Kleismic family: Difference between revisions

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{{interwiki
| de = Hanson Kleismisch
| en = Kleismic family
| es =
| 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 12:


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


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 }}
==== Other subgroup extensions ====
The main extension of note is a very remarkable extension to the [[2.3.5.13 subgroup]], as the hemitwelfth, reached by three generator steps, can be interpreted as [[26/15]]. Notice 15625/15552 = ([[325/324]])⋅([[625/624]]) and 325/324 = (625/624)⋅([[676/675]]). The [[S-expression]]-based comma list of the temperament is {{nowrap| {[[325/324|S10/S12 = S25⋅S26]], ([[625/624|S25]],) [[676/675|S13/S15 = S26]]} }}.


Optimal tunings:
See [[#Subgroup extensions]].
* CTE: ~2 = 1\1, ~6/5 = 317.1110
* POTE: ~2 = 1\1, ~6/5 = 317.0756


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


Badness: 0.394
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.  


== Keemun ==
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]])}.
{{Main| Keemun }}


=== 7-limit ===
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 49/48, 126/125
[[Comma list]]: 225/224, 4375/4374


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


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 316.473
[[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 }}


[[Tuning ranges]]:  
[[Tuning ranges]]:  
* 7-odd-limit [[diamond monotone]]: ~6/5 = [300.000, 327.273] (1\4 to 3\11)
* 7- and 9-odd-limit [[diamond monotone]]: ~6/5 = [315.789, 317.647] (5\19 to 9\34)
* 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 = [315.641, 317.263]
* 7- and 9-odd-limit [[diamond tradeoff]]: ~6/5 = [308.744, 322.942]
 
{{Optimal ET sequence|legend=1| 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: {{mapping| 1 0 1 -3 0 | 0 6 5 22 14 }}
 
Optimal tunings:
* WE: ~2 = 1200.7838{{c}}, ~6/5 = 316.9478{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.7939{{c}}


{{Optimal ET sequence|legend=1| 15, 19, 53d, 72dd, 91dd }}
{{Optimal ET sequence|legend=0| 19, 34d, 53, 72, 125f, 197f }}


[[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}}
Tuning ranges:
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0403{{c}}
* 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 }}
{{Optimal ET sequence|legend=0| 19e, 34d, 53 }}


Badness: 0.022749
Badness (Sintel): 1.32


==== Kumbaya ====
==== 13-limit ====
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)
* 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]


Comma list: 169/168, 225/224, 325/324
{{Optimal ET sequence|legend=1| 15, 19, 53d, 72dd }}


Sval mapping: {{mapping| 1 0 1 -3 0 | 0 6 5 22 14 }}
[[Badness]] (Sintel): 0.694
 
Optimal tuning (CTE): ~2 = 1\1, ~6/5 = 316.8865
 
{{Optimal ET sequence|legend=1| 19, 34d, 53, 72, 125f, 197f }}
 
Badness: 0.0118


=== 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=1| 19, 34de, 53, 72, 125f, 197ef, 269ceff }}
{{Optimal ET sequence|legend=0| 4, 15f, 19 }}


Badness: 0.016883
Badness (Sintel): 1.23


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


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


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


Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 317.042
Optimal tunings:
* WE: ~2 = 1199.7816{{c}}, ~6/5 = 317.3653{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.4070{{c}}


{{Optimal ET sequence|legend=1| 19e, 34d, 53 }}
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=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 442: Line 492:
{{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 448: Line 502:
* 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 459: Line 513:
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 465: Line 521:
* 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 476: Line 532:
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 483: Line 541:
* 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 494: Line 552:
{{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 507: Line 569:
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 520: Line 584:
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 532: Line 598:


{{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 546: Line 617:
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 559: Line 632:
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 570: Line 645:
[[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 592: Line 670:
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 605: Line 685:
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 619: Line 701:


{{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 646: Line 731:
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 653: Line 740:
: 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 664: Line 751:
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 677: Line 766:
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 742: Line 893:
[[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 757: Line 911:
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 770: Line 926:
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 major tenth (~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]] (Sintel): 0.993
 
[[Badness]]: 0.063448


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


: mapping generators: ~2, ~56/45
[[Optimal tuning]]s:  
 
* [[WE]]: ~2 = 1200.1357{{c}}, ~196/125 = 783.4853{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~56/45 = 379.252
: [[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


Mapping: {{mapping| 1 -6 -4 -19 -43 -14 | 0 24 20 69 147 56 }}
Comma list: 325/324, 364/363, 375/374, 400/399, 442/441, 595/594


Optimal tuning (POTE): ~2 = 1\1, ~56/45 = 379.256
Mapping: {{mapping| 1 -18 -14 -22 -22 -42 -39 16 | 0 30 25 38 39 70 66 -18 }}


{{Optimal ET sequence|legend=1| 19e, …, 193, 212, 405f, 617cff }}
Optimal tunings:
* WE: ~2 = 1199.8567{{c}}, ~11/7 = 783.3262{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 783.4176{{c}}


Badness: 0.043728
{{Optimal ET sequence|legend=0| 49fg, 72, 121, 193 }}
 
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 928: Line 1,141:


{{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 944: Line 1,160:
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 957: Line 1,175:
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 969: Line 1,189:


{{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 985: Line 1,208:
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 998: Line 1,223:
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 }}
For the high-limit version of cata with a 1\5 period, see [[thunderclysmic]].


The just value of sqrt (φ) is 416.545 cents.  
=== Kleismic (2.3.5.13) a.k.a. cata ===
The structure of the temperament as dividing 3/1 into 6 equal parts can be deduced completely from its [[S-expression]]-based comma list of {{nowrap| {[[325/324|S10/S12 = S25⋅S26]], ([[625/624|S25]],) [[676/675|S13/S15 = S26]]} }}. Specifically, dividing 3/1 into two halves of ~26/15 is equivalent to dividing 4/3 into two halves of ~15/13, hence the [[semiparticular]] S13/S15 = ([[4/3|16/12]])/([[15/13]])<sup>2</sup>. From here, we notice that (26/15)/(13/9) = 6/5, so all that remains is dividing 13/9 into two 6/5's via the semiparticular S10/S12 = (13/9)/([[6/5|12/10]])<sup>2</sup>, hence explaining the mapping of the entire 2.3.5.13 subgroup.


[[Subgroup]]: 2.3.5.7
Subgroup: 2.3.5.13


[[Comma list]]: 15625/15552, 16875/16807
Comma list: 325/324, 625/624


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


: mapping generators: ~2, 125/98
Optimal tunings:  
 
* WE: ~2 = 1200.1210{{c}}, ~6/5 = 317.1076{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~125/98 = 416.603
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0920{{c}}
 
{{Optimal ET sequence|legend=1| 49, 72, 193, 265 }}
 
[[Badness]]: 0.070378
 
=== 11-limit ===
Subgroup: 2.3.5.7.11


Comma list: 540/539, 1375/1372, 4375/4356
{{Optimal ET sequence|legend=0| 15, 19, 34, 53, 140, 193, 246 }}


Mapping: {{mapping| 1 12 11 16 17 | 0 -30 -25 -38 -39 }}
Badness (Sintel): 0.131


Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 416.604
==== 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.


{{Optimal ET sequence|legend=1| 49, 72, 193, 265 }}
Subgroup: 2.3.5.13.37


Badness: 0.025515
Comma list: 325/324, 481/480, 625/624


=== 13-limit ===
Subgroup-val mapping: {{mapping| 1 0 1 0 6 | 0 6 5 14 -3 }}
Subgroup: 2.3.5.7.11.13


Comma list: 325/324, 364/363, 625/624, 1375/1372
Optimal tunings:
* WE: ~2 = 1200.2924{{c}}, ~6/5 = 317.0998{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0452{{c}}


Mapping: {{mapping| 1 12 11 16 17 28 | 0 -30 -25 -38 -39 -70 }}
{{Optimal ET sequence|legend=0| 15, 19, 34, 53, 299l, 352fl, 405fl, 458fl, 511cfll, 564cffll }}


Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 416.585
Badness (Sintel): 0.167


{{Optimal ET sequence|legend=1| 49f, 72, 121, 193 }}
==== 2.3.5.13.37.41 subgroup ====
Subgroup: 2.3.5.13.37.41


Badness: 0.020040
Comma list: 325/324, 481/480, 625/624, 1025/1024


=== 17-limit ===
Subgroup-val mapping: {{mapping| 1 0 1 0 6 8 | 0 6 5 14 -3 -10 }}
Subgroup: 2.3.5.7.11.13.17


Comma list: 325/324, 364/363, 375/374, 540/539, 595/594
Optimal tunings:
* WE: ~2 = 1200.1651{{c}}, ~6/5 = 317.1126{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0748{{c}}


Mapping: {{mapping| 1 12 11 16 17 28 27 | 0 -30 -25 -38 -39 -70 -66 }}
{{Optimal ET sequence|legend=0| 15, 19, 34, 53, 140, 193, 246l }}


Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 416.585
Badness (Sintel): 0.223
 
{{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:Kleismic family| ]] <!-- main article -->
[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Kleismic family| ]] <!-- main article -->
[[Category:Catalogs of rank-2 temperaments]]
[[Category:Rank 2]]
[[Category:Listen]]
[[Category:Listen]]

Latest revision as of 13:29, 28 August 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

Full 7-limit extensions

The second comma of the normal comma list defines which 7-limit family member we are looking at. 4375/4374, the ragisma, gives catakleismic. 875/864, the keemic comma, gives keemun. 5120/5103, 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.

Other subgroup extensions

The main extension of note is a very remarkable extension to the 2.3.5.13 subgroup, as the hemitwelfth, reached by three generator steps, can be interpreted as 26/15. Notice 15625/15552 = (325/324)⋅(625/624) and 325/324 = (625/624)⋅(676/675). The S-expression-based comma list of the temperament is {S10/S12 = S25⋅S26, (S25,) S13/S15 = S26}.

See #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

For the high-limit version of cata with a 1\5 period, see thunderclysmic.

Kleismic (2.3.5.13) a.k.a. cata

The structure of the temperament as dividing 3/1 into 6 equal parts can be deduced completely from its S-expression-based comma list of {S10/S12 = S25⋅S26, (S25,) S13/S15 = S26}. Specifically, dividing 3/1 into two halves of ~26/15 is equivalent to dividing 4/3 into two halves of ~15/13, hence the semiparticular S13/S15 = (16/12)/(15/13)2. From here, we notice that (26/15)/(13/9) = 6/5, so all that remains is dividing 13/9 into two 6/5's via the semiparticular S10/S12 = (13/9)/(12/10)2, hence explaining the mapping of the entire 2.3.5.13 subgroup.

Subgroup: 2.3.5.13

Comma list: 325/324, 625/624

Subgroup-val mapping: [⟨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

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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