 0 6 -2 -1 }}
Optimal tunings:
* WE: ~2 = 1200.8295{{c}}, ~128/85 = 717.2597{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~128/85 = 716.7933{{c}}
{{Optimal ET sequence|legend=0| 5, 42*, 47, 52, 57, 62, 67, 72, 149*, 370d***, 519bdd***** }}
<nowiki/>* Wart for 85


== References ==
== References ==


[[Category:Gamelismic clan| ]] <!-- main article -->
[[Category:Temperament clans]]
[[Category:Temperament clans]]
[[Category:Gamelismic clan| ]] <!-- main article -->
[[Category:Catalogs of rank-2 temperaments]]
[[Category:Rank 2]]
[[Category:Listen]]
[[Category:Listen]]
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