Kleismic family: Difference between revisions

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<span style="display: block; text-align: right;">
{{Technical data page}}
[[de:Hanson_Kleismisch]]</span>
The [[5-limit]] parent comma for the '''kleismic family''' is [[15625/15552]], the kleisma, which is the amount by which a stack of six [[6/5|classical minor third]]s falls short of the [[3/1|3rd]] [[harmonic]].


The [[5-limit]] parent comma for the '''kleismic family''' is [[15625/15552]], the kleisma. Its monzo is {{monzo| -6 -5 6 }}, and flipping that yields &lt;&lt;6 5 -6|| for the wedgie. This tells us the generator is a minor third, and that to get to the interval class of major thirds will require five of these, and so to get to fifths will require six. In fact, (6/5)<sup>5</sup> = 5/2 × 15625/15552. This 5-limit temperament (virtually a [[microtemperament]]) is commonly called '''Hanson''', and 14\53 is about perfect as a hanson generator, though 9\34 also makes sense, and 5\19 and 4\15 are possible. Other tunings include [[72edo]], [[87edo]] and [[140edo]].
== Kleismic a.k.a. hanson ==
{{Main| Kleismic }}


[[Tuning Ranges of Regular Temperaments|valid range]]: [300.000, 327.273] (4 to 11b)
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]].


nice range: [315.641, 317.263]
[[Subgroup]]: 2.3.5


strict range: [315.641, 317.263]
[[Comma list]]: 15625/15552


[[POTE tuning|POTE generator]]: 317.007
{{Mapping|legend=1| 1 0 1 | 0 6 5 }}
: mapping generators: ~2, ~6/5


Map: [&lt;1 0 1|, &lt;0 6 5|]
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.1659{{c}}, ~6/5 = 317.0504{{c}}
: [[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 }}


EDOs: {{EDOs| 15, 19, 34, 53, 458, 882c }}
[[Tuning ranges]]:  
* [[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)


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


[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Parizek/Hanson%20%20Improv.mp3 Hanson Improv] by [[Petr Parizek]]
[[Badness]] (Sintel): 0.310


[http://clones.soonlabel.com/public/micro/Hanson/daily20110127-in-hanson11.mp3 In Hanson11] by [[Chris Vaisvil]]
=== Overview to extensions ===
==== Full 7-limit extensions ====
The second comma of the [[normal forms #Normal forms for commas|normal comma list]] defines which [[7-limit]] family member we are looking at. [[4375/4374]], the ragisma, gives catakleismic. [[875/864]], the keemic comma, gives keemun. [[5120/5103]], hemifamity, gives countercata. [[179200/177147]], the tolerant comma, gives metakleismic. [[64/63]], the archytas comma, gives catalan. Catakleismic, keemun, countercata, metakleismic, and catalan all have octave period and use the minor third as a generator; catakleismic, countercata, and metakleismic define the 7/4 more complexly but more accurately than keemun and catalan.


Analysis and diagrams:
[[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.


<span style="">[http://dkeenan.com/Music/ChainOfMinor3rds.htm 11 note chain-of-minor-thirds scale], by David Keenan</span>
Temperaments involving larger splits include [[#Sqrtphi|sqrtphi]], [[#Quartkeenlig|quartkeenlig]], [[#Novemkleismic|novemkleismic]]. Those split the kleismic structure into five to nine parts.  


== Seven limit children ==
==== Other subgroup extensions ====
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, and 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.
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]]} }}.


= Keemun =
See [[#Subgroup extensions]].
[[Comma]]s: 49/48, 126/125


valid range: [300.000, 327.273] (4 to 11b)
== Catakleismic ==
{{Main| Catakleismic }}


nice range: [308.744, 322.942]
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.


strict range: [308.744, 322.942]
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]])}.  


[[POTE tuning|POTE generator]]: ~6/5 = 316.473
=== 7-limit ===
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 0 1 2|, &lt;0 6 5 3|]
[[Comma list]]: 225/224, 4375/4374


[[Wedgie]]: &lt;&lt;6 5 3 -6 -12 -7||
{{Mapping|legend=1| 1 0 1 -3 | 0 6 5 22 }}


EDOs: {{EDOs| 15, 19, 53d, 72dd, 91dd }}
[[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 }}


[[Badness]]: 0.0274
[[Tuning ranges]]:  
* 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]


== 11-limit ==
{{Optimal ET sequence|legend=1| 19, 34d, 53, 72, 197, 269c }}
Commas: 49/48, 56/55, 100/99


valid range: [315.789, 320.000] (19 to 15)
[[Badness]] (Sintel): 0.544


nice range: [308.744, 324.341]
==== 2.3.5.7.13 subgroup ====
Subgroup: 2.3.5.7.13


strict range: [315.789, 320.000]
Comma list: 169/168, 225/224, 325/324


POTE generator: ~6/5 = 317.576
Subgroup-val mapping: {{mapping| 1 0 1 -3 0 | 0 6 5 22 14 }}


Map: [&lt;1 0 1 2 4|, &lt;0 6 5 3 -2|]
Optimal tunings:  
* WE: ~2 = 1200.7838{{c}}, ~6/5 = 316.9478{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.7939{{c}}


EDOs: {{EDOs| 4, 15, 19, 34 }}
{{Optimal ET sequence|legend=0| 19, 34d, 53, 72, 125f, 197f }}


Badness: 0.0274
Badness (Sintel): 0.410


=== 13-limit ===
=== 11-limit ===
Commas: 49/48, 56/55, 78/77, 100/99
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 385/384, 4375/4374
 
Mapping: {{mapping| 1 0 1 -3 9 | 0 6 5 22 -21 }}
 
Optimal tunings:
* WE: ~2 = 1200.6524{{c}}, ~6/5 = 316.8911{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.7267{{c}}
 
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|legend=0| 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: {{mapping| 1 0 1 -3 9 0 | 0 6 5 22 -21 14 }}
 
Optimal tunings:
* WE: ~2 = 1200.7982{{c}}, ~6/5 = 316.9482{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.7491{{c}}
 
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|legend=0| 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: {{mapping| 1 0 1 -3 -5 | 0 6 5 22 32 }}
 
Optimal tunings:
* WE: ~2 = 1199.9590{{c}}, ~6/5 = 317.0315{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0403{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 0 1 -3 -5 0 | 0 6 5 22 32 14 }}
 
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=0| 19e, 34d, 53 }}
 
Badness (Sintel): 0.932
 
=== Catalytic ===
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 441/440, 4375/4374
 
Mapping: {{mapping| 1 0 1 -3 -10 | 0 6 5 22 51 }}
 
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=0| 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: {{mapping| 1 0 1 -3 -10 0 | 0 6 5 22 51 14 }}
 
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=0| 19e, 53e, 72, 307bcdeeffff }}
 
Badness (Sintel): 0.923
 
=== Cataleptic ===
Subgroup: 2.3.5.7.11
 
Comma list: 100/99, 225/224, 864/847
 
Mapping: {{mapping| 1 0 1 -3 4 | 0 6 5 22 -2 }}
 
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=0| 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: {{mapping| 1 0 1 -3 4 0 | 0 6 5 22 -2 14 }}
 
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=0| 19, 34d, 53e }}
 
Badness (Sintel): 1.13
 
=== Bikleismic ===
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 243/242, 4375/4356
 
Mapping: {{mapping| 2 0 2 -6 -1 | 0 6 5 22 15 }}
: mapping generators: ~99/70, ~6/5
 
Optimal tunings:
* WE: ~99/70 = 600.2674{{c}}, ~6/5 = 316.8624{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~6/5 = 316.7575{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 2 0 2 -6 -1 0 | 0 6 5 22 15 14 }}
 
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 ET sequence|legend=0| 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: {{mapping| 2 0 2 -6 -1 0 5 | 0 6 5 22 15 14 6 }}
 
Optimal tunings:
* WE: ~17/12 = 600.4210{{c}}, ~6/5 = 316.9282{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~6/5 = 316.7578{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 2 0 2 -6 -1 0 5 -1 | 0 6 5 22 15 14 6 18 }}
 
Optimal tunings:
* WE: ~17/12 = 600.3763{{c}}, ~6/5 = 316.8720{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~6/5 = 316.7205{{c}}
 
{{Optimal ET sequence|legend=0| 34dh, 38df, 72 }}
 
Badness (Sintel): 0.959
 
== Keemun ==
{{Main| Keemun }}
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 49/48, 126/125
 
{{Mapping|legend=1| 1 0 1 2 | 0 6 5 3 }}
 
[[Optimal tuning]]s:
* [[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 }}
 
[[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|legend=1| 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: {{mapping| 1 0 1 2 4 | 0 6 5 3 -2 }}
 
Optimal tunings:
* WE: ~2 = 1199.7353{{c}}, ~6/5 = 317.5055{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.5546{{c}}
 
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|legend=0| 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: {{mapping| 1 0 1 2 4 5 | 0 6 5 3 -2 -5 }}
 
Optimal tunings:
* WE: ~2 = 1201.8360{{c}}, ~6/5 = 317.0958{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.6829{{c}}
 
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|legend=0| 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: {{mapping| 1 0 1 2 4 0 | 0 6 5 3 -2 14 }}
 
Optimal tunings:
* WE: ~2 = 1199.7816{{c}}, ~6/5 = 317.3653{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.4070{{c}}
 
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 (Sintel): 0.940
 
==== Kumbaya ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 40/39, 49/48, 56/55, 66/65
 
Mapping: {{mapping| 1 0 1 2 4 4 | 0 6 5 3 -2 -1 }}
 
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=0| 4, 11b, 15 }}
 
Badness (Sintel): 1.31
 
=== Qeema ===
Subgroup: 2.3.5.7.11
 
Comma list: 45/44, 49/48, 126/125
 
Mapping: {{mapping| 1 0 1 2 -1 | 0 6 5 3 17 }}
 
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=0| 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: {{mapping| 1 0 1 2 -1 0 | 0 6 5 3 17 14 }}
 
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=0| 4ef, 19 }}
 
Badness (Sintel): 1.22
 
=== Darjeeling ===
Subgroup: 2.3.5.7.11


valid range: 315.789 (19)
Comma list: 49/48, 55/54, 77/75


nice range: [303.597, 324.341]
Mapping: {{mapping| 1 0 1 2 0 | 0 6 5 3 13 }}


strict range: 315.789
Optimal tunings:  
* WE: ~2 = 1201.6569{{c}}, ~6/5 = 318.0942{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.8547{{c}}


POTE generator: ~6/5 = 316.611
{{Optimal ET sequence|legend=0| 15, 19e, 34e }}


Map: [&lt;1 0 1 2 4 5|, &lt;0 6 5 3 -2 -5|]
Badness (Sintel): 0.914


EDOs: {{EDOs| 4, 15f, 19, 53def, 72def }}
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Badness: 0.0297
Comma list: 49/48, 55/54, 66/65, 77/75


=== Kema ===
Mapping: {{mapping| 1 0 1 2 0 0 | 0 6 5 3 13 14 }}
Commas: 49/48, 56/55, 91/90, 100/99


valid range: [315.789, 320.000] (19 to 15)
Optimal tunings:  
* WE: ~2 = 1201.9324{{c}}, ~6/5 = 317.8090{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.5170{{c}}


nice range: [308.744, 324.341]
{{Optimal ET sequence|legend=0| 15, 19e, 34e }}


strict range: [315.789, 320.000]
Badness (Sintel): 0.886


POTE generator: ~6/5 = 317.423
== Catalan ==
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 0 1 2 4 0|, &lt;0 6 5 3 -2 14|]
[[Comma list]]: 64/63, 15625/15552


EDOs: {{EDOs| 15, 19, 34, 87ddee }}
{{Mapping|legend=1| 1 0 1 6 | 0 6 5 -12 }}


Badness: 0.0227
[[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 }}


=== Kumbaya ===
[[Tuning ranges]]:
Commas: 40/39, 49/48, 56/55, 66/65
* 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]


POTE generator: ~6/5 = 318.595
{{Optimal ET sequence|legend=1| 15, 34d, 49, 132bcdd, 181bbcddd }}


Map: [&lt;1 0 1 2 4 4|, &lt;0 6 5 3 -2 -1|]
[[Badness]] (Sintel): 2.40


EDOs: {{EDOs| 4, 15, 19f, 34ff }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.0316
Comma list: 64/63, 100/99, 1331/1323


== Qeema ==
Mapping: {{mapping| 1 0 1 6 4 | 0 6 5 -12 -2 }}
Commas: 45/44, 49/48, 126/125


POTE generator: ~6/5 = 314.730
Optimal tunings:  
* WE: ~2 = 1197.0368{{c}}, ~6/5 = 317.4956{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 318.2672{{c}}


Map: [&lt;1 0 1 2 -1|, &lt;0 6 5 3 17|]
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]


EDOs: {{EDOs| 4e, 19, 42bcd, 61bcdd }}
{{Optimal ET sequence|legend=0| 15, 34d, 49, 181bbcdddeee }}


Badness: 0.0401
Badness (Sintel): 1.22


=== 13-limit ===
=== 13-limit ===
Commas: 45/44, 49/48, 78/77, 126/125
Subgroup: 2.3.5.7.11.13
 
Comma list: 64/63, 100/99, 144/143, 275/273
 
Mapping: {{mapping| 1 0 1 6 4 0 | 0 6 5 -12 -2 14 }}
 
Optimal tunings:
* WE: ~2 = 1196.8961{{c}}, ~6/5 = 317.3837{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 318.1621{{c}}
 
{{Optimal ET sequence|legend=0| 15, 34d, 49f, 83def, 132bcddeefff }}
 
Badness (Sintel): 1.09
 
== Countercata ==
[[Subgroup]]: 2.3.5.7


POTE generator: ~6/5 = 315.044
[[Comma list]]: 5120/5103, 15625/15552


Map: [&lt;1 0 1 2 -1 0|, &lt;0 6 5 3 17 14|]
{{Mapping|legend=1| 1 0 1 11 | 0 6 5 -31 }}


EDOs: {{EDOs| 4ef, 19 }}
[[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 }}


Badness: 0.0294
[[Tuning ranges]]:  
* 7- and 9-odd-limit [[diamond monotone]]: ~6/5 = [316.667, 317.647] (19\72 to 9\34)
* 7- and 9-odd-limit [[diamond tradeoff]]: ~6/5 = [315.641, 317.263]


== Darjeeling ==
{{Optimal ET sequence|legend=1| 19d, 34, 53, 87, 140, 333, 473 }}
Commas: 49/48, 55/54, 77/75


POTE generator: ~6/5 = 317.656
[[Badness]] (Sintel): 1.32


Map: [&lt;1 0 1 2 0|, &lt;0 6 5 3 13|]
=== 11-limit ===
Subgroup: 2.3.5.7.11


EDOs: {{EDOs| 15, 19e, 34e }}
Comma list: 385/384, 2200/2187, 3388/3375


Badness: 0.0276
Mapping: {{mapping| 1 0 1 11 -5 | 0 6 5 -31 32 }}
 
Optimal tunings:
* WE: ~2 = 1200.0980{{c}}, ~6/5 = 317.1879{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.1623{{c}}
 
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|legend=0| 34, 53, 87, 140, 227, 367e }}
 
Badness (Sintel): 1.31


=== 13-limit ===
=== 13-limit ===
Commas: 49/48, 55/54, 66/65, 77/75
Subgroup: 2.3.5.7.11.13
 
Comma list: 325/324, 352/351, 385/384, 625/624
 
Mapping: {{mapping| 1 0 1 11 -5 0 | 0 6 5 -31 32 14 }}


POTE generator: ~6/5 = 317.298
Optimal tunings:  
* WE: ~2 = 1200.0936{{c}}, ~6/5 = 317.1864{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.1622{{c}}


Map: [&lt;1 0 1 2 0 0|, &lt;0 6 5 3 13 14|]
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]


EDOs: {{EDOs| 15, 19e, 34e, 53dee }}
{{Optimal ET sequence|legend=0| 34, 53, 87, 140, 367e, 507e }}


Badness: 0.02144
Badness (Sintel): 0.833


= Catalan =
== Metakleismic ==
Commas: 64/63, 15625/15552
[[Subgroup]]: 2.3.5.7


POTE generator: ~6/5 = 318.267
[[Comma list]]: 15625/15552, 179200/177147


Map: [&lt;1 0 1 6|, &lt;0 6 5 -12|]
{{Mapping|legend=1| 1 0 1 -12 | 0 6 5 56 }}


Wedgie: &lt;&lt;6 5 -12 -6 -36 -42||
[[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 }}


EDOs: {{EDOs| 15, 34d, 49, 132bcd }}
{{Optimal ET sequence|legend=1| 34d, 87, 121, 208, 537b }}


Badness: 0.0949
[[Badness]] (Sintel): 4.14


== 11-limit ==
=== 11-limit ===
Commas: 64/63, 100/99, 1331/1323
Subgroup: 2.3.5.7.11


POTE generator: ~6/5 = 318.282
Comma list: 896/891, 2200/2187, 14700/14641


Map: [&lt;1 0 1 6 4|, &lt;0 6 5 -12 -2|]
Mapping: {{mapping| 1 0 1 -12 -5 | 0 6 5 56 32 }}


EDOs: {{EDOs| 15, 34d, 49 }}
Optimal tunings:  
* WE: ~2 = 1199.5425{{c}}, ~6/5 = 317.1901{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.3020{{c}}


Badness: 0.0369
{{Optimal ET sequence|legend=0| 34d, 53d, 87, 121, 208 }}


= Catakleismic =
Badness (Sintel): 1.61
[[Comma|Commas]]: 225/224, 4375/4374


valid range: [315.789, 317.647] (19 to 34)
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


nice range: [315.641, 317.263]
Comma list: 325/324, 352/351, 364/363, 625/624


strict range: [315.789, 317.263]
Mapping: {{mapping| 1 0 1 -12 -5 0 | 0 6 5 56 32 14 }}


[[POTE tuning|POTE generator]]: 316.732
Optimal tunings:  
* WE: ~2 = 1199.5339{{c}}, ~6/5 = 317.1882{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.3028{{c}}


Map: [&lt;1 0 1 -3|, &lt;0 6 5 22|]
{{Optimal ET sequence|legend=0| 34d, 53d, 87, 121, 208 }}


[[Wedgie]]: &lt;&lt;6 5 22 -6 18 37||
Badness (Sintel): 1.01


EDOs: {{EDOs| 19, 53, 72, 197, 269c }}
== Hemikleismic ==
[[Subgroup]]: 2.3.5.7


[[Badness]]: 0.0215
[[Comma list]]: 4000/3969, 6144/6125


== 11-limit ==
{{Mapping|legend=1| 1 0 1 4 | 0 12 10 -9 }}
[[Comma|Commas]]: 225/224, 385/384, 4375/4374
: mapping generators: ~2, ~35/32


valid range: [315.789, 316.981] (19 to 53)
[[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 }}


nice range: [315.641, 317.263]
{{Optimal ET sequence|legend=1| 15, 38, 53, 121, 174d, 295d }}


strict range: [315.789, 316.981]
[[Badness]] (Sintel): 1.32


[[POTE tuning|POTE generator]]: 316.719
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 0 1 -3 9|, &lt;0 6 5 22 -21|]
Comma list: 121/120, 176/175, 4000/3969


EDOs: {{EDOs| 19, 53, 72, 197e, 269ce, 341ce, 610bce }}
Mapping: {{mapping| 1 0 1 4 2 | 0 12 10 -9 11 }}


[[Badness]]: 0.0218
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=0| 15, 38, 53, 68, 121e }}
 
Badness (Sintel): 1.26


=== 13-limit ===
=== 13-limit ===
[[Comma|Commas]]: 169/168, 225/224, 325/324, 540/539
Subgroup: 2.3.5.7.11.13


valid range: [315.789, 316.981] (19 to 53)
Comma list: 121/120, 176/175, 275/273, 325/324


nice range: [315.641, 318.309]
Mapping: {{mapping| 1 0 1 4 2 0 | 0 12 10 -9 11 28 }}


strict range: [315.789, 316.981]
Optimal tunings:  
* WE: ~2 = 1199.7952{{c}}, ~11/10 = 158.6279{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 158.6493{{c}}


[[POTE tuning|POTE generator]]: 316.738
{{Optimal ET sequence|legend=0| 15, 38f, 53, 121e }}


Map: [&lt;1 0 1 -3 9 0|, &lt;0 6 5 22 -21 14|]
Badness (Sintel): 1.07


EDOs: {{EDOs| 19, 53, 72, 125f, 197ef, 269cef }}
== Clyde ==
[[Subgroup]]: 2.3.5.7


[[Badness]]: 0.0169
[[Comma list]]: 245/243, 3136/3125


== Cataclysmic ==
{{Mapping|legend=1| 1 -6 -4 -13 | 0 12 10 25 }}
Commas: 99/98, 176/175, 2200/2187
: mapping generators: ~2, ~14/9


POTE generator: ~6/5 = 317.042
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.8369{{c}}, ~14/9 = 758.5621{{c}}
: [[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 }}


Map: [&lt;1 0 1 -3 -5|, &lt;0 6 5 22 32|]
[[Minimax tuning]]:  
* [[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 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7


EDOs: {{EDOs| 53, 87d, 140d, 171de, 181de, 193de, 224de, 246de, 277de }}
[[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.


Badness: 0.0400
{{Optimal ET sequence|legend=1| 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: {{mapping| 1 -6 -4 -13 18 | 0 12 10 25 -23 }}
 
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=0| 19, 49e, 68, 87 }}
 
Badness (Sintel): 1.57


=== 13-limit ===
=== 13-limit ===
Commas: 99/98, 169/168, 176/175, 275/273
Subgroup: 2.3.5.7.11.13
 
Comma list: 196/195, 245/243, 385/384, 625/624
 
Mapping: {{mapping| 1 -6 -4 -13 18 -14 | 0 12 10 25 -23 28 }}
 
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=0| 19, 68, 87 }}
 
Badness (Sintel): 1.11
 
== Tritikleismic ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 1029/1024, 15625/15552


POTE generator: ~6/5 = 317.036
{{Mapping|legend=1| 3 0 3 10 | 0 6 5 -2 }}
: mapping generators: ~63/50, ~6/5


Map: [&lt;1 0 1 -3 -5 0|, &lt;0 6 5 22 32 14|]
[[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 }}


EDOs: {{EDOs| 53, 87d, 140d, 193de, 246de }}
[[Minimax tuning]]:
* [[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 }}]
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5
* [[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 }}]
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7


Badness: 0.0226
{{Optimal ET sequence|legend=1| 15, 42bc, 57, 72, 159, 231, 765ccddd }}


== Catalytic ==
[[Badness]] (Sintel): 1.43
Commas: 225/224, 441/440, 4375/4374


POTE generator: ~6/5 = 316.653
; Music
* [https://www.youtube.com/watch?v=vdjhC9i5KF4 ''Four Short Experiments in Octave Stretched 42edo''] (2024) by [[Budjarn Lambeth]]


Map: [&lt;1 0 1 -3 -10|, &lt;0 6 5 22 51|]
=== 11-limit ===
Subgroup: 2.3.5.7.11


EDOs: {{EDOs| 53e, 72 }}
Comma list: 385/384, 441/440, 4000/3993


Badness: 0.0304
Mapping: {{mapping| 3 0 3 10 8 | 0 6 5 -2 3 }}
 
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:
* 11-odd-limit: ~6/5 = {{monzo| 5/21 1/7 0 -1/14 }}
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 10/7 6/7 0 -3/7 0 }}, {{monzo| 46/21 5/7 0 -5/14 0 }}, {{monzo| 20/7 -2/7 0 1/7 0 }}, {{monzo| 71/21 3/7 0 -3/14 0 }}]
: unchanged-interval (eigenmonzo) basis: 2.9/7
 
{{Optimal ET sequence|legend=0| 15, 42bc, 57, 72, 159, 231 }}
 
Badness (Sintel): 0.639


=== 13-limit ===
=== 13-limit ===
Commas: 169/168 225/224 325/324 1716/1715
Subgroup: 2.3.5.7.11.13
 
Comma list: 325/324, 364/363, 385/384, 625/624
 
Mapping: {{mapping| 3 0 3 10 8 0 | 0 6 5 -2 3 14 }}
 
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=0| 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: {{mapping| 3 0 3 10 8 0 -2 | 0 6 5 -2 3 14 18 }}
 
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 }}
 
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 {{nowrap| 19 & 140 }} temperament. It is generated by a marvel fourth of [[75/56]] (or a marvel fifth of [[112/75]]), three of which minus an octave make the hanson generator of ~6/5. Its [[ploidacot]] is zeta-18-cot.
 
[[Subgroup]]: 2.3.5.7
 
[[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


POTE generator: ~6/5 = 316.639
Comma list: 385/384, 6250/6237, 10976/10935


Map: [&lt;1 0 1 -3 -10 0|, &lt;0 6 5 22 51 14|]
Mapping: {{mapping| 1 -6 -4 -17 22 | 0 18 15 47 -44 }}


EDOs: {{EDOs| 19e, 53e, 72 }}
Optimal tunings:  
* WE: ~2 = 1200.2484{{c}}, ~75/56 = 505.7882{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~75/56 = 505.6853{{c}}


Badness: 0.0223
{{Optimal ET sequence|legend=0| 19, 121e, 140, 159, 299 }}


== Cataleptic ==
Badness (Sintel): 1.95
Commas: 100/99, 225/224, 864/847


POTE generator: ~6/5 = 317.083
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 0 1 -3 4|, &lt;0 6 5 22 -2|]
Comma list: 325/324, 385/384, 625/624, 10976/10935


EDOs: {{EDOs| 19, 34d, 53e }}
Mapping: {{mapping| 1 -6 -4 -17 22 -14 | 0 18 15 47 -44 42 }}


Badness: 0.0443
Optimal tunings:  
* WE: ~2 = 1200.2747{{c}}, ~75/56 = 505.8019{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~75/56 = 505.6883{{c}}


=== 13-limit ===
{{Optimal ET sequence|legend=0| 19, 121e, 140, 159, 299 }}
Commas: 78/77, 100/99, 144/143, 676/675
 
Badness (Sintel): 1.24
 
=== Diatessic ===
Diatessic may be described as {{nowrap| 121 & 140 }} and is closely related to the Diatess tuning (generator: 505.727281 cents).


POTE generator: ~6/5 = 317.118
Subgroup: 2.3.5.7.11


Map: [&lt;1 0 1 -3 4 0|, &lt;0 6 5 22 -2 14|]
Comma list: 1375/1372, 2200/2187, 5632/5625


EDOs: {{EDOs| 19, 34d, 53e, 87de }}
Mapping: {{mapping| 1 -6 -4 -17 -37 | 0 18 15 47 96 }}


Badness: 0.0273
Optimal tunings:  
* WE: ~2 = 1199.7886{{c}}, ~75/56 = 505.6513{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~75/56 = 505.7366{{c}}


= Countercata =
{{Optimal ET sequence|legend=0| 19e, …, 121, 140, 261, 401 }}
[[Comma|Commas]]: 5120/5103, 15625/15552


[[POTE tuning|POTE generator]]: 317.121
Badness (Sintel): 2.02


Map: [&lt;1 0 1 11|, &lt;0 6 5 -31|]
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


[[Wedgie]]: &lt;&lt;6 5 -31 -6 -66 -86||
Comma list: 325/324, 352/351, 625/624, 1375/1372


EDOs: {{EDOs| 19d, 34, 53, 87, 140, 333, 473, 806b }}
Mapping: {{mapping| 1 -6 -4 -17 -37 -14 | 0 18 15 47 96 42 }}


[[Badness]]: 0.0521
Optimal tunings:  
* WE: ~2 = 1199.7996{{c}}, ~75/56 = 505.6558{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~75/56 = 505.7366{{c}}


== 11-limit ==
{{Optimal ET sequence|legend=0| 19e, …, 121, 140, 261, 401 }}
[[Comma]]s: 385/384, 2200/2187, 3388/3375


POTE generator: ~6/5 = 317.162
Badness (Sintel): 1.18


Map: [&lt;1 0 1 11 -5|, &lt;0 6 5 -31 32|]
=== Marf ===
Marf may be described as {{nowrap| 19 & 121 }}. It has a POTE generator which strongly approximates the marvelous fifth interval of 112/75.


EDOs: {{EDOs| 34, 53, 87, 140, 227 }}
Subgroup: 2.3.5.7.11


[[Badness]]: 0.0398
Comma list: 540/539, 896/891, 15625/15552


== 13-limit ==
Mapping: {{mapping| 1 -6 -4 -17 14 | 0 18 15 47 -25 }}
[[Comma]]s: 325/324, 352/351, 385/384, 625/624


POTE generator: ~6/5 = 317.162
Optimal tunings:  
* WE: ~2 = 1199.3198{{c}}, ~75/56 = 505.4822{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~75/56 = 505.7607{{c}}


Map: [&lt;1 0 1 11 -5 0|, &lt;0 6 5 -31 32 14|]
{{Optimal ET sequence|legend=0| 19, 102d, 121 }}


EDOs: {{EDOs| 34, 53, 87, 140, 227, 367e, 507e }}
Badness (Sintel): 2.48


[[Badness]]: 0.0202
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


= Metakleismic =
Comma list: 325/324, 540/539, 625/624, 896/891
[[Comma]]s: 15625/15552, 179200/177147


POTE generator: ~6/5 = 317.314
Mapping: {{mapping| 1 -6 -4 -17 14 -14 | 0 18 15 47 -25 42 }}


Map: [&lt;1 0 1 -12|, &lt;0 6 5 56|]
Optimal tunings:  
* WE: ~2 = 1199.3368{{c}}, ~75/56 = 505.4919{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~75/56 = 505.7627{{c}}


Wedgie: &lt;&lt;6 5 56 -6 72 116||
{{Optimal ET sequence|legend=0| 19, 102df, 121 }}


EDOs: {{EDOs| 34d, 53d, 87, 121, 208 }}
Badness (Sintel): 1.58


[[Badness]]: 0.1635
== Kleiboh ==
[[Subgroup]]: 2.3.5.7


== 11-limit ==
[[Comma list]]: 1728/1715, 3125/3087
[[Comma]]s: 896/891, 2200/2187, 14700/14641


POTE generator: ~6/5 = 317.311
{{Mapping|legend=1| 1 -12 -9 -7 | 0 18 15 13 }}
: mapping generators: ~2, ~42/25


Map: [&lt;1 0 1 -12 -5|, &lt;0 6 5 56 32|]
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.5290{{c}}, ~42/25 = 905.3417{{c}}
: [[error map]]: {{val| -0.471 -0.152 -1.949 +3.914 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~42/25 = 905.6741{{c}}
: error map: {{val| 0.000 +0.178 -1.203 +4.937 }}


EDOs: {{EDOs| 34d, 53d, 87, 121, 208 }}
{{Optimal ET sequence|legend=1| 49, 53 }}


[[Badness]]: 0.0486
[[Badness]] (Sintel): 1.93


== 13-limit ==
=== 11-limit ===
[[Comma]]s: 325/324, 352/351, 364/363, 625/624
Subgroup: 2.3.5.7.11


POTE generator: ~6/5 = 317.311
Comma list: 176/175, 540/539, 3125/3087


Map: [&lt;1 0 1 -12 -5 0|, &lt;0 6 5 56 32 14|]
Mapping: {{mapping| 1 -12 -9 -7 -29 | 0 18 15 13 43 }}


EDOs: {{EDOs| 34d, 53d, 87, 121, 208 }}
Optimal tunings:  
* WE: ~2 = 1199.1389{{c}}, ~42/25 = 905.1688{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~42/25 = 905.7840{{c}}


[[Badness]]: 0.0244
{{Optimal ET sequence|legend=0| 49, 53, 102d }}


= Hemikleismic =
Badness (Sintel): 1.75
Commas: 4000/3969, 6144/6125


[[POTE tuning|POTE generator]]: 158.649
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 0 1 4|, &lt;0 12 10 -9|]
Comma list: 176/175, 275/273, 325/324, 540/539


EDOs: {{EDOs| 15, 53, 121 }}
Mapping: {{mapping| 1 -12 -9 -7 -29 -28 | 0 18 15 13 43 42 }}


Badness: 0.0521
Optimal tunings:  
* WE: ~2 = 1199.1517{{c}}, ~22/13 = 905.1727{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/13 = 905.7801{{c}}


== 11-limit ==
{{Optimal ET sequence|legend=0| 49f, 53, 102df }}
Commas: 121/120, 176/175, 4000/3969


POTE generator: ~11/10 = 158.677
Badness (Sintel): 1.28


Map: [&lt;1 0 1 4 2|, &lt;0 12 10 -9 11|]
== Quadritikleismic ==
[[Subgroup]]: 2.3.5.7


EDOs: {{EDOs| 15, 38, 53, 68, 121e }}
[[Comma list]]: 2401/2400, 15625/15552


Badness: 0.0380
{{Mapping|legend=1| 4 0 4 7 | 0 6 5 4 }}
: mapping generators: ~25/21, ~6/5


== 13-limit ==
[[Optimal tuning]]s:
Commas: 121/120, 176/175, 275/273, 325/324
* [[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 }}


POTE generator: ~11/10 = 158.655
{{Optimal ET sequence|legend=1| 68, 72, 140, 212, 776cd, 988ccd, 1200ccd }}


Map: [&lt;1 0 1 4 2 0|, &lt;0 12 10 -9 11 28|]
[[Badness]] (Sintel): 0.993


EDOs: {{EDOs| 15, 53, 121e }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.0260
Comma list: 385/384, 1375/1372, 6250/6237


= Clyde =
Mapping: {{mapping| 4 0 4 7 17 | 0 6 5 4 -3 }}
[[Comma|Commas]]: 245/243, 3136/3125


7 and 9 limit minimax
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}})


[|1 0 0 0&gt;, |6/25 0 0 12/25&gt;, |6/5 0 0 2/5&gt;, |0 0 0 1&gt;]
{{Optimal ET sequence|legend=0| 68, 72, 140, 212, 284, 496ce, 780ccdee }}


[[Eigenmonzo|Eigenmonzos]]: 2, 7
Badness (Sintel): 0.774


[[POTE tuning|POTE generator]]: ~9/7 = 441.335
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Algebraic generator: real root of 5x^3-6x-3, the Poussami generator. Approximately 441.309 [[cent|cents]]. Associated recurrence relationship quickly converges.
Comma list: 325/324, 385/384, 625/624, 1375/1372


Map: [&lt;1 6 6 12|, &lt;0 -12 -10 -25|]
Mapping: {{mapping| 4 0 4 7 17 0 | 0 6 5 4 -3 14 }}


[[Generator|Generators]]: 2, 9/7
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}})


[[EDO|EDOs]]: {{EDOs| 19, 49, 68, 87, 155 }}
{{Optimal ET sequence|legend=0| 68, 72, 140, 212 }}


Badness: 0.0473
Badness (Sintel): 0.774


== 11-limit ==
=== 17-limit ===
Commas: 245/243, 3136/3125, 385/384
Subgroup: 2.3.5.7.11.13.17


POTE generator: ~9/7 = 441.355
Comma list: 289/288, 325/324, 385/384, 442/441, 625/624


Map: [&lt;1 6 6 12 -5|, &lt;0 -12 -10 -25 23|]
Mapping: {{mapping| 4 0 4 7 17 0 10 | 0 6 5 4 -3 14 6 }}


EDOs: {{EDOs| 19, 68, 87, 329bd, 419bd, 503bd, 590bd }}
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}})


Badness: 0.0474
{{Optimal ET sequence|legend=0| 68, 72, 140, 212g }}


== 13-limit ==
Badness (Sintel): 0.651
Commas: 196/195, 245/243, 385/384, 625/624


POTE generator: ~9/7 = 441.363
== 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.  


Map: [&lt;1 6 6 12 -5 14|, &lt;0 -12 -10 -25 23 -28|]
[[Subgroup]]: 2.3.5.7


EDOs: {{EDOs| 19, 68, 87, 503bdf, 590bdf }}
[[Comma list]]: 15625/15552, 2460375/2458624


Badness: 0.0268
{{Mapping|legend=1| 1 -6 -4 -19 | 0 24 20 69 }}
: mapping generators: ~2, ~56/45


= Bikleismic =
[[Optimal tuning]]s:
Commas: 225/224, 243/242, 4375/4356
* [[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 }}


POTE generator: ~6/5 = 316.721
{{Optimal ET sequence|legend=1| 19, …, 193, 212, 617c, 829c }}


Map: [&lt;2 0 2 -6 -1|, &lt;0 6 5 22 15|]
[[Badness]] (Sintel): 2.64


EDOs: {{EDOs| 72, 106, 178, 250, 322c, 394c, 466bc, 538bc, 610bc }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.0293
Comma list: 1375/1372, 15625/15552, 19712/19683


== 13-limit ==
Mapping: {{mapping| 1 -6 -4 -19 -43 | 0 24 20 69 147 }}
Commas: 169/168, 225/224, 243/242, 325/324


POTE generator: ~6/5 = 316.726
Optimal tunings:  
* WE: ~2 = 1200.1189{{c}}, ~56/45 = 379.2942{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~56/45 = 379.2580{{c}}


Map: [&lt;2 0 2 -6 -1 0|, &lt;0 6 5 22 15 14|]
{{Optimal ET sequence|legend=0| 19e, …, 193, 212, 405, 617c }}


EDOs: {{EDOs| 72, 106, 322cff, 394cff, 466bcff, 538bcfff }}
Badness (Sintel): 2.50


Badness: 0.0218
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


= Tritikleismic =
Comma list: 325/324, 625/624, 1375/1372, 19712/19683
{{see also|Gamelismic clan #Tritikleismic}}


[[Comma]]s: 1029/1024, 15625/15552
Mapping: {{mapping| 1 -6 -4 -19 -43 -14 | 0 24 20 69 147 56 }}


[[POTE tuning|POTE generator]]: 316.872
Optimal tunings:  
* WE: ~2 = 1200.2154{{c}}, ~56/45 = 379.3236{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~56/45 = 379.2580{{c}}


Map: [&lt;3 0 3 10|, &lt;0 6 5 -2|]
{{Optimal ET sequence|legend=0| 19e, …, 193, 212, 405f, 617cff }}


[[Wedgie]]: &lt;&lt;18 15 -6 -18 -60 -56||
Badness (Sintel): 1.81


EDOs: {{EDOs| 15, 57, 72, 159, 231 }}
== Sqrtphi ==
{{Main| Sqrtphi }}


[[Badness]]: 0.0563
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.  


== 11-limit ==
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.
[[Comma]]s: 385/384, 441/440, 4000/3993


[[POTE tuning|POTE generator]]: 316.881
[[Subgroup]]: 2.3.5.7


Map: [&lt;3 0 3 10 8|, &lt;0 6 5 -2 3|]
[[Comma list]]: 15625/15552, 16875/16807


EDOs: {{EDOs| 15, 57, 72, 159, 231 }}
{{Mapping|legend=1| 1 -18 -14 -22 | 0 30 25 38 }}
: mapping generators: ~2, 196/125


[[Badness]]: 0.0193
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.1357{{c}}, ~196/125 = 783.4853{{c}}
: [[error map]]: {{val| +0.136 +0.163 -1.080 +0.632 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~196/125 = 783.4009{{c}}
: error map: {{val| 0.000 +0.072 -1.291 +0.408 }}


== 13-limit ==
{{Optimal ET sequence|legend=1| 23d, 49, 72, 193, 265 }}
[[Comma]]s: 325/324, 364/363, 441/440, 625/624


[[POTE tuning|POTE generator]]: 316.959
[[Badness]] (Sintel): 1.78


Map: [&lt;3 0 3 10 8 0|, &lt;0 6 5 -2 3 14|]
=== 11-limit ===
Subgroup: 2.3.5.7.11


EDOs: {{EDOs| 15, 72, 87, 159, 867, 1026 }}
Comma list: 540/539, 1375/1372, 4375/4356


[[Badness]]: 0.0157
Mapping: {{mapping| 1 -18 -14 -22 -22 | 0 30 25 38 39 }}


= Quadritikleismic =
Optimal tunings:
[[Comma]]s: 2401/2400, 15625/15552
* WE: ~2 = 1200.0514{{c}}, ~11/7 = 783.4294{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 783.3975{{c}}


[[POTE tuning|POTE generator]]: 316.9999
{{Optimal ET sequence|legend=0| 23de, 49, 72, 193, 265 }}


Map: [&lt;4 0 4 7|, &lt;0 6 5 4|]
Badness (Sintel): 0.844


Wedgie: &lt;&lt;24 20 16 -24 -42 -19||
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


EDOs: {{EDOs| 68, 72, 140, 212, 1200 }}
Comma list: 325/324, 364/363, 625/624, 1375/1372


[[Badness]]: 0.0392
Mapping: {{mapping| 1 -18 -14 -22 -22 -42 | 0 30 25 38 39 70 }}


== 11-limit ==
Optimal tunings:
Commas: 385/384, 1375/1372, 6250/6237
* WE: ~2 = 1199.9314{{c}}, ~11/7 = 783.3705{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 783.4134{{c}}


[[POTE tuning|POTE generator]]: 316.925
{{Optimal ET sequence|legend=0| 23deff, 49f, 72, 121, 193 }}


Map: [&lt;4 0 4 7 17|, &lt;0 6 5 4 -3|]
Badness (Sintel): 0.828


EDOs: {{EDOs| 68, 72, 140, 212, 284, 496, 780 }}
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


[[Badness]]: 0.0234
Comma list: 325/324, 364/363, 375/374, 540/539, 595/594


== 13-limit ==
Mapping: {{mapping| 1 -18 -14 -22 -22 -42 -39 | 0 30 25 38 39 70 66 }}
Commas: 325/324, 385/384, 625/624, 1573/1568


[[POTE tuning|POTE generator]]: 316.989
Optimal tunings:  
* WE: ~2 = 1199.9324{{c}}, ~11/7 = 783.3706{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 783.4129{{c}}


Map: [&lt;4 0 4 7 17 0|, &lt;0 6 5 4 -3 14|]
{{Optimal ET sequence|legend=0| 23deffgg, 49fg, 72, 121, 193 }}


EDOs: {{EDOs| 68, 72, 140, 212 }}
Badness (Sintel): 0.664


[[Badness]]: 0.0187
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


= Kleiboh =
Comma list: 325/324, 364/363, 375/374, 400/399, 442/441, 595/594
Commas: 1728/1715, 3125/3087


POTE generator: ~25/21 = 294.303
Mapping: {{mapping| 1 -18 -14 -22 -22 -42 -39 16 | 0 30 25 38 39 70 66 -18 }}


Map: [&lt;1 6 6 6|, &lt;0 -18 -15 -13|]
Optimal tunings:  
* WE: ~2 = 1199.8567{{c}}, ~11/7 = 783.3262{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 783.4176{{c}}


Wedgie: &lt;&lt;18 15 13 -18 -30 -12||
{{Optimal ET sequence|legend=0| 49fg, 72, 121, 193 }}


EDOs: {{EDOs| 49, 53, 314d }}
Badness (Sintel): 0.897


Badness: 0.0765
== 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 [[23edo and octave stretching|stretched 23edo]].


== 11-limit ==
[[Subgroup]]: 2.3.5.7
Commas: 176/175, 540/539, 3125/3087


POTE generator: ~25/21 = 294.181
[[Comma list]]: 15625/15552, 117649/116640


Map: [&lt;1 6 6 6 14|, &lt;0 -18 -15 -13 -43|]
{{Mapping|legend=1| 1 0 1 1 | 0 36 30 41 }}
: mapping generator: ~2, ~36/35


EDOs: {{EDOs| 49, 53, 102d, 155d }}
[[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 }}


Badness: 0.0528
{{Optimal ET sequence|legend=1| 68, 91, 159, 386d, 545dd }}


== 13-limit ==
[[Badness]] (Sintel): 3.69
Commas: 176/175, 275/273, 325/324, 540/539


POTE generator: ~13/11 = 294.187
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 6 6 6 14 14|, &lt;0 -18 -15 -13 -43 -42|]
Comma list: 385/384, 6250/6237, 67228/66825


EDOs: {{EDOs| 53, 102df, 155d }}
Mapping: {{mapping| 1 0 1 1 5 | 0 36 30 41 -35 }}


Badness: 0.0311
Optimal tunings:  
* WE: ~2 = 1200.2526{{c}}, ~36/35 = 52.8534{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 52.8446{{c}}


= Novemkleismic =
{{Optimal ET sequence|legend=0| 68, 91, 159, 386d, 545dd }}
Commas: 15625/15552, 40353607/40310784


POTE generator: ~6/5 = 317.005
Badness (Sintel): 2.86


Map: [&lt;9 0 9 11|, &lt;0 6 5 6|]
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Wedgie: &lt;&lt;54 45 54 -54 -66 -1||
Comma list: 325/324, 385/384, 625/624, 16807/16731


EDOs: {{EDOs| 72, 261, 333, 405, 477c, 882c }}
Mapping: {{mapping| 1 0 1 1 5 0 | 0 36 30 41 -35 84 }}


Badness: 0.1934
Optimal tunings:  
* WE: ~2 = 1200.2564{{c}}, ~36/35 = 52.8568{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 52.8479{{c}}


== 11-limit ==
{{Optimal ET sequence|legend=0| 68, 159, 386d, 545ddf }}
Commas: 1375/1372, 4000/3993, 15625/15552


POTE generator: ~6/5 = 317.010
Badness (Sintel): 1.97


Map: [&lt;9 0 9 11 24|, &lt;0 6 5 6 3|]
== Novemkleismic ==
[[Subgroup]]: 2.3.5.7


EDOs: {{EDOs| 72, 261, 333, 405, 882c }}
[[Comma list]]: 15625/15552, 40353607/40310784


Badness: 0.05172
{{Mapping|legend=1| 9 0 9 11 | 0 6 5 6 }}
: mapping generators: ~2592/2401, ~6/5


== 13-limit ==
[[Optimal tuning]]s:
Commas: 325/324, 625/624, 1375/1372, 4000/3993
* [[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 }}


POTE generator: ~6/5 = 317.086
{{Optimal ET sequence|legend=1| 72, 261, 333, 405, 477c, 882c }}


Map: [&lt;9 0 9 11 24 0|, &lt;0 6 5 6 3 14|]
[[Badness]] (Sintel): 4.90


EDOs: {{EDOs| 72, 261, 333, 738cf, 1071bcf }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.0391
Comma list: 1375/1372, 4000/3993, 15625/15552


= Sqrtphi =
Mapping: {{mapping| 9 0 9 11 24 | 0 6 5 6 3 }}
Commas: 15625/15552, 16875/16807


[[POTE tuning|POTE generator]]: ~125/98 = 416.603 cents
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}})


Sqrt(phi) = 416.545 cents
{{Optimal ET sequence|legend=0| 72, 261, 333, 405, 882c }}


Map: [&lt;1 12 11 16|, &lt;0 -30 -25 -38|]
Badness (Sintel): 1.71


EDOs: {{EDOs| 49, 72, 193, 265 }}
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Badness: 0.0704
Comma list: 325/324, 625/624, 1375/1372, 4000/3993


== 11-limit ==
Mapping: {{mapping| 9 0 9 11 24 0 | 0 6 5 6 3 14 }}
Commas: 540/539, 1375/1372, 4375/4356


POTE generator: ~14/11 = 416.604
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}})


Map: [&lt;1 12 11 16 17|, &lt;0 -30 -25 -38 -39|]
{{Optimal ET sequence|legend=0| 72, 189f, 261, 333, 738cf }}


EDOs: {{EDOs| 49, 72, 193, 265 }}
Badness (Sintel): 1.61


Badness: 0.0255
== Subgroup extensions ==
For the high-limit version of cata with a 1\5 period, see [[thunderclysmic]].


== 13-limit ==
=== Kleismic (2.3.5.13) a.k.a. cata ===
Commas: 325/324, 364/363, 625/624, 1375/1372
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.


POTE generator: ~14/11 = 416.585
Subgroup: 2.3.5.13


Map: [&lt;1 12 11 16 17 28|, &lt;0 -30 -25 -38 -39 -70|]
Comma list: 325/324, 625/624


EDOs: {{EDOs| 72, 121, 193 }}
Subgroup-val mapping: {{mapping| 1 0 1 0 | 0 6 5 14 }}


Badness: 0.0200
Optimal tunings:  
* WE: ~2 = 1200.1210{{c}}, ~6/5 = 317.1076{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0920{{c}}


== 17-limit ==
{{Optimal ET sequence|legend=0| 15, 19, 34, 53, 140, 193, 246 }}
Commas: 325/324, 364/363, 375/374, 540/539, 595/594


POTE generator: ~14/11 = 416.585
Badness (Sintel): 0.131


Map: [&lt;1 12 11 16 17 28 27|, &lt;0 -30 -25 -38 -39 -70 -66|]
==== 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.


EDOs: {{EDOs| 72, 121, 193 }}
Subgroup: 2.3.5.13.37


Badness: 0.0130
Comma list: 325/324, 481/480, 625/624


== 19-limit ==
Subgroup-val mapping: {{mapping| 1 0 1 0 6 | 0 6 5 14 -3 }}
Commas: 325/324, 364/363, 375/374, 400/399, 442/441, 595/594


POTE generator: ~14/11 = 416.580
Optimal tunings:
* WE: ~2 = 1200.2924{{c}}, ~6/5 = 317.0998{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0452{{c}}


Map: [&lt;1 12 11 16 17 28 27 -2|, &lt;0 -30 -25 -38 -39 -70 -66 18|]
{{Optimal ET sequence|legend=0| 15, 19, 34, 53, 299l, 352fl, 405fl, 458fl, 511cfll, 564cffll }}


EDOs: {{EDOs| 72, 121, 193 }}
Badness (Sintel): 0.167


Badness: 0.0147
==== 2.3.5.13.37.41 subgroup ====
Subgroup: 2.3.5.13.37.41


== Scales  ==
Comma list: 325/324, 481/480, 625/624, 1025/1024
* [[sqrtphi17]]
* [[sqrtphi23]]
* [[sqrtphi49]]


== Music  ==
Subgroup-val mapping: {{mapping| 1 0 1 0 6 8 | 0 6 5 14 -3 -10 }}
* [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:Theory]]
Optimal tunings:
[[Category:Temperament family]]
* WE: ~2 = 1200.1651{{c}}, ~6/5 = 317.1126{{c}}
[[Category:Kleismic]]
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0748{{c}}
[[Category:Rank 2]]
 
{{Optimal ET sequence|legend=0| 15, 19, 34, 53, 140, 193, 246l }}
 
Badness (Sintel): 0.223
 
[[Category:Kleismic family| ]] <!-- main article -->
[[Category:Temperament families]]
[[Category:Catalogs of rank-2 temperaments]]
[[Category:Listen]]
[[Category:Listen]]
[[Category:Todo:add definition]]
[[Category:Todo:review]]

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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