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Kleismic family: Difference between revisions

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{{interwiki
| en = Kleismic family
| de = Hanson-Kleismisch
| es =
| ja =
}}
{{Technical data page}}
{{Technical data page}}
The [[5-limit]] parent comma for the '''kleismic family''' is [[15625/15552]], the kleisma, which is the amount by which a stack of six [[6/5|classical minor third]]s falls short of the [[3/1|3rd]] [[harmonic]].  
The [[5-limit]] parent comma for the '''kleismic family''' is [[15625/15552]], the kleisma, which is the amount by which a stack of six [[6/5|classical minor third]]s falls short of the [[3/1|3rd harmonic]], or equivalently the amount by which a stack of three [[25/24]] overshoots the [[9/8]] whole tone. As a result, all kleismic temperaments trisect 9/8, for which one third-tone represents [[25/24]] and two represents [[27/25]].  


== Kleismic a.k.a. hanson ==
== Kleismic a.k.a. hanson ==
{{Main| Kleismic }}
{{Main| Kleismic }}


The [[generator]] of kleismic is a [[6/5|classical minor third]], and to get to the interval class of [[5/4|major thirds]] requires five of these, and so to get to [[3/2|fifths]] requires six. In fact, (6/5)<sup>5</sup> = (5/2)⋅(15625/15552). This 5-limit temperament (virtually a [[microtemperament]]) is sometimes called ''hanson'', and [[53edo|14\53]] is about perfect as a generator, though [[34edo|9\34]] also makes sense, and [[19edo|5\19]] and [[15edo|4\15]] are possible. Other tunings include [[72edo]], [[87edo]] and [[140edo]].
The [[generator]] of kleismic is a [[6/5|classical minor third]], and to get to the interval class of [[5/4|major thirds]] requires five of these, and so to get to [[3/2|perfect fifths]] requires six. In fact, (6/5)<sup>5</sup> = (5/2)⋅(15625/15552). This 5-limit temperament (virtually a [[microtemperament]]) is sometimes called ''hanson'', and [[53edo|14\53]] is about perfect as a generator, though [[34edo|9\34]] also makes sense, and [[19edo|5\19]] and [[15edo|4\15]] are possible. Other edo tunings include [[72edo]], [[87edo]] and [[140edo]].


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
Line 35: Line 29:


=== Overview to extensions ===
=== Overview to extensions ===
The second comma of the [[normal forms #Normal forms for commas|normal comma list]] defines which [[7-limit]] family member we are looking at. [[875/864]], the keemic comma, gives keemun. [[4375/4374]], the ragisma, gives catakleismic. [[5120/5103]], hemifamity, gives countercata. Keemun, catakleismic and countercata all have octave period and use the minor third as a generator; catakleismic and countercata define the 7/4 more complexly but more accurately than keemun.  
==== Full 7-limit extensions ====
The second comma of the [[normal forms #Normal forms for commas|normal comma list]] defines which [[7-limit]] family member we are looking at. [[4375/4374]], the ragisma, gives catakleismic. [[875/864]], the keemic comma, gives keemun. [[5120/5103]], the aberschisma, gives countercata. [[179200/177147]], the tolerant comma, gives metakleismic. [[64/63]], the archytas comma, gives catalan. Catakleismic, keemun, countercata, metakleismic, and catalan all have octave period and use the minor third as a generator; catakleismic, countercata, and metakleismic define the 7/4 more complexly but more accurately than keemun and catalan.  


[[6144/6125]], the porwell comma, gives hemikleismic. [[245/243]], sensamagic, gives clyde. [[1029/1024]], the gamelisma, gives tritikleismic. [[2401/2400]] the breedsma, gives quadritikleismic. Hemikleismic splits the 6/5 in half to get a neutral second generator of 35/32, and clyde similarly splits the 5/3 in half to get a 9/7 generator. Finally, tritikleismic has a 1/3-octave period with minor third generator, and quadritikleismic a 1/4-octave period with the minor third generator.
[[6144/6125]], the porwell comma, gives [[#Hemikleismic|hemikleismic]]. [[245/243]], sensamagic, gives [[#Clyde|clyde]]. [[1029/1024]], the gamelisma, gives [[#Tritikleismic|tritikleismic]]. [[10976/10935]], hemimage, gives [[#Marfifths|marfifths]]. [[1728/1715]], the orwellismia, gives [[#Kleiboh|kleiboh]]. [[2401/2400]], the breedsma, gives [[#Quadritikleismic|quadritikleismic]]. [[2460375/2458624]], the breeze comma, gives [[#Marthirds|marthirds]]. Hemikleismic splits the 6/5 in half to get a neutral second generator of ~35/32, and clyde similarly splits the 5/3 in half to get a ~9/7 generator. Marfifths splits the 12/5 into three. Kleiboh splits the 24/5 into three. Marthirds splits the 12/5 into four. Finally, tritikleismic has a 1/3-octave period with minor third generator, and quadritikleismic a 1/4-octave period with the minor third generator.  
 
Temperaments involving larger splits include [[#Sqrtphi|sqrtphi]], [[#Quartkeenlig|quartkeenlig]], [[#Novemkleismic|novemkleismic]]. Those split the kleismic structure into five to nine parts.
 
==== Subgroup extensions ====
Kleismic lends itself to a natural extension to the [[2.3.5.13 subgroup]], as the hemitwelfth, reached by three generator steps, can be interpreted as [[26/15]]. This is discussed immediately below.
 
=== 2.3.5.13 subgroup (cata) ===
The kleisma has the [[S-expression]] S25<sup>2</sup>⋅S26, which explains how the tone is trisected, and thus tempering out its constituents [[625/624]] ({{S|25}}) and [[676/675]] ({{S|26}}) is a readily natural subgroup extension. As a result of coincidental S-expression equivalences, [[325/324]] (S25⋅S26) is also tempered out, and two generator steps stand in for ~[[13/9]]. Finally, since this step is one third of a perfect twelfth, the comma [[2197/2187]] (S25⋅S26<sup>2</sup>) is tempered out. The third tone is now 25/24~26/25~27/26, which means 27/25 and [[13/12]] are made equal.
 
For a version of cata with a 1\5 period, see [[Thunderclysmic]].
 
Subgroup: 2.3.5.13
 
Comma list: 325/324, 625/624
 
{{Mapping|legend=2| 1 0 1 0 | 0 6 5 14 }}
 
Optimal tunings:
* WE: ~2 = 1200.1210{{c}}, ~6/5 = 317.1076{{c}}
: error map: {{val| +0.121 +0.690 -0.655 -1.022 }}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0920{{c}}
: error map: {{val| 0.000 +0.597 -0.854 -1.240 }}
 
{{Optimal ET sequence|legend=0| 15, 19, 34, 53, 140, 193, 246 }}
 
Badness (Sintel): 0.131


== Catakleismic ==
== Catakleismic ==
{{Main| Catakleismic }}
{{Main| Catakleismic }}
Catakleismic tempers out 225/224, the [[marvel comma]], and 4375/4374, the [[ragisma]], and may be described as the {{nowrap| 53 & 72 }} temperament. [[125edo]] and especially [[197edo]] make for excellent tunings. In here, the JI intervals 25/24–27/25–9/8–7/6 are made equidistant.


=== 7-limit ===
=== 7-limit ===
Line 64: Line 87:


==== 2.3.5.7.13 subgroup ====
==== 2.3.5.7.13 subgroup ====
The [[S-expression]]-based comma list of this temperament is {[[169/168|S13]], [[225/224|S15 = S25*S26*S27]], [[325/324|S10/S12 = S25*S26]], ([[625/624|S25]], [[676/675|S26 = S13/S15]], [[729/728|S27]])}.  
Catakleismic extends easily with [[prime interval|prime]] [[13/1|13]] as seen before with cata, which has the S-expression list {S25, S26}. The marvel comma has an equivalent S-expression S25⋅S26⋅S27 and the ragisma S25/S27, thus [[729/728]] ({{S|27}}) is another constituent that can be naturally tempered out. As a result, the third tone becomes 25/24~26/25~27/26~28/27, and the equivalence 27/25~13/12 is added.
 
Add-13 catakleismic is less accurate than cata or catakleismic alone as the error of 7 and 13 go in separate directions – cata wants a sharper 6/5 but catakleismic wants a flatter one. Nontheless, its structural properties make it quite notable.


Subgroup: 2.3.5.7.13
Subgroup: 2.3.5.7.13
Line 70: Line 95:
Comma list: 169/168, 225/224, 325/324
Comma list: 169/168, 225/224, 325/324


Subgroup-val mapping: {{mapping| 1 0 1 -3 0 | 0 6 5 22 14 }}
{{Mapping|legend=2| 1 0 1 -3 0 | 0 6 5 22 14 }}


Optimal tunings:  
Optimal tunings:  
Line 85: Line 110:
Comma list: 225/224, 385/384, 4375/4374
Comma list: 225/224, 385/384, 4375/4374


Mapping: {{mapping| 1 0 1 -3 9 | 0 6 5 22 -21 }}
{{Mapping|legend=0| 1 0 1 -3 9 | 0 6 5 22 -21 }}


Optimal tunings:  
Optimal tunings:  
Line 104: Line 129:
Comma list: 169/168, 225/224, 325/324, 385/384
Comma list: 169/168, 225/224, 325/324, 385/384


Mapping: {{mapping| 1 0 1 -3 9 0 | 0 6 5 22 -21 14 }}
{{Mapping|legend=0| 1 0 1 -3 9 0 | 0 6 5 22 -21 14 }}


Optimal tunings:  
Optimal tunings:  
Line 123: Line 148:
Comma list: 99/98, 176/175, 2200/2187
Comma list: 99/98, 176/175, 2200/2187


Mapping: {{mapping| 1 0 1 -3 -5 | 0 6 5 22 32 }}
{{Mapping|legend=0| 1 0 1 -3 -5 | 0 6 5 22 32 }}


Optimal tunings:  
Optimal tunings:  
Line 138: Line 163:
Comma list: 99/98, 169/168, 176/175, 275/273
Comma list: 99/98, 169/168, 176/175, 275/273


Mapping: {{mapping| 1 0 1 -3 -5 0 | 0 6 5 22 32 14 }}
{{Mapping|legend=0| 1 0 1 -3 -5 0 | 0 6 5 22 32 14 }}


Optimal tunings:  
Optimal tunings:  
Line 153: Line 178:
Comma list: 225/224, 441/440, 4375/4374
Comma list: 225/224, 441/440, 4375/4374


Mapping: {{mapping| 1 0 1 -3 -10 | 0 6 5 22 51 }}
{{Mapping|legend=0| 1 0 1 -3 -10 | 0 6 5 22 51 }}


Optimal tunings:  
Optimal tunings:  
Line 168: Line 193:
Comma list: 169/168, 225/224, 325/324, 1716/1715
Comma list: 169/168, 225/224, 325/324, 1716/1715


Mapping: {{mapping| 1 0 1 -3 -10 0 | 0 6 5 22 51 14 }}
{{Mapping|legend=0| 1 0 1 -3 -10 0 | 0 6 5 22 51 14 }}


Optimal tunings:  
Optimal tunings:  
Line 183: Line 208:
Comma list: 100/99, 225/224, 864/847
Comma list: 100/99, 225/224, 864/847


Mapping: {{mapping| 1 0 1 -3 4 | 0 6 5 22 -2 }}
{{Mapping|legend=0| 1 0 1 -3 4 | 0 6 5 22 -2 }}


Optimal tunings:  
Optimal tunings:  
Line 198: Line 223:
Comma list: 78/77, 100/99, 144/143, 676/675
Comma list: 78/77, 100/99, 144/143, 676/675


Mapping: {{mapping| 1 0 1 -3 4 0 | 0 6 5 22 -2 14 }}
{{Mapping|legend=0| 1 0 1 -3 4 0 | 0 6 5 22 -2 14 }}


Optimal tunings:  
Optimal tunings:  
Line 213: Line 238:
Comma list: 225/224, 243/242, 4375/4356
Comma list: 225/224, 243/242, 4375/4356


Mapping: {{mapping| 2 0 2 -6 -1 | 0 6 5 22 15 }}
{{Mapping|legend=0| 2 0 2 -6 -1 | 0 6 5 22 15 }}
: mapping generators: ~99/70, ~6/5
: mapping generators: ~99/70, ~6/5


Line 229: Line 254:
Comma list: 169/168, 225/224, 243/242, 325/324
Comma list: 169/168, 225/224, 243/242, 325/324


Mapping: {{mapping| 2 0 2 -6 -1 0 | 0 6 5 22 15 14 }}
{{Mapping|legend=0| 2 0 2 -6 -1 0 | 0 6 5 22 15 14 }}


Optimal tunings:  
Optimal tunings:  
Line 244: Line 269:
Comma list: 169/168, 221/220, 225/224, 243/242, 325/324
Comma list: 169/168, 221/220, 225/224, 243/242, 325/324


Mapping: {{mapping| 2 0 2 -6 -1 0 5 | 0 6 5 22 15 14 6 }}
{{Mapping|legend=0| 2 0 2 -6 -1 0 5 | 0 6 5 22 15 14 6 }}


Optimal tunings:  
Optimal tunings:  
Line 259: Line 284:
Comma list: 153/152, 169/168, 221/220, 225/224, 243/242, 325/324
Comma list: 153/152, 169/168, 221/220, 225/224, 243/242, 325/324


Mapping: {{mapping| 2 0 2 -6 -1 0 5 -1 | 0 6 5 22 15 14 6 18 }}
{{Mapping|legend=0| 2 0 2 -6 -1 0 5 -1 | 0 6 5 22 15 14 6 18 }}


Optimal tunings:  
Optimal tunings:  
Line 298: Line 323:
Comma list: 49/48, 56/55, 100/99
Comma list: 49/48, 56/55, 100/99


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


Optimal tunings:  
Optimal tunings:  
Line 317: Line 342:
Comma list: 49/48, 56/55, 65/64, 100/99
Comma list: 49/48, 56/55, 65/64, 100/99


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


Optimal tunings:  
Optimal tunings:  
Line 336: Line 361:
Comma list: 49/48, 56/55, 91/90, 100/99
Comma list: 49/48, 56/55, 91/90, 100/99


Mapping: {{mapping| 1 0 1 2 4 0 | 0 6 5 3 -2 14 }}
{{Mapping|legend=0| 1 0 1 2 4 0 | 0 6 5 3 -2 14 }}


Optimal tunings:  
Optimal tunings:  
Line 356: Line 381:
Comma list: 40/39, 49/48, 56/55, 66/65
Comma list: 40/39, 49/48, 56/55, 66/65


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


Optimal tunings:  
Optimal tunings:  
Line 371: Line 396:
Comma list: 45/44, 49/48, 126/125
Comma list: 45/44, 49/48, 126/125


Mapping: {{mapping| 1 0 1 2 -1 | 0 6 5 3 17 }}
{{Mapping|legend=0| 1 0 1 2 -1 | 0 6 5 3 17 }}


Optimal tunings:  
Optimal tunings:  
Line 386: Line 411:
Comma list: 45/44, 49/48, 78/77, 126/125
Comma list: 45/44, 49/48, 78/77, 126/125


Mapping: {{mapping| 1 0 1 2 -1 0 | 0 6 5 3 17 14 }}
{{Mapping|legend=0| 1 0 1 2 -1 0 | 0 6 5 3 17 14 }}


Optimal tunings:  
Optimal tunings:  
Line 401: Line 426:
Comma list: 49/48, 55/54, 77/75
Comma list: 49/48, 55/54, 77/75


Mapping: {{mapping| 1 0 1 2 0 | 0 6 5 3 13 }}
{{Mapping|legend=0| 1 0 1 2 0 | 0 6 5 3 13 }}


Optimal tunings:  
Optimal tunings:  
Line 416: Line 441:
Comma list: 49/48, 55/54, 66/65, 77/75
Comma list: 49/48, 55/54, 66/65, 77/75


Mapping: {{mapping| 1 0 1 2 0 0 | 0 6 5 3 13 14 }}
{{Mapping|legend=0| 1 0 1 2 0 0 | 0 6 5 3 13 14 }}


Optimal tunings:  
Optimal tunings:  
Line 452: Line 477:
Comma list: 64/63, 100/99, 1331/1323
Comma list: 64/63, 100/99, 1331/1323


Mapping: {{mapping| 1 0 1 6 4 | 0 6 5 -12 -2 }}
{{Mapping|legend=0| 1 0 1 6 4 | 0 6 5 -12 -2 }}


Optimal tunings:  
Optimal tunings:  
Line 471: Line 496:
Comma list: 64/63, 100/99, 144/143, 275/273
Comma list: 64/63, 100/99, 144/143, 275/273


Mapping: {{mapping| 1 0 1 6 4 0 | 0 6 5 -12 -2 14 }}
{{Mapping|legend=0| 1 0 1 6 4 0 | 0 6 5 -12 -2 14 }}


Optimal tunings:  
Optimal tunings:  
Line 507: Line 532:
Comma list: 385/384, 2200/2187, 3388/3375
Comma list: 385/384, 2200/2187, 3388/3375


Mapping: {{mapping| 1 0 1 11 -5 | 0 6 5 -31 32 }}
{{Mapping|legend=0| 1 0 1 11 -5 | 0 6 5 -31 32 }}


Optimal tunings:  
Optimal tunings:  
Line 526: Line 551:
Comma list: 325/324, 352/351, 385/384, 625/624
Comma list: 325/324, 352/351, 385/384, 625/624


Mapping: {{mapping| 1 0 1 11 -5 0 | 0 6 5 -31 32 14 }}
{{Mapping|legend=0| 1 0 1 11 -5 0 | 0 6 5 -31 32 14 }}


Optimal tunings:  
Optimal tunings:  
Line 563: Line 588:
Comma list: 896/891, 2200/2187, 14700/14641
Comma list: 896/891, 2200/2187, 14700/14641


Mapping: {{mapping| 1 0 1 -12 -5 | 0 6 5 56 32 }}
{{Mapping|legend=0| 1 0 1 -12 -5 | 0 6 5 56 32 }}


Optimal tunings:  
Optimal tunings:  
Line 578: Line 603:
Comma list: 325/324, 352/351, 364/363, 625/624
Comma list: 325/324, 352/351, 364/363, 625/624


Mapping: {{mapping| 1 0 1 -12 -5 0 | 0 6 5 56 32 14 }}
{{Mapping|legend=0| 1 0 1 -12 -5 0 | 0 6 5 56 32 14 }}


Optimal tunings:  
Optimal tunings:  
Line 611: Line 636:
Comma list: 121/120, 176/175, 4000/3969
Comma list: 121/120, 176/175, 4000/3969


Mapping: {{mapping| 1 0 1 4 2 | 0 12 10 -9 11 }}
{{Mapping|legend=0| 1 0 1 4 2 | 0 12 10 -9 11 }}


Optimal tunings:  
Optimal tunings:  
Line 626: Line 651:
Comma list: 121/120, 176/175, 275/273, 325/324
Comma list: 121/120, 176/175, 275/273, 325/324


Mapping: {{mapping| 1 0 1 4 2 0 | 0 12 10 -9 11 28 }}
{{Mapping|legend=0| 1 0 1 4 2 0 | 0 12 10 -9 11 28 }}


Optimal tunings:  
Optimal tunings:  
Line 666: Line 691:
Comma list: 245/243, 385/384, 3136/3125
Comma list: 245/243, 385/384, 3136/3125


Mapping: {{mapping| 1 -6 -4 -13 18 | 0 12 10 25 -23 }}
{{Mapping|legend=0| 1 -6 -4 -13 18 | 0 12 10 25 -23 }}


Optimal tunings:  
Optimal tunings:  
Line 681: Line 706:
Comma list: 196/195, 245/243, 385/384, 625/624
Comma list: 196/195, 245/243, 385/384, 625/624


Mapping: {{mapping| 1 -6 -4 -13 18 -14 | 0 12 10 25 -23 28 }}
{{Mapping|legend=0| 1 -6 -4 -13 18 -14 | 0 12 10 25 -23 28 }}


Optimal tunings:  
Optimal tunings:  
Line 717: Line 742:
[[Badness]] (Sintel): 1.43
[[Badness]] (Sintel): 1.43


Music:
; Music
* ''[https://www.youtube.com/watch?v=vdjhC9i5KF4 Four Short Experiments in Octave Stretched 42edo (Dec 2024)]'' by [[Budjarn Lambeth]]
* [https://www.youtube.com/watch?v=vdjhC9i5KF4 ''Four Short Experiments in Octave Stretched 42edo''] (2024) by [[Budjarn Lambeth]]


=== 11-limit ===
=== 11-limit ===
Line 725: Line 750:
Comma list: 385/384, 441/440, 4000/3993
Comma list: 385/384, 441/440, 4000/3993


Mapping: {{mapping| 3 0 3 10 8 | 0 6 5 -2 3 }}
{{Mapping|legend=0| 3 0 3 10 8 | 0 6 5 -2 3 }}


Optimal tunings:  
Optimal tunings:  
Line 745: Line 770:
Comma list: 325/324, 364/363, 385/384, 625/624
Comma list: 325/324, 364/363, 385/384, 625/624


Mapping: {{mapping| 3 0 3 10 8 0 | 0 6 5 -2 3 14 }}
{{Mapping|legend=0| 3 0 3 10 8 0 | 0 6 5 -2 3 14 }}


Optimal tunings:  
Optimal tunings:  
Line 760: Line 785:
Comma list: 273/272, 325/324, 364/363, 375/374, 385/384
Comma list: 273/272, 325/324, 364/363, 375/374, 385/384


Mapping: {{mapping| 3 0 3 10 8 0 -2 | 0 6 5 -2 3 14 18 }}
{{Mapping|legend=0| 3 0 3 10 8 0 -2 | 0 6 5 -2 3 14 18 }}


Optimal tunings:  
Optimal tunings:  
Line 769: Line 794:


Badness (Sintel): 0.690
Badness (Sintel): 0.690
== Quadritikleismic ==
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 2401/2400, 15625/15552
{{Mapping|legend=1| 4 0 4 7 | 0 6 5 4 }}
: mapping generators: ~25/21, ~6/5
[[Optimal tuning]]s:
* [[WE]]: ~25/21 = 300.0520{{c}}, ~6/5 = 317.0548{{c}} (~126/125 = 17.0029{{c}})
: [[error map]]: {{val| +0.208 +0.374 -0.832 -0.243 }}
* [[CWE]]: ~25/21 = 300.0000{{c}}, ~6/5 = 317.0301{{c}} (~126/125 = 17.0301{{c}})
: error map: {{val| 0.000 +0.225 -1.163 -0.706 }}
{{Optimal ET sequence|legend=1| 68, 72, 140, 212, 776cd, 988ccd, 1200ccd }}
[[Badness]] (Sintel): 0.993
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 385/384, 1375/1372, 6250/6237
Mapping: {{mapping| 4 0 4 7 17 | 0 6 5 4 -3 }}
Optimal tunings:
* WE: ~25/21 = 300.0995{{c}}, ~6/5 = 317.0298{{c}} (~100/99 = 16.9303{{c}})
* CWE: ~25/21 = 300.0000{{c}}, ~6/5 = 316.9540{{c}} (~100/99 = 16.9540{{c}})
{{Optimal ET sequence|legend=0| 68, 72, 140, 212, 284, 496ce, 780ccdee }}
Badness (Sintel): 0.774
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 325/324, 385/384, 625/624, 1375/1372
Mapping: {{mapping| 4 0 4 7 17 0 | 0 6 5 4 -3 14 }}
Optimal tunings:
* WE: ~25/21 = 300.0985{{c}}, ~6/5 = 317.0899{{c}} (~100/99 = 16.9941{{c}})
* CWE: ~25/21 = 300.0000{{c}}, ~6/5 = 317.0155{{c}} (~100/99 = 17.0155{{c}})
{{Optimal ET sequence|legend=0| 68, 72, 140, 212 }}
Badness (Sintel): 0.774
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Comma list: 289/288, 325/324, 385/384, 442/441, 625/624
Mapping: {{mapping| 4 0 4 7 17 0 10 | 0 6 5 4 -3 14 6 }}
Optimal tunings:
* WE: ~25/21 = 300.1102{{c}}, ~6/5 = 317.1011{{c}} (~100/99 = 16.9909{{c}})
* CWE: ~25/21 = 300.0000{{c}}, ~6/5 = 317.0155{{c}} (~100/99 = 17.0155{{c}})
{{Optimal ET sequence|legend=0| 68, 72, 140, 212g }}
Badness (Sintel): 0.651
== Kleiboh ==
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 1728/1715, 3125/3087
{{Mapping|legend=1| 1 -12 -9 -7 | 0 18 15 13 }}
: mapping generators: ~2, ~42/25
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.5290{{c}}, ~42/25 = 905.3417{{c}}
: [[error map]]: {{val| -0.471 -0.152 -1.949 +3.914 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~42/25 = 905.6741{{c}}
: error map: {{val| 0.000 +0.178 -1.203 +4.937 }}
{{Optimal ET sequence|legend=1| 49, 53 }}
[[Badness]] (Sintel): 1.93
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 176/175, 540/539, 3125/3087
Mapping: {{mapping| 1 -12 -9 -7 -29 | 0 18 15 13 43 }}
Optimal tunings:
* WE: ~2 = 1199.1389{{c}}, ~42/25 = 905.1688{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~42/25 = 905.7840{{c}}
{{Optimal ET sequence|legend=0| 49, 53, 102d }}
Badness (Sintel): 1.75
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 176/175, 275/273, 325/324, 540/539
Mapping: {{mapping| 1 -12 -9 -7 -29 -28 | 0 18 15 13 43 42 }}
Optimal tunings:
* WE: ~2 = 1199.1517{{c}}, ~22/13 = 905.1727{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/13 = 905.7801{{c}}
{{Optimal ET sequence|legend=0| 49f, 53, 102df }}
Badness (Sintel): 1.28


== Marfifths ==
== Marfifths ==
The ''marfifths'' temperament (19 & 140) tempers out the [[hemimage comma]], 10976/10935. It splits the interval of a major thirteenth (~10/3) into three marvelous fifth ([[112/75]]) intervals, and uses it for a generator.
Named by [[Xenllium]] in 2021, marfifths tempers out the 10976/10935, the [[hemimage comma]], and may be described as the {{nowrap| 19 & 140 }} temperament. It is generated by a marvel fourth of [[75/56]] (or a marvel fifth of [[112/75]]), three of which minus an octave make the hanson generator of ~6/5. Its [[ploidacot]] is zeta-18-cot.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 906: Line 820:
Comma list: 385/384, 6250/6237, 10976/10935
Comma list: 385/384, 6250/6237, 10976/10935


Mapping: {{mapping| 1 -6 -4 -17 22 | 0 18 15 47 -44 }}
{{Mapping|legend=0| 1 -6 -4 -17 22 | 0 18 15 47 -44 }}


Optimal tunings:  
Optimal tunings:  
Line 921: Line 835:
Comma list: 325/324, 385/384, 625/624, 10976/10935
Comma list: 325/324, 385/384, 625/624, 10976/10935


Mapping: {{mapping| 1 -6 -4 -17 22 -14 | 0 18 15 47 -44 42 }}
{{Mapping|legend=0| 1 -6 -4 -17 22 -14 | 0 18 15 47 -44 42 }}


Optimal tunings:  
Optimal tunings:  
Line 932: Line 846:


=== Diatessic ===
=== Diatessic ===
The ''diatessic'' temperament (121 & 140) is closely related to the '''diatess tuning''' (generator: 505.727281 cents).
Diatessic may be described as {{nowrap| 121 & 140 }} and is closely related to the Diatess tuning (generator: 505.727281 cents).


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 938: Line 852:
Comma list: 1375/1372, 2200/2187, 5632/5625
Comma list: 1375/1372, 2200/2187, 5632/5625


Mapping: {{mapping| 1 -6 -4 -17 -37 | 0 18 15 47 96 }}
{{Mapping|legend=0| 1 -6 -4 -17 -37 | 0 18 15 47 96 }}


Optimal tunings:  
Optimal tunings:  
Line 953: Line 867:
Comma list: 325/324, 352/351, 625/624, 1375/1372
Comma list: 325/324, 352/351, 625/624, 1375/1372


Mapping: {{mapping| 1 -6 -4 -17 -37 -14 | 0 18 15 47 96 42 }}
{{Mapping|legend=0| 1 -6 -4 -17 -37 -14 | 0 18 15 47 96 42 }}


Optimal tunings:  
Optimal tunings:  
Line 964: Line 878:


=== Marf ===
=== Marf ===
The ''marf'' temperament (19 & 121) has a POTE generator which strongly approximates the marvelous fifth interval of 112/75.
Marf may be described as {{nowrap| 19 & 121 }}. It has a POTE generator which strongly approximates the marvelous fifth interval of 112/75.


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 970: Line 884:
Comma list: 540/539, 896/891, 15625/15552
Comma list: 540/539, 896/891, 15625/15552


Mapping: {{mapping| 1 -6 -4 -17 14 | 0 18 15 47 -25 }}
{{Mapping|legend=0| 1 -6 -4 -17 14 | 0 18 15 47 -25 }}


Optimal tunings:  
Optimal tunings:  
Line 985: Line 899:
Comma list: 325/324, 540/539, 625/624, 896/891
Comma list: 325/324, 540/539, 625/624, 896/891


Mapping: {{mapping| 1 -6 -4 -17 14 -14 | 0 18 15 47 -25 42 }}
{{Mapping|legend=0| 1 -6 -4 -17 14 -14 | 0 18 15 47 -25 42 }}


Optimal tunings:  
Optimal tunings:  
Line 995: Line 909:
Badness (Sintel): 1.58
Badness (Sintel): 1.58


== Marthirds ==
== Kleiboh ==
The ''marthirds'' temperament (19 & 193) tempers out the breeze comma (laquadru-atruyo comma), [[2460375/2458624]]. It splits the interval of minor tenth (~12/5) into four marvelous major third ([[56/45]]) intervals, and uses it for a generator.
 
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 15625/15552, 2460375/2458624
[[Comma list]]: 1728/1715, 3125/3087


{{Mapping|legend=1| 1 -6 -4 -19 | 0 24 20 69 }}
{{Mapping|legend=1| 1 -12 -9 -7 | 0 18 15 13 }}
: mapping generators: ~2, ~56/45
: mapping generators: ~2, ~42/25


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.1662{{c}}, ~56/45 = 379.3041{{c}}
* [[WE]]: ~2 = 1199.5290{{c}}, ~42/25 = 905.3417{{c}}
: [[error map]]: {{val| +0.166 +0.347 -0.896 +0.000 }}
: [[error map]]: {{val| -0.471 -0.152 -1.949 +3.914 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~56/45 = 379.2552{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~42/25 = 905.6741{{c}}
: error map: {{val| 0.000 +0.171 -1.209 -0.214 }}
: error map: {{val| 0.000 +0.178 -1.203 +4.937 }}


{{Optimal ET sequence|legend=1| 19, …, 193, 212, 617c, 829c }}
{{Optimal ET sequence|legend=1| 49, 53 }}


[[Badness]] (Sintel): 2.64
[[Badness]] (Sintel): 1.93


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


Comma list: 1375/1372, 15625/15552, 19712/19683
Comma list: 176/175, 540/539, 3125/3087


Mapping: {{mapping| 1 -6 -4 -19 -43 | 0 24 20 69 147 }}
{{Mapping|legend=0| 1 -12 -9 -7 -29 | 0 18 15 13 43 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.1189{{c}}, ~56/45 = 379.2942{{c}}
* WE: ~2 = 1199.1389{{c}}, ~42/25 = 905.1688{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~56/45 = 379.2580{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~42/25 = 905.7840{{c}}


{{Optimal ET sequence|legend=0| 19e, …, 193, 212, 405, 617c }}
{{Optimal ET sequence|legend=0| 49, 53, 102d }}


Badness (Sintel): 2.50
Badness (Sintel): 1.75


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


Comma list: 325/324, 625/624, 1375/1372, 19712/19683
Comma list: 176/175, 275/273, 325/324, 540/539


Mapping: {{mapping| 1 -6 -4 -19 -43 -14 | 0 24 20 69 147 56 }}
{{Mapping|legend=0| 1 -12 -9 -7 -29 -28 | 0 18 15 13 43 42 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.2154{{c}}, ~56/45 = 379.3236{{c}}
* WE: ~2 = 1199.1517{{c}}, ~22/13 = 905.1727{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~56/45 = 379.2580{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/13 = 905.7801{{c}}


{{Optimal ET sequence|legend=0| 19e, …, 193, 212, 405f, 617cff }}
{{Optimal ET sequence|legend=0| 49f, 53, 102df }}


Badness (Sintel): 1.81
Badness (Sintel): 1.28
 
== Quartkeenlig ==
Quartkeenlig uses a generator in the 11-limit that is 33/32~36/35 tempered together, and is called so because it tempers out the [[quartisma]] by virtue of five 33/32's being with 7/6, keenanisma, 385/384, tempering 33/32 and 36/35 together, and liganellus comma (6250/6237). It can also be viewed as a regular temperament interpretation of [[23edo and octave stretching|stretched 23edo]].


== Quadritikleismic ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 15625/15552, 117649/116640
[[Comma list]]: 2401/2400, 15625/15552


{{Mapping|legend=1| 1 0 1 1 | 0 36 30 41 }}
{{Mapping|legend=1| 4 0 4 7 | 0 6 5 4 }}
: mapping generator: ~2, ~36/35
: mapping generators: ~25/21, ~6/5


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.2825{{c}}, ~36/35 = 52.8528{{c}}
* [[WE]]: ~25/21 = 300.0520{{c}}, ~6/5 = 317.0548{{c}} (~126/125 = 17.0029{{c}})
: [[error map]]: {{val| +0.282 +0.745 -0.448 -1.579 }}
: [[error map]]: {{val| +0.208 +0.374 -0.832 -0.243 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~36/35 = 52.8476{{c}}
* [[CWE]]: ~25/21 = 300.0000{{c}}, ~6/5 = 317.0301{{c}} (~126/125 = 17.0301{{c}})
: error map: {{val| 0.000 +0.558 -0.886 -2.074 }}
: error map: {{val| 0.000 +0.225 -1.163 -0.706 }}


{{Optimal ET sequence|legend=1| 68, 91, 159, 386d, 545dd }}
{{Optimal ET sequence|legend=1| 68, 72, 140, 212, 776cd, 988ccd, 1200ccd }}


[[Badness]] (Sintel): 3.69
[[Badness]] (Sintel): 0.993


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


Comma list: 385/384, 6250/6237, 67228/66825
Comma list: 385/384, 1375/1372, 6250/6237


Mapping: {{mapping| 1 0 1 1 5 | 0 36 30 41 -35 }}
{{Mapping|legend=0| 4 0 4 7 17 | 0 6 5 4 -3 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.2526{{c}}, ~36/35 = 52.8534{{c}}
* WE: ~25/21 = 300.0995{{c}}, ~6/5 = 317.0298{{c}} (~100/99 = 16.9303{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 52.8446{{c}}
* CWE: ~25/21 = 300.0000{{c}}, ~6/5 = 316.9540{{c}} (~100/99 = 16.9540{{c}})


{{Optimal ET sequence|legend=0| 68, 91, 159, 386d, 545dd }}
{{Optimal ET sequence|legend=0| 68, 72, 140, 212, 284, 496ce, 780ccdee }}


Badness (Sintel): 2.86
Badness (Sintel): 0.774


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


Comma list: 325/324, 385/384, 625/624, 16807/16731
Comma list: 325/324, 385/384, 625/624, 1375/1372
 
{{Mapping|legend=0| 4 0 4 7 17 0 | 0 6 5 4 -3 14 }}
 
Optimal tunings:
* WE: ~25/21 = 300.0985{{c}}, ~6/5 = 317.0899{{c}} (~100/99 = 16.9941{{c}})
* CWE: ~25/21 = 300.0000{{c}}, ~6/5 = 317.0155{{c}} (~100/99 = 17.0155{{c}})
 
{{Optimal ET sequence|legend=0| 68, 72, 140, 212 }}
 
Badness (Sintel): 0.774
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 289/288, 325/324, 385/384, 442/441, 625/624


Mapping: {{mapping| 1 0 1 1 5 0 | 0 36 30 41 -35 84 }}
{{Mapping|legend=0| 4 0 4 7 17 0 10 | 0 6 5 4 -3 14 6 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.2564{{c}}, ~36/35 = 52.8568{{c}}
* WE: ~25/21 = 300.1102{{c}}, ~6/5 = 317.1011{{c}} (~100/99 = 16.9909{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 52.8479{{c}}
* CWE: ~25/21 = 300.0000{{c}}, ~6/5 = 317.0155{{c}} (~100/99 = 17.0155{{c}})
 
{{Optimal ET sequence|legend=0| 68, 72, 140, 212g }}


{{Optimal ET sequence|legend=0| 68, 159, 386d, 545ddf }}
Badness (Sintel): 0.651


Badness (Sintel): 1.97
== Marthirds ==
Named by [[Xenllium]] in 2021, marthirds tempers out 2460375/2458624, the [[breeze comma]], and may be described as the {{nowrap| 19 & 193 }} temperament. It is generated by a marvel-comma-flat classical major third, [[56/45]], four of which minus an octave make the hanson generator of [[6/5]]. Its [[ploidacot]] is zeta-24-cot.  


== Novemkleismic ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 15625/15552, 40353607/40310784
[[Comma list]]: 15625/15552, 2460375/2458624


{{Mapping|legend=1| 9 0 9 11 | 0 6 5 6 }}
{{Mapping|legend=1| 1 -6 -4 -19 | 0 24 20 69 }}
: mapping generators: ~2592/2401, ~6/5
: mapping generators: ~2, ~56/45


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2592/2401 = 133.3488{{c}}, ~6/5 = 317.0413{{c}} (~36/35 = 50.3437{{c}})
* [[WE]]: ~2 = 1200.1662{{c}}, ~56/45 = 379.3041{{c}}
: [[error map]]: {{val| +0.139 +0.293 -0.968 +0.259 }}
: [[error map]]: {{val| +0.166 +0.347 -0.896 +0.000 }}
* [[CWE]]: ~2592/2401 = 133.3333{{c}}, ~6/5 = 317.0260{{c}} (~36/35 = 50.3593{{c}})
* [[CWE]]: ~2 = 1200.0000{{c}}, ~56/45 = 379.2552{{c}}
: error map: {{val| 0.000 +0.201 -1.184 -0.003 }}
: error map: {{val| 0.000 +0.171 -1.209 -0.214 }}


{{Optimal ET sequence|legend=1| 72, 261, 333, 405, 477c, 882c }}
{{Optimal ET sequence|legend=1| 19, …, 193, 212, 617c, 829c }}


[[Badness]] (Sintel): 4.90
[[Badness]] (Sintel): 2.64


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


Comma list: 1375/1372, 4000/3993, 15625/15552
Comma list: 1375/1372, 15625/15552, 19712/19683


Mapping: {{mapping| 9 0 9 11 24 | 0 6 5 6 3 }}
{{Mapping|legend=0| 1 -6 -4 -19 -43 | 0 24 20 69 147 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~250/231 = 133.3465{{c}}, ~6/5 = 317.0416{{c}} (~36/35 = 50.3486{{c}})
* WE: ~2 = 1200.1189{{c}}, ~56/45 = 379.2942{{c}}
* CWE: ~250/231 = 133.3333{{c}}, ~6/5 = 317.0264{{c}} (~36/35 = 50.3597{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~56/45 = 379.2580{{c}}


{{Optimal ET sequence|legend=0| 72, 261, 333, 405, 882c }}
{{Optimal ET sequence|legend=0| 19e, …, 193, 212, 405, 617c }}


Badness (Sintel): 1.71
Badness (Sintel): 2.50


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


Comma list: 325/324, 625/624, 1375/1372, 4000/3993
Comma list: 325/324, 625/624, 1375/1372, 19712/19683


Mapping: {{mapping| 9 0 9 11 24 0 | 0 6 5 6 3 14 }}
{{Mapping|legend=0| 1 -6 -4 -19 -43 -14 | 0 24 20 69 147 56 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~250/231 = 133.3385{{c}}, ~6/5 = 317.0978{{c}} (~36/35 = 50.4208{{c}})
* WE: ~2 = 1200.2154{{c}}, ~56/45 = 379.3236{{c}}
* CWE: ~250/231 = 133.3333{{c}}, ~6/5 = 317.0910{{c}} (~36/35 = 50.4243{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~56/45 = 379.2580{{c}}


{{Optimal ET sequence|legend=0| 72, 189f, 261, 333, 738cf }}
{{Optimal ET sequence|legend=0| 19e, …, 193, 212, 405f, 617cff }}


Badness (Sintel): 1.61
Badness (Sintel): 1.81


== Sqrtphi ==
== Sqrtphi ==
{{Main| Sqrtphi }}
{{Main| Sqrtphi }}


The just value of sqrt (φ) is 416.545 cents.  
Sqrtphi tempers out 16875/16807, the [[mirkwai comma]], and may be described as the {{nowrap| 49 & 72 }} temperament. The just value of sqrt(φ) is 416.545 cents, and this temperament gives a close approximation of it.
 
Note that in the data below, the generator is given as its [[octave complement]], which stands in for [[~]][[11/7]] from the [[11-limit]] onwards. Five generators octave reduced make the hanson generator of ~[[6/5]]. The [[ploidacot]] for this temperament is 19-sheared 30-cot.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 1,170: Line 1,099:
Comma list: 540/539, 1375/1372, 4375/4356
Comma list: 540/539, 1375/1372, 4375/4356


Mapping: {{mapping| 1 -18 -14 -22 -22 | 0 30 25 38 39 }}
{{Mapping|legend=0| 1 -18 -14 -22 -22 | 0 30 25 38 39 }}


Optimal tunings:  
Optimal tunings:  
Line 1,185: Line 1,114:
Comma list: 325/324, 364/363, 625/624, 1375/1372
Comma list: 325/324, 364/363, 625/624, 1375/1372


Mapping: {{mapping| 1 -18 -14 -22 -22 -42 | 0 30 25 38 39 70 }}
{{Mapping|legend=0| 1 -18 -14 -22 -22 -42 | 0 30 25 38 39 70 }}


Optimal tunings:  
Optimal tunings:  
Line 1,200: Line 1,129:
Comma list: 325/324, 364/363, 375/374, 540/539, 595/594
Comma list: 325/324, 364/363, 375/374, 540/539, 595/594


Mapping: {{mapping| 1 -18 -14 -22 -22 -42 -39 | 0 30 25 38 39 70 66 }}
{{Mapping|legend=0| 1 -18 -14 -22 -22 -42 -39 | 0 30 25 38 39 70 66 }}


Optimal tunings:  
Optimal tunings:  
Line 1,215: Line 1,144:
Comma list: 325/324, 364/363, 375/374, 400/399, 442/441, 595/594
Comma list: 325/324, 364/363, 375/374, 400/399, 442/441, 595/594


Mapping: {{mapping| 1 -18 -14 -22 -22 -42 -39 16 | 0 30 25 38 39 70 66 -18 }}
{{Mapping|legend=0| 1 -18 -14 -22 -22 -42 -39 16 | 0 30 25 38 39 70 66 -18 }}


Optimal tunings:  
Optimal tunings:  
Line 1,225: Line 1,154:
Badness (Sintel): 0.897
Badness (Sintel): 0.897


== Subgroup extensions ==
== Quartkeenlig ==
=== Kleismic (2.3.5.13) a.k.a. cata ===
Named by [[Eliora]] in 2022, quartkeenlig uses a generator that is a quartertone of [[33/32]][[~]][[36/35]] tempered together in the [[11-limit]], and is called so because it tempers out the [[quartisma]] by virtue of five 33/32's being with [[7/6]], keenanisma, [[385/384]], tempering 33/32 and 36/35 together, and liganellus comma (6250/6237). As six quartertones make the hanson generator of ~[[6/5]], its [[ploidacot]] is alpha-36-cot. It can also be viewed as a regular temperament interpretation of [[23edo and octave stretching|stretched 23edo]].
Hanson lends itself nicely to this extension in the 2.3.5.13 subgroup, as the hemitwelfth, reached by three generator steps, can be interpreted as [[26/15]]. Notice 15625/15552 = ([[325/324]])([[625/624]]) and 325/324 = (625/624)([[676/675]]). The [[S-expression]]-based comma list of the temperament is {[[325/324|S10/S12 = S25*S26]], ([[625/624|S25]],) [[676/675|S13/S15 = S26]]}. For the high-limit version of cata with a 1\5 period, see [[thunderclysmic]].
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 15625/15552, 117649/116640
 
{{Mapping|legend=1| 1 0 1 1 | 0 36 30 41 }}
: mapping generator: ~2, ~36/35


Subgroup: 2.3.5.13
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.2825{{c}}, ~36/35 = 52.8528{{c}}
: [[error map]]: {{val| +0.282 +0.745 -0.448 -1.579 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~36/35 = 52.8476{{c}}
: error map: {{val| 0.000 +0.558 -0.886 -2.074 }}
 
{{Optimal ET sequence|legend=1| 68, 91, 159, 386d, 545dd }}
 
[[Badness]] (Sintel): 3.69
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 385/384, 6250/6237, 67228/66825
 
{{Mapping|legend=0| 1 0 1 1 5 | 0 36 30 41 -35 }}
 
Optimal tunings:
* WE: ~2 = 1200.2526{{c}}, ~36/35 = 52.8534{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 52.8446{{c}}
 
{{Optimal ET sequence|legend=0| 68, 91, 159, 386d, 545dd }}
 
Badness (Sintel): 2.86
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Comma list: 325/324, 625/624
Comma list: 325/324, 385/384, 625/624, 16807/16731


Subgroup-val mapping: {{mapping| 1 0 1 0 | 0 6 5 14 }}
{{Mapping|legend=0| 1 0 1 1 5 0 | 0 36 30 41 -35 84 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.1210{{c}}, ~6/5 = 317.1076{{c}}
* WE: ~2 = 1200.2564{{c}}, ~36/35 = 52.8568{{c}}
* CWE: ~2 = 1200.1210{{c}}, ~6/5 = 317.0920{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 52.8479{{c}}
 
{{Optimal ET sequence|legend=0| 68, 159, 386d, 545ddf }}
 
Badness (Sintel): 1.97
 
== Novemkleismic ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 15625/15552, 40353607/40310784
 
{{Mapping|legend=1| 9 0 9 11 | 0 6 5 6 }}
: mapping generators: ~2592/2401, ~6/5


{{Optimal ET sequence|legend=0| 15, 19, 34, 53, 140, 193, 246 }}
[[Optimal tuning]]s:
* [[WE]]: ~2592/2401 = 133.3488{{c}}, ~6/5 = 317.0413{{c}} (~36/35 = 50.3437{{c}})
: [[error map]]: {{val| +0.139 +0.293 -0.968 +0.259 }}
* [[CWE]]: ~2592/2401 = 133.3333{{c}}, ~6/5 = 317.0260{{c}} (~36/35 = 50.3593{{c}})
: error map: {{val| 0.000 +0.201 -1.184 -0.003 }}


Badness (Sintel): 0.131
{{Optimal ET sequence|legend=1| 72, 261, 333, 405, 477c, 882c }}


==== 2.3.5.13.37 subgroup ====
[[Badness]] (Sintel): 4.90
Hanson can be extended even further to the 2.3.5.13.37.41 subgroup while maintaining a rather low complexity and high accuracy.


Subgroup: 2.3.5.13.37.41
=== 11-limit ===
Subgroup: 2.3.5.7.11


Comma list: 325/324, 481/480, 625/624
Comma list: 1375/1372, 4000/3993, 15625/15552


Subgroup-val mapping: {{mapping| 1 0 1 0 6 | 0 6 5 14 -3 }}
{{Mapping|legend=0| 9 0 9 11 24 | 0 6 5 6 3 }}


Optimal tunings:
Optimal tunings:  
* WE: ~2 = 1200.2924{{c}}, ~6/5 = 317.0998{{c}}
* WE: ~250/231 = 133.3465{{c}}, ~6/5 = 317.0416{{c}} (~36/35 = 50.3486{{c}})
* CWE: ~2 = 1200.000{{c}}, ~6/5 = 317.0452{{c}}
* CWE: ~250/231 = 133.3333{{c}}, ~6/5 = 317.0264{{c}} (~36/35 = 50.3597{{c}})


{{Optimal ET sequence|legend=0| 15, 19, 34, 53, 299l, 352fl, 405fl, 458fl, 511cfll, 564cffll }}
{{Optimal ET sequence|legend=0| 72, 261, 333, 405, 882c }}


Badness (Sintel): 0.167
Badness (Sintel): 1.71


==== 2.3.5.13.37.41 subgroup ====
=== 13-limit ===
Subgroup: 2.3.5.13.37.41
Subgroup: 2.3.5.7.11.13


Comma list: 325/324, 481/480, 625/624, 1025/1024
Comma list: 325/324, 625/624, 1375/1372, 4000/3993


Subgroup-val mapping: {{mapping| 1 0 1 0 6 8 | 0 6 5 14 -3 -10 }}
{{Mapping|legend=0| 9 0 9 11 24 0 | 0 6 5 6 3 14 }}


Optimal tunings:
Optimal tunings:  
* WE: ~2 = 1200.1651{{c}}, ~6/5 = 317.1126{{c}}
* WE: ~250/231 = 133.3385{{c}}, ~6/5 = 317.0978{{c}} (~36/35 = 50.4208{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0748{{c}}
* CWE: ~250/231 = 133.3333{{c}}, ~6/5 = 317.0910{{c}} (~36/35 = 50.4243{{c}})


{{Optimal ET sequence|legend=0| 15, 19, 34, 53, 140, 193, 246l }}
{{Optimal ET sequence|legend=0| 72, 189f, 261, 333, 738cf }}


Badness (Sintel): 0.223
Badness (Sintel): 1.61


[[Category:Kleismic family| ]] <!-- main article -->
[[Category:Kleismic]]
[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Kleismic family| ]] <!-- main article -->
[[Category:Catalogs of rank-2 temperaments]]
[[Category:Rank 2]]
[[Category:Listen]]
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