Kleismic family: Difference between revisions

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→Acrocata: Acrokleismic already exists, leaving this simply as 2.3.5.13.37.41-subgroup cata (for now)
→Catakleismic: move music to the dedicated article
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{{interwiki
{{interwiki
| de = Hanson Kleismisch
| en = Kleismic family
| en = Kleismic family
| de = Hanson-Kleismisch
| es =  
| es =  
| ja =  
| ja =  
}}
}}
{{Technical data page}}
{{Technical data page}}
The [[5-limit]] parent comma for the '''kleismic family''' is [[15625/15552]], the kleisma. The [[generator]] is a [[6/5|classical minor third (6/5)]], 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 14\53 is about perfect as a generator, though 9\34 also makes sense, and 5\19 and 4\15 are possible. Other tunings include [[72edo]], [[87edo]] and [[140edo]].
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 second comma of the [[normal lists|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. [[6144/6125]], the porwell comma, gives hemikleismic. [[245/243]], sensamagic, gives clyde. [[1029/1024]], the gamelisma, gives tritikleismic. [[2401/2400]] the breedsma, gives quadritikleismic. 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. 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.


== 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]].


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
Line 18: Line 18:


{{Mapping|legend=1| 1 0 1 | 0 6 5 }}
{{Mapping|legend=1| 1 0 1 | 0 6 5 }}
: mapping generators: ~2, ~6/5
: mapping generators: ~2, ~6/5


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* CTE: ~2 = 1\1, ~6/5 = 317.0308
* [[WE]]: ~2 = 1200.1659{{c}}, ~6/5 = 317.0504{{c}}
* POTE: ~2 = 1\1, ~6/5 = 317.007
: [[error map]]: {{val| +0.166 +0.347 -0.896 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 317.0308{{c}}
: error map: {{val| 0.000 +0.230 -1.160 }}


[[Tuning ranges]]:  
[[Tuning ranges]]:  
* 5-odd-limit [[diamond monotone]]: ~6/5 = [300.000, 327.273] (1\4 to 3\11)
* [[5-odd-limit]] [[diamond monotone]]: ~6/5 = [300.000, 327.273] (1\4 to 3\11)
* 5-odd-limit [[diamond tradeoff]]: ~6/5 = [315.641, 317.263] (untempered to 1/5-comma)
* 5-odd-limit [[diamond tradeoff]]: ~6/5 = [315.641, 317.263] (untempered to 1/5-comma)


{{Optimal ET sequence|legend=1| 15, 19, 34, 53, 458, 511c, …, 882c }}
{{Optimal ET sequence|legend=1| 15, 19, 34, 53, 458, 511c, …, 829c, 882c }}


[[Badness]]: 0.013234
[[Badness]] (Sintel): 0.310


=== 2.3.5.13 subgroup (cata) ===
=== Overview to extensions ===
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]].
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.  


Subgroup: 2.3.5.13
[[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.  


Comma list: 325/324, 625/624
Temperaments involving larger splits include [[#Sqrtphi|sqrtphi]], [[#Quartkeenlig|quartkeenlig]], [[#Novemkleismic|novemkleismic]]. Those split the kleismic structure into five to nine parts.


Sval mapping: {{mapping| 1 0 1 0 | 0 6 5 14 }}
The kleismic family boasts a very remarkable extension to the [[2.3.5.13 subgroup]], which has further extensions with higher primes. These are listed at the bottom of this page, in [[#Subgroup extensions]].


Optimal tunings:
== Catakleismic ==
* CTE: ~2 = 1\1, ~6/5 = 317.1110
{{Main| Catakleismic }}
* POTE: ~2 = 1\1, ~6/5 = 317.0756


{{Optimal ET sequence|legend=1| 15, 19, 34, 53, 140, 193, 246 }}
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.


Badness (Sintel): 0.131
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]])}.  


==== 2.3.5.13.37.41 subgroup ====
=== 7-limit ===
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]]: 2.3.5.7


Subgroup: 2.3.5.13.37.41
[[Comma list]]: 225/224, 4375/4374


Comma list: 325/324, 625/624, [[481/480]], [[1600/1599]]
{{Mapping|legend=1| 1 0 1 -3 | 0 6 5 22 }}


[[Mapping]]: {{mapping| 1 0 1 0 6 8 | 0 6 5 14 -3 -10 }}
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.5965{{c}}, ~6/5 = 316.8893{{c}}
: [[error map]]: {{val| +0.596 -0.619 -1.271 +0.948 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 316.7705{{c}}
: error map: {{val| 0.000 -1.332 -2.461 +0.126 }}


Optimal tunings:
[[Tuning ranges]]:  
* WE: ~2 = 1200.165, ~6/5 = 317.113
* 7- and 9-odd-limit [[diamond monotone]]: ~6/5 = [315.789, 317.647] (5\19 to 9\34)
* CWE: 2, ~6/5 = 317.075
* 7- and 9-odd-limit [[diamond tradeoff]]: ~6/5 = [315.641, 317.263]


Badness (Sintel): 0.223
{{Optimal ET sequence|legend=1| 19, 34d, 53, 72, 197, 269c }}


== Keemun ==
[[Badness]] (Sintel): 0.544
{{Main| Keemun }}


[[Subgroup]]: 2.3.5.7
==== 2.3.5.7.13 subgroup ====
Subgroup: 2.3.5.7.13


[[Comma list]]: 49/48, 126/125
Comma list: 169/168, 225/224, 325/324


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


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 316.473
Optimal tunings:
* WE: ~2 = 1200.7838{{c}}, ~6/5 = 316.9478{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.7939{{c}}


[[Tuning ranges]]:
{{Optimal ET sequence|legend=0| 19, 34d, 53, 72, 125f, 197f }}
* 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|legend=1| 15, 19, 53d, 72dd, 91dd }}


[[Badness]]: 0.027408
Badness (Sintel): 0.410


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


Comma list: 49/48, 56/55, 100/99
Comma list: 225/224, 385/384, 4375/4374


Mapping: {{mapping| 1 0 1 2 4 | 0 6 5 3 -2 }}
Mapping: {{mapping| 1 0 1 -3 9 | 0 6 5 22 -21 }}


Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 317.576
Optimal tunings:
* WE: ~2 = 1200.6524{{c}}, ~6/5 = 316.8911{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.7267{{c}}


Tuning ranges:  
Tuning ranges:  
* 11-odd-limit diamond monotone: ~6/5 = [315.789, 320.000] (5\19 to 4\15)
* 11-odd-limit diamond monotone range: ~6/5 = [315.789, 316.981] (5\19 to 14\53)
* 11-odd-limit diamond tradeoff: ~6/5 = [308.744, 324.341]
* 11-odd-limit diamond tradeoff range: ~6/5 = [315.641, 317.263]


{{Optimal ET sequence|legend=1| 4, 15, 19, 34 }}
{{Optimal ET sequence|legend=0| 19, 53, 72, 197e, 269ce, 341ce }}


Badness: 0.027410
Badness (Sintel): 0.722


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


Comma list: 49/48, 56/55, 65/64, 100/99
Comma list: 169/168, 225/224, 325/324, 385/384


Mapping: {{mapping| 1 0 1 2 4 5 | 0 6 5 3 -2 -5 }}
Mapping: {{mapping| 1 0 1 -3 9 0 | 0 6 5 22 -21 14 }}


Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.611
Optimal tunings:
* WE: ~2 = 1200.7982{{c}}, ~6/5 = 316.9482{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.7491{{c}}


Tuning ranges:  
Tuning ranges:  
* 13- and 15-odd-limit diamond monotone: ~6/5 = 315.789 (5\19)
* 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 = [303.597, 324.341]
* 13- and 15-odd-limit diamond tradeoff: ~6/5 = [315.641, 318.309]


{{Optimal ET sequence|legend=1| 4, 15f, 19, 53def, 72def }}
{{Optimal ET sequence|legend=0| 19, 53, 72, 125f, 197ef }}


Badness: 0.029749
Badness (Sintel): 0.698


==== Kema ====
=== Cataclysmic ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11


Comma list: 49/48, 56/55, 91/90, 100/99
Comma list: 99/98, 176/175, 2200/2187


Mapping: {{mapping| 1 0 1 2 4 0 | 0 6 5 3 -2 14 }}
Mapping: {{mapping| 1 0 1 -3 -5 | 0 6 5 22 32 }}


Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 317.423
Optimal tunings:
* WE: ~2 = 1199.9590{{c}}, ~6/5 = 317.0315{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0403{{c}}


Tuning ranges:
{{Optimal ET sequence|legend=0| 19e, 34d, 53 }}
* 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|legend=1| 15, 19, 34, 87ddee }}
Badness (Sintel): 1.32


Badness: 0.022749
==== 13-limit ====
 
==== Kumbaya ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 40/39, 49/48, 56/55, 66/65
Comma list: 99/98, 169/168, 176/175, 275/273


Mapping: {{mapping| 1 0 1 2 4 4 | 0 6 5 3 -2 -1 }}
Mapping: {{mapping| 1 0 1 -3 -5 0 | 0 6 5 22 32 14 }}


Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 318.595
Optimal tunings:
* WE: ~2 = 1200.0797{{c}}, ~6/5 = 317.0571{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0400{{c}}


{{Optimal ET sequence|legend=1| 4, 15, 19f, 34ff }}
{{Optimal ET sequence|legend=0| 19e, 34d, 53 }}


Badness: 0.031633
Badness (Sintel): 0.932


=== Qeema ===
=== Catalytic ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 45/44, 49/48, 126/125
Comma list: 225/224, 441/440, 4375/4374


Mapping: {{mapping| 1 0 1 2 -1 | 0 6 5 3 17 }}
Mapping: {{mapping| 1 0 1 -3 -10 | 0 6 5 22 51 }}


Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 314.730
Optimal tunings:
* WE: ~2 = 1200.8102{{c}}, ~6/5 = 316.8669{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.6768{{c}}


{{Optimal ET sequence|legend=1| 4e, 19, 42bcd, 61bcdd }}
{{Optimal ET sequence|legend=0| 19e, 53e, 72 }}


Badness: 0.040056
Badness (Sintel): 1.01


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


Comma list: 45/44, 49/48, 78/77, 126/125
Comma list: 169/168, 225/224, 325/324, 1716/1715


Mapping: {{mapping| 1 0 1 2 -1 0 | 0 6 5 3 17 14 }}
Mapping: {{mapping| 1 0 1 -3 -10 0 | 0 6 5 22 51 14 }}


Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.044
Optimal tunings:
* WE: ~2 = 1201.0807{{c}}, ~6/5 = 316.9246{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.6700{{c}}


{{Optimal ET sequence|legend=1| 4ef, 19 }}
{{Optimal ET sequence|legend=0| 19e, 53e, 72, 307bcdeeffff }}


Badness: 0.029419
Badness (Sintel): 0.923


=== Darjeeling ===
=== Cataleptic ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 49/48, 55/54, 77/75
Comma list: 100/99, 225/224, 864/847


Mapping: {{mapping| 1 0 1 2 0 | 0 6 5 3 13 }}
Mapping: {{mapping| 1 0 1 -3 4 | 0 6 5 22 -2 }}


Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 317.656
Optimal tunings:
* WE: ~2 = 1198.6575{{c}}, ~6/5 = 316.7282{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0257{{c}}


{{Optimal ET sequence|legend=1| 15, 19e, 34e }}
{{Optimal ET sequence|legend=0| 19, 34d, 53e }}


Badness: 0.027648
Badness (Sintel): 1.47


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


Comma list: 49/48, 55/54, 66/65, 77/75
Comma list: 78/77, 100/99, 144/143, 676/675


Mapping: {{mapping| 1 0 1 2 0 0 | 0 6 5 3 13 14 }}
Mapping: {{mapping| 1 0 1 -3 4 0 | 0 6 5 22 -2 14 }}


Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 317.298
Optimal tunings:
* WE: ~2 = 1198.8403{{c}}, ~6/5 = 316.8111{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0652{{c}}


{{Optimal ET sequence|legend=1| 15, 19e, 34e, 53dee }}
{{Optimal ET sequence|legend=0| 19, 34d, 53e }}


Badness: 0.021445
Badness (Sintel): 1.13


== Catalan ==
=== Bikleismic ===
[[Subgroup]]: 2.3.5.7
Subgroup: 2.3.5.7.11


[[Comma list]]: 64/63, 15625/15552
Comma list: 225/224, 243/242, 4375/4356


{{Mapping|legend=1| 1 0 1 6 | 0 6 5 -12 }}
Mapping: {{mapping| 2 0 2 -6 -1 | 0 6 5 22 15 }}
: mapping generators: ~99/70, ~6/5


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 318.267
Optimal tunings:
* WE: ~99/70 = 600.2674{{c}}, ~6/5 = 316.8624{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~6/5 = 316.7575{{c}}


[[Tuning ranges]]:
{{Optimal ET sequence|legend=0| 34d, 72, 322c, 394c }}
* 7- and 9-odd-limit [[diamond monotone]]: ~6/5 = [317.647, 320.000] (9\34 to 4\15)
* 7- and 9-odd-limit [[diamond tradeoff]]: ~6/5 = [315.641, 319.265]


{{Optimal ET sequence|legend=1| 15, 34d, 49, 132bcdd, 181bbcddd }}
Badness (Sintel): 0.969


[[Badness]]: 0.094872
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


=== 11-limit ===
Comma list: 169/168, 225/224, 243/242, 325/324
Subgroup: 2.3.5.7.11


Comma list: 64/63, 100/99, 1331/1323
Mapping: {{mapping| 2 0 2 -6 -1 0 | 0 6 5 22 15 14 }}


Mapping: {{mapping| 1 0 1 6 4 | 0 6 5 -12 -2 }}
Optimal tunings:  
* WE: ~55/39 = 600.3582{{c}}, ~6/5 = 316.9152{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~6/5 = 316.7759{{c}}


Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 318.282
{{Optimal ET sequence|legend=0| 34d, 72 }}


Tuning ranges:
Badness (Sintel): 0.901
* 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|legend=1| 15, 34d, 49, 181bbcdddeee }}
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


Badness: 0.036894
Comma list: 169/168, 221/220, 225/224, 243/242, 325/324


=== 13-limit ===
Mapping: {{mapping| 2 0 2 -6 -1 0 5 | 0 6 5 22 15 14 6 }}
Subgroup: 2.3.5.7.11.13


Comma list: 64/63, 100/99, 144/143, 275/273
Optimal tunings:  
* WE: ~17/12 = 600.4210{{c}}, ~6/5 = 316.9282{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~6/5 = 316.7578{{c}}


Mapping: {{mapping| 1 0 1 6 4 0 | 0 6 5 -12 -2 14 }}
{{Optimal ET sequence|legend=0| 34d, 38df, 72 }}


Optimal tuning (CTE): ~2 = 1\1, ~6/5 = 317.9159
Badness (Sintel): 0.798


{{Optimal ET sequence|legend=1| 15, 34d, 49f, 83def, 132bcddeefff }}
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


Badness: 0.0263
Comma list: 153/152, 169/168, 221/220, 225/224, 243/242, 325/324


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


=== 7-limit ===
Optimal tunings:
[[Subgroup]]: 2.3.5.7
* WE: ~17/12 = 600.3763{{c}}, ~6/5 = 316.8720{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~6/5 = 316.7205{{c}}


[[Comma list]]: 225/224, 4375/4374
{{Optimal ET sequence|legend=0| 34dh, 38df, 72 }}


{{Mapping|legend=1| 1 0 1 -3 | 0 6 5 22 }}
Badness (Sintel): 0.959


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 316.732
== Keemun ==
{{Main| Keemun }}


[[Tuning ranges]]:  
[[Subgroup]]: 2.3.5.7
* 7- and 9-odd-limit [[diamond monotone]]: ~6/5 = [315.789, 317.647] (5\19 to 9\34)
* 7- and 9-odd-limit [[diamond tradeoff]]: ~6/5 = [315.641, 317.263]


{{Optimal ET sequence|legend=1| 19, 34d, 53, 72, 197, 269c }}
[[Comma list]]: 49/48, 126/125


[[Badness]]: 0.021501
{{Mapping|legend=1| 1 0 1 2 | 0 6 5 3 }}


==== 2.3.5.7.13 subgroup ====
[[Optimal tuning]]s:
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]])}.
* [[WE]]: ~2 = 1202.6235{{c}}, ~6/5 = 317.1646{{c}}
: [[error map]]: {{val| +2.624 +1.033 +2.133 -12.085 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 316.8293{{c}}
: error map: {{val| 0.000 -0.979 -2.167 -18.388 }}


Subgroup: 2.3.5.7.13
[[Tuning ranges]]:  
 
* 7-odd-limit [[diamond monotone]]: ~6/5 = [300.000, 327.273] (1\4 to 3\11)
Comma list: 169/168, 225/224, 325/324
* 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]
Sval mapping: {{mapping| 1 0 1 -3 0 | 0 6 5 22 14 }}
 
Optimal tuning (CTE): ~2 = 1\1, ~6/5 = 316.8865


{{Optimal ET sequence|legend=1| 19, 34d, 53, 72, 125f, 197f }}
{{Optimal ET sequence|legend=1| 15, 19, 53d, 72dd }}


Badness: 0.0118
[[Badness]] (Sintel): 0.694


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


Comma list: 225/224, 385/384, 4375/4374
Comma list: 49/48, 56/55, 100/99


Mapping: {{mapping| 1 0 1 -3 9 | 0 6 5 22 -21 }}
Mapping: {{mapping| 1 0 1 2 4 | 0 6 5 3 -2 }}


Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.719
Optimal tunings:
* WE: ~2 = 1199.7353{{c}}, ~6/5 = 317.5055{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.5546{{c}}


Tuning ranges:  
Tuning ranges:  
* 11-odd-limit diamond monotone range: ~6/5 = [315.789, 316.981] (5\19 to 14\53)
* 11-odd-limit diamond monotone: ~6/5 = [315.789, 320.000] (5\19 to 4\15)
* 11-odd-limit diamond tradeoff range: ~6/5 = [315.641, 317.263]
* 11-odd-limit diamond tradeoff: ~6/5 = [308.744, 324.341]


{{Optimal ET sequence|legend=1| 19, 34de, 53, 72, 197e, 269ce, 341ce, 610bccee }}
{{Optimal ET sequence|legend=0| 15, 19, 34 }}


Badness: 0.021849
Badness (Sintel): 0.906


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


Comma list: 169/168, 225/224, 325/324, 385/384
Comma list: 49/48, 56/55, 65/64, 100/99


Mapping: {{mapping| 1 0 1 -3 9 0 | 0 6 5 22 -21 14 }}
Mapping: {{mapping| 1 0 1 2 4 5 | 0 6 5 3 -2 -5 }}


Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.738
Optimal tunings:
* WE: ~2 = 1201.8360{{c}}, ~6/5 = 317.0958{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.6829{{c}}


Tuning ranges:  
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 monotone: ~6/5 = 315.789 (5\19)
* 13- and 15-odd-limit diamond tradeoff: ~6/5 = [315.641, 318.309]
* 13- and 15-odd-limit diamond tradeoff: ~6/5 = [303.597, 324.341]
 
{{Optimal ET sequence|legend=0| 4, 15f, 19 }}


{{Optimal ET sequence|legend=1| 19, 34de, 53, 72, 125f, 197ef, 269ceff }}
Badness (Sintel): 1.23


Badness: 0.016883
==== Kema ====
Subgroup: 2.3.5.7.11.13


=== Cataclysmic ===
Comma list: 49/48, 56/55, 91/90, 100/99
Subgroup: 2.3.5.7.11


Comma list: 99/98, 176/175, 2200/2187
Mapping: {{mapping| 1 0 1 2 4 0 | 0 6 5 3 -2 14 }}


Mapping: {{mapping| 1 0 1 -3 -5 | 0 6 5 22 32 }}
Optimal tunings:  
* WE: ~2 = 1199.7816{{c}}, ~6/5 = 317.3653{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.4070{{c}}


Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 317.042
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|legend=1| 19e, 34d, 53 }}
{{Optimal ET sequence|legend=0| 15, 19, 34 }}


Badness: 0.039954
Badness (Sintel): 0.940


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


Comma list: 99/98, 169/168, 176/175, 275/273
Comma list: 40/39, 49/48, 56/55, 66/65


Mapping: {{mapping| 1 0 1 -3 -5 0 | 0 6 5 22 32 14 }}
Mapping: {{mapping| 1 0 1 2 4 4 | 0 6 5 3 -2 -1 }}


Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 317.036
Optimal tunings:
* WE: ~2 = 1196.7615{{c}}, ~6/5 = 317.7353{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 319.4059{{c}}


{{Optimal ET sequence|legend=1| 19e, 34d, 53 }}
{{Optimal ET sequence|legend=0| 4, 11b, 15 }}


Badness: 0.022555
Badness (Sintel): 1.31


=== Catalytic ===
=== Qeema ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 225/224, 441/440, 4375/4374
Comma list: 45/44, 49/48, 126/125


Mapping: {{mapping| 1 0 1 -3 -10 | 0 6 5 22 51 }}
Mapping: {{mapping| 1 0 1 2 -1 | 0 6 5 3 17 }}


Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.653
Optimal tunings:
* WE: ~2 = 1204.5534{{c}}, ~6/5 = 315.9247{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 315.1686{{c}}


{{Optimal ET sequence|legend=1| 19e, 53e, 72 }}
{{Optimal ET sequence|legend=0| 4e, 19, 42bcd }}


Badness: 0.030422
Badness (Sintel): 1.32


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


Comma list: 169/168, 225/224, 325/324, 1716/1715
Comma list: 45/44, 49/48, 78/77, 126/125


Mapping: {{mapping| 1 0 1 -3 -10 0 | 0 6 5 22 51 14 }}
Mapping: {{mapping| 1 0 1 2 -1 0 | 0 6 5 3 17 14 }}


Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.639
Optimal tunings:
* WE: ~2 = 1204.4937{{c}}, ~6/5 = 316.2241{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 315.4748{{c}}


{{Optimal ET sequence|legend=1| 19e, 53e, 72 }}
{{Optimal ET sequence|legend=0| 4ef, 19 }}


Badness: 0.022337
Badness (Sintel): 1.22


=== Cataleptic ===
=== Darjeeling ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 100/99, 225/224, 864/847
Comma list: 49/48, 55/54, 77/75


Mapping: {{mapping| 1 0 1 -3 4 | 0 6 5 22 -2 }}
Mapping: {{mapping| 1 0 1 2 0 | 0 6 5 3 13 }}


Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 317.083
Optimal tunings:
* WE: ~2 = 1201.6569{{c}}, ~6/5 = 318.0942{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.8547{{c}}


{{Optimal ET sequence|legend=1| 19, 34d, 53e }}
{{Optimal ET sequence|legend=0| 15, 19e, 34e }}


Badness: 0.044335
Badness (Sintel): 0.914


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


Comma list: 78/77, 100/99, 144/143, 676/675
Comma list: 49/48, 55/54, 66/65, 77/75


Mapping: {{mapping| 1 0 1 -3 4 0 | 0 6 5 22 -2 14 }}
Mapping: {{mapping| 1 0 1 2 0 0 | 0 6 5 3 13 14 }}


Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 317.118
Optimal tunings:
* WE: ~2 = 1201.9324{{c}}, ~6/5 = 317.8090{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.5170{{c}}


{{Optimal ET sequence|legend=1| 19, 34d, 53e, 87dee }}
{{Optimal ET sequence|legend=0| 15, 19e, 34e }}


Badness: 0.027343
Badness (Sintel): 0.886


=== Bikleismic ===
== Catalan ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7


Comma list: 225/224, 243/242, 4375/4356
[[Comma list]]: 64/63, 15625/15552


Mapping: {{mapping| 2 0 2 -6 -1 | 0 6 5 22 15 }}
{{Mapping|legend=1| 1 0 1 6 | 0 6 5 -12 }}


: mapping generators: ~99/70, ~6/5
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1197.1789{{c}}, ~6/5 = 317.5185{{c}}
: [[error map]]: {{val| -2.821 +3.156 -1.542 +4.025 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 318.2411{{c}}
: error map: {{val| 0.000 +7.492 +4.892 +12.281 }}


Optimal tuning (POTE): ~99/70 = 1\2, ~6/5 = 316.721
[[Tuning ranges]]:
* 7- and 9-odd-limit [[diamond monotone]]: ~6/5 = [317.647, 320.000] (9\34 to 4\15)
* 7- and 9-odd-limit [[diamond tradeoff]]: ~6/5 = [315.641, 319.265]


{{Optimal ET sequence|legend=1| 34d, 72, 322c, …, 610bcc }}
{{Optimal ET sequence|legend=1| 15, 34d, 49, 132bcdd, 181bbcddd }}


Badness: 0.029319
[[Badness]] (Sintel): 2.40


==== 13-limit ====
=== 11-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11


Comma list: 169/168, 225/224, 243/242, 325/324
Comma list: 64/63, 100/99, 1331/1323


Mapping: {{mapping| 2 0 2 -6 -1 0 | 0 6 5 22 15 14 }}
Mapping: {{mapping| 1 0 1 6 4 | 0 6 5 -12 -2 }}


Optimal tuning (POTE): ~55/39 = 1\2, ~6/5 = 316.726
Optimal tunings:
* WE: ~2 = 1197.0368{{c}}, ~6/5 = 317.4956{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 318.2672{{c}}


{{Optimal ET sequence|legend=1| 34d, 72 }}
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]


Badness: 0.021814
{{Optimal ET sequence|legend=0| 15, 34d, 49, 181bbcdddeee }}


==== 17-limit ====
Badness (Sintel): 1.22
Subgroup: 2.3.5.7.11.13.17


Comma list: 169/168, 221/220, 225/224, 243/242, 325/324
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Mapping: {{mapping| 2 0 2 -6 -1 0 5 | 0 6 5 22 15 14 6 }}
Comma list: 64/63, 100/99, 144/143, 275/273


Optimal tuning (POTE): ~17/12 = 1\2, ~6/5 = 316.726
Mapping: {{mapping| 1 0 1 6 4 0 | 0 6 5 -12 -2 14 }}


{{Optimal ET sequence|legend=1| 34d, 38df, 72 }}
Optimal tunings:  
 
* WE: ~2 = 1196.8961{{c}}, ~6/5 = 317.3837{{c}}
Badness: 0.015656
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 318.1621{{c}}
 
==== 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: {{mapping| 2 0 2 -6 -1 0 5 -1 | 0 6 5 22 15 14 6 18 }}
 
Optimal tuning (POTE): ~17/12 = 1\2, ~6/5 = 316.726


{{Optimal ET sequence|legend=1| 34dh, 38df, 72 }}
{{Optimal ET sequence|legend=0| 15, 34d, 49f, 83def, 132bcddeefff }}


Badness: 0.015771
Badness (Sintel): 1.09


== Countercata ==
== Countercata ==
Line 457: Line 494:
{{Mapping|legend=1| 1 0 1 11 | 0 6 5 -31 }}
{{Mapping|legend=1| 1 0 1 11 | 0 6 5 -31 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 317.121
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9172{{c}}, ~6/5 = 317.0995{{c}}
: [[error map]]: {{val| -0.083 +0.642 -0.899 +0.178 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 317.1220{{c}}
: error map: {{val| 0.000 +0.777 -0.704 +0.391 }}


[[Tuning ranges]]:  
[[Tuning ranges]]:  
Line 463: Line 504:
* 7- and 9-odd-limit [[diamond tradeoff]]: ~6/5 = [315.641, 317.263]
* 7- and 9-odd-limit [[diamond tradeoff]]: ~6/5 = [315.641, 317.263]


{{Optimal ET sequence|legend=1| 19d, 34, 53, 87, 140, 333, 473, 806b }}
{{Optimal ET sequence|legend=1| 19d, 34, 53, 87, 140, 333, 473 }}


[[Badness]]: 0.052129
[[Badness]] (Sintel): 1.32


=== 11-limit ===
=== 11-limit ===
Line 474: Line 515:
Mapping: {{mapping| 1 0 1 11 -5 | 0 6 5 -31 32 }}
Mapping: {{mapping| 1 0 1 11 -5 | 0 6 5 -31 32 }}


Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 317.162
Optimal tunings:
* WE: ~2 = 1200.0980{{c}}, ~6/5 = 317.1879{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.1623{{c}}


Tuning ranges:  
Tuning ranges:  
Line 480: Line 523:
* 11-odd-limit diamond tradeoff: ~6/5 = [315.641, 317.370]
* 11-odd-limit diamond tradeoff: ~6/5 = [315.641, 317.370]


{{Optimal ET sequence|legend=1| 34, 53, 87, 140, 227, 367e, 507e }}
{{Optimal ET sequence|legend=0| 34, 53, 87, 140, 227, 367e }}


Badness: 0.039770
Badness (Sintel): 1.31


=== 13-limit ===
=== 13-limit ===
Line 491: Line 534:
Mapping: {{mapping| 1 0 1 11 -5 0 | 0 6 5 -31 32 14 }}
Mapping: {{mapping| 1 0 1 11 -5 0 | 0 6 5 -31 32 14 }}


Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 317.162
Optimal tunings:
* WE: ~2 = 1200.0936{{c}}, ~6/5 = 317.1864{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.1622{{c}}


Tuning ranges:  
Tuning ranges:  
Line 498: Line 543:
* 13- and 15-odd-limit diamond tradeoff: ~6/5 = [315.641, 318.309]
* 13- and 15-odd-limit diamond tradeoff: ~6/5 = [315.641, 318.309]


{{Optimal ET sequence|legend=1| 34, 53, 87, 140, 367e, 507e }}
{{Optimal ET sequence|legend=0| 34, 53, 87, 140, 367e, 507e }}


Badness: 0.020156
Badness (Sintel): 0.833


== Metakleismic ==
== Metakleismic ==
Line 509: Line 554:
{{Mapping|legend=1| 1 0 1 -12 | 0 6 5 56 }}
{{Mapping|legend=1| 1 0 1 -12 | 0 6 5 56 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 317.314
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.5969{{c}}, ~6/5 = 317.2079{{c}}
: [[error map]]: {{val| -0.403 +1.292 -0.678 -0.349 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 317.3071{{c}}
: error map: {{val| 0.000 +1.887 +0.222 +0.370 }}


{{Optimal ET sequence|legend=1| 34d, 87, 121, 208 }}
{{Optimal ET sequence|legend=1| 34d, 87, 121, 208, 537b }}


[[Badness]]: 0.163519
[[Badness]] (Sintel): 4.14


=== 11-limit ===
=== 11-limit ===
Line 522: Line 571:
Mapping: {{mapping| 1 0 1 -12 -5 | 0 6 5 56 32 }}
Mapping: {{mapping| 1 0 1 -12 -5 | 0 6 5 56 32 }}


Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 317.311
Optimal tunings:
* WE: ~2 = 1199.5425{{c}}, ~6/5 = 317.1901{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.3020{{c}}


{{Optimal ET sequence|legend=1| 34d, 53d, 87, 121, 208 }}
{{Optimal ET sequence|legend=0| 34d, 53d, 87, 121, 208 }}


Badness: 0.048570
Badness (Sintel): 1.61


=== 13-limit ===
=== 13-limit ===
Line 535: Line 586:
Mapping: {{mapping| 1 0 1 -12 -5 0 | 0 6 5 56 32 14 }}
Mapping: {{mapping| 1 0 1 -12 -5 0 | 0 6 5 56 32 14 }}


Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 317.311
Optimal tunings:
* WE: ~2 = 1199.5339{{c}}, ~6/5 = 317.1882{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.3028{{c}}


{{Optimal ET sequence|legend=1| 34d, 53d, 87, 121, 208 }}
{{Optimal ET sequence|legend=0| 34d, 53d, 87, 121, 208 }}


Badness: 0.024371
Badness (Sintel): 1.01


== Hemikleismic ==
== Hemikleismic ==
Line 547: Line 600:


{{Mapping|legend=1| 1 0 1 4 | 0 12 10 -9 }}
{{Mapping|legend=1| 1 0 1 4 | 0 12 10 -9 }}
: mapping generators: ~2, ~35/32


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~35/32 = 158.649
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.3950{{c}}, ~35/32 = 158.5686{{c}}
: [[error map]]: {{val| -0.605 +0.868 -1.233 +1.637 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~35/32 = 158.6338{{c}}
: error map: {{val| 0.000 +1.651 +0.024 +3.470 }}


{{Optimal ET sequence|legend=1| 15, 38, 53, 121 }}
{{Optimal ET sequence|legend=1| 15, 38, 53, 121, 174d, 295d }}


[[Badness]]: 0.052054
[[Badness]] (Sintel): 1.32


=== 11-limit ===
=== 11-limit ===
Line 561: Line 619:
Mapping: {{mapping| 1 0 1 4 2 | 0 12 10 -9 11 }}
Mapping: {{mapping| 1 0 1 4 2 | 0 12 10 -9 11 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 158.677
Optimal tunings:
* WE: ~2 = 1199.8009{{c}}, ~11/10 = 158.6508{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 158.6717{{c}}


{{Optimal ET sequence|legend=1| 15, 38, 53, 68, 121e }}
{{Optimal ET sequence|legend=0| 15, 38, 53, 68, 121e }}


Badness: 0.038023
Badness (Sintel): 1.26


=== 13-limit ===
=== 13-limit ===
Line 574: Line 634:
Mapping: {{mapping| 1 0 1 4 2 0 | 0 12 10 -9 11 28 }}
Mapping: {{mapping| 1 0 1 4 2 0 | 0 12 10 -9 11 28 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 158.655
Optimal tunings:
* WE: ~2 = 1199.7952{{c}}, ~11/10 = 158.6279{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 158.6493{{c}}


{{Optimal ET sequence|legend=1| 15, 38f, 53, 121e }}
{{Optimal ET sequence|legend=0| 15, 38f, 53, 121e }}


Badness: 0.026005
Badness (Sintel): 1.07


== Clyde ==
== Clyde ==
Line 585: Line 647:
[[Comma list]]: 245/243, 3136/3125
[[Comma list]]: 245/243, 3136/3125


{{Mapping|legend=1| 1 6 6 12 | 0 -12 -10 -25 }}
{{Mapping|legend=1| 1 -6 -4 -13 | 0 12 10 25 }}
: mapping generators: ~2, ~14/9


: mapping generators: ~2, ~9/7
[[Optimal tuning]]s:  
 
* [[WE]]: ~2 = 1199.8369{{c}}, ~14/9 = 758.5621{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/7 = 441.335
: [[error map]]: {{val| -0.163 +1.769 -0.040 -2.652 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~14/9 = 758.6554{{c}}
: error map: {{val| 0.000 +1.910 +0.240 -2.441 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~9/7 = {{monzo| 12/25 0 0 -1/25 }}
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~14/9 = {{monzo| 13/25 0 0 1/25 }}
: {{monzo list| 1 0 0 0 | 6/25 0 0 12/25 | 6/5 0 0 2/5 | 0 0 0 1 }}
: {{monzo list| 1 0 0 0 | 6/25 0 0 12/25 | 6/5 0 0 2/5 | 0 0 0 1 }}
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7


[[Algebraic generator]]: real root of 5''x''<sup>3</sup> - 6''x'' - 3, the Poussami generator. Approximately 441.309 [[cent]]s. Associated recurrence relationship quickly converges.
[[Algebraic generator]]: real root of 5''x''<sup>3</sup> - 6''x'' - 3, the Poussami generator. Approximately 441.309 [[cent]]s. Associated recurrence relationship quickly converges.


{{Optimal ET sequence|legend=1| 19, 49, 68, 87, 155 }}
{{Optimal ET sequence|legend=1| 19, 49, 68, 87, 155, 242 }}


[[Badness]]: 0.047261
[[Badness]] (Sintel): 1.20


=== 11-limit ===
=== 11-limit ===
Line 607: Line 672:
Comma list: 245/243, 385/384, 3136/3125
Comma list: 245/243, 385/384, 3136/3125


Mapping: {{mapping| 1 6 6 12 -5 | 0 -12 -10 -25 23 }}
Mapping: {{mapping| 1 -6 -4 -13 18 | 0 12 10 25 -23 }}


Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 441.355
Optimal tunings:
* WE: ~2 = 1199.9620{{c}}, ~14/9 = 758.6210{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/9 = 758.6445{{c}}


{{Optimal ET sequence|legend=1| 19, 49e, 68, 87, 329bd, 419bd, 503bd, 590bd }}
{{Optimal ET sequence|legend=0| 19, 49e, 68, 87 }}


Badness: 0.047417
Badness (Sintel): 1.57


=== 13-limit ===
=== 13-limit ===
Line 620: Line 687:
Comma list: 196/195, 245/243, 385/384, 625/624
Comma list: 196/195, 245/243, 385/384, 625/624


Mapping: {{mapping| 1 6 6 12 -5 14 | 0 -12 -10 -25 23 -28 }}
Mapping: {{mapping| 1 -6 -4 -13 18 -14 | 0 12 10 25 -23 28 }}


Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 441.363
Optimal tunings:
* WE: ~2 = 1199.9292{{c}}, ~14/9 = 758.5919{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/9 = 758.6355{{c}}


{{Optimal ET sequence|legend=1| 19, 49ef, 68, 87, 503bdf, 590bdf }}
{{Optimal ET sequence|legend=0| 19, 68, 87 }}


Badness: 0.026842
Badness (Sintel): 1.11


== Tritikleismic ==
== Tritikleismic ==
Line 634: Line 703:


{{Mapping|legend=1| 3 0 3 10 | 0 6 5 -2 }}
{{Mapping|legend=1| 3 0 3 10 | 0 6 5 -2 }}
: mapping generators: ~63/50, ~6/5
: mapping generators: ~63/50, ~6/5


[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~6/5 = 316.872 (~21/20 = 83.128)
[[Optimal tuning]]s:
* [[WE]]: ~63/50 = 400.1845{{c}}, ~6/5 = 317.0178{{c}} (~21/20 = 83.1667{{c}})
: [[error map]]: {{val| +0.553 +0.152 -0.671 -1.017 }}
* [[CWE]]: ~63/50 = 400.0000{{c}}, ~6/5 = 316.9129{{c}} (~21/20 = 83.0871{{c}})
: error map: {{val| 0.000 -0.478 -1.749 -2.652 }}


[[Minimax tuning]]:
[[Minimax tuning]]:
* [[7-odd-limit]]: ~6/5 = {{monzo| 1/3 0 1/7 -1/7 }}
* [[7-odd-limit]]: ~6/5 = {{monzo| 1/3 0 1/7 -1/7 }}
: [{{monzo| 1 0 0 0 }}, {{monzo| 2 0 6/7 -6/7 }}, {{monzo| 8/3 0 5/7 -5/7 }}, {{monzo| 8/3 0 -2/7 2/7 }}]
: [{{monzo| 1 0 0 0 }}, {{monzo| 2 0 6/7 -6/7 }}, {{monzo| 8/3 0 5/7 -5/7 }}, {{monzo| 8/3 0 -2/7 2/7 }}]
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5
* [[9-odd-limit]]: ~6/5 = {{monzo| 5/21 1/7 0 -1/14 }}
* [[9-odd-limit]]: ~6/5 = {{monzo| 5/21 1/7 0 -1/14 }}
: [{{monzo| 1 0 0 0 }}, {{monzo| 10/7 6/7 0 -3/7 }}, {{monzo| 46/21 5/7 0 -5/14 }}, {{monzo| 20/7 -2/7 0 1/7 }}]
: [{{monzo| 1 0 0 0 }}, {{monzo| 10/7 6/7 0 -3/7 }}, {{monzo| 46/21 5/7 0 -5/14 }}, {{monzo| 20/7 -2/7 0 1/7 }}]
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7


{{Optimal ET sequence|legend=1| 15, 42bc, 57, 72, 159, 231 }}
{{Optimal ET sequence|legend=1| 15, 42bc, 57, 72, 159, 231, 765ccddd }}


[[Badness]]: 0.056337
[[Badness]] (Sintel): 1.43


Music:
; Music
* ''[https://www.youtube.com/watch?v=vdjhC9i5KF4 Four Short Experiments in Octave Stretched 42edo (Dec 2024)]'' - [[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 661: Line 733:
Mapping: {{mapping| 3 0 3 10 8 | 0 6 5 -2 3 }}
Mapping: {{mapping| 3 0 3 10 8 | 0 6 5 -2 3 }}


Optimal tuning (POTE): ~44/35 = 1\3, ~6/5 = 316.881 (~21/20 = 83.119)
Optimal tunings:
* WE: ~44/35 = 400.1571{{c}}, ~6/5 = 317.0058{{c}} (~21/20 = 83.1514{{c}})
* CWE: ~44/35 = 400.0000{{c}}, ~6/5 = 316.9154{{c}} (~21/20 = 83.0846{{c}})


Minimax tuning:
Minimax tuning:
Line 668: Line 742:
: unchanged-interval (eigenmonzo) basis: 2.9/7
: unchanged-interval (eigenmonzo) basis: 2.9/7


{{Optimal ET sequence|legend=1| 15, 42bc, 57, 72, 159, 231 }}
{{Optimal ET sequence|legend=0| 15, 42bc, 57, 72, 159, 231 }}


Badness: 0.019333
Badness (Sintel): 0.639


=== 13-limit ===
=== 13-limit ===
Line 679: Line 753:
Mapping: {{mapping| 3 0 3 10 8 0 | 0 6 5 -2 3 14 }}
Mapping: {{mapping| 3 0 3 10 8 0 | 0 6 5 -2 3 14 }}


Optimal tuning (POTE): ~44/35 = 1\3, ~6/5 = 316.9585 (~21/20 = 83.0415)
Optimal tunings:
* WE: ~44/35 = 400.1514{{c}}, ~6/5 = 317.0785{{c}} (~21/20 = 83.0729{{c}})
* CWE: ~44/35 = 400.0000{{c}}, ~6/5 = 316.9896{{c}} (~21/20 = 83.0104{{c}})


{{Optimal ET sequence|legend=1| 72, 87, 159 }}
{{Optimal ET sequence|legend=0| 15, 57f, 72, 87, 159 }}


Badness: 0.015652
Badness (Sintel): 0.647


=== 17-limit ===
=== 17-limit ===
Line 692: Line 768:
Mapping: {{mapping| 3 0 3 10 8 0 -2 | 0 6 5 -2 3 14 18 }}
Mapping: {{mapping| 3 0 3 10 8 0 -2 | 0 6 5 -2 3 14 18 }}


Optimal tuning (POTE): ~34/27 = 1\3, ~6/5 = 316.9082 (~21/20 = 83.0918)
Optimal tunings:
* WE: ~34/27 = 400.1604{{c}}, ~6/5 = 317.0353{{c}} (~21/20 = 83.1251{{c}})
* CWE: ~34/27 = 400.0000{{c}}, ~6/5 = 316.9384{{c}} (~21/20 = 83.0616{{c}})
 
{{Optimal ET sequence|legend=0| 15g, 57fg, 72, 159, 231f }}


{{Optimal ET sequence|legend=1| 72, 159, 231f }}
Badness (Sintel): 0.690


Badness: 0.013551
== Marfifths ==
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.  


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


[[Comma list]]: 2401/2400, 15625/15552
[[Comma list]]: 10976/10935, 15625/15552
 
{{Mapping|legend=1| 1 -6 -4 -17 | 0 18 15 47 }}
: mapping generators: ~2, ~75/56
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0223{{c}}, ~75/56 = 505.7147{{c}}
: [[error map]]: {{val| +0.022 +0.775 -0.683 -0.615 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~75/56 = 505.7060{{c}}
: error map: {{val| 0.000 +0.753 -0.724 -0.643 }}
 
{{Optimal ET sequence|legend=1| 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: {{mapping| 1 -6 -4 -17 22 | 0 18 15 47 -44 }}
 
Optimal tunings:
* WE: ~2 = 1200.2484{{c}}, ~75/56 = 505.7882{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~75/56 = 505.6853{{c}}
 
{{Optimal ET sequence|legend=0| 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|legend=1| 4 0 4 7 | 0 6 5 4 }}
Mapping: {{mapping| 1 -6 -4 -17 22 -14 | 0 18 15 47 -44 42 }}


: mapping generators: ~25/21, ~6/5
Optimal tunings:  
* WE: ~2 = 1200.2747{{c}}, ~75/56 = 505.8019{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~75/56 = 505.6883{{c}}


[[Optimal tuning]] ([[POTE]]): ~25/21 = 1\4, ~6/5 = 316.9999 (~126/125 = 16.9999)
{{Optimal ET sequence|legend=0| 19, 121e, 140, 159, 299 }}


{{Optimal ET sequence|legend=1| 68, 72, 140, 212, 776cd, 988ccd, 1200ccd }}
Badness (Sintel): 1.24


[[Badness]]: 0.039231
=== Diatessic ===
Diatessic may be described as {{nowrap| 121 & 140 }} and is closely related to the Diatess tuning (generator: 505.727281 cents).


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


Comma list: 385/384, 1375/1372, 6250/6237
Comma list: 1375/1372, 2200/2187, 5632/5625


Mapping: {{mapping| 4 0 4 7 17 | 0 6 5 4 -3 }}
Mapping: {{mapping| 1 -6 -4 -17 -37 | 0 18 15 47 96 }}


Optimal tuning (POTE): ~25/21 = 1\4, ~6/5 = 316.9247 (~100/99 = 16.9247)
Optimal tunings:
* WE: ~2 = 1199.7886{{c}}, ~75/56 = 505.6513{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~75/56 = 505.7366{{c}}


{{Optimal ET sequence|legend=1| 68, 72, 140, 212, 284, 496ce, 780ccdee }}
{{Optimal ET sequence|legend=0| 19e, …, 121, 140, 261, 401 }}


Badness: 0.023406
Badness (Sintel): 2.02


=== 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, 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 }}
 
Optimal tunings:
* WE: ~2 = 1199.7996{{c}}, ~75/56 = 505.6558{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~75/56 = 505.7366{{c}}
 
{{Optimal ET sequence|legend=0| 19e, …, 121, 140, 261, 401 }}
 
Badness (Sintel): 1.18
 
=== Marf ===
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
 
Comma list: 540/539, 896/891, 15625/15552


Mapping: {{mapping| 4 0 4 7 17 0 | 0 6 5 4 -3 14 }}
Mapping: {{mapping| 1 -6 -4 -17 14 | 0 18 15 47 -25 }}


Optimal tuning (POTE): ~25/21 = 1\4, ~6/5 = 316.9887 (~100/99 = 16.9887)
Optimal tunings:
* WE: ~2 = 1199.3198{{c}}, ~75/56 = 505.4822{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~75/56 = 505.7607{{c}}


{{Optimal ET sequence|legend=1| 68, 72, 140, 212 }}
{{Optimal ET sequence|legend=0| 19, 102d, 121 }}


Badness: 0.018731
Badness (Sintel): 2.48


=== 17-limit ===
==== 13-limit ====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13


Comma list: 289/288, 325/324, 385/384, 442/441, 625/624
Comma list: 325/324, 540/539, 625/624, 896/891


Mapping: {{mapping| 4 0 4 7 17 0 10 | 0 6 5 4 -3 14 6 }}
Mapping: {{mapping| 1 -6 -4 -17 14 -14 | 0 18 15 47 -25 42 }}


Optimal tuning (POTE): ~25/21 = 1\4, ~6/5 = 316.9846 (~100/99 = 16.9846)
Optimal tunings:
* WE: ~2 = 1199.3368{{c}}, ~75/56 = 505.4919{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~75/56 = 505.7627{{c}}


{{Optimal ET sequence|legend=1| 68, 72, 140, 212g }}
{{Optimal ET sequence|legend=0| 19, 102df, 121 }}


Badness: 0.012784
Badness (Sintel): 1.58


== Kleiboh ==
== Kleiboh ==
Line 757: Line 895:
[[Comma list]]: 1728/1715, 3125/3087
[[Comma list]]: 1728/1715, 3125/3087


{{Mapping|legend=1| 1 6 6 6 | 0 -18 -15 -13 }}
{{Mapping|legend=1| 1 -12 -9 -7 | 0 18 15 13 }}
: mapping generators: ~2, ~42/25


: mapping generators: ~2, ~25/21
[[Optimal tuning]]s:  
 
* [[WE]]: ~2 = 1199.5290{{c}}, ~42/25 = 905.3417{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~25/21 = 294.303
: [[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, 314d }}
{{Optimal ET sequence|legend=1| 49, 53 }}


[[Badness]]: 0.076460
[[Badness]] (Sintel): 1.93


=== 11-limit ===
=== 11-limit ===
Line 772: Line 913:
Comma list: 176/175, 540/539, 3125/3087
Comma list: 176/175, 540/539, 3125/3087


Mapping: {{mapping| 1 6 6 6 14 | 0 -18 -15 -13 -43 }}
Mapping: {{mapping| 1 -12 -9 -7 -29 | 0 18 15 13 43 }}


Optimal tuning (POTE): ~2 = 1\1, ~25/21 = 294.181
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=1| 49, 53, 102d, 155d }}
{{Optimal ET sequence|legend=0| 49, 53, 102d }}


Badness: 0.052805
Badness (Sintel): 1.75


=== 13-limit ===
=== 13-limit ===
Line 785: Line 928:
Comma list: 176/175, 275/273, 325/324, 540/539
Comma list: 176/175, 275/273, 325/324, 540/539


Mapping: {{mapping| 1 6 6 6 14 14 | 0 -18 -15 -13 -43 -42 }}
Mapping: {{mapping| 1 -12 -9 -7 -29 -28 | 0 18 15 13 43 42 }}


Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 294.187
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=1| 49f, 53, 102df, 155d }}
{{Optimal ET sequence|legend=0| 49f, 53, 102df }}


Badness: 0.031074
Badness (Sintel): 1.28
 
== Marfifths ==
The ''marfifths'' temperament (19&amp;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.


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


[[Comma list]]: 10976/10935, 15625/15552
[[Comma list]]: 2401/2400, 15625/15552


{{Mapping|legend=1| 1 -6 -4 -17 | 0 18 15 47 }}
{{Mapping|legend=1| 4 0 4 7 | 0 6 5 4 }}
: mapping generators: ~25/21, ~6/5


: mapping generators: ~2, ~75/5
[[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 tuning]] ([[POTE]]): ~2 = 1\1, ~75/56 = 505.705
{{Optimal ET sequence|legend=1| 68, 72, 140, 212, 776cd, 988ccd, 1200ccd }}
 
{{Optimal ET sequence|legend=1| 19, …, 121, 140, 579, 719, 859bcd, 999bcd, 1858bbccdd }}


[[Badness]]: 0.063448
[[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, 10976/10935
Comma list: 385/384, 1375/1372, 6250/6237


Mapping: {{mapping| 1 -6 -4 -17 22 | 0 18 15 47 -44 }}
Mapping: {{mapping| 4 0 4 7 17 | 0 6 5 4 -3 }}


Optimal tuning (POTE): ~2 = 1\1, ~75/56 = 505.684
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=1| 19, 121e, 140, 159, 299 }}
{{Optimal ET sequence|legend=0| 68, 72, 140, 212, 284, 496ce, 780ccdee }}


Badness: 0.058902
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, 10976/10935
Comma list: 325/324, 385/384, 625/624, 1375/1372


Mapping: {{mapping| 1 -6 -4 -17 22 -14 | 0 18 15 47 -44 42 }}
Mapping: {{mapping| 4 0 4 7 17 0 | 0 6 5 4 -3 14 }}


Optimal tuning (POTE): ~2 = 1\1, ~75/56 = 505.686
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=1| 19, 121e, 140, 159, 299 }}
{{Optimal ET sequence|legend=0| 68, 72, 140, 212 }}


Badness: 0.030082
Badness (Sintel): 0.774


=== Diatessic ===
=== 17-limit ===
The ''diatessic'' temperament (121 &amp; 140) is closely related to the '''diatess tuning''' (generator: 505.727281 cents).
Subgroup: 2.3.5.7.11.13.17


Subgroup: 2.3.5.7.11
Comma list: 289/288, 325/324, 385/384, 442/441, 625/624


Comma list: 1375/1372, 2200/2187, 5632/5625
Mapping: {{mapping| 4 0 4 7 17 0 10 | 0 6 5 4 -3 14 6 }}


Mapping: {{mapping| 1 -6 -4 -17 -37 | 0 18 15 47 96 }}
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 tuning (POTE): ~2 = 1\1, ~75/56 = 505.740
{{Optimal ET sequence|legend=0| 68, 72, 140, 212g }}


{{Optimal ET sequence|legend=1| 19e, …, 121, 140, 261, 401 }}
Badness (Sintel): 0.651


Badness: 0.061172
== 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.  


==== 13-limit ====
[[Subgroup]]: 2.3.5.7
Subgroup: 2.3.5.7.11.13


Comma list: 325/324, 352/351, 625/624, 1375/1372
[[Comma list]]: 15625/15552, 2460375/2458624


Mapping: {{mapping| 1 -6 -4 -17 -37 -14 | 0 18 15 47 96 42 }}
{{Mapping|legend=1| 1 -6 -4 -19 | 0 24 20 69 }}
: mapping generators: ~2, ~56/45


Optimal tuning (POTE): ~2 = 1\1, ~75/56 = 505.740
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.1662{{c}}, ~56/45 = 379.3041{{c}}
: [[error map]]: {{val| +0.166 +0.347 -0.896 +0.000 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~56/45 = 379.2552{{c}}
: error map: {{val| 0.000 +0.171 -1.209 -0.214 }}


{{Optimal ET sequence|legend=1| 19e, …, 121, 140, 261, 401 }}
{{Optimal ET sequence|legend=1| 19, …, 193, 212, 617c, 829c }}


Badness: 0.028671
[[Badness]] (Sintel): 2.64
 
=== Marf ===
The ''marf'' temperament (19 &amp; 121) has a POTE generator which strongly approximates the marvelous fifth interval of 112/75.


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


Comma list: 540/539, 896/891, 15625/15552
Comma list: 1375/1372, 15625/15552, 19712/19683


Mapping: {{mapping| 1 -6 -4 -17 14 | 0 18 15 47 -25 }}
Mapping: {{mapping| 1 -6 -4 -19 -43 | 0 24 20 69 147 }}


Optimal tuning (POTE): ~2 = 1\1, ~75/56 = 505.769
Optimal tunings:
* WE: ~2 = 1200.1189{{c}}, ~56/45 = 379.2942{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~56/45 = 379.2580{{c}}


{{Optimal ET sequence|legend=1| 19, 102d, 121 }}
{{Optimal ET sequence|legend=0| 19e, …, 193, 212, 405, 617c }}


Badness: 0.075112
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, 540/539, 625/624, 896/891
Comma list: 325/324, 625/624, 1375/1372, 19712/19683
 
Mapping: {{mapping| 1 -6 -4 -19 -43 -14 | 0 24 20 69 147 56 }}
 
Optimal tunings:
* WE: ~2 = 1200.2154{{c}}, ~56/45 = 379.3236{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~56/45 = 379.2580{{c}}


Mapping: {{mapping| 1 -6 -4 -17 14 -14 | 0 18 15 47 -25 42 }}
{{Optimal ET sequence|legend=0| 19e, …, 193, 212, 405f, 617cff }}


Optimal tuning (POTE): ~2 = 1\1, ~75/56 = 505.771
Badness (Sintel): 1.81


{{Optimal ET sequence|legend=1| 19, 102df, 121 }}
== Sqrtphi ==
{{Main| Sqrtphi }}


Badness: 0.038317
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.  


== Marthirds ==
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.  
The ''marthirds'' temperament (19 &amp; 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]]: 15625/15552, 16875/16807
 
{{Mapping|legend=1| 1 -6 -4 -19 | 0 24 20 69 }}


: mapping generators: ~2, ~56/45
{{Mapping|legend=1| 1 -18 -14 -22 | 0 30 25 38 }}
: mapping generators: ~2, 196/125


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~56/45 = 379.252
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.1357{{c}}, ~196/125 = 783.4853{{c}}
: [[error map]]: {{val| +0.136 +0.163 -1.080 +0.632 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~196/125 = 783.4009{{c}}
: error map: {{val| 0.000 +0.072 -1.291 +0.408 }}


{{Optimal ET sequence|legend=1| 19, …, 193, 212, 617c, 829c }}
{{Optimal ET sequence|legend=1| 23d, 49, 72, 193, 265 }}


[[Badness]]: 0.104253
[[Badness]] (Sintel): 1.78


=== 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: 540/539, 1375/1372, 4375/4356


Mapping: {{mapping| 1 -6 -4 -19 -43 | 0 24 20 69 147 }}
Mapping: {{mapping| 1 -18 -14 -22 -22 | 0 30 25 38 39 }}


Optimal tuning (POTE): ~2 = 1\1, ~56/45 = 379.257
Optimal tunings:
* WE: ~2 = 1200.0514{{c}}, ~11/7 = 783.4294{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 783.3975{{c}}


{{Optimal ET sequence|legend=1| 19e, …, 193, 212, 405, 617c, 1022cce }}
{{Optimal ET sequence|legend=0| 23de, 49, 72, 193, 265 }}


Badness: 0.075624
Badness (Sintel): 0.844


=== 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: 325/324, 364/363, 625/624, 1375/1372
 
Mapping: {{mapping| 1 -18 -14 -22 -22 -42 | 0 30 25 38 39 70 }}
 
Optimal tunings:
* WE: ~2 = 1199.9314{{c}}, ~11/7 = 783.3705{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 783.4134{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 -18 -14 -22 -22 -42 -39 | 0 30 25 38 39 70 66 }}
 
Optimal tunings:
* WE: ~2 = 1199.9324{{c}}, ~11/7 = 783.3706{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 783.4129{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 -6 -4 -19 -43 -14 | 0 24 20 69 147 56 }}
Mapping: {{mapping| 1 -18 -14 -22 -22 -42 -39 16 | 0 30 25 38 39 70 66 -18 }}


Optimal tuning (POTE): ~2 = 1\1, ~56/45 = 379.256
Optimal tunings:
* WE: ~2 = 1199.8567{{c}}, ~11/7 = 783.3262{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 783.4176{{c}}


{{Optimal ET sequence|legend=1| 19e, …, 193, 212, 405f, 617cff }}
{{Optimal ET sequence|legend=0| 49fg, 72, 121, 193 }}


Badness: 0.043728
Badness (Sintel): 0.897


== Quartkeenlig ==
== 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]].
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]].


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 943: Line 1,143:


{{Mapping|legend=1| 1 0 1 1 | 0 36 30 41 }}
{{Mapping|legend=1| 1 0 1 1 | 0 36 30 41 }}
: mapping generator: ~2, ~36/35
: mapping generator: ~2, ~36/35


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~36/35 = 52.8562
[[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 }}
{{Optimal ET sequence|legend=1| 68, 91, 159, 386d, 545dd }}


[[Badness]]: 0.146
[[Badness]] (Sintel): 3.69


=== 11-limit ===
=== 11-limit ===
Line 959: Line 1,162:
Mapping: {{mapping| 1 0 1 1 5 | 0 36 30 41 -35 }}
Mapping: {{mapping| 1 0 1 1 5 | 0 36 30 41 -35 }}


Optimal tuning (CTE): ~2 = 1\1, ~33/32 = 52.8524
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=1| 68, 91, 159, 386d, 545dd }}
{{Optimal ET sequence|legend=0| 68, 91, 159, 386d, 545dd }}


Badness: 0.0865
Badness (Sintel): 2.86


=== 13-limit ===
=== 13-limit ===
Line 972: Line 1,177:
Mapping: {{mapping| 1 0 1 1 5 0 | 0 36 30 41 -35 84 }}
Mapping: {{mapping| 1 0 1 1 5 0 | 0 36 30 41 -35 84 }}


Optimal tuning (CTE): ~2 = 1\1, ~33/32 = 52.8562
Optimal tunings:
* WE: ~2 = 1200.2564{{c}}, ~36/35 = 52.8568{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 52.8479{{c}}


{{Optimal ET sequence|legend=1| 68, 159, 386d, 545ddf }}
{{Optimal ET sequence|legend=0| 68, 159, 386d, 545ddf }}


Badness: 0.0477
Badness (Sintel): 1.97


== Novemkleismic ==
== Novemkleismic ==
Line 984: Line 1,191:


{{Mapping|legend=1| 9 0 9 11 | 0 6 5 6 }}
{{Mapping|legend=1| 9 0 9 11 | 0 6 5 6 }}
: mapping generators: ~2592/2401, ~6/5
: mapping generators: ~2592/2401, ~6/5


[[Optimal tuning]] ([[POTE]]): ~2592/2401 = 1\9, ~6/5 = 317.005 (~36/35 = 50.338)
[[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 }}


{{Optimal ET sequence|legend=1| 72, 261, 333, 405, 477c, 882c }}
{{Optimal ET sequence|legend=1| 72, 261, 333, 405, 477c, 882c }}


[[Badness]]: 0.193429
[[Badness]] (Sintel): 4.90


=== 11-limit ===
=== 11-limit ===
Line 1,000: Line 1,210:
Mapping: {{mapping| 9 0 9 11 24 | 0 6 5 6 3 }}
Mapping: {{mapping| 9 0 9 11 24 | 0 6 5 6 3 }}


Optimal tuning (POTE): ~250/231 = 1\9, ~6/5 = 317.010 (~36/35 = 50.343)
Optimal tunings:
* WE: ~250/231 = 133.3465{{c}}, ~6/5 = 317.0416{{c}} (~36/35 = 50.3486{{c}})
* CWE: ~250/231 = 133.3333{{c}}, ~6/5 = 317.0264{{c}} (~36/35 = 50.3597{{c}})


{{Optimal ET sequence|legend=1| 72, 261, 333, 405, 882c }}
{{Optimal ET sequence|legend=0| 72, 261, 333, 405, 882c }}


Badness: 0.051730
Badness (Sintel): 1.71


=== 13-limit ===
=== 13-limit ===
Line 1,013: Line 1,225:
Mapping: {{mapping| 9 0 9 11 24 0 | 0 6 5 6 3 14 }}
Mapping: {{mapping| 9 0 9 11 24 0 | 0 6 5 6 3 14 }}


Optimal tuning (POTE): ~250/231 = 1\9, ~6/5 = 317.086 (~36/35 = 50.419)
Optimal tunings:
* WE: ~250/231 = 133.3385{{c}}, ~6/5 = 317.0978{{c}} (~36/35 = 50.4208{{c}})
* CWE: ~250/231 = 133.3333{{c}}, ~6/5 = 317.0910{{c}} (~36/35 = 50.4243{{c}})


{{Optimal ET sequence|legend=1| 72, 189f, 261, 333, 738cf }}
{{Optimal ET sequence|legend=0| 72, 189f, 261, 333, 738cf }}


Badness: 0.039072
Badness (Sintel): 1.61


== Sqrtphi ==
== Subgroup extensions ==
{{Main| Sqrtphi }}
=== Kleismic (2.3.5.13) a.k.a. cata ===
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]].


The just value of sqrt (φ) is 416.545 cents.  
Subgroup: 2.3.5.13


[[Subgroup]]: 2.3.5.7
Comma list: 325/324, 625/624


[[Comma list]]: 15625/15552, 16875/16807
Subgroup-val mapping: {{mapping| 1 0 1 0 | 0 6 5 14 }}


{{Mapping|legend=1| 1 12 11 16 | 0 -30 -25 -38 }}
Optimal tunings:
* WE: ~2 = 1200.1210{{c}}, ~6/5 = 317.1076{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0920{{c}}


: mapping generators: ~2, 125/98
{{Optimal ET sequence|legend=0| 15, 19, 34, 53, 140, 193, 246 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~125/98 = 416.603
Badness (Sintel): 0.131


{{Optimal ET sequence|legend=1| 49, 72, 193, 265 }}
==== 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.


[[Badness]]: 0.070378
Subgroup: 2.3.5.13.37.41
 
=== 11-limit ===
Subgroup: 2.3.5.7.11


Comma list: 540/539, 1375/1372, 4375/4356
Comma list: 325/324, 481/480, 625/624


Mapping: {{mapping| 1 12 11 16 17 | 0 -30 -25 -38 -39 }}
Subgroup-val mapping: {{mapping| 1 0 1 0 6 | 0 6 5 14 -3 }}


Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 416.604
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=1| 49, 72, 193, 265 }}
{{Optimal ET sequence|legend=0| 15, 19, 34, 53, 299l, 352fl, 405fl, 458fl, 511cfll, 564cffll }}


Badness: 0.025515
Badness (Sintel): 0.167


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


Comma list: 325/324, 364/363, 625/624, 1375/1372
Comma list: 325/324, 481/480, 625/624, 1025/1024


Mapping: {{mapping| 1 12 11 16 17 28 | 0 -30 -25 -38 -39 -70 }}
Subgroup-val mapping: {{mapping| 1 0 1 0 6 8 | 0 6 5 14 -3 -10 }}


Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 416.585
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=1| 49f, 72, 121, 193 }}
{{Optimal ET sequence|legend=0| 15, 19, 34, 53, 140, 193, 246l }}


Badness: 0.020040
Badness (Sintel): 0.223
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 325/324, 364/363, 375/374, 540/539, 595/594
 
Mapping: {{mapping| 1 12 11 16 17 28 27 | 0 -30 -25 -38 -39 -70 -66 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 416.585
 
{{Optimal ET sequence|legend=1| 49fg, 72, 121, 193 }}
 
Badness: 0.013028
 
=== 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: {{mapping| 1 12 11 16 17 28 27 -2 | 0 -30 -25 -38 -39 -70 -66 18 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 416.580
 
{{Optimal ET sequence|legend=1| 49fg, 72, 121, 193 }}
 
Badness: 0.014748
 
; Scales
* [[Sqrtphi17]]
* [[Sqrtphi23]]
* [[Sqrtphi49]]
 
; Music
* [http://micro.soonlabel.com/sqrt_phi/daily20111123a-sqrt-phi-17.mp3 ''Prelude for Piano in Square root of Phi Tuning''] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Sicurella/A%20Fight%20For%20Phi.mp3 ''A Fight for Phi''] by [[Vito Sicurella]]


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Kleismic family| ]] <!-- main article -->
[[Category:Kleismic family| ]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
[[Category:Listen]]
[[Category:Listen]]

Latest revision as of 09:24, 16 February 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.

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 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 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

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, 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.

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.

The kleismic family boasts a very remarkable extension to the 2.3.5.13 subgroup, which has further extensions with higher primes. These are listed at the bottom of this page, in #Subgroup extensions.

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.

Catakleismic extends easily with prime 13. The S-expression-based comma list of this extension is {S13, S15 = S25⋅S26⋅S27, S10/S12 = S25⋅S26, (S25, S26 = S13/S15, S27)}.

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

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

Subgroup extensions

Kleismic (2.3.5.13) a.k.a. cata

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 {S10/S12 = S25⋅S26, (S25), S13/S15 = S26}. For the high-limit 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 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 317.0920 ¢

Optimal ET sequence: 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: 2.3.5.13.37.41

Comma list: 325/324, 481/480, 625/624

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

Optimal tunings:

  • WE: ~2 = 1200.2924 ¢, ~6/5 = 317.0998 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 317.0452 ¢

Optimal ET sequence: 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: [⟨1 0 1 0 6 8], ⟨0 6 5 14 -3 -10]]

Optimal tunings:

  • WE: ~2 = 1200.1651 ¢, ~6/5 = 317.1126 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 317.0748 ¢

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

Badness (Sintel): 0.223

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