Kleismic family: Difference between revisions

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~~{{interwiki~~

~~| en = Kleismic family~~

~~| de = Hanson-Kleismisch~~

~~| es =~~

~~| ja =~~

}}

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]].

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* [[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.

==== 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.

[[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.

==== Other subgroup extensions ====

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

See [[#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}}

~~~~

Line 94:

* 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:

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* 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}}

~~~~

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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}}

~~~~

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

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

~~~~

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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 }}

~~~~

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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}}

~~~~

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: mapping generators: ~2, ~35/32

[[Optimal tuning]] ([[~~POTE~~]]): ~2 = 1200.~~000~~{{c}}, ~35/32 = 158.~~649~~{{c}}

[[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 }}

[[Badness]] (~~Smith~~): 0.~~052054~~

[[Badness]] (Sintel): 1.32

=== 11-limit ===

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Mapping: {{mapping| 1 0 1 4 2 | 0 12 10 -9 11 }}

Optimal ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~11/10 = 158.~~677~~{{c}}

Optimal tunings:

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

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

Badness (~~Smith~~): 0.~~038023~~

Badness (Sintel): 1.26

=== 13-limit ===

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Mapping: {{mapping| 1 0 1 4 2 0 | 0 12 10 -9 11 28 }}

Optimal ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~11/10 = 158.~~655~~{{c}}

Optimal tunings:

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

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

Badness (~~Smith~~): 0.~~026005~~

Badness (Sintel): 1.07

== Clyde ==

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

: mapping generators: ~2, ~9/7

: mapping generators: ~2, ~14/9

[[Optimal tuning]] ([[~~POTE~~]]): ~2 = 1200.~~000~~{{c}}, ~9/7 = ~~441~~.~~335~~{{c}}

[[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 }}

[[Minimax tuning]]:

* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~9/7 = {{monzo| 12/25 0 0 -1/25 }}

* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~14/9 = {{monzo| 13/25 0 0 1/25 }}

: {{monzo list| 1 0 0 0 | 6/25 0 0 12/25 | 6/5 0 0 2/5 | 0 0 0 1 }}

: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7

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

[[Badness]] (~~Smith~~): 0.~~047261~~

[[Badness]] (Sintel): 1.20

=== 11-limit ===

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

Mapping: {{mapping| 1 6 ~~6 12~~ -5 | 0 -12 -10 -25 23 }}

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

Optimal ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~9/7 = ~~441~~.~~355~~{{c}}

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~~, 329bd, 419bd, 503bd, 590bd~~ }}

Badness (~~Smith~~): 0.~~047417~~

Badness (Sintel): 1.57

=== 13-limit ===

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

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

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

Optimal ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~9/7 = ~~441~~.~~363~~{{c}}

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~~, 49ef~~, 68, 87~~, 503bdf, 590bdf~~ }}

Badness (~~Smith~~): 0.~~026842~~

Badness (Sintel): 1.11

== Tritikleismic ==

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

[[Optimal tuning]] ([[~~POTE~~]]): ~63/50 = 400.~~000~~{{c}}, ~6/5 = 316.~~872~~{{c}} (~21/20 = 83.~~128~~{{c}})

[[Optimal tuning]]s:

* [[WE]]: ~63/50 = 400.1845{{c}}, ~6/5 = 317.0178{{c}} (~21/20 = 83.1667{{c}})

: [[error map]]: {{val| +0.553 +0.152 -0.671 -1.017 }}

* [[CWE]]: ~63/50 = 400.0000{{c}}, ~6/5 = 316.9129{{c}} (~21/20 = 83.0871{{c}})

: error map: {{val| 0.000 -0.478 -1.749 -2.652 }}

[[Minimax tuning]]:

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: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7

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

[[Badness]] (~~Smith~~): 0.~~056337~~

[[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 740:

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Mapping: {{mapping| 3 0 3 10 8 | 0 6 5 -2 3 }}

Optimal ~~tuning~~ (~~POTE~~): ~44/35 = 400.~~000~~{{c}}, ~6/5 = 316.~~881~~{{c}} (~21/20 = 83.~~119~~{{c}})

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:

Line 749:

Line 742:

Badness (~~Smith~~): 0.~~019333~~

Badness (Sintel): 0.639

=== 13-limit ===

Line 758:

Line 751:

Mapping: {{mapping| 3 0 3 10 8 0 | 0 6 5 -2 3 14 }}

Optimal ~~tuning~~ (~~POTE~~): ~44/35 = 400.0000{{c}}, ~6/5 = 316.~~9585~~{{c}} (~21/20 = 83.~~0415~~{{c}})

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}})

Badness (~~Smith~~): 0.~~015652~~

Badness (Sintel): 0.647

=== 17-limit ===

Line 771:

Line 766:

Mapping: {{mapping| 3 0 3 10 8 0 -2 | 0 6 5 -2 3 14 18 }}

Optimal ~~tuning~~ (~~POTE~~): ~34/27 = 400.0000{{c}}, ~6/5 = 316.~~9082~~{{c}} (~21/20 = 83.~~0918~~{{c}})

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~~| 72, 159, 231f }}~~

Badness (Sintel): 0.690

~~Badness~~ (~~Smith~~)~~: 0~~.~~013551~~

== Marfifths ==

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

~~== Quadritikleismic ==~~

[[Subgroup]]: 2.3.5.7

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

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

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

: mapping generators: ~2, ~75/56

[[Optimal tuning]] ([[~~POTE~~]]): ~~~25/21~~ = ~~300~~.~~0000~~{{c}}, ~6/5 = ~~316~~.~~9999~~{{c}} (~~~126~~/~~125~~ = 16.~~9999~~{{c}})

[[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| 68, 72, 140, ~~212, 776cd, 988ccd~~, ~~1200ccd~~ }}

[[Badness]] (~~Smith~~): 0.~~039231~~

[[Badness]] (Sintel): 1.61

=== 11-limit ===

Subgroup: 2.3.5.7.11

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

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

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

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

Optimal ~~tuning (POTE)~~: ~~~25/21~~ = ~~300~~.~~0000~~{{c}}, ~6/5 = ~~316~~.~~9247~~{{c}} (~~~100~~/99 = 16.~~9247~~{{c}})

Optimal tunings:

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

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

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

Badness (~~Smith~~): 0.~~023406~~

Badness (Sintel): 1.95

=== 13-limit ===

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

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

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

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

Optimal tunings:

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

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

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

~~Optimal tuning~~ (~~POTE~~): ~~~25/21 = 300~~.~~0000{{c}}, ~6/5 = 316.9887{{c}} (~100/99 = 16.9887{{c}})~~

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).

~~Badness (Smith)~~: 0.~~018731~~

Subgroup: 2.3.5.7.11

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

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

~~Subgroup~~: ~~2.3.5.7.11.13.17~~

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

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

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

Optimal tunings:

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

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

Optimal ~~tuning (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): 2.02

~~Badness (Smith)~~: 0.~~012784~~

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

~~== Kleiboh ==~~

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

~~[[Subgroup]]~~: ~~2.3.5.7~~

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

Mapping: {{mapping| 1 -6 -4 -17 -37 -14 | 0 18 15 47 96 42 }}

Optimal tunings:

~~: mapping generators~~: ~2, ~25/21

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

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

[[Optimal ~~tuning]] ([[POTE]]): ~2~~ = ~~1200.000{{c}}~~, ~~~25/21 = 294.303{{c~~}}

~~{{Optimal ET sequence|legend=~~1~~| 49, 53, 314d }}~~

Badness (Sintel): 1.18

~~[[Badness]] (Smith): 0~~.~~076460~~

=== Marf ===

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

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

Subgroup: 2.3.5.7.11

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

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

Mapping: {{mapping| 1 6 ~~6 6~~ 14 | 0 -18 -15 -~~13 -43~~ }}

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

Optimal ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~25/21 = ~~294~~.~~181~~{{c}}

Optimal tunings:

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

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

Badness (~~Smith~~): 0.~~052805~~

Badness (Sintel): 2.48

=== 13-limit ===

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

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

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

Mapping: {{mapping| 1 6 ~~6 6~~ 14 14 | 0 -18 -15 -~~13 -43 -~~42 }}

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

Optimal ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~13/11 = ~~294~~.~~187~~{{c}}

Optimal tunings:

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

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

Badness (~~Smith~~): ~~0.031074~~

Badness (Sintel): 1.58

~~== 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.

== Kleiboh ==

[[Subgroup]]: 2.3.5.7

[[Comma list]]: ~~10976~~/~~10935~~, ~~15625~~/~~15552~~

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

: mapping generators: ~2, ~75/5

: mapping generators: ~2, ~42/25

[[Optimal tuning]] ([[~~POTE~~]]): ~2 = 1200.~~000~~{{c}}, ~75/56 = ~~505~~.~~705~~{{c}}

[[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| 19, ~~…, 121, 140, 579, 719, 859bcd, 999bcd, 1858bbccdd~~ }}

[[Badness]] (~~Smith~~): 0.~~063448~~

[[Badness]] (Sintel): 1.93

=== 11-limit ===

Subgroup: 2.3.5.7.11

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

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

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

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

Optimal ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~75/56 = ~~505~~.~~684~~{{c}}

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| 19, ~~121e~~, ~~140, 159, 299~~ }}

Badness (~~Smith~~): 0.~~058902~~

Badness (Sintel): 1.75

==== 13-limit ====

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

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

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

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

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

Optimal ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~75/56 = ~~505~~.~~686~~{{c}}

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| 19, ~~121e~~, ~~140, 159, 299~~ }}

Badness (~~Smith~~): 0.~~030082~~

Badness (Sintel): 1.28

==~~= Diatessic =~~==

== Quadritikleismic ==

~~The ''diatessic'' temperament (121 & 140) is closely related to the '''diatess tuning''' (generator~~: ~~505~~.~~727281 cents)~~.

[[Subgroup]]: 2.3.5.7

~~Subgroup~~: ~~2.3.5.7.11~~

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

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

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

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

[[Optimal tuning]]s:

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

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

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

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

Optimal ~~tuning (POTE): ~2~~ = ~~1200.000{{c}}~~, ~~~75/56 = 505.740{{c~~}}

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

~~{{Optimal ET sequence|legend=~~0~~| 19e, …, 121, 140, 261, 401 }}~~

[[Badness]] (Sintel): 0.993

~~Badness (Smith)~~: 0.~~061172~~

=== 11-limit ===

Subgroup: 2.3.5.7.11

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

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

~~Subgroup~~: ~~2.3.5.7.11.13~~

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

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

~~Mapping~~: {{~~mapping| 1 -~~6 ~~-4 -17 -37 -14 | 0 18 15 47 96 42~~ }}

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 ~~tuning (POTE): ~2~~ = ~~1200.000{{c}}~~, ~~~75/56 = 505.740{{c~~}}

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

~~{{Optimal ET sequence|legend=~~0~~| 19e, …, 121, 140, 261, 401 }}~~

Badness (Sintel): 0.774

~~Badness (Smith)~~: 0.~~028671~~

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

~~=== Marf ===~~

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

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

~~Subgroup~~: 2.3~~.5.7.11~~

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

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

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}})

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

~~Optimal tuning~~ (~~POTE~~): ~~~2 = 1200~~.~~000{{c}}, ~75/56 = 505.769{{c}}~~

Badness (Sintel): 0.774

~~{{Optimal ET sequence|legend~~=~~0| 19, 102d, 121 }}~~

=== 17-limit ===

Subgroup: 2.3.5.7.11.13.17

~~Badness (Smith)~~: ~~0.075112~~

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

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

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

~~Subgroup: 2.~~3~~.5.7.11.13~~

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

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}})

~~Mapping:~~ {{~~mapping~~| ~~1 -6 -4 -17 14 -14~~ | ~~0 18 15 47 -25 42~~ }}

~~Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~75/56 = 505.771{{c}}~~

Badness (Sintel): 0.651

~~{{Optimal ET sequence|legend=0| 19, 102df, 121 }}~~

Badness (~~Smith~~): 0.~~038317~~

== Marthirds ==

~~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~~.

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.

[[Subgroup]]: 2.3.5.7

Line 978:

Line 1,009:

: mapping generators: ~2, ~56/45

[[Optimal tuning]] ([[~~POTE~~]]): ~2 = 1200.000{{c}}, ~56/45 = 379.~~252~~{{c}}

[[Optimal tuning]]s:

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

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

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

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

[[Badness]] (~~Smith~~): 0.~~104253~~

[[Badness]] (Sintel): 2.64

=== 11-limit ===

Line 991:

Line 1,026:

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

Optimal ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~56/45 = 379.~~257~~{{c}}

Optimal tunings:

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

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

{{Optimal ET sequence|legend=0| 19e, …, 193, 212, 405, 617c~~, 1022cce~~ }}

Badness (~~Smith~~): 0.~~075624~~

Badness (Sintel): 2.50

=== 13-limit ===

Line 1,004:

Line 1,041:

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

Optimal ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~56/45 = 379.~~256~~{{c}}

Optimal tunings:

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

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

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

Badness (~~Smith~~): 0.~~043728~~

Badness (Sintel): 1.81

== ~~Quartkeenlig~~ ==

== Sqrtphi ==

~~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~~]].

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

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

[[Comma list]]: 15625/15552, 16875/16807

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

: mapping generators: ~2, 196/125

[[Optimal tuning]] ([[~~CTE~~]]): ~2 = 1200.0000{{c}}, ~36/35 = 52.~~8562~~{{c}}

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.1357{{c}}, ~196/125 = 783.4853{{c}}

: [[error map]]: {{val| +0.136 +0.163 -1.080 +0.632 }}

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

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

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

[[Badness]] (~~Smith~~): 0.~~146~~

[[Badness]] (Sintel): 1.78

=== 11-limit ===

Subgroup: 2.3.5.7.11

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

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

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

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

Optimal ~~tuning (CTE)~~: ~2 = 1200.0000{{c}}, ~33/32 = 52.~~8524~~{{c}}

Optimal tunings:

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

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

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

Badness (~~Smith~~): 0.~~0865~~

Badness (Sintel): 0.844

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

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

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

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

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

Optimal ~~tuning (CTE)~~: ~2 = 1200.0000{{c}}, ~33/32 = 52.~~8562~~{{c}}

Optimal tunings:

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

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

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

Badness (~~Smith~~): 0.~~0477~~

Badness (Sintel): 0.828

== ~~Novemkleismic~~ ==

=== 17-limit ===

[[Subgroup]]: 2.3.5.7

Subgroup: 2.3.5.7.11.13.17

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

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

Mapping: {{mapping| 1 -18 -14 -22 -22 -42 -39 | 0 30 25 38 39 70 66 }}

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

[[Optimal ~~tuning]] ([[POTE]])~~: ~~~2592/2401~~ = ~~133~~.~~333~~{{c}}, ~6/5 = ~~317~~.~~005~~{{c}} (~36/35 = 50.~~338~~{{c}})

Optimal tunings:

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

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

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

[[Badness]] (~~Smith~~): 0.~~193429~~

Badness (Sintel): 0.664

=== 11-limit ===

=== 19-limit ===

Subgroup: 2.3.5.7.11

Subgroup: 2.3.5.7.11.13.17.19

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

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

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

Mapping: {{mapping| 1 -18 -14 -22 -22 -42 -39 16 | 0 30 25 38 39 70 66 -18 }}

Optimal ~~tuning (POTE)~~: ~~~250/231~~ = ~~133~~.~~333~~{{c}}, ~6/5 = ~~317~~.~~010~~{{c}} (~36/35 = 50.~~343~~{{c}})

Optimal tunings:

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

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

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

Badness (~~Smith~~): 0.~~051730~~

Badness (Sintel): 0.897

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

== Quartkeenlig ==

~~Subgroup: 2.3~~.5.~~7.11~~.13

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]].

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

[[Subgroup]]: 2.3.5.7

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

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

~~Optimal tuning (POTE): ~250/231 = 133.333~~{{c}}, ~~~6/5 = 317.086{{c}} (~~~36/35 ~~= 50.419{{c}})~~

: mapping generator: ~2, ~36/35

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

[[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 (Smith): 0.039072~~

~~== Sqrtphi ==~~

[[Badness]] (Sintel): 3.69

~~{{Main| Sqrtphi }}~~

~~The just value of sqrt (φ) is 416~~.~~545 cents~~.

=== 11-limit ===

Subgroup: 2.3.5.7.11

~~[[Subgroup]]~~: ~~2.3.5.7~~

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

~~[[Comma list]]~~: ~~15625/15552, 16875/16807~~

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

Optimal tunings:

~~: mapping generators~~: ~2, ~~125~~/98

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

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

[[Optimal ~~tuning]] ([[POTE]]): ~2~~ = ~~1200.000{{c}}~~, ~~~125/98 = 416.603{{c~~}}

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

Badness (Sintel): 2.86

~~[[Badness]] (Smith)~~: 0.~~070378~~

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

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

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

~~Subgroup~~: ~~2.3.5.7.11~~

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

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

~~Mapping~~: {{~~mapping| 1 12 11 16 17 | 0 -30 -25 -38 -39~~ }}

Optimal tunings:

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

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

Optimal ~~tuning (POTE): ~2~~ = ~~1200.000{{c}}~~, ~~~14/11 = 416.604{{c~~}}

~~{{Optimal ET sequence|legend=0| 49, 72, 193, 265 }}~~

Badness (Sintel): 1.97

~~Badness (Smith)~~: 0.~~025515~~

== Novemkleismic ==

[[Subgroup]]: 2.3.5.7

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

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

~~Subgroup~~: ~~2.3.5.7.11.13~~

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

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

~~Mapping~~: {{~~mapping~~| ~~1 12 11 16 17 28~~ | 0 -30 -~~25 -38 -39 -70~~ }}

[[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 ~~tuning (POTE): ~2~~ = ~~1200.000{{c}}~~, ~~~14/11 = 416.585{{c~~}}

~~{{Optimal ET sequence|legend=0| 49f, 72, 121, 193 }}~~

[[Badness]] (Sintel): 4.90

~~Badness (Smith)~~: 0.~~020040~~

=== 11-limit ===

Subgroup: 2.3.5.7.11

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

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

~~Subgroup~~: ~~2.3.5.7.11.13.17~~

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

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

~~Mapping~~: {{~~mapping| 1 12 11 16 17 28 27 | 0 -30 -25 -38 -39 -70 -66~~ }}

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 ~~tuning (POTE): ~2~~ = ~~1200.000{{c}}~~, ~~~14/11 = 416.585{{c~~}}

~~{{Optimal ET sequence|legend=0| 49fg, 72, 121, 193 }}~~

Badness (Sintel): 1.71

~~Badness (Smith)~~: 0.~~013028~~

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

~~=== 19-limit ===~~

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

~~Subgroup~~: ~~2.3.5.7.11.13.17.19~~

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

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

~~Mapping~~: {{~~mapping| 1 12 11 16 17 28 27 -2 | 0 -30 -25 -38 -39 -70 -66 18~~ }}

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 ~~tuning (POTE): ~2~~ = ~~1200.000{{c}}~~, ~~~14/11 = 416.580{{c~~}}

~~{{Optimal ET sequence|legend=0| 49fg, 72, 121, 193 }}~~

Badness (Sintel): 1.61

~~Badness (Smith): 0~~.~~014748~~

== Subgroup extensions ==

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

~~== 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~~]].

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]])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)/([[6/5|12/10]])2, hence explaining the mapping of the entire 2.3.5.13 subgroup.

Subgroup: 2.3.5.13

Line 1,173:

Line 1,244:

Optimal tunings:

* ~~CTE~~: ~2 = 1200.~~0000~~{{c}}, ~6/5 = 317.~~1110~~{{c}}

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

* ~~POTE~~: ~2 = 1200.0000{{c}}, ~6/5 = 317.~~0756~~{{c}}

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

Badness (Sintel): 0.131

==== 2.3.5.13.37~~.41~~ subgroup ====

==== 2.3.5.13.37 subgroup ====

Hanson can be extended even further to the 2.3.5.13.37.41 subgroup while maintaining a rather low complexity and high accuracy.

Subgroup: 2.3.5.13.37

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

Subgroup-val mapping: {{mapping| 1 0 1 0 6 | 0 6 5 14 -3 }}

Optimal tunings:

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

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

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

Badness (Sintel): 0.167

==== 2.3.5.13.37.41 subgroup ====

Subgroup: 2.3.5.13.37.41

Comma list: 325/324, 625/624, ~~[[481/480]], [[1600~~/~~1599]]~~

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

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

Optimal tunings:

* WE: ~2 = 1200.~~165~~{{c}}, ~6/5 = 317.~~113~~{{c}}

* WE: ~2 = 1200.1651{{c}}, ~6/5 = 317.1126{{c}}

* CWE: ~2 = 1200.~~000~~{{c}}, ~6/5 = 317.~~075~~{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0748{{c}}

{{Optimal ET sequence|legend=0| 15, 19, 34, 53, 140, 193, 246l }}

Badness (Sintel): 0.223

[[Category:Kleismic family| ]]

[[Category:Temperament families]]

[[Category:~~Kleismic family| ]] <!~~-~~- main article -->~~

[[Category:Catalogs of rank-2 temperaments]]

~~[[Category:Rank~~ 2]]

[[Category:Listen]]

@@ Line 1: / Line 1: @@
-{{interwiki
-| en = Kleismic family
-| de = Hanson-Kleismisch
-| es =
-| ja =
-}}
 {{Technical data page}}
 The [[5-limit]] parent comma for the '''kleismic family''' is [[15625/15552]], the kleisma, which is the amount by which a stack of six [[6/5|classical minor third]]s falls short of the [[3/1|3rd]] [[harmonic]].
@@ Line 25: / Line 19: @@
 * [[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 29: @@
 === 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.
+==== 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.
-[[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.
+==== Other subgroup extensions ====
+The main extension of note is a very remarkable extension to the [[2.3.5.13 subgroup]], as the hemitwelfth, reached by three generator steps, can be interpreted as [[26/15]]. Notice 15625/15552 = ([[325/324]])⋅([[625/624]]) and 325/324 = (625/624)⋅([[676/675]]). The [[S-expression]]-based comma list of the temperament is {{nowrap| {[[325/324|S10/S12 = S25⋅S26]], ([[625/624|S25]],) [[676/675|S13/S15 = S26]]} }}.
+See [[#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 60: @@
 * [[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 70: @@
 ==== 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 79: @@
 * 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 94: @@
 * 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 113: @@
 * 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 132: @@
 * 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 147: @@
 * 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 162: @@
 * 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 177: @@
 * 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 192: @@
 * 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 207: @@
 * 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 223: @@
 * 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 238: @@
 * 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 253: @@
 * 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 268: @@
 * 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 287: @@
 * [[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 307: @@
 * 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 326: @@
 * 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 345: @@
 * 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 365: @@
 * 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 380: @@
 * 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 395: @@
 * 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 410: @@
 * 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 425: @@
 * 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 442: @@
 * [[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 461: @@
 * 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 480: @@
 * 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 497: @@
 * [[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 516: @@
 * 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 535: @@
 * 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 557: @@
 * [[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 572: @@
 * 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 587: @@
 * 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 628: / Line 598: @@
 {{Mapping|legend=1| 1 0 1 4 | 0 12 10 -9 }}
+: mapping generators: ~2, ~35/32
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~35/32 = 158.649{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.3950{{c}}, ~35/32 = 158.5686{{c}}
+: [[error map]]: {{val| -0.605 +0.868 -1.233 +1.637 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~35/32 = 158.6338{{c}}
+: error map: {{val| 0.000 +1.651 +0.024 +3.470 }}
-{{Optimal ET sequence|legend=1| 15, 38, 53, 121 }}
+{{Optimal ET sequence|legend=1| 15, 38, 53, 121, 174d, 295d }}
-[[Badness]] (Smith): 0.052054
+[[Badness]] (Sintel): 1.32
 === 11-limit ===
@@ Line 642: / Line 617: @@
 Mapping: {{mapping| 1 0 1 4 2 | 0 12 10 -9 11 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~11/10 = 158.677{{c}}
+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 (Smith): 0.038023
+Badness (Sintel): 1.26
 === 13-limit ===
@@ Line 655: / Line 632: @@
 Mapping: {{mapping| 1 0 1 4 2 0 | 0 12 10 -9 11 28 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~11/10 = 158.655{{c}}
+Optimal tunings:
+* WE: ~2 = 1199.7952{{c}}, ~11/10 = 158.6279{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 158.6493{{c}}
 {{Optimal ET sequence|legend=0| 15, 38f, 53, 121e }}
-Badness (Smith): 0.026005
+Badness (Sintel): 1.07
 == Clyde ==
@@ Line 666: / Line 645: @@
 [[Comma list]]: 245/243, 3136/3125
-{{Mapping|legend=1| 1 6 6 12 | 0 -12 -10 -25 }}
+{{Mapping|legend=1| 1 -6 -4 -13 | 0 12 10 25 }}
-: mapping generators: ~2, ~9/7
+: mapping generators: ~2, ~14/9
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~9/7 = 441.335{{c}}
+[[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 }}
 [[Minimax tuning]]:
-* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~9/7 = {{monzo| 12/25 0 0 -1/25 }}
+* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~14/9 = {{monzo| 13/25 0 0 1/25 }}
 : {{monzo list| 1 0 0 0 | 6/25 0 0 12/25 | 6/5 0 0 2/5 | 0 0 0 1 }}
 : [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7
@@ Line 678: / Line 661: @@
 [[Algebraic generator]]: real root of 5''x''<sup>3</sup> - 6''x'' - 3, the Poussami generator. Approximately 441.309 [[cent]]s. Associated recurrence relationship quickly converges.
-{{Optimal ET sequence|legend=1| 19, 49, 68, 87, 155 }}
+{{Optimal ET sequence|legend=1| 19, 49, 68, 87, 155, 242 }}
-[[Badness]] (Smith): 0.047261
+[[Badness]] (Sintel): 1.20
 === 11-limit ===
@@ Line 687: / Line 670: @@
 Comma list: 245/243, 385/384, 3136/3125
-Mapping: {{mapping| 1 6 6 12 -5 | 0 -12 -10 -25 23 }}
+Mapping: {{mapping| 1 -6 -4 -13 18 | 0 12 10 25 -23 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~9/7 = 441.355{{c}}
+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, 329bd, 419bd, 503bd, 590bd }}
+{{Optimal ET sequence|legend=0| 19, 49e, 68, 87 }}
-Badness (Smith): 0.047417
+Badness (Sintel): 1.57
 === 13-limit ===
@@ Line 700: / Line 685: @@
 Comma list: 196/195, 245/243, 385/384, 625/624
-Mapping: {{mapping| 1 6 6 12 -5 14 | 0 -12 -10 -25 23 -28 }}
+Mapping: {{mapping| 1 -6 -4 -13 18 -14 | 0 12 10 25 -23 28 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~9/7 = 441.363{{c}}
+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, 49ef, 68, 87, 503bdf, 590bdf }}
+{{Optimal ET sequence|legend=0| 19, 68, 87 }}
-Badness (Smith): 0.026842
+Badness (Sintel): 1.11
 == Tritikleismic ==
@@ Line 716: / Line 703: @@
 : mapping generators: ~63/50, ~6/5
-[[Optimal tuning]] ([[POTE]]): ~63/50 = 400.000{{c}}, ~6/5 = 316.872{{c}} (~21/20 = 83.128{{c}})
+[[Optimal tuning]]s:
+* [[WE]]: ~63/50 = 400.1845{{c}}, ~6/5 = 317.0178{{c}} (~21/20 = 83.1667{{c}})
+: [[error map]]: {{val| +0.553 +0.152 -0.671 -1.017 }}
+* [[CWE]]: ~63/50 = 400.0000{{c}}, ~6/5 = 316.9129{{c}} (~21/20 = 83.0871{{c}})
+: error map: {{val| 0.000 -0.478 -1.749 -2.652 }}
 [[Minimax tuning]]:
@@ Line 726: / Line 717: @@
 : [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7
-{{Optimal ET sequence|legend=1| 15, 42bc, 57, 72, 159, 231 }}
+{{Optimal ET sequence|legend=1| 15, 42bc, 57, 72, 159, 231, 765ccddd }}
-[[Badness]] (Smith): 0.056337
+[[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 740: / Line 731: @@
 Mapping: {{mapping| 3 0 3 10 8 | 0 6 5 -2 3 }}
-Optimal tuning (POTE): ~44/35 = 400.000{{c}}, ~6/5 = 316.881{{c}} (~21/20 = 83.119{{c}})
+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:
@@ Line 749: / Line 742: @@
 {{Optimal ET sequence|legend=0| 15, 42bc, 57, 72, 159, 231 }}
-Badness (Smith): 0.019333
+Badness (Sintel): 0.639
 === 13-limit ===
@@ Line 758: / Line 751: @@
 Mapping: {{mapping| 3 0 3 10 8 0 | 0 6 5 -2 3 14 }}
-Optimal tuning (POTE): ~44/35 = 400.0000{{c}}, ~6/5 = 316.9585{{c}} (~21/20 = 83.0415{{c}})
+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| 72, 87, 159 }}
+{{Optimal ET sequence|legend=0| 15, 57f, 72, 87, 159 }}
-Badness (Smith): 0.015652
+Badness (Sintel): 0.647
 === 17-limit ===
@@ Line 771: / Line 766: @@
 Mapping: {{mapping| 3 0 3 10 8 0 -2 | 0 6 5 -2 3 14 18 }}
-Optimal tuning (POTE): ~34/27 = 400.0000{{c}}, ~6/5 = 316.9082{{c}} (~21/20 = 83.0918{{c}})
+Optimal tunings:
+* WE: ~34/27 = 400.1604{{c}}, ~6/5 = 317.0353{{c}} (~21/20 = 83.1251{{c}})
+* CWE: ~34/27 = 400.0000{{c}}, ~6/5 = 316.9384{{c}} (~21/20 = 83.0616{{c}})
+{{Optimal ET sequence|legend=0| 15g, 57fg, 72, 159, 231f }}
-{{Optimal ET sequence|legend=0| 72, 159, 231f }}
+Badness (Sintel): 0.690
-Badness (Smith): 0.013551
+== Marfifths ==
+Named by [[Xenllium]] in 2021, marfifths tempers out the 10976/10935, the [[hemimage comma]], and may be described as the {{nowrap| 19 & 140 }} temperament. It is generated by a marvel fourth of [[75/56]] (or a marvel fifth of [[112/75]]), three of which minus an octave make the hanson generator of ~6/5. Its [[ploidacot]] is zeta-18-cot.
-== Quadritikleismic ==
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 2401/2400, 15625/15552
+[[Comma list]]: 10976/10935, 15625/15552
-{{Mapping|legend=1| 4 0 4 7 | 0 6 5 4 }}
+{{Mapping|legend=1| 1 -6 -4 -17 | 0 18 15 47 }}
-: mapping generators: ~25/21, ~6/5
+: mapping generators: ~2, ~75/56
-[[Optimal tuning]] ([[POTE]]): ~25/21 = 300.0000{{c}}, ~6/5 = 316.9999{{c}} (~126/125 = 16.9999{{c}})
+[[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| 68, 72, 140, 212, 776cd, 988ccd, 1200ccd }}
+{{Optimal ET sequence|legend=1| 19, …, 121, 140, 579, 719 }}
-[[Badness]] (Smith): 0.039231
+[[Badness]] (Sintel): 1.61
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 385/384, 1375/1372, 6250/6237
+Comma list: 385/384, 6250/6237, 10976/10935
-Mapping: {{mapping| 4 0 4 7 17 | 0 6 5 4 -3 }}
+Mapping: {{mapping| 1 -6 -4 -17 22 | 0 18 15 47 -44 }}
-Optimal tuning (POTE): ~25/21 = 300.0000{{c}}, ~6/5 = 316.9247{{c}} (~100/99 = 16.9247{{c}})
+Optimal tunings:
+* WE: ~2 = 1200.2484{{c}}, ~75/56 = 505.7882{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~75/56 = 505.6853{{c}}
-{{Optimal ET sequence|legend=0| 68, 72, 140, 212, 284, 496ce, 780ccdee }}
+{{Optimal ET sequence|legend=0| 19, 121e, 140, 159, 299 }}
-Badness (Smith): 0.023406
+Badness (Sintel): 1.95
-=== 13-limit ===
+==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 325/324, 385/384, 625/624, 1375/1372
+Comma list: 325/324, 385/384, 625/624, 10976/10935
+Mapping: {{mapping| 1 -6 -4 -17 22 -14 | 0 18 15 47 -44 42 }}
+Optimal tunings:
+* WE: ~2 = 1200.2747{{c}}, ~75/56 = 505.8019{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~75/56 = 505.6883{{c}}
-Mapping: {{mapping| 4 0 4 7 17 0 | 0 6 5 4 -3 14 }}
+{{Optimal ET sequence|legend=0| 19, 121e, 140, 159, 299 }}
-Optimal tuning (POTE): ~25/21 = 300.0000{{c}}, ~6/5 = 316.9887{{c}} (~100/99 = 16.9887{{c}})
+Badness (Sintel): 1.24
-{{Optimal ET sequence|legend=0| 68, 72, 140, 212 }}
+=== Diatessic ===
+Diatessic may be described as {{nowrap| 121 & 140 }} and is closely related to the Diatess tuning (generator: 505.727281 cents).
-Badness (Smith): 0.018731
+Subgroup: 2.3.5.7.11
-=== 17-limit ===
+Comma list: 1375/1372, 2200/2187, 5632/5625
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 289/288, 325/324, 385/384, 442/441, 625/624
+Mapping: {{mapping| 1 -6 -4 -17 -37 | 0 18 15 47 96 }}
-Mapping: {{mapping| 4 0 4 7 17 0 10 | 0 6 5 4 -3 14 6 }}
+Optimal tunings:
+* WE: ~2 = 1199.7886{{c}}, ~75/56 = 505.6513{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~75/56 = 505.7366{{c}}
-Optimal tuning (POTE): ~25/21 = 300.0000{{c}}, ~6/5 = 316.9846{{c}} (~100/99 = 16.9846{{c}})
+{{Optimal ET sequence|legend=0| 19e, …, 121, 140, 261, 401 }}
-{{Optimal ET sequence|legend=0| 68, 72, 140, 212g }}
+Badness (Sintel): 2.02
-Badness (Smith): 0.012784
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
-== Kleiboh ==
+Comma list: 325/324, 352/351, 625/624, 1375/1372
-[[Subgroup]]: 2.3.5.7
-[[Comma list]]: 1728/1715, 3125/3087
+Mapping: {{mapping| 1 -6 -4 -17 -37 -14 | 0 18 15 47 96 42 }}
-{{Mapping|legend=1| 1 6 6 6 | 0 -18 -15 -13 }}
+Optimal tunings:
-: mapping generators: ~2, ~25/21
+* WE: ~2 = 1199.7996{{c}}, ~75/56 = 505.6558{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~75/56 = 505.7366{{c}}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~25/21 = 294.303{{c}}
+{{Optimal ET sequence|legend=0| 19e, …, 121, 140, 261, 401 }}
-{{Optimal ET sequence|legend=1| 49, 53, 314d }}
+Badness (Sintel): 1.18
-[[Badness]] (Smith): 0.076460
+=== Marf ===
+Marf may be described as {{nowrap| 19 & 121 }}. It has a POTE generator which strongly approximates the marvelous fifth interval of 112/75.
-=== 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 176/175, 540/539, 3125/3087
+Comma list: 540/539, 896/891, 15625/15552
-Mapping: {{mapping| 1 6 6 6 14 | 0 -18 -15 -13 -43 }}
+Mapping: {{mapping| 1 -6 -4 -17 14 | 0 18 15 47 -25 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~25/21 = 294.181{{c}}
+Optimal tunings:
+* WE: ~2 = 1199.3198{{c}}, ~75/56 = 505.4822{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~75/56 = 505.7607{{c}}
-{{Optimal ET sequence|legend=0| 49, 53, 102d, 155d }}
+{{Optimal ET sequence|legend=0| 19, 102d, 121 }}
-Badness (Smith): 0.052805
+Badness (Sintel): 2.48
-=== 13-limit ===
+==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 176/175, 275/273, 325/324, 540/539
+Comma list: 325/324, 540/539, 625/624, 896/891
-Mapping: {{mapping| 1 6 6 6 14 14 | 0 -18 -15 -13 -43 -42 }}
+Mapping: {{mapping| 1 -6 -4 -17 14 -14 | 0 18 15 47 -25 42 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~13/11 = 294.187{{c}}
+Optimal tunings:
+* WE: ~2 = 1199.3368{{c}}, ~75/56 = 505.4919{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~75/56 = 505.7627{{c}}
-{{Optimal ET sequence|legend=0| 49f, 53, 102df, 155d }}
+{{Optimal ET sequence|legend=0| 19, 102df, 121 }}
-Badness (Smith): 0.031074
+Badness (Sintel): 1.58
-== 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.
+== Kleiboh ==
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 10976/10935, 15625/15552
+[[Comma list]]: 1728/1715, 3125/3087
-{{Mapping|legend=1| 1 -6 -4 -17 | 0 18 15 47 }}
+{{Mapping|legend=1| 1 -12 -9 -7 | 0 18 15 13 }}
-: mapping generators: ~2, ~75/5
+: mapping generators: ~2, ~42/25
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~75/56 = 505.705{{c}}
+[[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| 19, …, 121, 140, 579, 719, 859bcd, 999bcd, 1858bbccdd }}
+{{Optimal ET sequence|legend=1| 49, 53 }}
-[[Badness]] (Smith): 0.063448
+[[Badness]] (Sintel): 1.93
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 385/384, 6250/6237, 10976/10935
+Comma list: 176/175, 540/539, 3125/3087
-Mapping: {{mapping| 1 -6 -4 -17 22 | 0 18 15 47 -44 }}
+Mapping: {{mapping| 1 -12 -9 -7 -29 | 0 18 15 13 43 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~75/56 = 505.684{{c}}
+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| 19, 121e, 140, 159, 299 }}
+{{Optimal ET sequence|legend=0| 49, 53, 102d }}
-Badness (Smith): 0.058902
+Badness (Sintel): 1.75
-==== 13-limit ====
+=== 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 325/324, 385/384, 625/624, 10976/10935
+Comma list: 176/175, 275/273, 325/324, 540/539
-Mapping: {{mapping| 1 -6 -4 -17 22 -14 | 0 18 15 47 -44 42 }}
+Mapping: {{mapping| 1 -12 -9 -7 -29 -28 | 0 18 15 13 43 42 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~75/56 = 505.686{{c}}
+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| 19, 121e, 140, 159, 299 }}
+{{Optimal ET sequence|legend=0| 49f, 53, 102df }}
-Badness (Smith): 0.030082
+Badness (Sintel): 1.28
-=== Diatessic ===
+== Quadritikleismic ==
-The ''diatessic'' temperament (121 & 140) is closely related to the '''diatess tuning''' (generator: 505.727281 cents).
+[[Subgroup]]: 2.3.5.7
-Subgroup: 2.3.5.7.11
+[[Comma list]]: 2401/2400, 15625/15552
-Comma list: 1375/1372, 2200/2187, 5632/5625
+{{Mapping|legend=1| 4 0 4 7 | 0 6 5 4 }}
+: mapping generators: ~25/21, ~6/5
-Mapping: {{mapping| 1 -6 -4 -17 -37 | 0 18 15 47 96 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~25/21 = 300.0520{{c}}, ~6/5 = 317.0548{{c}} (~126/125 = 17.0029{{c}})
+: [[error map]]: {{val| +0.208 +0.374 -0.832 -0.243 }}
+* [[CWE]]: ~25/21 = 300.0000{{c}}, ~6/5 = 317.0301{{c}} (~126/125 = 17.0301{{c}})
+: error map: {{val| 0.000 +0.225 -1.163 -0.706 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~75/56 = 505.740{{c}}
+{{Optimal ET sequence|legend=1| 68, 72, 140, 212, 776cd, 988ccd, 1200ccd }}
-{{Optimal ET sequence|legend=0| 19e, …, 121, 140, 261, 401 }}
+[[Badness]] (Sintel): 0.993
-Badness (Smith): 0.061172
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-==== 13-limit ====
+Comma list: 385/384, 1375/1372, 6250/6237
-Subgroup: 2.3.5.7.11.13
-Comma list: 325/324, 352/351, 625/624, 1375/1372
+Mapping: {{mapping| 4 0 4 7 17 | 0 6 5 4 -3 }}
-Mapping: {{mapping| 1 -6 -4 -17 -37 -14 | 0 18 15 47 96 42 }}
+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 tuning (POTE): ~2 = 1200.000{{c}}, ~75/56 = 505.740{{c}}
+{{Optimal ET sequence|legend=0| 68, 72, 140, 212, 284, 496ce, 780ccdee }}
-{{Optimal ET sequence|legend=0| 19e, …, 121, 140, 261, 401 }}
+Badness (Sintel): 0.774
-Badness (Smith): 0.028671
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
-=== Marf ===
+Comma list: 325/324, 385/384, 625/624, 1375/1372
-The ''marf'' temperament (19 & 121) has a POTE generator which strongly approximates the marvelous fifth interval of 112/75.
-Subgroup: 2.3.5.7.11
+Mapping: {{mapping| 4 0 4 7 17 0 | 0 6 5 4 -3 14 }}
-Comma list: 540/539, 896/891, 15625/15552
+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}})
-Mapping: {{mapping| 1 -6 -4 -17 14 | 0 18 15 47 -25 }}
+{{Optimal ET sequence|legend=0| 68, 72, 140, 212 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~75/56 = 505.769{{c}}
+Badness (Sintel): 0.774
-{{Optimal ET sequence|legend=0| 19, 102d, 121 }}
+=== 17-limit ===
+Subgroup: 2.3.5.7.11.13.17
-Badness (Smith): 0.075112
+Comma list: 289/288, 325/324, 385/384, 442/441, 625/624
-==== 13-limit ====
+Mapping: {{mapping| 4 0 4 7 17 0 10 | 0 6 5 4 -3 14 6 }}
-Subgroup: 2.3.5.7.11.13
-Comma list: 325/324, 540/539, 625/624, 896/891
+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}})
-Mapping: {{mapping| 1 -6 -4 -17 14 -14 | 0 18 15 47 -25 42 }}
+{{Optimal ET sequence|legend=0| 68, 72, 140, 212g }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~75/56 = 505.771{{c}}
+Badness (Sintel): 0.651
-{{Optimal ET sequence|legend=0| 19, 102df, 121 }}
-Badness (Smith): 0.038317
 == Marthirds ==
-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.
+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.
 [[Subgroup]]: 2.3.5.7
@@ Line 978: / Line 1,009: @@
 : mapping generators: ~2, ~56/45
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~56/45 = 379.252{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.1662{{c}}, ~56/45 = 379.3041{{c}}
+: [[error map]]: {{val| +0.166 +0.347 -0.896 +0.000 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~56/45 = 379.2552{{c}}
+: error map: {{val| 0.000 +0.171 -1.209 -0.214 }}
 {{Optimal ET sequence|legend=1| 19, …, 193, 212, 617c, 829c }}
-[[Badness]] (Smith): 0.104253
+[[Badness]] (Sintel): 2.64
 === 11-limit ===
@@ Line 991: / Line 1,026: @@
 Mapping: {{mapping| 1 -6 -4 -19 -43 | 0 24 20 69 147 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~56/45 = 379.257{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.1189{{c}}, ~56/45 = 379.2942{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~56/45 = 379.2580{{c}}
-{{Optimal ET sequence|legend=0| 19e, …, 193, 212, 405, 617c, 1022cce }}
+{{Optimal ET sequence|legend=0| 19e, …, 193, 212, 405, 617c }}
-Badness (Smith): 0.075624
+Badness (Sintel): 2.50
 === 13-limit ===
@@ Line 1,004: / Line 1,041: @@
 Mapping: {{mapping| 1 -6 -4 -19 -43 -14 | 0 24 20 69 147 56 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~56/45 = 379.256{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.2154{{c}}, ~56/45 = 379.3236{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~56/45 = 379.2580{{c}}
 {{Optimal ET sequence|legend=0| 19e, …, 193, 212, 405f, 617cff }}
-Badness (Smith): 0.043728
+Badness (Sintel): 1.81
-== Quartkeenlig ==
+== Sqrtphi ==
-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]].
+{{Main| Sqrtphi }}
+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
-[[Comma list]]: 15625/15552, 117649/116640
+[[Comma list]]: 15625/15552, 16875/16807
-{{Mapping|legend=1| 1 0 1 1 | 0 36 30 41 }}
+{{Mapping|legend=1| 1 -18 -14 -22 | 0 30 25 38 }}
-: mapping generator: ~2, ~36/35
+: mapping generators: ~2, 196/125
-[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~36/35 = 52.8562{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.1357{{c}}, ~196/125 = 783.4853{{c}}
+: [[error map]]: {{val| +0.136 +0.163 -1.080 +0.632 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~196/125 = 783.4009{{c}}
+: error map: {{val| 0.000 +0.072 -1.291 +0.408 }}
-{{Optimal ET sequence|legend=1| 68, 91, 159, 386d, 545dd }}
+{{Optimal ET sequence|legend=1| 23d, 49, 72, 193, 265 }}
-[[Badness]] (Smith): 0.146
+[[Badness]] (Sintel): 1.78
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 385/384, 6250/6237, 67228/66825
+Comma list: 540/539, 1375/1372, 4375/4356
-Mapping: {{mapping| 1 0 1 1 5 | 0 36 30 41 -35 }}
+Mapping: {{mapping| 1 -18 -14 -22 -22 | 0 30 25 38 39 }}
-Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~33/32 = 52.8524{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.0514{{c}}, ~11/7 = 783.4294{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 783.3975{{c}}
-{{Optimal ET sequence|legend=0| 68, 91, 159, 386d, 545dd }}
+{{Optimal ET sequence|legend=0| 23de, 49, 72, 193, 265 }}
-Badness (Smith): 0.0865
+Badness (Sintel): 0.844
 === 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 325/324, 385/384, 625/624, 16807/16731
+Comma list: 325/324, 364/363, 625/624, 1375/1372
-Mapping: {{mapping| 1 0 1 1 5 0 | 0 36 30 41 -35 84 }}
+Mapping: {{mapping| 1 -18 -14 -22 -22 -42 | 0 30 25 38 39 70 }}
-Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~33/32 = 52.8562{{c}}
+Optimal tunings:
+* WE: ~2 = 1199.9314{{c}}, ~11/7 = 783.3705{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 783.4134{{c}}
-{{Optimal ET sequence|legend=0| 68, 159, 386d, 545ddf }}
+{{Optimal ET sequence|legend=0| 23deff, 49f, 72, 121, 193 }}
-Badness (Smith): 0.0477
+Badness (Sintel): 0.828
-== Novemkleismic ==
+=== 17-limit ===
-[[Subgroup]]: 2.3.5.7
+Subgroup: 2.3.5.7.11.13.17
-[[Comma list]]: 15625/15552, 40353607/40310784
+Comma list: 325/324, 364/363, 375/374, 540/539, 595/594
-{{Mapping|legend=1| 9 0 9 11 | 0 6 5 6 }}
+Mapping: {{mapping| 1 -18 -14 -22 -22 -42 -39 | 0 30 25 38 39 70 66 }}
-: mapping generators: ~2592/2401, ~6/5
-[[Optimal tuning]] ([[POTE]]): ~2592/2401 = 133.333{{c}}, ~6/5 = 317.005{{c}} (~36/35 = 50.338{{c}})
+Optimal tunings:
+* WE: ~2 = 1199.9324{{c}}, ~11/7 = 783.3706{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 783.4129{{c}}
-{{Optimal ET sequence|legend=1| 72, 261, 333, 405, 477c, 882c }}
+{{Optimal ET sequence|legend=0| 23deffgg, 49fg, 72, 121, 193 }}
-[[Badness]] (Smith): 0.193429
+Badness (Sintel): 0.664
-=== 11-limit ===
+=== 19-limit ===
-Subgroup: 2.3.5.7.11
+Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 1375/1372, 4000/3993, 15625/15552
+Comma list: 325/324, 364/363, 375/374, 400/399, 442/441, 595/594
-Mapping: {{mapping| 9 0 9 11 24 | 0 6 5 6 3 }}
+Mapping: {{mapping| 1 -18 -14 -22 -22 -42 -39 16 | 0 30 25 38 39 70 66 -18 }}
-Optimal tuning (POTE): ~250/231 = 133.333{{c}}, ~6/5 = 317.010{{c}} (~36/35 = 50.343{{c}})
+Optimal tunings:
+* WE: ~2 = 1199.8567{{c}}, ~11/7 = 783.3262{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 783.4176{{c}}
-{{Optimal ET sequence|legend=0| 72, 261, 333, 405, 882c }}
+{{Optimal ET sequence|legend=0| 49fg, 72, 121, 193 }}
-Badness (Smith): 0.051730
+Badness (Sintel): 0.897
-=== 13-limit ===
+== Quartkeenlig ==
-Subgroup: 2.3.5.7.11.13
+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]].
-Comma list: 325/324, 625/624, 1375/1372, 4000/3993
+[[Subgroup]]: 2.3.5.7
-Mapping: {{mapping| 9 0 9 11 24 0 | 0 6 5 6 3 14 }}
+[[Comma list]]: 15625/15552, 117649/116640
-Optimal tuning (POTE): ~250/231 = 133.333{{c}}, ~6/5 = 317.086{{c}} (~36/35 = 50.419{{c}})
+{{Mapping|legend=1| 1 0 1 1 | 0 36 30 41 }}
+: mapping generator: ~2, ~36/35
-{{Optimal ET sequence|legend=0| 72, 189f, 261, 333, 738cf }}
+[[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 (Smith): 0.039072
+{{Optimal ET sequence|legend=1| 68, 91, 159, 386d, 545dd }}
-== Sqrtphi ==
+[[Badness]] (Sintel): 3.69
-{{Main| Sqrtphi }}
-The just value of sqrt (φ) is 416.545 cents.
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-[[Subgroup]]: 2.3.5.7
+Comma list: 385/384, 6250/6237, 67228/66825
-[[Comma list]]: 15625/15552, 16875/16807
+Mapping: {{mapping| 1 0 1 1 5 | 0 36 30 41 -35 }}
-{{Mapping|legend=1| 1 12 11 16 | 0 -30 -25 -38 }}
+Optimal tunings:
-: mapping generators: ~2, 125/98
+* WE: ~2 = 1200.2526{{c}}, ~36/35 = 52.8534{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 52.8446{{c}}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~125/98 = 416.603{{c}}
+{{Optimal ET sequence|legend=0| 68, 91, 159, 386d, 545dd }}
-{{Optimal ET sequence|legend=1| 49, 72, 193, 265 }}
+Badness (Sintel): 2.86
-[[Badness]] (Smith): 0.070378
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
-=== 11-limit ===
+Comma list: 325/324, 385/384, 625/624, 16807/16731
-Subgroup: 2.3.5.7.11
-Comma list: 540/539, 1375/1372, 4375/4356
+Mapping: {{mapping| 1 0 1 1 5 0 | 0 36 30 41 -35 84 }}
-Mapping: {{mapping| 1 12 11 16 17 | 0 -30 -25 -38 -39 }}
+Optimal tunings:
+* WE: ~2 = 1200.2564{{c}}, ~36/35 = 52.8568{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 52.8479{{c}}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~14/11 = 416.604{{c}}
+{{Optimal ET sequence|legend=0| 68, 159, 386d, 545ddf }}
-{{Optimal ET sequence|legend=0| 49, 72, 193, 265 }}
+Badness (Sintel): 1.97
-Badness (Smith): 0.025515
+== Novemkleismic ==
+[[Subgroup]]: 2.3.5.7
-=== 13-limit ===
+[[Comma list]]: 15625/15552, 40353607/40310784
-Subgroup: 2.3.5.7.11.13
-Comma list: 325/324, 364/363, 625/624, 1375/1372
+{{Mapping|legend=1| 9 0 9 11 | 0 6 5 6 }}
+: mapping generators: ~2592/2401, ~6/5
-Mapping: {{mapping| 1 12 11 16 17 28 | 0 -30 -25 -38 -39 -70 }}
+[[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 tuning (POTE): ~2 = 1200.000{{c}}, ~14/11 = 416.585{{c}}
+{{Optimal ET sequence|legend=1| 72, 261, 333, 405, 477c, 882c }}
-{{Optimal ET sequence|legend=0| 49f, 72, 121, 193 }}
+[[Badness]] (Sintel): 4.90
-Badness (Smith): 0.020040
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-=== 17-limit ===
+Comma list: 1375/1372, 4000/3993, 15625/15552
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 325/324, 364/363, 375/374, 540/539, 595/594
+Mapping: {{mapping| 9 0 9 11 24 | 0 6 5 6 3 }}
-Mapping: {{mapping| 1 12 11 16 17 28 27 | 0 -30 -25 -38 -39 -70 -66 }}
+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 tuning (POTE): ~2 = 1200.000{{c}}, ~14/11 = 416.585{{c}}
+{{Optimal ET sequence|legend=0| 72, 261, 333, 405, 882c }}
-{{Optimal ET sequence|legend=0| 49fg, 72, 121, 193 }}
+Badness (Sintel): 1.71
-Badness (Smith): 0.013028
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
-=== 19-limit ===
+Comma list: 325/324, 625/624, 1375/1372, 4000/3993
-Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 325/324, 364/363, 375/374, 400/399, 442/441, 595/594
+Mapping: {{mapping| 9 0 9 11 24 0 | 0 6 5 6 3 14 }}
-Mapping: {{mapping| 1 12 11 16 17 28 27 -2 | 0 -30 -25 -38 -39 -70 -66 18 }}
+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 tuning (POTE): ~2 = 1200.000{{c}}, ~14/11 = 416.580{{c}}
+{{Optimal ET sequence|legend=0| 72, 189f, 261, 333, 738cf }}
-{{Optimal ET sequence|legend=0| 49fg, 72, 121, 193 }}
+Badness (Sintel): 1.61
-Badness (Smith): 0.014748
+== Subgroup extensions ==
+For the high-limit version of cata with a 1\5 period, see [[thunderclysmic]].
-== 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]].
+The structure of the temperament as dividing 3/1 into 6 equal parts can be deduced completely from its [[S-expression]]-based comma list of {{nowrap| {[[325/324|S10/S12 = S25⋅S26]], ([[625/624|S25]],) [[676/675|S13/S15 = S26]]} }}. Specifically, dividing 3/1 into two halves of ~26/15 is equivalent to dividing 4/3 into two halves of ~15/13, hence the [[semiparticular]] S13/S15 = ([[4/3|16/12]])/([[15/13]])<sup>2</sup>. From here, we notice that (26/15)/(13/9) = 6/5, so all that remains is dividing 13/9 into two 6/5's via the semiparticular S10/S12 = (13/9)/([[6/5|12/10]])<sup>2</sup>, hence explaining the mapping of the entire 2.3.5.13 subgroup.
 Subgroup: 2.3.5.13
@@ Line 1,173: / Line 1,244: @@
 Optimal tunings:
-* CTE: ~2 = 1200.0000{{c}}, ~6/5 = 317.1110{{c}}
+* WE: ~2 = 1200.1210{{c}}, ~6/5 = 317.1076{{c}}
-* POTE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0756{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0920{{c}}
-{{Optimal ET sequence|legend=1| 15, 19, 34, 53, 140, 193, 246 }}
+{{Optimal ET sequence|legend=0| 15, 19, 34, 53, 140, 193, 246 }}
 Badness (Sintel): 0.131
-==== 2.3.5.13.37.41 subgroup ====
+==== 2.3.5.13.37 subgroup ====
 Hanson can be extended even further to the 2.3.5.13.37.41 subgroup while maintaining a rather low complexity and high accuracy.
+Subgroup: 2.3.5.13.37
+Comma list: 325/324, 481/480, 625/624
+Subgroup-val mapping: {{mapping| 1 0 1 0 6 | 0 6 5 14 -3 }}
+Optimal tunings:
+* WE: ~2 = 1200.2924{{c}}, ~6/5 = 317.0998{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0452{{c}}
+{{Optimal ET sequence|legend=0| 15, 19, 34, 53, 299l, 352fl, 405fl, 458fl, 511cfll, 564cffll }}
+Badness (Sintel): 0.167
+==== 2.3.5.13.37.41 subgroup ====
 Subgroup: 2.3.5.13.37.41
-Comma list: 325/324, 625/624, [[481/480]], [[1600/1599]]
+Comma list: 325/324, 481/480, 625/624, 1025/1024
 Subgroup-val mapping: {{mapping| 1 0 1 0 6 8 | 0 6 5 14 -3 -10 }}
 Optimal tunings:
-* WE: ~2 = 1200.165{{c}}, ~6/5 = 317.113{{c}}
+* WE: ~2 = 1200.1651{{c}}, ~6/5 = 317.1126{{c}}
-* CWE: ~2 = 1200.000{{c}}, ~6/5 = 317.075{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0748{{c}}
+{{Optimal ET sequence|legend=0| 15, 19, 34, 53, 140, 193, 246l }}
 Badness (Sintel): 0.223
+[[Category:Kleismic family| ]] <!-- main article -->
 [[Category:Temperament families]]
-[[Category:Kleismic family| ]] <!-- main article -->
+[[Category:Catalogs of rank-2 temperaments]]
-[[Category:Rank 2]]
 [[Category:Listen]]