Kleismic family: Difference between revisions

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{{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 30: Line 30:
=== Overview to extensions ===
=== Overview to extensions ===
==== Full 7-limit extensions ====
==== 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]], hemifamity, 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.  
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|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.  
[[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.  
Temperaments involving larger splits include [[#Sqrtphi|sqrtphi]], [[#Quartkeenlig|quartkeenlig]], [[#Novemkleismic|novemkleismic]]. Those split the kleismic structure into five to nine parts.


==== Other subgroup extensions ====
==== Subgroup extensions ====
The main extension of note is a very remarkable extension to 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 {{nowrap| {[[325/324|S10/S12 = S25⋅S26]], ([[625/624|S25]]), [[676/675|S13/S15 = S26]]} }}.
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.  


See [[#Subgroup extensions]].
=== 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.  
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.  
 
Catakleismic extends easily with [[prime interval|prime]] [[13/1|13]]. The [[S-expression]]-based comma list of this extension 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]])}.  


=== 7-limit ===
=== 7-limit ===
Line 70: Line 87:


==== 2.3.5.7.13 subgroup ====
==== 2.3.5.7.13 subgroup ====
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


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 89: 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 108: 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 127: 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 142: 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 157: 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 172: 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 187: 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 202: 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 217: 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 233: 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 248: 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 263: 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 302: 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 321: 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 340: 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 360: 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 375: 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 390: 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 405: 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 420: 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 456: 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 475: 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 511: 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 530: 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 567: 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 582: 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 615: 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 630: 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 670: 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 685: 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 729: 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 749: 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 764: 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 799: 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 814: 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 831: 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 846: 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 863: 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 878: 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 911: Line 932:
Comma list: 176/175, 540/539, 3125/3087
Comma list: 176/175, 540/539, 3125/3087


Mapping: {{mapping| 1 -12 -9 -7 -29 | 0 18 15 13 43 }}
{{Mapping|legend=0| 1 -12 -9 -7 -29 | 0 18 15 13 43 }}


Optimal tunings:  
Optimal tunings:  
Line 926: Line 947:
Comma list: 176/175, 275/273, 325/324, 540/539
Comma list: 176/175, 275/273, 325/324, 540/539


Mapping: {{mapping| 1 -12 -9 -7 -29 -28 | 0 18 15 13 43 42 }}
{{Mapping|legend=0| 1 -12 -9 -7 -29 -28 | 0 18 15 13 43 42 }}


Optimal tunings:  
Optimal tunings:  
Line 959: Line 980:
Comma list: 385/384, 1375/1372, 6250/6237
Comma list: 385/384, 1375/1372, 6250/6237


Mapping: {{mapping| 4 0 4 7 17 | 0 6 5 4 -3 }}
{{Mapping|legend=0| 4 0 4 7 17 | 0 6 5 4 -3 }}


Optimal tunings:  
Optimal tunings:  
Line 974: Line 995:
Comma list: 325/324, 385/384, 625/624, 1375/1372
Comma list: 325/324, 385/384, 625/624, 1375/1372


Mapping: {{mapping| 4 0 4 7 17 0 | 0 6 5 4 -3 14 }}
{{Mapping|legend=0| 4 0 4 7 17 0 | 0 6 5 4 -3 14 }}


Optimal tunings:  
Optimal tunings:  
Line 989: Line 1,010:
Comma list: 289/288, 325/324, 385/384, 442/441, 625/624
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 }}
{{Mapping|legend=0| 4 0 4 7 17 0 10 | 0 6 5 4 -3 14 6 }}


Optimal tunings:  
Optimal tunings:  
Line 1,024: Line 1,045:
Comma list: 1375/1372, 15625/15552, 19712/19683
Comma list: 1375/1372, 15625/15552, 19712/19683


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


Optimal tunings:  
Optimal tunings:  
Line 1,039: Line 1,060:
Comma list: 325/324, 625/624, 1375/1372, 19712/19683
Comma list: 325/324, 625/624, 1375/1372, 19712/19683


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


Optimal tunings:  
Optimal tunings:  
Line 1,078: 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 1,093: 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 1,108: 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 1,123: 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 1,158: Line 1,179:
Comma list: 385/384, 6250/6237, 67228/66825
Comma list: 385/384, 6250/6237, 67228/66825


Mapping: {{mapping| 1 0 1 1 5 | 0 36 30 41 -35 }}
{{Mapping|legend=0| 1 0 1 1 5 | 0 36 30 41 -35 }}


Optimal tunings:  
Optimal tunings:  
Line 1,173: Line 1,194:
Comma list: 325/324, 385/384, 625/624, 16807/16731
Comma list: 325/324, 385/384, 625/624, 16807/16731


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


Optimal tunings:  
Optimal tunings:  
Line 1,206: Line 1,227:
Comma list: 1375/1372, 4000/3993, 15625/15552
Comma list: 1375/1372, 4000/3993, 15625/15552


Mapping: {{mapping| 9 0 9 11 24 | 0 6 5 6 3 }}
{{Mapping|legend=0| 9 0 9 11 24 | 0 6 5 6 3 }}


Optimal tunings:  
Optimal tunings:  
Line 1,221: Line 1,242:
Comma list: 325/324, 625/624, 1375/1372, 4000/3993
Comma list: 325/324, 625/624, 1375/1372, 4000/3993


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


Optimal tunings:  
Optimal tunings:  
Line 1,230: Line 1,251:


Badness (Sintel): 1.61
Badness (Sintel): 1.61
== Subgroup extensions ==
For the high-limit version of cata with a 1\5 period, see [[thunderclysmic]].
=== Kleismic (2.3.5.13) a.k.a. cata ===
The structure of the temperament as dividing 3/1 into 6 equal parts can be deduced completely from its [[S-expression]]-based comma list of {{nowrap| {[[325/324|S10/S12 = S25⋅S26]], ([[625/624|S25]]), [[676/675|S13/S15 = S26]]} }}. Specifically, dividing 3/1 into two halves of ~26/15 is equivalent to dividing 4/3 into two halves of ~15/13, hence the [[semiparticular]] S13/S15 = ([[4/3|16/12]])/([[15/13]])<sup>2</sup>. From here, we notice that (26/15)/(13/9) = 6/5, so all that remains is dividing 13/9 into two 6/5's via the semiparticular S10/S12 = (13/9)/([[6/5|12/10]])<sup>2</sup>, hence explaining the mapping of the entire 2.3.5.13 subgroup.
Subgroup: 2.3.5.13
Comma list: 325/324, 625/624
Subgroup-val mapping: {{mapping| 1 0 1 0 | 0 6 5 14 }}
Optimal tunings:
* WE: ~2 = 1200.1210{{c}}, ~6/5 = 317.1076{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0920{{c}}
{{Optimal ET sequence|legend=0| 15, 19, 34, 53, 140, 193, 246 }}
Badness (Sintel): 0.131
==== 2.3.5.13.37 subgroup ====
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
Comma list: 325/324, 481/480, 625/624
Subgroup-val mapping: {{mapping| 1 0 1 0 6 | 0 6 5 14 -3 }}
Optimal tunings:
* WE: ~2 = 1200.2924{{c}}, ~6/5 = 317.0998{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0452{{c}}
{{Optimal ET sequence|legend=0| 15, 19, 34, 53, 299l, 352fl, 405fl, 458fl, 511cfll, 564cffll }}
Badness (Sintel): 0.167
==== 2.3.5.13.37.41 subgroup ====
Subgroup: 2.3.5.13.37.41
Comma list: 325/324, 481/480, 625/624, 1025/1024
Subgroup-val mapping: {{mapping| 1 0 1 0 6 8 | 0 6 5 14 -3 -10 }}
Optimal tunings:
* WE: ~2 = 1200.1651{{c}}, ~6/5 = 317.1126{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0748{{c}}
{{Optimal ET sequence|legend=0| 15, 19, 34, 53, 140, 193, 246l }}
Badness (Sintel): 0.223


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

Latest revision as of 17:44, 30 September 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.

The 5-limit parent comma for the kleismic family is 15625/15552, the kleisma, which is the amount by which a stack of six classical minor thirds falls short of the 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

Main article: Kleismic

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

Subgroup: 2.3.5

Comma list: 15625/15552

Mapping: [⟨1 0 1], ⟨0 6 5]]

mapping generators: ~2, ~6/5

Optimal tunings:

  • WE: ~2 = 1200.1659 ¢, ~6/5 = 317.0504 ¢
error map: ⟨+0.166 +0.347 -0.896]
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 317.0308 ¢
error map: ⟨0.000 +0.230 -1.160]

Tuning ranges:

Optimal ET sequence: 15, 19, 34, 53, 458, 511c, …, 829c, 882c

Badness (Sintel): 0.310

Overview to extensions

Full 7-limit extensions

The second comma of the 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. 10976/10935, hemimage, gives marfifths. 1728/1715, the orwellismia, gives kleiboh. 2401/2400, the breedsma, gives quadritikleismic. 2460375/2458624, the breeze comma, gives 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, quartkeenlig, 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 S252⋅S26, which explains how the tone is trisected, and thus tempering out its constituents 625/624 (S25) and 676/675 (S26) 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⋅S262) 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

Subgroup-val mapping: [⟨1 0 1 0], ⟨0 6 5 14]]

Optimal tunings:

  • WE: ~2 = 1200.1210 ¢, ~6/5 = 317.1076 ¢
error map: ⟨+0.121 +0.690 -0.655 -1.022]
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 317.0920 ¢
error map: ⟨0.000 +0.597 -0.854 -1.240]

Optimal ET sequence: 15, 19, 34, 53, 140, 193, 246

Badness (Sintel): 0.131

Catakleismic

Main article: Catakleismic

Catakleismic tempers out 225/224, the marvel comma, and 4375/4374, the ragisma, and may be described as the 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

Subgroup: 2.3.5.7

Comma list: 225/224, 4375/4374

Mapping: [⟨1 0 1 -3], ⟨0 6 5 22]]

Optimal tunings:

  • WE: ~2 = 1200.5965 ¢, ~6/5 = 316.8893 ¢
error map: ⟨+0.596 -0.619 -1.271 +0.948]
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 316.7705 ¢
error map: ⟨0.000 -1.332 -2.461 +0.126]

Tuning ranges:

Optimal ET sequence: 19, 34d, 53, 72, 197, 269c

Badness (Sintel): 0.544

2.3.5.7.13 subgroup

Catakleismic extends easily with prime 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 (S27) 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

Comma list: 169/168, 225/224, 325/324

Subgroup-val mapping: [⟨1 0 1 -3 0], ⟨0 6 5 22 14]]

Optimal tunings:

  • WE: ~2 = 1200.7838 ¢, ~6/5 = 316.9478 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 316.7939 ¢

Optimal ET sequence: 19, 34d, 53, 72, 125f, 197f

Badness (Sintel): 0.410

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 4375/4374

Mapping: [⟨1 0 1 -3 9], ⟨0 6 5 22 -21]]

Optimal tunings:

  • WE: ~2 = 1200.6524 ¢, ~6/5 = 316.8911 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 316.7267 ¢

Tuning ranges:

  • 11-odd-limit diamond monotone range: ~6/5 = [315.789, 316.981] (5\19 to 14\53)
  • 11-odd-limit diamond tradeoff range: ~6/5 = [315.641, 317.263]

Optimal ET sequence: 19, 53, 72, 197e, 269ce, 341ce

Badness (Sintel): 0.722

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 225/224, 325/324, 385/384

Mapping: [⟨1 0 1 -3 9 0], ⟨0 6 5 22 -21 14]]

Optimal tunings:

  • WE: ~2 = 1200.7982 ¢, ~6/5 = 316.9482 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 316.7491 ¢

Tuning ranges:

  • 13- and 15-odd-limit diamond monotone: ~6/5 = [315.789, 316.981] (5\19 to 14\53)
  • 13- and 15-odd-limit diamond tradeoff: ~6/5 = [315.641, 318.309]

Optimal ET sequence: 19, 53, 72, 125f, 197ef

Badness (Sintel): 0.698

Cataclysmic

Subgroup: 2.3.5.7.11

Comma list: 99/98, 176/175, 2200/2187

Mapping: [⟨1 0 1 -3 -5], ⟨0 6 5 22 32]]

Optimal tunings:

  • WE: ~2 = 1199.9590 ¢, ~6/5 = 317.0315 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 317.0403 ¢

Optimal ET sequence: 19e, 34d, 53

Badness (Sintel): 1.32

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 99/98, 169/168, 176/175, 275/273

Mapping: [⟨1 0 1 -3 -5 0], ⟨0 6 5 22 32 14]]

Optimal tunings:

  • WE: ~2 = 1200.0797 ¢, ~6/5 = 317.0571 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 317.0400 ¢

Optimal ET sequence: 19e, 34d, 53

Badness (Sintel): 0.932

Catalytic

Subgroup: 2.3.5.7.11

Comma list: 225/224, 441/440, 4375/4374

Mapping: [⟨1 0 1 -3 -10], ⟨0 6 5 22 51]]

Optimal tunings:

  • WE: ~2 = 1200.8102 ¢, ~6/5 = 316.8669 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 316.6768 ¢

Optimal ET sequence: 19e, 53e, 72

Badness (Sintel): 1.01

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 225/224, 325/324, 1716/1715

Mapping: [⟨1 0 1 -3 -10 0], ⟨0 6 5 22 51 14]]

Optimal tunings:

  • WE: ~2 = 1201.0807 ¢, ~6/5 = 316.9246 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 316.6700 ¢

Optimal ET sequence: 19e, 53e, 72, 307bcdeeffff

Badness (Sintel): 0.923

Cataleptic

Subgroup: 2.3.5.7.11

Comma list: 100/99, 225/224, 864/847

Mapping: [⟨1 0 1 -3 4], ⟨0 6 5 22 -2]]

Optimal tunings:

  • WE: ~2 = 1198.6575 ¢, ~6/5 = 316.7282 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 317.0257 ¢

Optimal ET sequence: 19, 34d, 53e

Badness (Sintel): 1.47

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 78/77, 100/99, 144/143, 676/675

Mapping: [⟨1 0 1 -3 4 0], ⟨0 6 5 22 -2 14]]

Optimal tunings:

  • WE: ~2 = 1198.8403 ¢, ~6/5 = 316.8111 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 317.0652 ¢

Optimal ET sequence: 19, 34d, 53e

Badness (Sintel): 1.13

Bikleismic

Subgroup: 2.3.5.7.11

Comma list: 225/224, 243/242, 4375/4356

Mapping: [⟨2 0 2 -6 -1], ⟨0 6 5 22 15]]

mapping generators: ~99/70, ~6/5

Optimal tunings:

  • WE: ~99/70 = 600.2674 ¢, ~6/5 = 316.8624 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~6/5 = 316.7575 ¢

Optimal ET sequence: 34d, 72, 322c, 394c

Badness (Sintel): 0.969

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨2 0 2 -6 -1 0], ⟨0 6 5 22 15 14]]

Optimal tunings:

  • WE: ~55/39 = 600.3582 ¢, ~6/5 = 316.9152 ¢
  • CWE: ~55/39 = 600.0000 ¢, ~6/5 = 316.7759 ¢

Optimal ET sequence: 34d, 72

Badness (Sintel): 0.901

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 169/168, 221/220, 225/224, 243/242, 325/324

Mapping: [⟨2 0 2 -6 -1 0 5], ⟨0 6 5 22 15 14 6]]

Optimal tunings:

  • WE: ~17/12 = 600.4210 ¢, ~6/5 = 316.9282 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~6/5 = 316.7578 ¢

Optimal ET sequence: 34d, 38df, 72

Badness (Sintel): 0.798

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 153/152, 169/168, 221/220, 225/224, 243/242, 325/324

Mapping: [⟨2 0 2 -6 -1 0 5 -1], ⟨0 6 5 22 15 14 6 18]]

Optimal tunings:

  • WE: ~17/12 = 600.3763 ¢, ~6/5 = 316.8720 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~6/5 = 316.7205 ¢

Optimal ET sequence: 34dh, 38df, 72

Badness (Sintel): 0.959

Keemun

Main article: Keemun

Subgroup: 2.3.5.7

Comma list: 49/48, 126/125

Mapping: [⟨1 0 1 2], ⟨0 6 5 3]]

Optimal tunings:

  • WE: ~2 = 1202.6235 ¢, ~6/5 = 317.1646 ¢
error map: ⟨+2.624 +1.033 +2.133 -12.085]
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 316.8293 ¢
error map: ⟨0.000 -0.979 -2.167 -18.388]

Tuning ranges:

  • 7-odd-limit diamond monotone: ~6/5 = [300.000, 327.273] (1\4 to 3\11)
  • 9-odd-limit diamond monotone: ~6/5 = [315.789, 320.000] (5\19 to 4\15)
  • 7- and 9-odd-limit diamond tradeoff: ~6/5 = [308.744, 322.942]

Optimal ET sequence: 15, 19, 53d, 72dd

Badness (Sintel): 0.694

11-limit

Subgroup: 2.3.5.7.11

Comma list: 49/48, 56/55, 100/99

Mapping: [⟨1 0 1 2 4], ⟨0 6 5 3 -2]]

Optimal tunings:

  • WE: ~2 = 1199.7353 ¢, ~6/5 = 317.5055 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 317.5546 ¢

Tuning ranges:

  • 11-odd-limit diamond monotone: ~6/5 = [315.789, 320.000] (5\19 to 4\15)
  • 11-odd-limit diamond tradeoff: ~6/5 = [308.744, 324.341]

Optimal ET sequence: 15, 19, 34

Badness (Sintel): 0.906

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 56/55, 65/64, 100/99

Mapping: [⟨1 0 1 2 4 5], ⟨0 6 5 3 -2 -5]]

Optimal tunings:

  • WE: ~2 = 1201.8360 ¢, ~6/5 = 317.0958 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 316.6829 ¢

Tuning ranges:

  • 13- and 15-odd-limit diamond monotone: ~6/5 = 315.789 (5\19)
  • 13- and 15-odd-limit diamond tradeoff: ~6/5 = [303.597, 324.341]

Optimal ET sequence: 4, 15f, 19

Badness (Sintel): 1.23

Kema

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 56/55, 91/90, 100/99

Mapping: [⟨1 0 1 2 4 0], ⟨0 6 5 3 -2 14]]

Optimal tunings:

  • WE: ~2 = 1199.7816 ¢, ~6/5 = 317.3653 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 317.4070 ¢

Tuning ranges:

  • 13-odd-limit diamond monotone: ~6/5 = [315.789, 320.000] (5\19 to 4\15)
  • 15-odd-limit diamond monotone: ~6/5 = 315.789 (5\19)
  • 13- and 15-odd-limit diamond tradeoff: ~6/5 = [308.744, 324.341]

Optimal ET sequence: 15, 19, 34

Badness (Sintel): 0.940

Kumbaya

Subgroup: 2.3.5.7.11.13

Comma list: 40/39, 49/48, 56/55, 66/65

Mapping: [⟨1 0 1 2 4 4], ⟨0 6 5 3 -2 -1]]

Optimal tunings:

  • WE: ~2 = 1196.7615 ¢, ~6/5 = 317.7353 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 319.4059 ¢

Optimal ET sequence: 4, 11b, 15

Badness (Sintel): 1.31

Qeema

Subgroup: 2.3.5.7.11

Comma list: 45/44, 49/48, 126/125

Mapping: [⟨1 0 1 2 -1], ⟨0 6 5 3 17]]

Optimal tunings:

  • WE: ~2 = 1204.5534 ¢, ~6/5 = 315.9247 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 315.1686 ¢

Optimal ET sequence: 4e, 19, 42bcd

Badness (Sintel): 1.32

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 49/48, 78/77, 126/125

Mapping: [⟨1 0 1 2 -1 0], ⟨0 6 5 3 17 14]]

Optimal tunings:

  • WE: ~2 = 1204.4937 ¢, ~6/5 = 316.2241 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 315.4748 ¢

Optimal ET sequence: 4ef, 19

Badness (Sintel): 1.22

Darjeeling

Subgroup: 2.3.5.7.11

Comma list: 49/48, 55/54, 77/75

Mapping: [⟨1 0 1 2 0], ⟨0 6 5 3 13]]

Optimal tunings:

  • WE: ~2 = 1201.6569 ¢, ~6/5 = 318.0942 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 317.8547 ¢

Optimal ET sequence: 15, 19e, 34e

Badness (Sintel): 0.914

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 55/54, 66/65, 77/75

Mapping: [⟨1 0 1 2 0 0], ⟨0 6 5 3 13 14]]

Optimal tunings:

  • WE: ~2 = 1201.9324 ¢, ~6/5 = 317.8090 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 317.5170 ¢

Optimal ET sequence: 15, 19e, 34e

Badness (Sintel): 0.886

Catalan

Subgroup: 2.3.5.7

Comma list: 64/63, 15625/15552

Mapping: [⟨1 0 1 6], ⟨0 6 5 -12]]

Optimal tunings:

  • WE: ~2 = 1197.1789 ¢, ~6/5 = 317.5185 ¢
error map: ⟨-2.821 +3.156 -1.542 +4.025]
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 318.2411 ¢
error map: ⟨0.000 +7.492 +4.892 +12.281]

Tuning ranges:

Optimal ET sequence: 15, 34d, 49, 132bcdd, 181bbcddd

Badness (Sintel): 2.40

11-limit

Subgroup: 2.3.5.7.11

Comma list: 64/63, 100/99, 1331/1323

Mapping: [⟨1 0 1 6 4], ⟨0 6 5 -12 -2]]

Optimal tunings:

  • WE: ~2 = 1197.0368 ¢, ~6/5 = 317.4956 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 318.2672 ¢

Tuning ranges:

  • 11-odd-limit diamond monotone: ~6/5 = [317.647, 320.000] (9\34 to 4\15)
  • 11-odd-limit diamond tradeoff: ~6/5 = [315.641, 324.341]

Optimal ET sequence: 15, 34d, 49, 181bbcdddeee

Badness (Sintel): 1.22

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 64/63, 100/99, 144/143, 275/273

Mapping: [⟨1 0 1 6 4 0], ⟨0 6 5 -12 -2 14]]

Optimal tunings:

  • WE: ~2 = 1196.8961 ¢, ~6/5 = 317.3837 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 318.1621 ¢

Optimal ET sequence: 15, 34d, 49f, 83def, 132bcddeefff

Badness (Sintel): 1.09

Countercata

Subgroup: 2.3.5.7

Comma list: 5120/5103, 15625/15552

Mapping: [⟨1 0 1 11], ⟨0 6 5 -31]]

Optimal tunings:

  • WE: ~2 = 1199.9172 ¢, ~6/5 = 317.0995 ¢
error map: ⟨-0.083 +0.642 -0.899 +0.178]
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 317.1220 ¢
error map: ⟨0.000 +0.777 -0.704 +0.391]

Tuning ranges:

Optimal ET sequence: 19d, 34, 53, 87, 140, 333, 473

Badness (Sintel): 1.32

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 2200/2187, 3388/3375

Mapping: [⟨1 0 1 11 -5], ⟨0 6 5 -31 32]]

Optimal tunings:

  • WE: ~2 = 1200.0980 ¢, ~6/5 = 317.1879 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 317.1623 ¢

Tuning ranges:

  • 11-odd-limit diamond monotone: ~6/5 = [316.981, 317.647] (14\53 to 9\34)
  • 11-odd-limit diamond tradeoff: ~6/5 = [315.641, 317.370]

Optimal ET sequence: 34, 53, 87, 140, 227, 367e

Badness (Sintel): 1.31

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 352/351, 385/384, 625/624

Mapping: [⟨1 0 1 11 -5 0], ⟨0 6 5 -31 32 14]]

Optimal tunings:

  • WE: ~2 = 1200.0936 ¢, ~6/5 = 317.1864 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 317.1622 ¢

Tuning ranges:

  • 13-odd-limit diamond monotone: ~6/5 = [316.981, 317.647] (14\53 to 9\34)
  • 15-odd-limit diamond monotone: ~6/5 = [316.981, 317.241] (14\53 to 23\87)
  • 13- and 15-odd-limit diamond tradeoff: ~6/5 = [315.641, 318.309]

Optimal ET sequence: 34, 53, 87, 140, 367e, 507e

Badness (Sintel): 0.833

Metakleismic

Subgroup: 2.3.5.7

Comma list: 15625/15552, 179200/177147

Mapping: [⟨1 0 1 -12], ⟨0 6 5 56]]

Optimal tunings:

  • WE: ~2 = 1199.5969 ¢, ~6/5 = 317.2079 ¢
error map: ⟨-0.403 +1.292 -0.678 -0.349]
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 317.3071 ¢
error map: ⟨0.000 +1.887 +0.222 +0.370]

Optimal ET sequence: 34d, 87, 121, 208, 537b

Badness (Sintel): 4.14

11-limit

Subgroup: 2.3.5.7.11

Comma list: 896/891, 2200/2187, 14700/14641

Mapping: [⟨1 0 1 -12 -5], ⟨0 6 5 56 32]]

Optimal tunings:

  • WE: ~2 = 1199.5425 ¢, ~6/5 = 317.1901 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 317.3020 ¢

Optimal ET sequence: 34d, 53d, 87, 121, 208

Badness (Sintel): 1.61

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 352/351, 364/363, 625/624

Mapping: [⟨1 0 1 -12 -5 0], ⟨0 6 5 56 32 14]]

Optimal tunings:

  • WE: ~2 = 1199.5339 ¢, ~6/5 = 317.1882 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 317.3028 ¢

Optimal ET sequence: 34d, 53d, 87, 121, 208

Badness (Sintel): 1.01

Hemikleismic

Subgroup: 2.3.5.7

Comma list: 4000/3969, 6144/6125

Mapping: [⟨1 0 1 4], ⟨0 12 10 -9]]

mapping generators: ~2, ~35/32

Optimal tunings:

  • WE: ~2 = 1199.3950 ¢, ~35/32 = 158.5686 ¢
error map: ⟨-0.605 +0.868 -1.233 +1.637]
  • CWE: ~2 = 1200.0000 ¢, ~35/32 = 158.6338 ¢
error map: ⟨0.000 +1.651 +0.024 +3.470]

Optimal ET sequence: 15, 38, 53, 121, 174d, 295d

Badness (Sintel): 1.32

11-limit

Subgroup: 2.3.5.7.11

Comma list: 121/120, 176/175, 4000/3969

Mapping: [⟨1 0 1 4 2], ⟨0 12 10 -9 11]]

Optimal tunings:

  • WE: ~2 = 1199.8009 ¢, ~11/10 = 158.6508 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 158.6717 ¢

Optimal ET sequence: 15, 38, 53, 68, 121e

Badness (Sintel): 1.26

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 176/175, 275/273, 325/324

Mapping: [⟨1 0 1 4 2 0], ⟨0 12 10 -9 11 28]]

Optimal tunings:

  • WE: ~2 = 1199.7952 ¢, ~11/10 = 158.6279 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 158.6493 ¢

Optimal ET sequence: 15, 38f, 53, 121e

Badness (Sintel): 1.07

Clyde

Subgroup: 2.3.5.7

Comma list: 245/243, 3136/3125

Mapping: [⟨1 -6 -4 -13], ⟨0 12 10 25]]

mapping generators: ~2, ~14/9

Optimal tunings:

  • WE: ~2 = 1199.8369 ¢, ~14/9 = 758.5621 ¢
error map: ⟨-0.163 +1.769 -0.040 -2.652]
  • CWE: ~2 = 1200.0000 ¢, ~14/9 = 758.6554 ¢
error map: ⟨0.000 +1.910 +0.240 -2.441]

Minimax tuning:

[[1 0 0 0⟩, [6/25 0 0 12/25⟩, [6/5 0 0 2/5⟩, [0 0 0 1⟩]
unchanged-interval (eigenmonzo) basis: 2.7

Algebraic generator: real root of 5x3 - 6x - 3, the Poussami generator. Approximately 441.309 cents. Associated recurrence relationship quickly converges.

Optimal ET sequence: 19, 49, 68, 87, 155, 242

Badness (Sintel): 1.20

11-limit

Subgroup: 2.3.5.7.11

Comma list: 245/243, 385/384, 3136/3125

Mapping: [⟨1 -6 -4 -13 18], ⟨0 12 10 25 -23]]

Optimal tunings:

  • WE: ~2 = 1199.9620 ¢, ~14/9 = 758.6210 ¢
  • CWE: ~2 = 1200.0000 ¢, ~14/9 = 758.6445 ¢

Optimal ET sequence: 19, 49e, 68, 87

Badness (Sintel): 1.57

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 245/243, 385/384, 625/624

Mapping: [⟨1 -6 -4 -13 18 -14], ⟨0 12 10 25 -23 28]]

Optimal tunings:

  • WE: ~2 = 1199.9292 ¢, ~14/9 = 758.5919 ¢
  • CWE: ~2 = 1200.0000 ¢, ~14/9 = 758.6355 ¢

Optimal ET sequence: 19, 68, 87

Badness (Sintel): 1.11

Tritikleismic

Subgroup: 2.3.5.7

Comma list: 1029/1024, 15625/15552

Mapping: [⟨3 0 3 10], ⟨0 6 5 -2]]

mapping generators: ~63/50, ~6/5

Optimal tunings:

  • WE: ~63/50 = 400.1845 ¢, ~6/5 = 317.0178 ¢ (~21/20 = 83.1667 ¢)
error map: ⟨+0.553 +0.152 -0.671 -1.017]
  • CWE: ~63/50 = 400.0000 ¢, ~6/5 = 316.9129 ¢ (~21/20 = 83.0871 ¢)
error map: ⟨0.000 -0.478 -1.749 -2.652]

Minimax tuning:

[[1 0 0 0⟩, [2 0 6/7 -6/7⟩, [8/3 0 5/7 -5/7⟩, [8/3 0 -2/7 2/7⟩]
unchanged-interval (eigenmonzo) basis: 2.7/5
[[1 0 0 0⟩, [10/7 6/7 0 -3/7⟩, [46/21 5/7 0 -5/14⟩, [20/7 -2/7 0 1/7⟩]
unchanged-interval (eigenmonzo) basis: 2.9/7

Optimal ET sequence: 15, 42bc, 57, 72, 159, 231, 765ccddd

Badness (Sintel): 1.43

Music

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 441/440, 4000/3993

Mapping: [⟨3 0 3 10 8], ⟨0 6 5 -2 3]]

Optimal tunings:

  • WE: ~44/35 = 400.1571 ¢, ~6/5 = 317.0058 ¢ (~21/20 = 83.1514 ¢)
  • CWE: ~44/35 = 400.0000 ¢, ~6/5 = 316.9154 ¢ (~21/20 = 83.0846 ¢)

Minimax tuning:

  • 11-odd-limit: ~6/5 = [5/21 1/7 0 -1/14⟩
[[1 0 0 0 0⟩, [10/7 6/7 0 -3/7 0⟩, [46/21 5/7 0 -5/14 0⟩, [20/7 -2/7 0 1/7 0⟩, [71/21 3/7 0 -3/14 0⟩]
unchanged-interval (eigenmonzo) basis: 2.9/7

Optimal ET sequence: 15, 42bc, 57, 72, 159, 231

Badness (Sintel): 0.639

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 364/363, 385/384, 625/624

Mapping: [⟨3 0 3 10 8 0], ⟨0 6 5 -2 3 14]]

Optimal tunings:

  • WE: ~44/35 = 400.1514 ¢, ~6/5 = 317.0785 ¢ (~21/20 = 83.0729 ¢)
  • CWE: ~44/35 = 400.0000 ¢, ~6/5 = 316.9896 ¢ (~21/20 = 83.0104 ¢)

Optimal ET sequence: 15, 57f, 72, 87, 159

Badness (Sintel): 0.647

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 273/272, 325/324, 364/363, 375/374, 385/384

Mapping: [⟨3 0 3 10 8 0 -2], ⟨0 6 5 -2 3 14 18]]

Optimal tunings:

  • WE: ~34/27 = 400.1604 ¢, ~6/5 = 317.0353 ¢ (~21/20 = 83.1251 ¢)
  • CWE: ~34/27 = 400.0000 ¢, ~6/5 = 316.9384 ¢ (~21/20 = 83.0616 ¢)

Optimal ET sequence: 15g, 57fg, 72, 159, 231f

Badness (Sintel): 0.690

Marfifths

Named by Xenllium in 2021, marfifths tempers out the 10976/10935, the hemimage comma, and may be described as the 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

Comma list: 10976/10935, 15625/15552

Mapping: [⟨1 -6 -4 -17], ⟨0 18 15 47]]

mapping generators: ~2, ~75/56

Optimal tunings:

  • WE: ~2 = 1200.0223 ¢, ~75/56 = 505.7147 ¢
error map: ⟨+0.022 +0.775 -0.683 -0.615]
  • CWE: ~2 = 1200.0000 ¢, ~75/56 = 505.7060 ¢
error map: ⟨0.000 +0.753 -0.724 -0.643]

Optimal ET sequence: 19, …, 121, 140, 579, 719

Badness (Sintel): 1.61

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 6250/6237, 10976/10935

Mapping: [⟨1 -6 -4 -17 22], ⟨0 18 15 47 -44]]

Optimal tunings:

  • WE: ~2 = 1200.2484 ¢, ~75/56 = 505.7882 ¢
  • CWE: ~2 = 1200.0000 ¢, ~75/56 = 505.6853 ¢

Optimal ET sequence: 19, 121e, 140, 159, 299

Badness (Sintel): 1.95

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 385/384, 625/624, 10976/10935

Mapping: [⟨1 -6 -4 -17 22 -14], ⟨0 18 15 47 -44 42]]

Optimal tunings:

  • WE: ~2 = 1200.2747 ¢, ~75/56 = 505.8019 ¢
  • CWE: ~2 = 1200.0000 ¢, ~75/56 = 505.6883 ¢

Optimal ET sequence: 19, 121e, 140, 159, 299

Badness (Sintel): 1.24

Diatessic

Diatessic may be described as 121 & 140 and is closely related to the Diatess tuning (generator: 505.727281 cents).

Subgroup: 2.3.5.7.11

Comma list: 1375/1372, 2200/2187, 5632/5625

Mapping: [⟨1 -6 -4 -17 -37], ⟨0 18 15 47 96]]

Optimal tunings:

  • WE: ~2 = 1199.7886 ¢, ~75/56 = 505.6513 ¢
  • CWE: ~2 = 1200.0000 ¢, ~75/56 = 505.7366 ¢

Optimal ET sequence: 19e, …, 121, 140, 261, 401

Badness (Sintel): 2.02

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 352/351, 625/624, 1375/1372

Mapping: [⟨1 -6 -4 -17 -37 -14], ⟨0 18 15 47 96 42]]

Optimal tunings:

  • WE: ~2 = 1199.7996 ¢, ~75/56 = 505.6558 ¢
  • CWE: ~2 = 1200.0000 ¢, ~75/56 = 505.7366 ¢

Optimal ET sequence: 19e, …, 121, 140, 261, 401

Badness (Sintel): 1.18

Marf

Marf may be described as 19 & 121. It has a POTE generator which strongly approximates the marvelous fifth interval of 112/75.

Subgroup: 2.3.5.7.11

Comma list: 540/539, 896/891, 15625/15552

Mapping: [⟨1 -6 -4 -17 14], ⟨0 18 15 47 -25]]

Optimal tunings:

  • WE: ~2 = 1199.3198 ¢, ~75/56 = 505.4822 ¢
  • CWE: ~2 = 1200.0000 ¢, ~75/56 = 505.7607 ¢

Optimal ET sequence: 19, 102d, 121

Badness (Sintel): 2.48

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 540/539, 625/624, 896/891

Mapping: [⟨1 -6 -4 -17 14 -14], ⟨0 18 15 47 -25 42]]

Optimal tunings:

  • WE: ~2 = 1199.3368 ¢, ~75/56 = 505.4919 ¢
  • CWE: ~2 = 1200.0000 ¢, ~75/56 = 505.7627 ¢

Optimal ET sequence: 19, 102df, 121

Badness (Sintel): 1.58

Kleiboh

Subgroup: 2.3.5.7

Comma list: 1728/1715, 3125/3087

Mapping: [⟨1 -12 -9 -7], ⟨0 18 15 13]]

mapping generators: ~2, ~42/25

Optimal tunings:

  • WE: ~2 = 1199.5290 ¢, ~42/25 = 905.3417 ¢
error map: ⟨-0.471 -0.152 -1.949 +3.914]
  • CWE: ~2 = 1200.0000 ¢, ~42/25 = 905.6741 ¢
error map: ⟨0.000 +0.178 -1.203 +4.937]

Optimal ET sequence: 49, 53

Badness (Sintel): 1.93

11-limit

Subgroup: 2.3.5.7.11

Comma list: 176/175, 540/539, 3125/3087

Mapping: [⟨1 -12 -9 -7 -29], ⟨0 18 15 13 43]]

Optimal tunings:

  • WE: ~2 = 1199.1389 ¢, ~42/25 = 905.1688 ¢
  • CWE: ~2 = 1200.0000 ¢, ~42/25 = 905.7840 ¢

Optimal ET sequence: 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: [⟨1 -12 -9 -7 -29 -28], ⟨0 18 15 13 43 42]]

Optimal tunings:

  • WE: ~2 = 1199.1517 ¢, ~22/13 = 905.1727 ¢
  • CWE: ~2 = 1200.0000 ¢, ~22/13 = 905.7801 ¢

Optimal ET sequence: 49f, 53, 102df

Badness (Sintel): 1.28

Quadritikleismic

Subgroup: 2.3.5.7

Comma list: 2401/2400, 15625/15552

Mapping: [⟨4 0 4 7], ⟨0 6 5 4]]

mapping generators: ~25/21, ~6/5

Optimal tunings:

  • WE: ~25/21 = 300.0520 ¢, ~6/5 = 317.0548 ¢ (~126/125 = 17.0029 ¢)
error map: ⟨+0.208 +0.374 -0.832 -0.243]
  • CWE: ~25/21 = 300.0000 ¢, ~6/5 = 317.0301 ¢ (~126/125 = 17.0301 ¢)
error map: ⟨0.000 +0.225 -1.163 -0.706]

Optimal ET sequence: 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: [⟨4 0 4 7 17], ⟨0 6 5 4 -3]]

Optimal tunings:

  • WE: ~25/21 = 300.0995 ¢, ~6/5 = 317.0298 ¢ (~100/99 = 16.9303 ¢)
  • CWE: ~25/21 = 300.0000 ¢, ~6/5 = 316.9540 ¢ (~100/99 = 16.9540 ¢)

Optimal ET sequence: 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: [⟨4 0 4 7 17 0], ⟨0 6 5 4 -3 14]]

Optimal tunings:

  • WE: ~25/21 = 300.0985 ¢, ~6/5 = 317.0899 ¢ (~100/99 = 16.9941 ¢)
  • CWE: ~25/21 = 300.0000 ¢, ~6/5 = 317.0155 ¢ (~100/99 = 17.0155 ¢)

Optimal ET sequence: 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: [⟨4 0 4 7 17 0 10], ⟨0 6 5 4 -3 14 6]]

Optimal tunings:

  • WE: ~25/21 = 300.1102 ¢, ~6/5 = 317.1011 ¢ (~100/99 = 16.9909 ¢)
  • CWE: ~25/21 = 300.0000 ¢, ~6/5 = 317.0155 ¢ (~100/99 = 17.0155 ¢)

Optimal ET sequence: 68, 72, 140, 212g

Badness (Sintel): 0.651

Marthirds

Named by Xenllium in 2021, marthirds tempers out 2460375/2458624, the breeze comma, and may be described as the 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.

Subgroup: 2.3.5.7

Comma list: 15625/15552, 2460375/2458624

Mapping: [⟨1 -6 -4 -19], ⟨0 24 20 69]]

mapping generators: ~2, ~56/45

Optimal tunings:

  • WE: ~2 = 1200.1662 ¢, ~56/45 = 379.3041 ¢
error map: ⟨+0.166 +0.347 -0.896 +0.000]
  • CWE: ~2 = 1200.0000 ¢, ~56/45 = 379.2552 ¢
error map: ⟨0.000 +0.171 -1.209 -0.214]

Optimal ET sequence: 19, …, 193, 212, 617c, 829c

Badness (Sintel): 2.64

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -6 -4 -19 -43], ⟨0 24 20 69 147]]

Optimal tunings:

  • WE: ~2 = 1200.1189 ¢, ~56/45 = 379.2942 ¢
  • CWE: ~2 = 1200.0000 ¢, ~56/45 = 379.2580 ¢

Optimal ET sequence: 19e, …, 193, 212, 405, 617c

Badness (Sintel): 2.50

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -6 -4 -19 -43 -14], ⟨0 24 20 69 147 56]]

Optimal tunings:

  • WE: ~2 = 1200.2154 ¢, ~56/45 = 379.3236 ¢
  • CWE: ~2 = 1200.0000 ¢, ~56/45 = 379.2580 ¢

Optimal ET sequence: 19e, …, 193, 212, 405f, 617cff

Badness (Sintel): 1.81

Sqrtphi

Main article: Sqrtphi

Sqrtphi tempers out 16875/16807, the mirkwai comma, and may be described as the 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

Comma list: 15625/15552, 16875/16807

Mapping: [⟨1 -18 -14 -22], ⟨0 30 25 38]]

mapping generators: ~2, 196/125

Optimal tunings:

  • WE: ~2 = 1200.1357 ¢, ~196/125 = 783.4853 ¢
error map: ⟨+0.136 +0.163 -1.080 +0.632]
  • CWE: ~2 = 1200.0000 ¢, ~196/125 = 783.4009 ¢
error map: ⟨0.000 +0.072 -1.291 +0.408]

Optimal ET sequence: 23d, 49, 72, 193, 265

Badness (Sintel): 1.78

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372, 4375/4356

Mapping: [⟨1 -18 -14 -22 -22], ⟨0 30 25 38 39]]

Optimal tunings:

  • WE: ~2 = 1200.0514 ¢, ~11/7 = 783.4294 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/7 = 783.3975 ¢

Optimal ET sequence: 23de, 49, 72, 193, 265

Badness (Sintel): 0.844

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 364/363, 625/624, 1375/1372

Mapping: [⟨1 -18 -14 -22 -22 -42], ⟨0 30 25 38 39 70]]

Optimal tunings:

  • WE: ~2 = 1199.9314 ¢, ~11/7 = 783.3705 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/7 = 783.4134 ¢

Optimal ET sequence: 23deff, 49f, 72, 121, 193

Badness (Sintel): 0.828

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 325/324, 364/363, 375/374, 540/539, 595/594

Mapping: [⟨1 -18 -14 -22 -22 -42 -39], ⟨0 30 25 38 39 70 66]]

Optimal tunings:

  • WE: ~2 = 1199.9324 ¢, ~11/7 = 783.3706 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/7 = 783.4129 ¢

Optimal ET sequence: 23deffgg, 49fg, 72, 121, 193

Badness (Sintel): 0.664

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 325/324, 364/363, 375/374, 400/399, 442/441, 595/594

Mapping: [⟨1 -18 -14 -22 -22 -42 -39 16], ⟨0 30 25 38 39 70 66 -18]]

Optimal tunings:

  • WE: ~2 = 1199.8567 ¢, ~11/7 = 783.3262 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/7 = 783.4176 ¢

Optimal ET sequence: 49fg, 72, 121, 193

Badness (Sintel): 0.897

Quartkeenlig

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 stretched 23edo.

Subgroup: 2.3.5.7

Comma list: 15625/15552, 117649/116640

Mapping: [⟨1 0 1 1], ⟨0 36 30 41]]

mapping generator: ~2, ~36/35

Optimal tunings:

  • WE: ~2 = 1200.2825 ¢, ~36/35 = 52.8528 ¢
error map: ⟨+0.282 +0.745 -0.448 -1.579]
  • CWE: ~2 = 1200.0000 ¢, ~36/35 = 52.8476 ¢
error map: ⟨0.000 +0.558 -0.886 -2.074]

Optimal ET sequence: 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: [⟨1 0 1 1 5], ⟨0 36 30 41 -35]]

Optimal tunings:

  • WE: ~2 = 1200.2526 ¢, ~36/35 = 52.8534 ¢
  • CWE: ~2 = 1200.0000 ¢, ~36/35 = 52.8446 ¢

Optimal ET sequence: 68, 91, 159, 386d, 545dd

Badness (Sintel): 2.86

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 0 1 1 5 0], ⟨0 36 30 41 -35 84]]

Optimal tunings:

  • WE: ~2 = 1200.2564 ¢, ~36/35 = 52.8568 ¢
  • CWE: ~2 = 1200.0000 ¢, ~36/35 = 52.8479 ¢

Optimal ET sequence: 68, 159, 386d, 545ddf

Badness (Sintel): 1.97

Novemkleismic

Subgroup: 2.3.5.7

Comma list: 15625/15552, 40353607/40310784

Mapping: [⟨9 0 9 11], ⟨0 6 5 6]]

mapping generators: ~2592/2401, ~6/5

Optimal tunings:

  • WE: ~2592/2401 = 133.3488 ¢, ~6/5 = 317.0413 ¢ (~36/35 = 50.3437 ¢)
error map: ⟨+0.139 +0.293 -0.968 +0.259]
  • CWE: ~2592/2401 = 133.3333 ¢, ~6/5 = 317.0260 ¢ (~36/35 = 50.3593 ¢)
error map: ⟨0.000 +0.201 -1.184 -0.003]

Optimal ET sequence: 72, 261, 333, 405, 477c, 882c

Badness (Sintel): 4.90

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨9 0 9 11 24], ⟨0 6 5 6 3]]

Optimal tunings:

  • WE: ~250/231 = 133.3465 ¢, ~6/5 = 317.0416 ¢ (~36/35 = 50.3486 ¢)
  • CWE: ~250/231 = 133.3333 ¢, ~6/5 = 317.0264 ¢ (~36/35 = 50.3597 ¢)

Optimal ET sequence: 72, 261, 333, 405, 882c

Badness (Sintel): 1.71

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨9 0 9 11 24 0], ⟨0 6 5 6 3 14]]

Optimal tunings:

  • WE: ~250/231 = 133.3385 ¢, ~6/5 = 317.0978 ¢ (~36/35 = 50.4208 ¢)
  • CWE: ~250/231 = 133.3333 ¢, ~6/5 = 317.0910 ¢ (~36/35 = 50.4243 ¢)

Optimal ET sequence: 72, 189f, 261, 333, 738cf

Badness (Sintel): 1.61

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