Kleismic family: Difference between revisions

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Line 25:

* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 317.0308{{c}}

: error map: {{val| 0.000 +0.230 -1.160 }}

~~<!-- * CTE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0586{{c}}~~

* POTE: ~2 = 1200.000{{c}}, ~6/5 = 317.007{{c}} -->

[[Tuning ranges]]:

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=== Overview to extensions ===

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

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.

[[6144/6125]], the porwell comma, gives hemikleismic. [[245/243]], sensamagic, gives clyde. [[1029/1024]], the gamelisma, gives tritikleismic. [[2401/2400]] the breedsma, gives quadritikleismic. Hemikleismic splits the 6/5 in half to get a neutral second generator of 35/32, and clyde similarly splits the 5/3 in half to get a 9/7 generator. Finally, tritikleismic has a 1/3-octave period with minor third generator, and quadritikleismic a 1/4-octave period with the minor third generator.

[[6144/6125]], the porwell comma, gives [[#Hemikleismic|hemikleismic]]. [[245/243]], sensamagic, gives [[#Clyde|clyde]]. [[1029/1024]], the gamelisma, gives [[#Tritikleismic|tritikleismic]]. [[10976/10935]], hemimage, gives [[#Marfifths|marfifths]]. [[1728/1715]], the orwellismia, gives [[#Kleiboh|kleiboh]]. [[2401/2400]], the breedsma, gives [[#Quadritikleismic|quadritikleismic]]. [[2460375/2458624]], the breeze comma, gives [[#Marthirds|marthirds]]. Hemikleismic splits the 6/5 in half to get a neutral second generator of ~35/32, and clyde similarly splits the 5/3 in half to get a ~9/7 generator. Marfifths splits the 12/5 into three. Kleiboh splits the 24/5 into three. Marthirds splits the 12/5 into four. Finally, tritikleismic has a 1/3-octave period with minor third generator, and quadritikleismic a 1/4-octave period with the minor third generator.

Temperaments involving larger splits include [[#Sqrtphi|sqrtphi]], [[#Quartkeenlig|quartkeenlig]], [[#Novemkleismic|novemkleismic]]. Those split the kleismic structure into five to nine parts.

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

== Catakleismic ==

Catakleismic tempers out 225/224, the [[marvel comma]], and 4375/4374, the [[ragisma]], and may be described as the {{nowrap| 53 & 72 }} temperament. [[125edo]] and especially [[197edo]] make for excellent tunings.

Catakleismic extends easily with [[prime interval|prime]] [[13/1|13]]. The [[S-expression]]-based comma list of this extension is {[[169/168|S13]], [[225/224|S15 = S25⋅S26⋅S27]], [[325/324|S10/S12 = S25⋅S26]], ([[625/624|S25]], [[676/675|S26 = S13/S15]], [[729/728|S27]])}.

=== 7-limit ===

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* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 316.7705{{c}}

: error map: {{val| 0.000 -1.332 -2.461 +0.126 }}

~~~~

[[Tuning ranges]]:

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==== 2.3.5.7.13 subgroup ====

The [[S-expression]]-based comma list of this temperament is {[[169/168|S13]], [[225/224|S15 = S25*S26*S27]], [[325/324|S10/S12 = S25*S26]], ([[625/624|S25]], [[676/675|S26 = S13/S15]], [[729/728|S27]])}.

Subgroup: 2.3.5.7.13

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* WE: ~2 = 1200.7838{{c}}, ~6/5 = 316.9478{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.7939{{c}}

~~~~

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* WE: ~2 = 1200.6524{{c}}, ~6/5 = 316.8911{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.7267{{c}}

~~~~

Tuning ranges:

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* WE: ~2 = 1200.7982{{c}}, ~6/5 = 316.9482{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.7491{{c}}

~~~~

Tuning ranges:

Line 134:

* WE: ~2 = 1199.9590{{c}}, ~6/5 = 317.0315{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0403{{c}}

~~~~

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* WE: ~2 = 1200.0797{{c}}, ~6/5 = 317.0571{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0400{{c}}

~~~~

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* WE: ~2 = 1200.8102{{c}}, ~6/5 = 316.8669{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.6768{{c}}

~~~~

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* WE: ~2 = 1201.0807{{c}}, ~6/5 = 316.9246{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.6700{{c}}

~~~~

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* WE: ~2 = 1198.6575{{c}}, ~6/5 = 316.7282{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0257{{c}}

~~~~

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* WE: ~2 = 1198.8403{{c}}, ~6/5 = 316.8111{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0652{{c}}

~~~~

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* WE: ~99/70 = 600.2674{{c}}, ~6/5 = 316.8624{{c}}

* CWE: ~99/70 = 600.0000{{c}}, ~6/5 = 316.7575{{c}}

~~~~

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* WE: ~55/39 = 600.3582{{c}}, ~6/5 = 316.9152{{c}}

* CWE: ~55/39 = 600.0000{{c}}, ~6/5 = 316.7759{{c}}

~~~~

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* WE: ~17/12 = 600.4210{{c}}, ~6/5 = 316.9282{{c}}

* CWE: ~17/12 = 600.0000{{c}}, ~6/5 = 316.7578{{c}}

~~~~

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* WE: ~17/12 = 600.3763{{c}}, ~6/5 = 316.8720{{c}}

* CWE: ~17/12 = 600.0000{{c}}, ~6/5 = 316.7205{{c}}

~~~~

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* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 316.8293{{c}}

: error map: {{val| 0.000 -0.979 -2.167 -18.388 }}

~~~~

[[Tuning ranges]]:

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* WE: ~2 = 1199.7353{{c}}, ~6/5 = 317.5055{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.5546{{c}}

~~~~

Tuning ranges:

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* WE: ~2 = 1201.8360{{c}}, ~6/5 = 317.0958{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.6829{{c}}

~~~~

Tuning ranges:

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* WE: ~2 = 1199.7816{{c}}, ~6/5 = 317.3653{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.4070{{c}}

~~~~

Tuning ranges:

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* WE: ~2 = 1196.7615{{c}}, ~6/5 = 317.7353{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 319.4059{{c}}

~~~~

Line 397:

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* WE: ~2 = 1204.5534{{c}}, ~6/5 = 315.9247{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 315.1686{{c}}

~~~~

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* WE: ~2 = 1204.4937{{c}}, ~6/5 = 316.2241{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 315.4748{{c}}

~~~~

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* WE: ~2 = 1201.6569{{c}}, ~6/5 = 318.0942{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.8547{{c}}

~~~~

Line 445:

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* WE: ~2 = 1201.9324{{c}}, ~6/5 = 317.8090{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.5170{{c}}

~~~~

Line 463:

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* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 318.2411{{c}}

: error map: {{val| 0.000 +7.492 +4.892 +12.281 }}

~~~~

[[Tuning ranges]]:

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* WE: ~2 = 1197.0368{{c}}, ~6/5 = 317.4956{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 318.2672{{c}}

~~~~

Tuning ranges:

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* 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 }}

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* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 317.1220{{c}}

: error map: {{val| 0.000 +0.777 -0.704 +0.391 }}

~~~~

[[Tuning ranges]]:

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* WE: ~2 = 1200.0980{{c}}, ~6/5 = 317.1879{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.1623{{c}}

~~~~

Tuning ranges:

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* WE: ~2 = 1200.0936{{c}}, ~6/5 = 317.1864{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.1622{{c}}

~~~~

Tuning ranges:

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* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 317.3071{{c}}

: error map: {{val| 0.000 +1.887 +0.222 +0.370 }}

~~~~

Line 600:

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* WE: ~2 = 1199.5425{{c}}, ~6/5 = 317.1901{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.3020{{c}}

~~~~

Line 616:

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* WE: ~2 = 1199.5339{{c}}, ~6/5 = 317.1882{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.3028{{c}}

~~~~

Line 635:

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* [[CWE]]: ~2 = 1200.0000{{c}}, ~35/32 = 158.6338{{c}}

: error map: {{val| 0.000 +1.651 +0.024 +3.470 }}

~~~~

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* WE: ~2 = 1199.8009{{c}}, ~11/10 = 158.6508{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 158.6717{{c}}

~~~~

Line 667:

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* WE: ~2 = 1199.7952{{c}}, ~11/10 = 158.6279{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 158.6493{{c}}

~~~~

Line 686:

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* [[CWE]]: ~2 = 1200.0000{{c}}, ~14/9 = 758.6554{{c}}

: error map: {{val| 0.000 +1.910 +0.240 -2.441 }}

~~~~

[[Minimax tuning]]:

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* WE: ~2 = 1199.9620{{c}}, ~14/9 = 758.6210{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~14/9 = 758.6445{{c}}

~~~~

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* WE: ~2 = 1199.9292{{c}}, ~14/9 = 758.5919{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~14/9 = 758.6355{{c}}

~~~~

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* [[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]]:

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[[Badness]] (Sintel): 1.43

Music:

; Music

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

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

=== 11-limit ===

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* 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:

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* 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}})

~~~~

Line 808:

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* 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}})

~~~~

Badness (Sintel): 0.690

~~== Quadritikleismic ==~~

~~[[Subgroup]]: 2.3.5.7~~

~~[[Comma list]]: 2401/2400, 15625/15552~~

~~{{Mapping|legend=1| 4 0 4 7 | 0 6 5 4 }}~~

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

~~[[Optimal tuning]]s:~~

* [[WE]]: ~25/21 = 300.0520{{c}}, ~6/5 = 317.0548{{c}} (~126/125 = 17.0029{{c}})

~~: [[error map]]: {{val| +0.208 +0.374 -0.832 -0.243 }}~~

* [[CWE]]: ~25/21 = 300.0000{{c}}, ~6/5 = 317.0301{{c}} (~126/125 = 17.0301{{c}})

~~: error map: {{val| 0.000 +0.225 -1.163 -0.706 }}~~

~~~~

~~{{Optimal ET sequence|legend=1| 68, 72, 140, 212, 776cd, 988ccd, 1200ccd }}~~

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

~~=== 11-limit ===~~

~~Subgroup: 2.3.5.7.11~~

~~Comma list: 385/384, 1375/1372, 6250/6237~~

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

~~Optimal tunings:~~

* WE: ~25/21 = 300.0995{{c}}, ~6/5 = 317.0298{{c}} (~100/99 = 16.9303{{c}})

* CWE: ~25/21 = 300.0000{{c}}, ~6/5 = 316.9540{{c}} (~100/99 = 16.9540{{c}})

~~~~

~~{{Optimal ET sequence|legend=0| 68, 72, 140, 212, 284, 496ce, 780ccdee }}~~

~~Badness (Sintel): 0.774~~

~~=== 13-limit ===~~

~~Subgroup: 2.3.5.7.11.13~~

~~Comma list: 325/324, 385/384, 625/624, 1375/1372~~

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

~~Optimal tunings:~~

* WE: ~25/21 = 300.0985{{c}}, ~6/5 = 317.0899{{c}} (~100/99 = 16.9941{{c}})

* CWE: ~25/21 = 300.0000{{c}}, ~6/5 = 317.0155{{c}} (~100/99 = 17.0155{{c}})

~~~~

~~{{Optimal ET sequence|legend=0| 68, 72, 140, 212 }}~~

~~Badness (Sintel): 0.774~~

~~=== 17-limit ===~~

~~Subgroup: 2.3.5.7.11.13.17~~

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

~~Mapping: {{mapping| 4 0 4 7 17 0 10 | 0 6 5 4 -3 14 6 }}~~

~~Optimal tunings:~~

* WE: ~25/21 = 300.1102{{c}}, ~6/5 = 317.1011{{c}} (~100/99 = 16.9909{{c}})

* CWE: ~25/21 = 300.0000{{c}}, ~6/5 = 317.0155{{c}} (~100/99 = 17.0155{{c}})

~~~~

~~{{Optimal ET sequence|legend=0| 68, 72, 140, 212g }}~~

~~Badness (Sintel): 0.651~~

~~== Kleiboh ==~~

~~[[Subgroup]]: 2.3.5.7~~

~~[[Comma list]]: 1728/1715, 3125/3087~~

~~{{Mapping|legend=1| 1 -12 -9 -7 | 0 18 15 13 }}~~

~~: mapping generators: ~2, ~42/25~~

~~[[Optimal tuning]]s:~~

* [[WE]]: ~2 = 1199.5290{{c}}, ~42/25 = 905.3417{{c}}

~~: [[error map]]: {{val| -0.471 -0.152 -1.949 +3.914 }}~~

* [[CWE]]: ~2 = 1200.0000{{c}}, ~42/25 = 905.6741{{c}}

~~: error map: {{val| 0.000 +0.178 -1.203 +4.937 }}~~

~~~~

~~{{Optimal ET sequence|legend=1| 49, 53 }}~~

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

~~=== 11-limit ===~~

~~Subgroup: 2.3.5.7.11~~

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

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

~~Optimal tunings:~~

* WE: ~2 = 1199.1389{{c}}, ~42/25 = 905.1688{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~42/25 = 905.7840{{c}}

~~~~

~~{{Optimal ET sequence|legend=0| 49, 53, 102d }}~~

~~Badness (Sintel): 1.75~~

~~=== 13-limit ===~~

~~Subgroup: 2.3.5.7.11.13~~

~~Comma list: 176/175, 275/273, 325/324, 540/539~~

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

~~Optimal tunings:~~

* WE: ~2 = 1199.1517{{c}}, ~22/13 = 905.1727{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~22/13 = 905.7801{{c}}

~~~~

~~{{Optimal ET sequence|legend=0| 49f, 53, 102df }}~~

~~Badness (Sintel): 1.28~~

== Marfifths ==

~~The ''~~marfifths~~'' temperament (19 & 140)~~ tempers out the [[hemimage comma]], ~~10976/10935~~. It ~~splits the interval~~ of a ~~major thirteenth (~10/3) into three marvelous~~ fifth ([[112/75]]) ~~intervals~~, ~~and uses it for a~~ generator.

Named by [[Xenllium]] in 2021, marfifths tempers out the 10976/10935, the [[hemimage comma]], and may be described as the {{nowrap| 19 & 140 }} temperament. It is generated by a marvel fourth of [[75/56]] (or a marvel fifth of [[112/75]]), three of which minus an octave make the hanson generator of ~6/5. Its [[ploidacot]] is zeta-18-cot.

[[Subgroup]]: 2.3.5.7

Line 947:

Line 791:

* [[CWE]]: ~2 = 1200.0000{{c}}, ~75/56 = 505.7060{{c}}

: error map: {{val| 0.000 +0.753 -0.724 -0.643 }}

~~~~

Line 963:

Line 806:

* WE: ~2 = 1200.2484{{c}}, ~75/56 = 505.7882{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~75/56 = 505.6853{{c}}

~~~~

Line 979:

Line 821:

* WE: ~2 = 1200.2747{{c}}, ~75/56 = 505.8019{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~75/56 = 505.6883{{c}}

~~~~

Line 986:

Line 827:

=== Diatessic ===

~~The ''diatessic'' temperament (~~121 & 140) is closely related to the ~~'''diatess~~ tuning~~'''~~ (generator: 505.727281 cents).

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

Subgroup: 2.3.5.7.11

Line 997:

Line 838:

* WE: ~2 = 1199.7886{{c}}, ~75/56 = 505.6513{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~75/56 = 505.7366{{c}}

~~~~

Line 1,013:

Line 853:

* WE: ~2 = 1199.7996{{c}}, ~75/56 = 505.6558{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~75/56 = 505.7366{{c}}

~~~~

Line 1,020:

Line 859:

=== Marf ===

~~The ''marf'' temperament (~~19 & 121) has a POTE generator which strongly approximates the marvelous fifth interval of 112/75.

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

Subgroup: 2.3.5.7.11

Line 1,031:

Line 870:

* WE: ~2 = 1199.3198{{c}}, ~75/56 = 505.4822{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~75/56 = 505.7607{{c}}

~~~~

Line 1,047:

Line 885:

* WE: ~2 = 1199.3368{{c}}, ~75/56 = 505.4919{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~75/56 = 505.7627{{c}}

~~~~

Line 1,053:

Line 890:

Badness (Sintel): 1.58

== ~~Marthirds~~ ==

== Kleiboh ==

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

[[Subgroup]]: 2.3.5.7

[[Comma list]]: ~~15625~~/~~15552~~, ~~2460375~~/~~2458624~~

[[Comma list]]: 1728/1715, 3125/3087

: mapping generators: ~2, ~56/45

: mapping generators: ~2, ~42/25

[[Optimal tuning]]s:

* [[WE]]: ~2 = ~~1200~~.~~1662~~{{c}}, ~56/45 = ~~379~~.~~3041~~{{c}}

* [[WE]]: ~2 = 1199.5290{{c}}, ~42/25 = 905.3417{{c}}

: [[error map]]: {{val| +0.~~166 +~~0.~~347~~ -0.~~896~~ +0.~~000~~ }}

: [[error map]]: {{val| -0.471 -0.152 -1.949 +3.914 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~56/45 = ~~379~~.~~2552~~{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~42/25 = 905.6741{{c}}

: error map: {{val| 0.000 +0.~~171~~ -1.~~209 -0.214 }}~~

: error map: {{val| 0.000 +0.178 -1.203 +4.937 }}

~~~~

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

[[Badness]] (Sintel): 2.64

[[Badness]] (Sintel): 1.93

=== 11-limit ===

Subgroup: 2.3.5.7.11

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

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

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

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

Optimal tunings:

* WE: ~2 = ~~1200~~.~~1189~~{{c}}, ~56/45 = ~~379~~.~~2942~~{{c}}

* WE: ~2 = 1199.1389{{c}}, ~42/25 = 905.1688{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~~~56/45 = 379.2580{{c}}~~

* CWE: ~2 = 1200.0000{{c}}, ~42/25 = 905.7840{{c}}

~~~~

{{Optimal ET sequence|legend=0| ~~19e~~, …, ~~193, 212, 405, 617c~~ }}

Badness (Sintel): 2.50

Badness (Sintel): 1.75

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

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

Comma list: 176/175, 275/273, 325/324, 540/539

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

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

Optimal tunings:

* WE: ~2 = ~~1200~~.~~2154~~{{c}}, ~56/45 = ~~379~~.~~3236~~{{c}}

* WE: ~2 = 1199.1517{{c}}, ~22/13 = 905.1727{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~~~56/45 = 379.2580{{c}}~~

* CWE: ~2 = 1200.0000{{c}}, ~22/13 = 905.7801{{c}}

~~~~

{{Optimal ET sequence|legend=0| ~~19e~~, …, ~~193, 212, 405f, 617cff~~ }}

Badness (Sintel): 1.81

Badness (Sintel): 1.28

~~== Quartkeenlig ==~~

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

== Quadritikleismic ==

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 15625/15552~~, 117649/116640~~

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

: mapping ~~generator~~: ~2, ~36/35

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

[[Optimal tuning]]s:

* [[WE]]: ~2 = ~~1200~~.~~2825~~{{c}}, ~36/35 = 52.~~8528~~{{c}}

* [[WE]]: ~25/21 = 300.0520{{c}}, ~6/5 = 317.0548{{c}} (~126/125 = 17.0029{{c}})

: [[error map]]: {{val| +0.~~282~~ +0.~~745~~ -0.~~448~~ -1.~~579~~ }}

: [[error map]]: {{val| +0.208 +0.374 -0.832 -0.243 }}

* [[CWE]]: ~2 = ~~1200~~.0000{{c}}, ~36/35 = 52.~~8476~~{{c}}

* [[CWE]]: ~25/21 = 300.0000{{c}}, ~6/5 = 317.0301{{c}} (~126/125 = 17.0301{{c}})

: error map: {{val| 0.000 +0.~~558~~ -0.~~886~~ -2.~~074~~ }}

: error map: {{val| 0.000 +0.225 -1.163 -0.706 }}

~~~~

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

{{Optimal ET sequence|legend=1| 68, 72, 140, 212, 776cd, 988ccd, 1200ccd }}

[[Badness]] (Sintel): 3.69

[[Badness]] (Sintel): 0.993

=== 11-limit ===

Subgroup: 2.3.5.7.11

Comma list: 385/384, 6250/6237~~, 67228/66825~~

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

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

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

Optimal tunings:

* WE: ~2 = ~~1200~~.~~2526~~{{c}}, ~36/35 = 52.~~8534~~{{c}}

* WE: ~25/21 = 300.0995{{c}}, ~6/5 = 317.0298{{c}} (~100/99 = 16.9303{{c}})

* CWE: ~2 = ~~1200~~.~~0000~~{{c}}, ~36/35 = 52.~~8446~~{{c}}

* CWE: ~25/21 = 300.0000{{c}}, ~6/5 = 316.9540{{c}} (~100/99 = 16.9540{{c}})

~~~~

{{Optimal ET sequence|legend=0| 68, 91, ~~159~~, ~~386d~~, ~~545dd~~ }}

{{Optimal ET sequence|legend=0| 68, 72, 140, 212, 284, 496ce, 780ccdee }}

Badness (Sintel): 2.86

Badness (Sintel): 0.774

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

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

Comma list: 325/324, 385/384, 625/624, 1375/1372

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

Optimal tunings:

* WE: ~25/21 = 300.0985{{c}}, ~6/5 = 317.0899{{c}} (~100/99 = 16.9941{{c}})

* CWE: ~25/21 = 300.0000{{c}}, ~6/5 = 317.0155{{c}} (~100/99 = 17.0155{{c}})

Badness (Sintel): 0.774

=== 17-limit ===

Subgroup: 2.3.5.7.11.13.17

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

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

Mapping: {{mapping| 4 0 4 7 17 0 10 | 0 6 5 4 -3 14 6 }}

Optimal tunings:

* WE: ~2 = ~~1200~~.~~2564~~{{c}}, ~36/35 = 52.~~8568~~{{c}}

* WE: ~25/21 = 300.1102{{c}}, ~6/5 = 317.1011{{c}} (~100/99 = 16.9909{{c}})

* CWE: ~2 = ~~1200~~.0000{{c}}, ~36/35 = 52.~~8479~~{{c}}

* CWE: ~25/21 = 300.0000{{c}}, ~6/5 = 317.0155{{c}} (~100/99 = 17.0155{{c}})

~~~~

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

Badness (Sintel): 0.651

~~Badness (Sintel): 1~~.97

== Marthirds ==

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

~~== Novemkleismic ==~~

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 15625/15552, ~~40353607~~/~~40310784~~

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

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

: mapping generators: ~2, ~56/45

[[Optimal tuning]]s:

* [[WE]]: ~~~2592/2401~~ = ~~133~~.~~3488~~{{c}}, ~6/5 = ~~317~~.~~0413~~{{c}} ~~(~36/35 = 50.3437{{c}})~~

* [[WE]]: ~2 = 1200.1662{{c}}, ~56/45 = 379.3041{{c}}

: [[error map]]: {{val| +0.~~139~~ +0.~~293~~ -0.~~968~~ +0.~~259~~ }}

: [[error map]]: {{val| +0.166 +0.347 -0.896 +0.000 }}

* [[CWE]]: ~~~2592/2401~~ = ~~133~~.~~3333~~{{c}}, ~~~6/5 = 317.0260{{c}} (~36~~/35 = 50.~~3593~~{{c}})

* [[CWE]]: ~2 = 1200.0000{{c}}, ~56/45 = 379.2552{{c}}

: error map: {{val| 0.000 +0.~~201~~ -1.~~184~~ -0.~~003 }}~~

: error map: {{val| 0.000 +0.171 -1.209 -0.214 }}

~~~~

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

[[Badness]] (Sintel): 4.90

[[Badness]] (Sintel): 2.64

=== 11-limit ===

Subgroup: 2.3.5.7.11

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

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

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

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

Optimal tunings:

* WE: ~~~250/231~~ = ~~133~~.~~3465~~{{c}}, ~6/5 = ~~317~~.~~0416~~{{c}} ~~(~36/35 = 50.3486{{c}})~~

* WE: ~2 = 1200.1189{{c}}, ~56/45 = 379.2942{{c}}

* CWE: ~~~250/231~~ = ~~133~~.~~3333~~{{c}}, ~~~6/5 = 317.0264{{c}} (~36/35 = 50.3597{{c}})~~

* CWE: ~2 = 1200.0000{{c}}, ~56/45 = 379.2580{{c}}

~~~~

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

Badness (Sintel): 1.71

Badness (Sintel): 2.50

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

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

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

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

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

Optimal tunings:

* WE: ~~~250/231~~ = ~~133~~.~~3385~~{{c}}, ~6/5 = ~~317~~.~~0978~~{{c}} ~~(~36/35 = 50.4208{{c}})~~

* WE: ~2 = 1200.2154{{c}}, ~56/45 = 379.3236{{c}}

* CWE: ~~~250/231~~ = ~~133~~.~~3333~~{{c}}, ~~~6/5 = 317.0910{{c}} (~36/35 = 50.4243{{c}})~~

* CWE: ~2 = 1200.0000{{c}}, ~56/45 = 379.2580{{c}}

~~~~

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

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

Badness (Sintel): 1.61

Badness (Sintel): 1.81

== Sqrtphi ==

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

Sqrtphi tempers out 16875/16807, the [[mirkwai comma]], and may be described as the {{nowrap| 49 & 72 }} temperament. The just value of sqrt(φ) is 416.545 cents, and this temperament gives a close approximation of it.

Note that in the data below, the generator is given as its [[octave complement]], which stands in for [[~]][[11/7]] from the [[11-limit]] onwards. Five generators octave reduced make the hanson generator of ~[[6/5]]. The [[ploidacot]] for this temperament is 19-sheared 30-cot.

[[Subgroup]]: 2.3.5.7

Line 1,227:

Line 1,070:

* [[CWE]]: ~2 = 1200.0000{{c}}, ~196/125 = 783.4009{{c}}

: error map: {{val| 0.000 +0.072 -1.291 +0.408 }}

~~~~

Line 1,243:

Line 1,085:

* WE: ~2 = 1200.0514{{c}}, ~11/7 = 783.4294{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 783.3975{{c}}

~~~~

Line 1,259:

Line 1,100:

* WE: ~2 = 1199.9314{{c}}, ~11/7 = 783.3705{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 783.4134{{c}}

~~~~

Line 1,275:

Line 1,115:

* WE: ~2 = 1199.9324{{c}}, ~11/7 = 783.3706{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 783.4129{{c}}

~~~~

Line 1,291:

Line 1,130:

* WE: ~2 = 1199.8567{{c}}, ~11/7 = 783.3262{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 783.4176{{c}}

~~~~

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 [[23edo and octave stretching|stretched 23edo]].

[[Subgroup]]: 2.3.5.7

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

: mapping generator: ~2, ~36/35

[[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]] (Sintel): 3.69

=== 11-limit ===

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

* WE: ~2 = 1200.2526{{c}}, ~36/35 = 52.8534{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 52.8446{{c}}

Badness (Sintel): 2.86

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

* WE: ~2 = 1200.2564{{c}}, ~36/35 = 52.8568{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 52.8479{{c}}

Badness (Sintel): 1.97

== Novemkleismic ==

[[Subgroup]]: 2.3.5.7

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

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

[[Optimal tuning]]s:

* [[WE]]: ~2592/2401 = 133.3488{{c}}, ~6/5 = 317.0413{{c}} (~36/35 = 50.3437{{c}})

: [[error map]]: {{val| +0.139 +0.293 -0.968 +0.259 }}

* [[CWE]]: ~2592/2401 = 133.3333{{c}}, ~6/5 = 317.0260{{c}} (~36/35 = 50.3593{{c}})

: error map: {{val| 0.000 +0.201 -1.184 -0.003 }}

[[Badness]] (Sintel): 4.90

=== 11-limit ===

Subgroup: 2.3.5.7.11

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

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

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}})

Badness (Sintel): 1.71

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

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

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

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}})

Badness (Sintel): 1.61

== Subgroup extensions ==

=== Kleismic (2.3.5.13) a.k.a. cata ===

Hanson lends itself nicely to this extension in the 2.3.5.13 subgroup, as the hemitwelfth, reached by three generator steps, can be interpreted as [[26/15]]. Notice 15625/15552 = ([[325/324]])([[625/624]]) and 325/324 = (625/624)([[676/675]]). The [[S-expression]]-based comma list of the temperament is {[[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]].

Hanson lends itself nicely to this extension in the 2.3.5.13 subgroup, as the hemitwelfth, reached by three generator steps, can be interpreted as [[26/15]]. Notice 15625/15552 = ([[325/324]])⋅([[625/624]]) and 325/324 = (625/624)⋅([[676/675]]). The [[S-expression]]-based comma list of the temperament is {[[325/324|S10/S12 = S25⋅S26]], ([[625/624|S25]]), [[676/675|S13/S15 = S26]]}. For the high-limit version of cata with a 1\5 period, see [[thunderclysmic]].

Subgroup: 2.3.5.13

Line 1,309:

Line 1,245:

Optimal tunings:

* WE: ~2 = 1200.1210{{c}}, ~6/5 = 317.1076{{c}}

* CWE: ~2 = 1200.~~1210~~{{c}}, ~6/5 = 317.0920{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0920{{c}}

~~<!-- * CTE: ~2 = 1200.0000{{c}}, ~6/5 = 317.1110{{c}}~~

* POTE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0756{{c}} -->

Line 1,328:

Line 1,262:

Optimal tunings:

* WE: ~2 = 1200.2924{{c}}, ~6/5 = 317.0998{{c}}

* CWE: ~2 = 1200.~~000~~{{c}}, ~6/5 = 317.0452{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0452{{c}}

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

@@ Line 25: / Line 25: @@
 * [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 317.0308{{c}}
 : error map: {{val| 0.000 +0.230 -1.160 }}
-<!-- * CTE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0586{{c}}
-* POTE: ~2 = 1200.000{{c}}, ~6/5 = 317.007{{c}} -->
 [[Tuning ranges]]:
@@ Line 37: / Line 35: @@
 === Overview to extensions ===
-The second comma of the [[normal forms #Normal forms for commas|normal comma list]] defines which [[7-limit]] family member we are looking at. [[875/864]], the keemic comma, gives keemun. [[4375/4374]], the ragisma, gives catakleismic. [[5120/5103]], hemifamity, gives countercata. Keemun, catakleismic and countercata all have octave period and use the minor third as a generator; catakleismic and countercata define the 7/4 more complexly but more accurately than keemun.
+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.
-[[6144/6125]], the porwell comma, gives hemikleismic. [[245/243]], sensamagic, gives clyde. [[1029/1024]], the gamelisma, gives tritikleismic. [[2401/2400]] the breedsma, gives quadritikleismic. Hemikleismic splits the 6/5 in half to get a neutral second generator of 35/32, and clyde similarly splits the 5/3 in half to get a 9/7 generator. Finally, tritikleismic has a 1/3-octave period with minor third generator, and quadritikleismic a 1/4-octave period with the minor third generator.
+[[6144/6125]], the porwell comma, gives [[#Hemikleismic|hemikleismic]]. [[245/243]], sensamagic, gives [[#Clyde|clyde]]. [[1029/1024]], the gamelisma, gives [[#Tritikleismic|tritikleismic]]. [[10976/10935]], hemimage, gives [[#Marfifths|marfifths]]. [[1728/1715]], the orwellismia, gives [[#Kleiboh|kleiboh]]. [[2401/2400]], the breedsma, gives [[#Quadritikleismic|quadritikleismic]]. [[2460375/2458624]], the breeze comma, gives [[#Marthirds|marthirds]]. Hemikleismic splits the 6/5 in half to get a neutral second generator of ~35/32, and clyde similarly splits the 5/3 in half to get a ~9/7 generator. Marfifths splits the 12/5 into three. Kleiboh splits the 24/5 into three. Marthirds splits the 12/5 into four. Finally, tritikleismic has a 1/3-octave period with minor third generator, and quadritikleismic a 1/4-octave period with the minor third generator.
+Temperaments involving larger splits include [[#Sqrtphi|sqrtphi]], [[#Quartkeenlig|quartkeenlig]], [[#Novemkleismic|novemkleismic]]. Those split the kleismic structure into five to nine parts.
+The kleismic family boasts a very remarkable extension to the [[2.3.5.13 subgroup]], which has further extensions with higher primes. These are listed at the bottom of this page, in [[#Subgroup extensions]].
 == Catakleismic ==
 {{Main| Catakleismic }}
+Catakleismic tempers out 225/224, the [[marvel comma]], and 4375/4374, the [[ragisma]], and may be described as the {{nowrap| 53 & 72 }} temperament. [[125edo]] and especially [[197edo]] make for excellent tunings.
+Catakleismic extends easily with [[prime interval|prime]] [[13/1|13]]. The [[S-expression]]-based comma list of this extension is {[[169/168|S13]], [[225/224|S15 = S25⋅S26⋅S27]], [[325/324|S10/S12 = S25⋅S26]], ([[625/624|S25]], [[676/675|S26 = S13/S15]], [[729/728|S27]])}.
 === 7-limit ===
@@ Line 56: / Line 62: @@
 * [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 316.7705{{c}}
 : error map: {{val| 0.000 -1.332 -2.461 +0.126 }}
-<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~6/5 = 316.732{{c}} -->
 [[Tuning ranges]]:
@@ Line 67: / Line 72: @@
 ==== 2.3.5.7.13 subgroup ====
-The [[S-expression]]-based comma list of this temperament is {[[169/168|S13]], [[225/224|S15 = S25*S26*S27]], [[325/324|S10/S12 = S25*S26]], ([[625/624|S25]], [[676/675|S26 = S13/S15]], [[729/728|S27]])}.
 Subgroup: 2.3.5.7.13
@@ Line 78: / Line 81: @@
 * WE: ~2 = 1200.7838{{c}}, ~6/5 = 316.9478{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.7939{{c}}
-<!-- * CTE: ~2 = 1200.0000{{c}}, ~6/5 = 316.8865{{c}} -->
 {{Optimal ET sequence|legend=0| 19, 34d, 53, 72, 125f, 197f }}
@@ Line 94: / Line 96: @@
 * WE: ~2 = 1200.6524{{c}}, ~6/5 = 316.8911{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.7267{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~6/5 = 316.719{{c}} -->
 Tuning ranges:
@@ Line 114: / Line 115: @@
 * WE: ~2 = 1200.7982{{c}}, ~6/5 = 316.9482{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.7491{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~6/5 = 316.738{{c}} -->
 Tuning ranges:
@@ Line 134: / Line 134: @@
 * WE: ~2 = 1199.9590{{c}}, ~6/5 = 317.0315{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0403{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~6/5 = 317.042{{c}} -->
 {{Optimal ET sequence|legend=0| 19e, 34d, 53 }}
@@ Line 150: / Line 149: @@
 * WE: ~2 = 1200.0797{{c}}, ~6/5 = 317.0571{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0400{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~6/5 = 317.036{{c}} -->
 {{Optimal ET sequence|legend=0| 19e, 34d, 53 }}
@@ Line 166: / Line 164: @@
 * WE: ~2 = 1200.8102{{c}}, ~6/5 = 316.8669{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.6768{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~6/5 = 316.653{{c}} -->
 {{Optimal ET sequence|legend=0| 19e, 53e, 72 }}
@@ Line 182: / Line 179: @@
 * WE: ~2 = 1201.0807{{c}}, ~6/5 = 316.9246{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.6700{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~6/5 = 316.639{{c}} -->
 {{Optimal ET sequence|legend=0| 19e, 53e, 72, 307bcdeeffff }}
@@ Line 198: / Line 194: @@
 * WE: ~2 = 1198.6575{{c}}, ~6/5 = 316.7282{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0257{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~6/5 = 317.083{{c}} -->
 {{Optimal ET sequence|legend=0| 19, 34d, 53e }}
@@ Line 214: / Line 209: @@
 * WE: ~2 = 1198.8403{{c}}, ~6/5 = 316.8111{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0652{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~6/5 = 317.118{{c}} -->
 {{Optimal ET sequence|legend=0| 19, 34d, 53e }}
@@ Line 231: / Line 225: @@
 * WE: ~99/70 = 600.2674{{c}}, ~6/5 = 316.8624{{c}}
 * CWE: ~99/70 = 600.0000{{c}}, ~6/5 = 316.7575{{c}}
-<!-- * POTE: ~99/70 = 600.000{{c}}, ~6/5 = 316.721{{c}} -->
 {{Optimal ET sequence|legend=0| 34d, 72, 322c, 394c }}
@@ Line 247: / Line 240: @@
 * WE: ~55/39 = 600.3582{{c}}, ~6/5 = 316.9152{{c}}
 * CWE: ~55/39 = 600.0000{{c}}, ~6/5 = 316.7759{{c}}
-<!-- * POTE: ~55/39 = 600.000{{c}}, ~6/5 = 316.726{{c}} -->
 {{Optimal ET sequence|legend=0| 34d, 72 }}
@@ Line 263: / Line 255: @@
 * WE: ~17/12 = 600.4210{{c}}, ~6/5 = 316.9282{{c}}
 * CWE: ~17/12 = 600.0000{{c}}, ~6/5 = 316.7578{{c}}
-<!-- * POTE: ~17/12 = 600.000{{c}}, ~6/5 = 316.726{{c}} -->
 {{Optimal ET sequence|legend=0| 34d, 38df, 72 }}
@@ Line 279: / Line 270: @@
 * WE: ~17/12 = 600.3763{{c}}, ~6/5 = 316.8720{{c}}
 * CWE: ~17/12 = 600.0000{{c}}, ~6/5 = 316.7205{{c}}
-<!-- * POTE: ~17/12 = 600.000{{c}}, ~6/5 = 316.726{{c}} -->
 {{Optimal ET sequence|legend=0| 34dh, 38df, 72 }}
@@ Line 299: / Line 289: @@
 * [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 316.8293{{c}}
 : error map: {{val| 0.000 -0.979 -2.167 -18.388 }}
-<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~6/5 = 316.473{{c}} -->
 [[Tuning ranges]]:
@@ Line 320: / Line 309: @@
 * WE: ~2 = 1199.7353{{c}}, ~6/5 = 317.5055{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.5546{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~6/5 = 317.576{{c}} -->
 Tuning ranges:
@@ Line 340: / Line 328: @@
 * WE: ~2 = 1201.8360{{c}}, ~6/5 = 317.0958{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.6829{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~6/5 = 316.611{{c}} -->
 Tuning ranges:
@@ Line 360: / Line 347: @@
 * WE: ~2 = 1199.7816{{c}}, ~6/5 = 317.3653{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.4070{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~6/5 = 317.423{{c}} -->
 Tuning ranges:
@@ Line 381: / Line 367: @@
 * WE: ~2 = 1196.7615{{c}}, ~6/5 = 317.7353{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~6/5 = 319.4059{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~6/5 = 318.595{{c}} -->
 {{Optimal ET sequence|legend=0| 4, 11b, 15 }}
@@ Line 397: / Line 382: @@
 * WE: ~2 = 1204.5534{{c}}, ~6/5 = 315.9247{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~6/5 = 315.1686{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~6/5 = 314.730{{c}} -->
 {{Optimal ET sequence|legend=0| 4e, 19, 42bcd }}
@@ Line 413: / Line 397: @@
 * WE: ~2 = 1204.4937{{c}}, ~6/5 = 316.2241{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~6/5 = 315.4748{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~6/5 = 315.044{{c}} -->
 {{Optimal ET sequence|legend=0| 4ef, 19 }}
@@ Line 429: / Line 412: @@
 * WE: ~2 = 1201.6569{{c}}, ~6/5 = 318.0942{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.8547{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~6/5 = 317.656{{c}} -->
 {{Optimal ET sequence|legend=0| 15, 19e, 34e }}
@@ Line 445: / Line 427: @@
 * WE: ~2 = 1201.9324{{c}}, ~6/5 = 317.8090{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.5170{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~6/5 = 317.298{{c}} -->
 {{Optimal ET sequence|legend=0| 15, 19e, 34e }}
@@ Line 463: / Line 444: @@
 * [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 318.2411{{c}}
 : error map: {{val| 0.000 +7.492 +4.892 +12.281 }}
-<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~6/5 = 318.267{{c}} -->
 [[Tuning ranges]]:
@@ Line 483: / Line 463: @@
 * WE: ~2 = 1197.0368{{c}}, ~6/5 = 317.4956{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~6/5 = 318.2672{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~6/5 = 318.282{{c}} -->
 Tuning ranges:
@@ Line 503: / Line 482: @@
 * WE: ~2 = 1196.8961{{c}}, ~6/5 = 317.3837{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~6/5 = 318.1621{{c}}
-<!-- * CTE: ~2 = 1200.0000{{c}}, ~6/5 = 317.9159{{c}} -->
 {{Optimal ET sequence|legend=0| 15, 34d, 49f, 83def, 132bcddeefff }}
@@ Line 521: / Line 499: @@
 * [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 317.1220{{c}}
 : error map: {{val| 0.000 +0.777 -0.704 +0.391 }}
-<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~6/5 = 317.121{{c}} -->
 [[Tuning ranges]]:
@@ Line 541: / Line 518: @@
 * WE: ~2 = 1200.0980{{c}}, ~6/5 = 317.1879{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.1623{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~6/5 = 317.162{{c}} -->
 Tuning ranges:
@@ Line 561: / Line 537: @@
 * WE: ~2 = 1200.0936{{c}}, ~6/5 = 317.1864{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.1622{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~6/5 = 317.162{{c}} -->
 Tuning ranges:
@@ Line 584: / Line 559: @@
 * [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 317.3071{{c}}
 : error map: {{val| 0.000 +1.887 +0.222 +0.370 }}
-<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~6/5 = 317.314{{c}} -->
 {{Optimal ET sequence|legend=1| 34d, 87, 121, 208, 537b }}
@@ Line 600: / Line 574: @@
 * WE: ~2 = 1199.5425{{c}}, ~6/5 = 317.1901{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.3020{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~6/5 = 317.311{{c}} -->
 {{Optimal ET sequence|legend=0| 34d, 53d, 87, 121, 208 }}
@@ Line 616: / Line 589: @@
 * WE: ~2 = 1199.5339{{c}}, ~6/5 = 317.1882{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.3028{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~6/5 = 317.311{{c}} -->
 {{Optimal ET sequence|legend=0| 34d, 53d, 87, 121, 208 }}
@@ Line 635: / Line 607: @@
 * [[CWE]]: ~2 = 1200.0000{{c}}, ~35/32 = 158.6338{{c}}
 : error map: {{val| 0.000 +1.651 +0.024 +3.470 }}
-<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~35/32 = 158.649{{c}} -->
 {{Optimal ET sequence|legend=1| 15, 38, 53, 121, 174d, 295d }}
@@ Line 651: / Line 622: @@
 * WE: ~2 = 1199.8009{{c}}, ~11/10 = 158.6508{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~11/10 = 158.6717{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~11/10 = 158.677{{c}} -->
 {{Optimal ET sequence|legend=0| 15, 38, 53, 68, 121e }}
@@ Line 667: / Line 637: @@
 * WE: ~2 = 1199.7952{{c}}, ~11/10 = 158.6279{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~11/10 = 158.6493{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~11/10 = 158.655{{c}} -->
 {{Optimal ET sequence|legend=0| 15, 38f, 53, 121e }}
@@ Line 686: / Line 655: @@
 * [[CWE]]: ~2 = 1200.0000{{c}}, ~14/9 = 758.6554{{c}}
 : error map: {{val| 0.000 +1.910 +0.240 -2.441 }}
-<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~14/9 = 758.665{{c}} -->
 [[Minimax tuning]]:
@@ Line 709: / Line 677: @@
 * WE: ~2 = 1199.9620{{c}}, ~14/9 = 758.6210{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~14/9 = 758.6445{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~14/9 = 758.645{{c}} -->
 {{Optimal ET sequence|legend=0| 19, 49e, 68, 87 }}
@@ Line 725: / Line 692: @@
 * WE: ~2 = 1199.9292{{c}}, ~14/9 = 758.5919{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~14/9 = 758.6355{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~14/9 = 758.637{{c}} -->
 {{Optimal ET sequence|legend=0| 19, 68, 87 }}
@@ Line 744: / Line 710: @@
 * [[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 }}
-<!-- * [[POTE]]: ~63/50 = 400.000{{c}}, ~6/5 = 316.872{{c}} (~21/20 = 83.128{{c}}) -->
 [[Minimax tuning]]:
@@ Line 758: / Line 723: @@
 [[Badness]] (Sintel): 1.43
-Music:
+; Music
-* ''[https://www.youtube.com/watch?v=vdjhC9i5KF4 Four Short Experiments in Octave Stretched 42edo (Dec 2024)]'' by [[Budjarn Lambeth]]
+* [https://www.youtube.com/watch?v=vdjhC9i5KF4 ''Four Short Experiments in Octave Stretched 42edo''] (2024) by [[Budjarn Lambeth]]
 === 11-limit ===
@@ Line 771: / Line 736: @@
 * 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}})
-<!-- * POTE: ~44/35 = 400.000{{c}}, ~6/5 = 316.881{{c}} (~21/20 = 83.119{{c}}) -->
 Minimax tuning:
@@ Line 792: / Line 756: @@
 * 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}})
-<!-- * POTE: ~44/35 = 400.0000{{c}}, ~6/5 = 316.9585{{c}} (~21/20 = 83.0415{{c}}) -->
 {{Optimal ET sequence|legend=0| 15, 57f, 72, 87, 159 }}
@@ Line 808: / Line 771: @@
 * 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}})
-<!-- * POTE: ~34/27 = 400.0000{{c}}, ~6/5 = 316.9082{{c}} (~21/20 = 83.0918{{c}}) -->
 {{Optimal ET sequence|legend=0| 15g, 57fg, 72, 159, 231f }}
 Badness (Sintel): 0.690
-== Quadritikleismic ==
-[[Subgroup]]: 2.3.5.7
-[[Comma list]]: 2401/2400, 15625/15552
-{{Mapping|legend=1| 4 0 4 7 | 0 6 5 4 }}
-: mapping generators: ~25/21, ~6/5
-[[Optimal tuning]]s:
-* [[WE]]: ~25/21 = 300.0520{{c}}, ~6/5 = 317.0548{{c}} (~126/125 = 17.0029{{c}})
-: [[error map]]: {{val| +0.208 +0.374 -0.832 -0.243 }}
-* [[CWE]]: ~25/21 = 300.0000{{c}}, ~6/5 = 317.0301{{c}} (~126/125 = 17.0301{{c}})
-: error map: {{val| 0.000 +0.225 -1.163 -0.706 }}
-<!-- * [[POTE]]: ~25/21 = 300.0000{{c}}, ~6/5 = 316.9999{{c}} (~126/125 = 16.9999{{c}}) -->
-{{Optimal ET sequence|legend=1| 68, 72, 140, 212, 776cd, 988ccd, 1200ccd }}
-[[Badness]] (Sintel): 0.993
-=== 11-limit ===
-Subgroup: 2.3.5.7.11
-Comma list: 385/384, 1375/1372, 6250/6237
-Mapping: {{mapping| 4 0 4 7 17 | 0 6 5 4 -3 }}
-Optimal tunings:
-* WE: ~25/21 = 300.0995{{c}}, ~6/5 = 317.0298{{c}} (~100/99 = 16.9303{{c}})
-* CWE: ~25/21 = 300.0000{{c}}, ~6/5 = 316.9540{{c}} (~100/99 = 16.9540{{c}})
-<!-- * POTE: ~25/21 = 300.0000{{c}}, ~6/5 = 316.9247{{c}} (~100/99 = 16.9247{{c}}) -->
-{{Optimal ET sequence|legend=0| 68, 72, 140, 212, 284, 496ce, 780ccdee }}
-Badness (Sintel): 0.774
-=== 13-limit ===
-Subgroup: 2.3.5.7.11.13
-Comma list: 325/324, 385/384, 625/624, 1375/1372
-Mapping: {{mapping| 4 0 4 7 17 0 | 0 6 5 4 -3 14 }}
-Optimal tunings:
-* WE: ~25/21 = 300.0985{{c}}, ~6/5 = 317.0899{{c}} (~100/99 = 16.9941{{c}})
-* CWE: ~25/21 = 300.0000{{c}}, ~6/5 = 317.0155{{c}} (~100/99 = 17.0155{{c}})
-<!-- * POTE: ~25/21 = 300.0000{{c}}, ~6/5 = 316.9887{{c}} (~100/99 = 16.9887{{c}}) -->
-{{Optimal ET sequence|legend=0| 68, 72, 140, 212 }}
-Badness (Sintel): 0.774
-=== 17-limit ===
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 289/288, 325/324, 385/384, 442/441, 625/624
-Mapping: {{mapping| 4 0 4 7 17 0 10 | 0 6 5 4 -3 14 6 }}
-Optimal tunings:
-* WE: ~25/21 = 300.1102{{c}}, ~6/5 = 317.1011{{c}} (~100/99 = 16.9909{{c}})
-* CWE: ~25/21 = 300.0000{{c}}, ~6/5 = 317.0155{{c}} (~100/99 = 17.0155{{c}})
-<!-- * POTE: ~25/21 = 300.0000{{c}}, ~6/5 = 316.9846{{c}} (~100/99 = 16.9846{{c}}) -->
-{{Optimal ET sequence|legend=0| 68, 72, 140, 212g }}
-Badness (Sintel): 0.651
-== Kleiboh ==
-[[Subgroup]]: 2.3.5.7
-[[Comma list]]: 1728/1715, 3125/3087
-{{Mapping|legend=1| 1 -12 -9 -7 | 0 18 15 13 }}
-: mapping generators: ~2, ~42/25
-[[Optimal tuning]]s:
-* [[WE]]: ~2 = 1199.5290{{c}}, ~42/25 = 905.3417{{c}}
-: [[error map]]: {{val| -0.471 -0.152 -1.949 +3.914 }}
-* [[CWE]]: ~2 = 1200.0000{{c}}, ~42/25 = 905.6741{{c}}
-: error map: {{val| 0.000 +0.178 -1.203 +4.937 }}
-<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~42/25 = 905.697{{c}} -->
-{{Optimal ET sequence|legend=1| 49, 53 }}
-[[Badness]] (Sintel): 1.93
-=== 11-limit ===
-Subgroup: 2.3.5.7.11
-Comma list: 176/175, 540/539, 3125/3087
-Mapping: {{mapping| 1 -12 -9 -7 -29 | 0 18 15 13 43 }}
-Optimal tunings:
-* WE: ~2 = 1199.1389{{c}}, ~42/25 = 905.1688{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~42/25 = 905.7840{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~42/25 = 905.819{{c}} -->
-{{Optimal ET sequence|legend=0| 49, 53, 102d }}
-Badness (Sintel): 1.75
-=== 13-limit ===
-Subgroup: 2.3.5.7.11.13
-Comma list: 176/175, 275/273, 325/324, 540/539
-Mapping: {{mapping| 1 -12 -9 -7 -29 -28 | 0 18 15 13 43 42 }}
-Optimal tunings:
-* WE: ~2 = 1199.1517{{c}}, ~22/13 = 905.1727{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~22/13 = 905.7801{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~22/13 = 905.813{{c}} -->
-{{Optimal ET sequence|legend=0| 49f, 53, 102df }}
-Badness (Sintel): 1.28
 == Marfifths ==
-The ''marfifths'' temperament (19 & 140) tempers out the [[hemimage comma]], 10976/10935. It splits the interval of a major thirteenth (~10/3) into three marvelous fifth ([[112/75]]) intervals, and uses it for a generator.
+Named by [[Xenllium]] in 2021, marfifths tempers out the 10976/10935, the [[hemimage comma]], and may be described as the {{nowrap| 19 & 140 }} temperament. It is generated by a marvel fourth of [[75/56]] (or a marvel fifth of [[112/75]]), three of which minus an octave make the hanson generator of ~6/5. Its [[ploidacot]] is zeta-18-cot.
 [[Subgroup]]: 2.3.5.7
@@ Line 947: / Line 791: @@
 * [[CWE]]: ~2 = 1200.0000{{c}}, ~75/56 = 505.7060{{c}}
 : error map: {{val| 0.000 +0.753 -0.724 -0.643 }}
-<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~75/56 = 505.705{{c}} -->
 {{Optimal ET sequence|legend=1| 19, …, 121, 140, 579, 719 }}
@@ Line 963: / Line 806: @@
 * WE: ~2 = 1200.2484{{c}}, ~75/56 = 505.7882{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~75/56 = 505.6853{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~75/56 = 505.684{{c}} -->
 {{Optimal ET sequence|legend=0| 19, 121e, 140, 159, 299 }}
@@ Line 979: / Line 821: @@
 * WE: ~2 = 1200.2747{{c}}, ~75/56 = 505.8019{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~75/56 = 505.6883{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~75/56 = 505.686{{c}} -->
 {{Optimal ET sequence|legend=0| 19, 121e, 140, 159, 299 }}
@@ Line 986: / Line 827: @@
 === Diatessic ===
-The ''diatessic'' temperament (121 & 140) is closely related to the '''diatess tuning''' (generator: 505.727281 cents).
+Diatessic may be described as {{nowrap| 121 & 140 }} and is closely related to the Diatess tuning (generator: 505.727281 cents).
 Subgroup: 2.3.5.7.11
@@ Line 997: / Line 838: @@
 * WE: ~2 = 1199.7886{{c}}, ~75/56 = 505.6513{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~75/56 = 505.7366{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~75/56 = 505.740{{c}} -->
 {{Optimal ET sequence|legend=0| 19e, …, 121, 140, 261, 401 }}
@@ Line 1,013: / Line 853: @@
 * WE: ~2 = 1199.7996{{c}}, ~75/56 = 505.6558{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~75/56 = 505.7366{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~75/56 = 505.740{{c}} -->
 {{Optimal ET sequence|legend=0| 19e, …, 121, 140, 261, 401 }}
@@ Line 1,020: / Line 859: @@
 === Marf ===
-The ''marf'' temperament (19 & 121) has a POTE generator which strongly approximates the marvelous fifth interval of 112/75.
+Marf may be described as {{nowrap| 19 & 121 }}. It has a POTE generator which strongly approximates the marvelous fifth interval of 112/75.
 Subgroup: 2.3.5.7.11
@@ Line 1,031: / Line 870: @@
 * WE: ~2 = 1199.3198{{c}}, ~75/56 = 505.4822{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~75/56 = 505.7607{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~75/56 = 505.769{{c}} -->
 {{Optimal ET sequence|legend=0| 19, 102d, 121 }}
@@ Line 1,047: / Line 885: @@
 * WE: ~2 = 1199.3368{{c}}, ~75/56 = 505.4919{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~75/56 = 505.7627{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~75/56 = 505.771{{c}} -->
 {{Optimal ET sequence|legend=0| 19, 102df, 121 }}
@@ Line 1,053: / Line 890: @@
 Badness (Sintel): 1.58
-== Marthirds ==
+== Kleiboh ==
-The ''marthirds'' temperament (19 & 193) tempers out the breeze comma (laquadru-atruyo comma), [[2460375/2458624]]. It splits the interval of minor tenth (~12/5) into four marvelous major third ([[56/45]]) intervals, and uses it for a generator.
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 15625/15552, 2460375/2458624
+[[Comma list]]: 1728/1715, 3125/3087
-{{Mapping|legend=1| 1 -6 -4 -19 | 0 24 20 69 }}
+{{Mapping|legend=1| 1 -12 -9 -7 | 0 18 15 13 }}
-: mapping generators: ~2, ~56/45
+: mapping generators: ~2, ~42/25
 [[Optimal tuning]]s:
-* [[WE]]: ~2 = 1200.1662{{c}}, ~56/45 = 379.3041{{c}}
+* [[WE]]: ~2 = 1199.5290{{c}}, ~42/25 = 905.3417{{c}}
-: [[error map]]: {{val| +0.166 +0.347 -0.896 +0.000 }}
+: [[error map]]: {{val| -0.471 -0.152 -1.949 +3.914 }}
-* [[CWE]]: ~2 = 1200.0000{{c}}, ~56/45 = 379.2552{{c}}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~42/25 = 905.6741{{c}}
-: error map: {{val| 0.000 +0.171 -1.209 -0.214 }}
+: error map: {{val| 0.000 +0.178 -1.203 +4.937 }}
-<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~56/45 = 379.252{{c}} -->
-{{Optimal ET sequence|legend=1| 19, …, 193, 212, 617c, 829c }}
+{{Optimal ET sequence|legend=1| 49, 53 }}
-[[Badness]] (Sintel): 2.64
+[[Badness]] (Sintel): 1.93
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 1375/1372, 15625/15552, 19712/19683
+Comma list: 176/175, 540/539, 3125/3087
-Mapping: {{mapping| 1 -6 -4 -19 -43 | 0 24 20 69 147 }}
+Mapping: {{mapping| 1 -12 -9 -7 -29 | 0 18 15 13 43 }}
 Optimal tunings:
-* WE: ~2 = 1200.1189{{c}}, ~56/45 = 379.2942{{c}}
+* WE: ~2 = 1199.1389{{c}}, ~42/25 = 905.1688{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~56/45 = 379.2580{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~42/25 = 905.7840{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~56/45 = 379.257{{c}} -->
-{{Optimal ET sequence|legend=0| 19e, …, 193, 212, 405, 617c }}
+{{Optimal ET sequence|legend=0| 49, 53, 102d }}
-Badness (Sintel): 2.50
+Badness (Sintel): 1.75
 === 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 325/324, 625/624, 1375/1372, 19712/19683
+Comma list: 176/175, 275/273, 325/324, 540/539
-Mapping: {{mapping| 1 -6 -4 -19 -43 -14 | 0 24 20 69 147 56 }}
+Mapping: {{mapping| 1 -12 -9 -7 -29 -28 | 0 18 15 13 43 42 }}
 Optimal tunings:
-* WE: ~2 = 1200.2154{{c}}, ~56/45 = 379.3236{{c}}
+* WE: ~2 = 1199.1517{{c}}, ~22/13 = 905.1727{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~56/45 = 379.2580{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~22/13 = 905.7801{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~56/45 = 379.256{{c}} -->
-{{Optimal ET sequence|legend=0| 19e, …, 193, 212, 405f, 617cff }}
+{{Optimal ET sequence|legend=0| 49f, 53, 102df }}
-Badness (Sintel): 1.81
+Badness (Sintel): 1.28
-== Quartkeenlig ==
-Quartkeenlig uses a generator in the 11-limit that is 33/32~36/35 tempered together, and is called so because it tempers out the [[quartisma]] by virtue of five 33/32's being with 7/6, keenanisma, 385/384, tempering 33/32 and 36/35 together, and liganellus comma (6250/6237). It can also be viewed as a regular temperament interpretation of [[23edo and octave stretching|stretched 23edo]].
+== Quadritikleismic ==
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 15625/15552, 117649/116640
+[[Comma list]]: 2401/2400, 15625/15552
-{{Mapping|legend=1| 1 0 1 1 | 0 36 30 41 }}
+{{Mapping|legend=1| 4 0 4 7 | 0 6 5 4 }}
-: mapping generator: ~2, ~36/35
+: mapping generators: ~25/21, ~6/5
 [[Optimal tuning]]s:
-* [[WE]]: ~2 = 1200.2825{{c}}, ~36/35 = 52.8528{{c}}
+* [[WE]]: ~25/21 = 300.0520{{c}}, ~6/5 = 317.0548{{c}} (~126/125 = 17.0029{{c}})
-: [[error map]]: {{val| +0.282 +0.745 -0.448 -1.579 }}
+: [[error map]]: {{val| +0.208 +0.374 -0.832 -0.243 }}
-* [[CWE]]: ~2 = 1200.0000{{c}}, ~36/35 = 52.8476{{c}}
+* [[CWE]]: ~25/21 = 300.0000{{c}}, ~6/5 = 317.0301{{c}} (~126/125 = 17.0301{{c}})
-: error map: {{val| 0.000 +0.558 -0.886 -2.074 }}
+: error map: {{val| 0.000 +0.225 -1.163 -0.706 }}
-<!-- * [[CTE]]: ~2 = 1200.0000{{c}}, ~36/35 = 52.8562{{c}} -->
-{{Optimal ET sequence|legend=1| 68, 91, 159, 386d, 545dd }}
+{{Optimal ET sequence|legend=1| 68, 72, 140, 212, 776cd, 988ccd, 1200ccd }}
-[[Badness]] (Sintel): 3.69
+[[Badness]] (Sintel): 0.993
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 385/384, 6250/6237, 67228/66825
+Comma list: 385/384, 1375/1372, 6250/6237
-Mapping: {{mapping| 1 0 1 1 5 | 0 36 30 41 -35 }}
+Mapping: {{mapping| 4 0 4 7 17 | 0 6 5 4 -3 }}
 Optimal tunings:
-* WE: ~2 = 1200.2526{{c}}, ~36/35 = 52.8534{{c}}
+* WE: ~25/21 = 300.0995{{c}}, ~6/5 = 317.0298{{c}} (~100/99 = 16.9303{{c}})
-* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 52.8446{{c}}
+* CWE: ~25/21 = 300.0000{{c}}, ~6/5 = 316.9540{{c}} (~100/99 = 16.9540{{c}})
-<!-- * CTE: ~2 = 1200.0000{{c}}, ~33/32 = 52.8524{{c}} -->
-{{Optimal ET sequence|legend=0| 68, 91, 159, 386d, 545dd }}
+{{Optimal ET sequence|legend=0| 68, 72, 140, 212, 284, 496ce, 780ccdee }}
-Badness (Sintel): 2.86
+Badness (Sintel): 0.774
 === 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 325/324, 385/384, 625/624, 16807/16731
+Comma list: 325/324, 385/384, 625/624, 1375/1372
+Mapping: {{mapping| 4 0 4 7 17 0 | 0 6 5 4 -3 14 }}
+Optimal tunings:
+* WE: ~25/21 = 300.0985{{c}}, ~6/5 = 317.0899{{c}} (~100/99 = 16.9941{{c}})
+* CWE: ~25/21 = 300.0000{{c}}, ~6/5 = 317.0155{{c}} (~100/99 = 17.0155{{c}})
+{{Optimal ET sequence|legend=0| 68, 72, 140, 212 }}
+Badness (Sintel): 0.774
+=== 17-limit ===
+Subgroup: 2.3.5.7.11.13.17
+Comma list: 289/288, 325/324, 385/384, 442/441, 625/624
-Mapping: {{mapping| 1 0 1 1 5 0 | 0 36 30 41 -35 84 }}
+Mapping: {{mapping| 4 0 4 7 17 0 10 | 0 6 5 4 -3 14 6 }}
 Optimal tunings:
-* WE: ~2 = 1200.2564{{c}}, ~36/35 = 52.8568{{c}}
+* WE: ~25/21 = 300.1102{{c}}, ~6/5 = 317.1011{{c}} (~100/99 = 16.9909{{c}})
-* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 52.8479{{c}}
+* CWE: ~25/21 = 300.0000{{c}}, ~6/5 = 317.0155{{c}} (~100/99 = 17.0155{{c}})
-<!-- * CTE: ~2 = 1200.0000{{c}}, ~33/32 = 52.8562{{c}} -->
+{{Optimal ET sequence|legend=0| 68, 72, 140, 212g }}
-{{Optimal ET sequence|legend=0| 68, 159, 386d, 545ddf }}
+Badness (Sintel): 0.651
-Badness (Sintel): 1.97
+== Marthirds ==
+Named by [[Xenllium]] in 2021, marthirds tempers out 2460375/2458624, the [[breeze comma]], and may be described as the {{nowrap| 19 & 193 }} temperament. It is generated by a marvel-comma-flat classical major third, [[56/45]], four of which minus an octave make the hanson generator of [[6/5]]. Its [[ploidacot]] is zeta-24-cot.
-== Novemkleismic ==
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 15625/15552, 40353607/40310784
+[[Comma list]]: 15625/15552, 2460375/2458624
-{{Mapping|legend=1| 9 0 9 11 | 0 6 5 6 }}
+{{Mapping|legend=1| 1 -6 -4 -19 | 0 24 20 69 }}
-: mapping generators: ~2592/2401, ~6/5
+: mapping generators: ~2, ~56/45
 [[Optimal tuning]]s:
-* [[WE]]: ~2592/2401 = 133.3488{{c}}, ~6/5 = 317.0413{{c}} (~36/35 = 50.3437{{c}})
+* [[WE]]: ~2 = 1200.1662{{c}}, ~56/45 = 379.3041{{c}}
-: [[error map]]: {{val| +0.139 +0.293 -0.968 +0.259 }}
+: [[error map]]: {{val| +0.166 +0.347 -0.896 +0.000 }}
-* [[CWE]]: ~2592/2401 = 133.3333{{c}}, ~6/5 = 317.0260{{c}} (~36/35 = 50.3593{{c}})
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~56/45 = 379.2552{{c}}
-: error map: {{val| 0.000 +0.201 -1.184 -0.003 }}
+: error map: {{val| 0.000 +0.171 -1.209 -0.214 }}
-<!-- * [[POTE]]: ~2592/2401 = 133.333{{c}}, ~6/5 = 317.005{{c}} (~36/35 = 50.338{{c}}) -->
-{{Optimal ET sequence|legend=1| 72, 261, 333, 405, 477c, 882c }}
+{{Optimal ET sequence|legend=1| 19, …, 193, 212, 617c, 829c }}
-[[Badness]] (Sintel): 4.90
+[[Badness]] (Sintel): 2.64
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 1375/1372, 4000/3993, 15625/15552
+Comma list: 1375/1372, 15625/15552, 19712/19683
-Mapping: {{mapping| 9 0 9 11 24 | 0 6 5 6 3 }}
+Mapping: {{mapping| 1 -6 -4 -19 -43 | 0 24 20 69 147 }}
 Optimal tunings:
-* WE: ~250/231 = 133.3465{{c}}, ~6/5 = 317.0416{{c}} (~36/35 = 50.3486{{c}})
+* WE: ~2 = 1200.1189{{c}}, ~56/45 = 379.2942{{c}}
-* CWE: ~250/231 = 133.3333{{c}}, ~6/5 = 317.0264{{c}} (~36/35 = 50.3597{{c}})
+* CWE: ~2 = 1200.0000{{c}}, ~56/45 = 379.2580{{c}}
-<!-- * POTE: ~250/231 = 133.333{{c}}, ~6/5 = 317.010{{c}} (~36/35 = 50.343{{c}}) -->
-{{Optimal ET sequence|legend=0| 72, 261, 333, 405, 882c }}
+{{Optimal ET sequence|legend=0| 19e, …, 193, 212, 405, 617c }}
-Badness (Sintel): 1.71
+Badness (Sintel): 2.50
 === 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 325/324, 625/624, 1375/1372, 4000/3993
+Comma list: 325/324, 625/624, 1375/1372, 19712/19683
-Mapping: {{mapping| 9 0 9 11 24 0 | 0 6 5 6 3 14 }}
+Mapping: {{mapping| 1 -6 -4 -19 -43 -14 | 0 24 20 69 147 56 }}
 Optimal tunings:
-* WE: ~250/231 = 133.3385{{c}}, ~6/5 = 317.0978{{c}} (~36/35 = 50.4208{{c}})
+* WE: ~2 = 1200.2154{{c}}, ~56/45 = 379.3236{{c}}
-* CWE: ~250/231 = 133.3333{{c}}, ~6/5 = 317.0910{{c}} (~36/35 = 50.4243{{c}})
+* CWE: ~2 = 1200.0000{{c}}, ~56/45 = 379.2580{{c}}
-<!-- * POTE: ~250/231 = 133.333{{c}}, ~6/5 = 317.086{{c}} (~36/35 = 50.419{{c}}) -->
-{{Optimal ET sequence|legend=0| 72, 189f, 261, 333, 738cf }}
+{{Optimal ET sequence|legend=0| 19e, …, 193, 212, 405f, 617cff }}
-Badness (Sintel): 1.61
+Badness (Sintel): 1.81
 == Sqrtphi ==
 {{Main| Sqrtphi }}
-The just value of sqrt (φ) is 416.545 cents.
+Sqrtphi tempers out 16875/16807, the [[mirkwai comma]], and may be described as the {{nowrap| 49 & 72 }} temperament. The just value of sqrt(φ) is 416.545 cents, and this temperament gives a close approximation of it.
+Note that in the data below, the generator is given as its [[octave complement]], which stands in for [[~]][[11/7]] from the [[11-limit]] onwards. Five generators octave reduced make the hanson generator of ~[[6/5]]. The [[ploidacot]] for this temperament is 19-sheared 30-cot.
 [[Subgroup]]: 2.3.5.7
@@ Line 1,227: / Line 1,070: @@
 * [[CWE]]: ~2 = 1200.0000{{c}}, ~196/125 = 783.4009{{c}}
 : error map: {{val| 0.000 +0.072 -1.291 +0.408 }}
-<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~196/125 = 783.397{{c}} -->
 {{Optimal ET sequence|legend=1| 23d, 49, 72, 193, 265 }}
@@ Line 1,243: / Line 1,085: @@
 * WE: ~2 = 1200.0514{{c}}, ~11/7 = 783.4294{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~11/7 = 783.3975{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~11/7 = 783.396{{c}} -->
 {{Optimal ET sequence|legend=0| 23de, 49, 72, 193, 265 }}
@@ Line 1,259: / Line 1,100: @@
 * WE: ~2 = 1199.9314{{c}}, ~11/7 = 783.3705{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~11/7 = 783.4134{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~11/7 = 783.415{{c}} -->
 {{Optimal ET sequence|legend=0| 23deff, 49f, 72, 121, 193 }}
@@ Line 1,275: / Line 1,115: @@
 * WE: ~2 = 1199.9324{{c}}, ~11/7 = 783.3706{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~11/7 = 783.4129{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~11/7 = 783.415{{c}} -->
 {{Optimal ET sequence|legend=0| 23deffgg, 49fg, 72, 121, 193 }}
@@ Line 1,291: / Line 1,130: @@
 * WE: ~2 = 1199.8567{{c}}, ~11/7 = 783.3262{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~11/7 = 783.4176{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~11/7 = 783.420{{c}} -->
 {{Optimal ET sequence|legend=0| 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 [[23edo and octave stretching|stretched 23edo]].
+[[Subgroup]]: 2.3.5.7
+[[Comma list]]: 15625/15552, 117649/116640
+{{Mapping|legend=1| 1 0 1 1 | 0 36 30 41 }}
+: mapping generator: ~2, ~36/35
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.2825{{c}}, ~36/35 = 52.8528{{c}}
+: [[error map]]: {{val| +0.282 +0.745 -0.448 -1.579 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~36/35 = 52.8476{{c}}
+: error map: {{val| 0.000 +0.558 -0.886 -2.074 }}
+{{Optimal ET sequence|legend=1| 68, 91, 159, 386d, 545dd }}
+[[Badness]] (Sintel): 3.69
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
+Comma list: 385/384, 6250/6237, 67228/66825
+Mapping: {{mapping| 1 0 1 1 5 | 0 36 30 41 -35 }}
+Optimal tunings:
+* WE: ~2 = 1200.2526{{c}}, ~36/35 = 52.8534{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 52.8446{{c}}
+{{Optimal ET sequence|legend=0| 68, 91, 159, 386d, 545dd }}
+Badness (Sintel): 2.86
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
+Comma list: 325/324, 385/384, 625/624, 16807/16731
+Mapping: {{mapping| 1 0 1 1 5 0 | 0 36 30 41 -35 84 }}
+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=0| 68, 159, 386d, 545ddf }}
+Badness (Sintel): 1.97
+== Novemkleismic ==
+[[Subgroup]]: 2.3.5.7
+[[Comma list]]: 15625/15552, 40353607/40310784
+{{Mapping|legend=1| 9 0 9 11 | 0 6 5 6 }}
+: mapping generators: ~2592/2401, ~6/5
+[[Optimal 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 }}
+[[Badness]] (Sintel): 4.90
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
+Comma list: 1375/1372, 4000/3993, 15625/15552
+Mapping: {{mapping| 9 0 9 11 24 | 0 6 5 6 3 }}
+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=0| 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: {{mapping| 9 0 9 11 24 0 | 0 6 5 6 3 14 }}
+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=0| 72, 189f, 261, 333, 738cf }}
+Badness (Sintel): 1.61
 == Subgroup extensions ==
 === Kleismic (2.3.5.13) a.k.a. cata ===
-Hanson lends itself nicely to this extension in the 2.3.5.13 subgroup, as the hemitwelfth, reached by three generator steps, can be interpreted as [[26/15]]. Notice 15625/15552 = ([[325/324]])([[625/624]]) and 325/324 = (625/624)([[676/675]]). The [[S-expression]]-based comma list of the temperament is {[[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]].
+Hanson lends itself nicely to this extension in the 2.3.5.13 subgroup, as the hemitwelfth, reached by three generator steps, can be interpreted as [[26/15]]. Notice 15625/15552 = ([[325/324]])⋅([[625/624]]) and 325/324 = (625/624)⋅([[676/675]]). The [[S-expression]]-based comma list of the temperament is {[[325/324|S10/S12 = S25⋅S26]], ([[625/624|S25]]), [[676/675|S13/S15 = S26]]}. For the high-limit version of cata with a 1\5 period, see [[thunderclysmic]].
 Subgroup: 2.3.5.13
@@ Line 1,309: / Line 1,245: @@
 Optimal tunings:
 * WE: ~2 = 1200.1210{{c}}, ~6/5 = 317.1076{{c}}
-* CWE: ~2 = 1200.1210{{c}}, ~6/5 = 317.0920{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0920{{c}}
-<!-- * CTE: ~2 = 1200.0000{{c}}, ~6/5 = 317.1110{{c}}
-* POTE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0756{{c}} -->
 {{Optimal ET sequence|legend=0| 15, 19, 34, 53, 140, 193, 246 }}
@@ Line 1,328: / Line 1,262: @@
 Optimal tunings:
 * WE: ~2 = 1200.2924{{c}}, ~6/5 = 317.0998{{c}}
-* CWE: ~2 = 1200.000{{c}}, ~6/5 = 317.0452{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0452{{c}}
 {{Optimal ET sequence|legend=0| 15, 19, 34, 53, 299l, 352fl, 405fl, 458fl, 511cfll, 564cffll }}