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~~]].

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]], or equivalently the amount by which a stack of three [[25/24]] overshoots the [[9/8]] whole tone. As a result, all kleismic temperaments trisect 9/8, for which one third-tone represents [[25/24]] and two represents [[27/25]].

== Kleismic a.k.a. hanson ==

The [[generator]] of kleismic is a [[6/5|classical minor third]], and to get to the interval class of [[5/4|major thirds]] requires five of these, and so to get to [[3/2|fifths]] requires six. In fact, (6/5)5 = (5/2)⋅(15625/15552). This 5-limit temperament (virtually a [[microtemperament]]) is sometimes called ''hanson'', and [[53edo|14\53]] is about perfect as a generator, though [[34edo|9\34]] also makes sense, and [[19edo|5\19]] and [[15edo|4\15]] are possible. Other tunings include [[72edo]], [[87edo]] and [[140edo]].

The [[generator]] of kleismic is a [[6/5|classical minor third]], and to get to the interval class of [[5/4|major thirds]] requires five of these, and so to get to [[3/2|perfect fifths]] requires six. In fact, (6/5)5 = (5/2)⋅(15625/15552). This 5-limit temperament (virtually a [[microtemperament]]) is sometimes called ''hanson'', and [[53edo|14\53]] is about perfect as a generator, though [[34edo|9\34]] also makes sense, and [[19edo|5\19]] and [[15edo|4\15]] are possible. Other edo tunings include [[72edo]], [[87edo]] and [[140edo]].

[[Subgroup]]: 2.3.5

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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. [[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.

==== 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]], the aberschisma, 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|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~~]].

==== Subgroup extensions ====

Kleismic lends itself to a natural extension to the [[2.3.5.13 subgroup]], as the hemitwelfth, reached by three generator steps, can be interpreted as [[26/15]]. This is discussed immediately below.

=== 2.3.5.13 subgroup (cata) ===

The kleisma has the [[S-expression]] S252⋅S26, which explains how the tone is trisected, and thus tempering out its constituents [[625/624]] ({{S|25}}) and [[676/675]] ({{S|26}}) is a readily natural subgroup extension. As a result of coincidental S-expression equivalences, [[325/324]] (S25⋅S26) is also tempered out, and two generator steps stand in for ~[[13/9]]. Finally, since this step is one third of a perfect twelfth, the comma [[2197/2187]] (S25⋅S262) is tempered out. The third tone is now 25/24~26/25~27/26, which means 27/25 and [[13/12]] are made equal.

For a version of cata with a 1\5 period, see [[Thunderclysmic]].

Subgroup: 2.3.5.13

Comma list: 325/324, 625/624

Optimal tunings:

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

: error map: {{val| +0.121 +0.690 -0.655 -1.022 }}

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

: error map: {{val| 0.000 +0.597 -0.854 -1.240 }}

Badness (Sintel): 0.131

== 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 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. In here, the JI intervals 25/24–27/25–9/8–7/6 are made equidistant.

~~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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==== 2.3.5.7.13 subgroup ====

Catakleismic extends easily with [[prime interval|prime]] [[13/1|13]] as seen before with cata, which has the S-expression list {S25, S26}. The marvel comma has an equivalent S-expression S25⋅S26⋅S27 and the ragisma S25/S27, thus [[729/728]] ({{S|27}}) is another constituent that can be naturally tempered out. As a result, the third tone becomes 25/24~26/25~27/26~28/27, and the equivalence 27/25~13/12 is added.

Add-13 catakleismic is less accurate than cata or catakleismic alone as the error of 7 and 13 go in separate directions – cata wants a sharper 6/5 but catakleismic wants a flatter one. Nontheless, its structural properties make it quite notable.

Subgroup: 2.3.5.7.13

Comma list: 169/168, 225/224, 325/324

~~Subgroup-val mapping:~~ {{~~mapping~~| 1 0 1 -3 0 | 0 6 5 22 14 }}

Optimal tunings:

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Comma list: 225/224, 385/384, 4375/4374

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

Optimal tunings:

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Line 129:

Comma list: 169/168, 225/224, 325/324, 385/384

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

Optimal tunings:

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Line 148:

Comma list: 99/98, 176/175, 2200/2187

~~Mapping:~~ {{~~mapping~~| 1 0 1 -3 -5 | 0 6 5 22 32 }}

Optimal tunings:

Line 144:

Line 163:

Comma list: 99/98, 169/168, 176/175, 275/273

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

Optimal tunings:

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Comma list: 225/224, 441/440, 4375/4374

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

Optimal tunings:

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Line 193:

Comma list: 169/168, 225/224, 325/324, 1716/1715

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

Optimal tunings:

Line 189:

Line 208:

Comma list: 100/99, 225/224, 864/847

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

Optimal tunings:

Line 204:

Line 223:

Comma list: 78/77, 100/99, 144/143, 676/675

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

Optimal tunings:

Line 219:

Line 238:

Comma list: 225/224, 243/242, 4375/4356

~~Mapping:~~ {{~~mapping~~| 2 0 2 -6 -1 | 0 6 5 22 15 }}

: mapping generators: ~99/70, ~6/5

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Line 254:

Comma list: 169/168, 225/224, 243/242, 325/324

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

Optimal tunings:

Line 250:

Line 269:

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

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

Optimal tunings:

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Line 284:

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

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

Optimal tunings:

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Line 323:

Comma list: 49/48, 56/55, 100/99

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

Optimal tunings:

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Line 342:

Comma list: 49/48, 56/55, 65/64, 100/99

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

Optimal tunings:

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Comma list: 49/48, 56/55, 91/90, 100/99

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

Optimal tunings:

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Line 381:

Comma list: 40/39, 49/48, 56/55, 66/65

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

Optimal tunings:

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Comma list: 45/44, 49/48, 126/125

~~Mapping:~~ {{~~mapping~~| 1 0 1 2 -1 | 0 6 5 3 17 }}

Optimal tunings:

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Line 411:

Comma list: 45/44, 49/48, 78/77, 126/125

~~Mapping:~~ {{~~mapping~~| 1 0 1 2 -1 0 | 0 6 5 3 17 14 }}

Optimal tunings:

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Line 426:

Comma list: 49/48, 55/54, 77/75

~~Mapping:~~ {{~~mapping~~| 1 0 1 2 0 | 0 6 5 3 13 }}

Optimal tunings:

Line 422:

Line 441:

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

~~Mapping:~~ {{~~mapping~~| 1 0 1 2 0 0 | 0 6 5 3 13 14 }}

Optimal tunings:

Line 458:

Line 477:

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

~~Mapping:~~ {{~~mapping~~| 1 0 1 6 4 | 0 6 5 -12 -2 }}

Optimal tunings:

Line 477:

Line 496:

Comma list: 64/63, 100/99, 144/143, 275/273

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

Optimal tunings:

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Line 532:

Comma list: 385/384, 2200/2187, 3388/3375

~~Mapping:~~ {{~~mapping~~| 1 0 1 11 -5 | 0 6 5 -31 32 }}

Optimal tunings:

Line 532:

Line 551:

Comma list: 325/324, 352/351, 385/384, 625/624

~~Mapping:~~ {{~~mapping~~| 1 0 1 11 -5 0 | 0 6 5 -31 32 14 }}

Optimal tunings:

Line 569:

Line 588:

Comma list: 896/891, 2200/2187, 14700/14641

~~Mapping:~~ {{~~mapping~~| 1 0 1 -12 -5 | 0 6 5 56 32 }}

Optimal tunings:

Line 584:

Line 603:

Comma list: 325/324, 352/351, 364/363, 625/624

~~Mapping:~~ {{~~mapping~~| 1 0 1 -12 -5 0 | 0 6 5 56 32 14 }}

Optimal tunings:

Line 617:

Line 636:

Comma list: 121/120, 176/175, 4000/3969

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

Optimal tunings:

Line 632:

Line 651:

Comma list: 121/120, 176/175, 275/273, 325/324

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

Optimal tunings:

Line 672:

Line 691:

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

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

Optimal tunings:

Line 687:

Line 706:

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

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

Optimal tunings:

Line 731:

Line 750:

Comma list: 385/384, 441/440, 4000/3993

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

Optimal tunings:

Line 751:

Line 770:

Comma list: 325/324, 364/363, 385/384, 625/624

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

Optimal tunings:

Line 766:

Line 785:

Comma list: 273/272, 325/324, 364/363, 375/374, 385/384

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

Optimal tunings:

Line 801:

Line 820:

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

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

Optimal tunings:

Line 816:

Line 835:

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:

Line 833:

Line 852:

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

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

Optimal tunings:

Line 848:

Line 867:

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

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

Optimal tunings:

Line 865:

Line 884:

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

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

Optimal tunings:

Line 880:

Line 899:

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

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

Optimal tunings:

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Line 932:

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

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

Optimal tunings:

Line 928:

Line 947:

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:

Line 961:

Line 980:

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

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

Optimal tunings:

Line 976:

Line 995:

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:

Line 991:

Line 1,010:

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:

Line 1,026:

Line 1,045:

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

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

Optimal tunings:

Line 1,041:

Line 1,060:

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

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

Optimal tunings:

Line 1,080:

Line 1,099:

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

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

Optimal tunings:

Line 1,095:

Line 1,114:

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

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

Optimal tunings:

Line 1,110:

Line 1,129:

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|legend=0| 1 -18 -14 -22 -22 -42 -39 | 0 30 25 38 39 70 66 }}

Optimal tunings:

Line 1,125:

Line 1,144:

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

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

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

Optimal tunings:

Line 1,160:

Line 1,179:

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

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

Optimal tunings:

Line 1,175:

Line 1,194:

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:

Line 1,208:

Line 1,227:

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

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

Optimal tunings:

Line 1,223:

Line 1,242:

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:

Line 1,233:

Line 1,252:

Badness (Sintel): 1.61

~~== Subgroup extensions ==~~

[[Category:Kleismic family| ]]

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

[[Category:Kleismic]]

~~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~~

~~Comma list: 325/324, 625/624~~

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

~~Optimal tunings:~~

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

* CWE: ~2 = 1200.1210{{c}}, ~6/5 = 317.0920{{c}}

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

~~Badness (Sintel): 0.131~~

~~==== 2.3.5.13.37 subgroup ====~~

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

~~Subgroup: 2.3.5.13.37.41~~

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

~~Subgroup-val mapping: {{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 = 1200.000{{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, 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 = 1200.1651{{c}}, ~6/5 = 317.1126{{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: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]].
+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]], or equivalently the amount by which a stack of three [[25/24]] overshoots the [[9/8]] whole tone. As a result, all kleismic temperaments trisect 9/8, for which one third-tone represents [[25/24]] and two represents [[27/25]].
 == Kleismic a.k.a. hanson ==
 {{Main| Kleismic }}
-The [[generator]] of kleismic is a [[6/5|classical minor third]], and to get to the interval class of [[5/4|major thirds]] requires five of these, and so to get to [[3/2|fifths]] requires six. In fact, (6/5)<sup>5</sup> = (5/2)⋅(15625/15552). This 5-limit temperament (virtually a [[microtemperament]]) is sometimes called ''hanson'', and [[53edo|14\53]] is about perfect as a generator, though [[34edo|9\34]] also makes sense, and [[19edo|5\19]] and [[15edo|4\15]] are possible. Other tunings include [[72edo]], [[87edo]] and [[140edo]].
+The [[generator]] of kleismic is a [[6/5|classical minor third]], and to get to the interval class of [[5/4|major thirds]] requires five of these, and so to get to [[3/2|perfect fifths]] requires six. In fact, (6/5)<sup>5</sup> = (5/2)⋅(15625/15552). This 5-limit temperament (virtually a [[microtemperament]]) is sometimes called ''hanson'', and [[53edo|14\53]] is about perfect as a generator, though [[34edo|9\34]] also makes sense, and [[19edo|5\19]] and [[15edo|4\15]] are possible. Other edo tunings include [[72edo]], [[87edo]] and [[140edo]].
 [[Subgroup]]: 2.3.5
@@ Line 35: / 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. [[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.
+==== 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]], the aberschisma, 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|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]].
+==== Subgroup extensions ====
+Kleismic lends itself to a natural extension to the [[2.3.5.13 subgroup]], as the hemitwelfth, reached by three generator steps, can be interpreted as [[26/15]]. This is discussed immediately below.
+=== 2.3.5.13 subgroup (cata) ===
+The kleisma has the [[S-expression]] S25<sup>2</sup>⋅S26, which explains how the tone is trisected, and thus tempering out its constituents [[625/624]] ({{S|25}}) and [[676/675]] ({{S|26}}) is a readily natural subgroup extension. As a result of coincidental S-expression equivalences, [[325/324]] (S25⋅S26) is also tempered out, and two generator steps stand in for ~[[13/9]]. Finally, since this step is one third of a perfect twelfth, the comma [[2197/2187]] (S25⋅S26<sup>2</sup>) is tempered out. The third tone is now 25/24~26/25~27/26, which means 27/25 and [[13/12]] are made equal.
+For a version of cata with a 1\5 period, see [[Thunderclysmic]].
+Subgroup: 2.3.5.13
+Comma list: 325/324, 625/624
+{{Mapping|legend=2| 1 0 1 0 | 0 6 5 14 }}
+Optimal tunings:
+* WE: ~2 = 1200.1210{{c}}, ~6/5 = 317.1076{{c}}
+: error map: {{val| +0.121 +0.690 -0.655 -1.022 }}
+* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0920{{c}}
+: error map: {{val| 0.000 +0.597 -0.854 -1.240 }}
+{{Optimal ET sequence|legend=0| 15, 19, 34, 53, 140, 193, 246 }}
+Badness (Sintel): 0.131
 == 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 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. In here, the JI intervals 25/24–27/25–9/8–7/6 are made equidistant.
-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 72: / Line 87: @@
 ==== 2.3.5.7.13 subgroup ====
+Catakleismic extends easily with [[prime interval|prime]] [[13/1|13]] as seen before with cata, which has the S-expression list {S25, S26}. The marvel comma has an equivalent S-expression S25⋅S26⋅S27 and the ragisma S25/S27, thus [[729/728]] ({{S|27}}) is another constituent that can be naturally tempered out. As a result, the third tone becomes 25/24~26/25~27/26~28/27, and the equivalence 27/25~13/12 is added.
+Add-13 catakleismic is less accurate than cata or catakleismic alone as the error of 7 and 13 go in separate directions – cata wants a sharper 6/5 but catakleismic wants a flatter one. Nontheless, its structural properties make it quite notable.
 Subgroup: 2.3.5.7.13
 Comma list: 169/168, 225/224, 325/324
-Subgroup-val mapping: {{mapping| 1 0 1 -3 0 | 0 6 5 22 14 }}
+{{Mapping|legend=2| 1 0 1 -3 0 | 0 6 5 22 14 }}
 Optimal tunings:
@@ Line 91: / Line 110: @@
 Comma list: 225/224, 385/384, 4375/4374
-Mapping: {{mapping| 1 0 1 -3 9 | 0 6 5 22 -21 }}
+{{Mapping|legend=0| 1 0 1 -3 9 | 0 6 5 22 -21 }}
 Optimal tunings:
@@ Line 110: / Line 129: @@
 Comma list: 169/168, 225/224, 325/324, 385/384
-Mapping: {{mapping| 1 0 1 -3 9 0 | 0 6 5 22 -21 14 }}
+{{Mapping|legend=0| 1 0 1 -3 9 0 | 0 6 5 22 -21 14 }}
 Optimal tunings:
@@ Line 129: / Line 148: @@
 Comma list: 99/98, 176/175, 2200/2187
-Mapping: {{mapping| 1 0 1 -3 -5 | 0 6 5 22 32 }}
+{{Mapping|legend=0| 1 0 1 -3 -5 | 0 6 5 22 32 }}
 Optimal tunings:
@@ Line 144: / Line 163: @@
 Comma list: 99/98, 169/168, 176/175, 275/273
-Mapping: {{mapping| 1 0 1 -3 -5 0 | 0 6 5 22 32 14 }}
+{{Mapping|legend=0| 1 0 1 -3 -5 0 | 0 6 5 22 32 14 }}
 Optimal tunings:
@@ Line 159: / Line 178: @@
 Comma list: 225/224, 441/440, 4375/4374
-Mapping: {{mapping| 1 0 1 -3 -10 | 0 6 5 22 51 }}
+{{Mapping|legend=0| 1 0 1 -3 -10 | 0 6 5 22 51 }}
 Optimal tunings:
@@ Line 174: / Line 193: @@
 Comma list: 169/168, 225/224, 325/324, 1716/1715
-Mapping: {{mapping| 1 0 1 -3 -10 0 | 0 6 5 22 51 14 }}
+{{Mapping|legend=0| 1 0 1 -3 -10 0 | 0 6 5 22 51 14 }}
 Optimal tunings:
@@ Line 189: / Line 208: @@
 Comma list: 100/99, 225/224, 864/847
-Mapping: {{mapping| 1 0 1 -3 4 | 0 6 5 22 -2 }}
+{{Mapping|legend=0| 1 0 1 -3 4 | 0 6 5 22 -2 }}
 Optimal tunings:
@@ Line 204: / Line 223: @@
 Comma list: 78/77, 100/99, 144/143, 676/675
-Mapping: {{mapping| 1 0 1 -3 4 0 | 0 6 5 22 -2 14 }}
+{{Mapping|legend=0| 1 0 1 -3 4 0 | 0 6 5 22 -2 14 }}
 Optimal tunings:
@@ Line 219: / Line 238: @@
 Comma list: 225/224, 243/242, 4375/4356
-Mapping: {{mapping| 2 0 2 -6 -1 | 0 6 5 22 15 }}
+{{Mapping|legend=0| 2 0 2 -6 -1 | 0 6 5 22 15 }}
 : mapping generators: ~99/70, ~6/5
@@ Line 235: / Line 254: @@
 Comma list: 169/168, 225/224, 243/242, 325/324
-Mapping: {{mapping| 2 0 2 -6 -1 0 | 0 6 5 22 15 14 }}
+{{Mapping|legend=0| 2 0 2 -6 -1 0 | 0 6 5 22 15 14 }}
 Optimal tunings:
@@ Line 250: / Line 269: @@
 Comma list: 169/168, 221/220, 225/224, 243/242, 325/324
-Mapping: {{mapping| 2 0 2 -6 -1 0 5 | 0 6 5 22 15 14 6 }}
+{{Mapping|legend=0| 2 0 2 -6 -1 0 5 | 0 6 5 22 15 14 6 }}
 Optimal tunings:
@@ Line 265: / Line 284: @@
 Comma list: 153/152, 169/168, 221/220, 225/224, 243/242, 325/324
-Mapping: {{mapping| 2 0 2 -6 -1 0 5 -1 | 0 6 5 22 15 14 6 18 }}
+{{Mapping|legend=0| 2 0 2 -6 -1 0 5 -1 | 0 6 5 22 15 14 6 18 }}
 Optimal tunings:
@@ Line 304: / Line 323: @@
 Comma list: 49/48, 56/55, 100/99
-Mapping: {{mapping| 1 0 1 2 4 | 0 6 5 3 -2 }}
+{{Mapping|legend=0| 1 0 1 2 4 | 0 6 5 3 -2 }}
 Optimal tunings:
@@ Line 323: / Line 342: @@
 Comma list: 49/48, 56/55, 65/64, 100/99
-Mapping: {{mapping| 1 0 1 2 4 5 | 0 6 5 3 -2 -5 }}
+{{Mapping|legend=0| 1 0 1 2 4 5 | 0 6 5 3 -2 -5 }}
 Optimal tunings:
@@ Line 342: / Line 361: @@
 Comma list: 49/48, 56/55, 91/90, 100/99
-Mapping: {{mapping| 1 0 1 2 4 0 | 0 6 5 3 -2 14 }}
+{{Mapping|legend=0| 1 0 1 2 4 0 | 0 6 5 3 -2 14 }}
 Optimal tunings:
@@ Line 362: / Line 381: @@
 Comma list: 40/39, 49/48, 56/55, 66/65
-Mapping: {{mapping| 1 0 1 2 4 4 | 0 6 5 3 -2 -1 }}
+{{Mapping|legend=0| 1 0 1 2 4 4 | 0 6 5 3 -2 -1 }}
 Optimal tunings:
@@ Line 377: / Line 396: @@
 Comma list: 45/44, 49/48, 126/125
-Mapping: {{mapping| 1 0 1 2 -1 | 0 6 5 3 17 }}
+{{Mapping|legend=0| 1 0 1 2 -1 | 0 6 5 3 17 }}
 Optimal tunings:
@@ Line 392: / Line 411: @@
 Comma list: 45/44, 49/48, 78/77, 126/125
-Mapping: {{mapping| 1 0 1 2 -1 0 | 0 6 5 3 17 14 }}
+{{Mapping|legend=0| 1 0 1 2 -1 0 | 0 6 5 3 17 14 }}
 Optimal tunings:
@@ Line 407: / Line 426: @@
 Comma list: 49/48, 55/54, 77/75
-Mapping: {{mapping| 1 0 1 2 0 | 0 6 5 3 13 }}
+{{Mapping|legend=0| 1 0 1 2 0 | 0 6 5 3 13 }}
 Optimal tunings:
@@ Line 422: / Line 441: @@
 Comma list: 49/48, 55/54, 66/65, 77/75
-Mapping: {{mapping| 1 0 1 2 0 0 | 0 6 5 3 13 14 }}
+{{Mapping|legend=0| 1 0 1 2 0 0 | 0 6 5 3 13 14 }}
 Optimal tunings:
@@ Line 458: / Line 477: @@
 Comma list: 64/63, 100/99, 1331/1323
-Mapping: {{mapping| 1 0 1 6 4 | 0 6 5 -12 -2 }}
+{{Mapping|legend=0| 1 0 1 6 4 | 0 6 5 -12 -2 }}
 Optimal tunings:
@@ Line 477: / Line 496: @@
 Comma list: 64/63, 100/99, 144/143, 275/273
-Mapping: {{mapping| 1 0 1 6 4 0 | 0 6 5 -12 -2 14 }}
+{{Mapping|legend=0| 1 0 1 6 4 0 | 0 6 5 -12 -2 14 }}
 Optimal tunings:
@@ Line 513: / Line 532: @@
 Comma list: 385/384, 2200/2187, 3388/3375
-Mapping: {{mapping| 1 0 1 11 -5 | 0 6 5 -31 32 }}
+{{Mapping|legend=0| 1 0 1 11 -5 | 0 6 5 -31 32 }}
 Optimal tunings:
@@ Line 532: / Line 551: @@
 Comma list: 325/324, 352/351, 385/384, 625/624
-Mapping: {{mapping| 1 0 1 11 -5 0 | 0 6 5 -31 32 14 }}
+{{Mapping|legend=0| 1 0 1 11 -5 0 | 0 6 5 -31 32 14 }}
 Optimal tunings:
@@ Line 569: / Line 588: @@
 Comma list: 896/891, 2200/2187, 14700/14641
-Mapping: {{mapping| 1 0 1 -12 -5 | 0 6 5 56 32 }}
+{{Mapping|legend=0| 1 0 1 -12 -5 | 0 6 5 56 32 }}
 Optimal tunings:
@@ Line 584: / Line 603: @@
 Comma list: 325/324, 352/351, 364/363, 625/624
-Mapping: {{mapping| 1 0 1 -12 -5 0 | 0 6 5 56 32 14 }}
+{{Mapping|legend=0| 1 0 1 -12 -5 0 | 0 6 5 56 32 14 }}
 Optimal tunings:
@@ Line 617: / Line 636: @@
 Comma list: 121/120, 176/175, 4000/3969
-Mapping: {{mapping| 1 0 1 4 2 | 0 12 10 -9 11 }}
+{{Mapping|legend=0| 1 0 1 4 2 | 0 12 10 -9 11 }}
 Optimal tunings:
@@ Line 632: / Line 651: @@
 Comma list: 121/120, 176/175, 275/273, 325/324
-Mapping: {{mapping| 1 0 1 4 2 0 | 0 12 10 -9 11 28 }}
+{{Mapping|legend=0| 1 0 1 4 2 0 | 0 12 10 -9 11 28 }}
 Optimal tunings:
@@ Line 672: / Line 691: @@
 Comma list: 245/243, 385/384, 3136/3125
-Mapping: {{mapping| 1 -6 -4 -13 18 | 0 12 10 25 -23 }}
+{{Mapping|legend=0| 1 -6 -4 -13 18 | 0 12 10 25 -23 }}
 Optimal tunings:
@@ Line 687: / Line 706: @@
 Comma list: 196/195, 245/243, 385/384, 625/624
-Mapping: {{mapping| 1 -6 -4 -13 18 -14 | 0 12 10 25 -23 28 }}
+{{Mapping|legend=0| 1 -6 -4 -13 18 -14 | 0 12 10 25 -23 28 }}
 Optimal tunings:
@@ Line 731: / Line 750: @@
 Comma list: 385/384, 441/440, 4000/3993
-Mapping: {{mapping| 3 0 3 10 8 | 0 6 5 -2 3 }}
+{{Mapping|legend=0| 3 0 3 10 8 | 0 6 5 -2 3 }}
 Optimal tunings:
@@ Line 751: / Line 770: @@
 Comma list: 325/324, 364/363, 385/384, 625/624
-Mapping: {{mapping| 3 0 3 10 8 0 | 0 6 5 -2 3 14 }}
+{{Mapping|legend=0| 3 0 3 10 8 0 | 0 6 5 -2 3 14 }}
 Optimal tunings:
@@ Line 766: / Line 785: @@
 Comma list: 273/272, 325/324, 364/363, 375/374, 385/384
-Mapping: {{mapping| 3 0 3 10 8 0 -2 | 0 6 5 -2 3 14 18 }}
+{{Mapping|legend=0| 3 0 3 10 8 0 -2 | 0 6 5 -2 3 14 18 }}
 Optimal tunings:
@@ Line 801: / Line 820: @@
 Comma list: 385/384, 6250/6237, 10976/10935
-Mapping: {{mapping| 1 -6 -4 -17 22 | 0 18 15 47 -44 }}
+{{Mapping|legend=0| 1 -6 -4 -17 22 | 0 18 15 47 -44 }}
 Optimal tunings:
@@ Line 816: / Line 835: @@
 Comma list: 325/324, 385/384, 625/624, 10976/10935
-Mapping: {{mapping| 1 -6 -4 -17 22 -14 | 0 18 15 47 -44 42 }}
+{{Mapping|legend=0| 1 -6 -4 -17 22 -14 | 0 18 15 47 -44 42 }}
 Optimal tunings:
@@ Line 833: / Line 852: @@
 Comma list: 1375/1372, 2200/2187, 5632/5625
-Mapping: {{mapping| 1 -6 -4 -17 -37 | 0 18 15 47 96 }}
+{{Mapping|legend=0| 1 -6 -4 -17 -37 | 0 18 15 47 96 }}
 Optimal tunings:
@@ Line 848: / Line 867: @@
 Comma list: 325/324, 352/351, 625/624, 1375/1372
-Mapping: {{mapping| 1 -6 -4 -17 -37 -14 | 0 18 15 47 96 42 }}
+{{Mapping|legend=0| 1 -6 -4 -17 -37 -14 | 0 18 15 47 96 42 }}
 Optimal tunings:
@@ Line 865: / Line 884: @@
 Comma list: 540/539, 896/891, 15625/15552
-Mapping: {{mapping| 1 -6 -4 -17 14 | 0 18 15 47 -25 }}
+{{Mapping|legend=0| 1 -6 -4 -17 14 | 0 18 15 47 -25 }}
 Optimal tunings:
@@ Line 880: / Line 899: @@
 Comma list: 325/324, 540/539, 625/624, 896/891
-Mapping: {{mapping| 1 -6 -4 -17 14 -14 | 0 18 15 47 -25 42 }}
+{{Mapping|legend=0| 1 -6 -4 -17 14 -14 | 0 18 15 47 -25 42 }}
 Optimal tunings:
@@ Line 913: / Line 932: @@
 Comma list: 176/175, 540/539, 3125/3087
-Mapping: {{mapping| 1 -12 -9 -7 -29 | 0 18 15 13 43 }}
+{{Mapping|legend=0| 1 -12 -9 -7 -29 | 0 18 15 13 43 }}
 Optimal tunings:
@@ Line 928: / Line 947: @@
 Comma list: 176/175, 275/273, 325/324, 540/539
-Mapping: {{mapping| 1 -12 -9 -7 -29 -28 | 0 18 15 13 43 42 }}
+{{Mapping|legend=0| 1 -12 -9 -7 -29 -28 | 0 18 15 13 43 42 }}
 Optimal tunings:
@@ Line 961: / Line 980: @@
 Comma list: 385/384, 1375/1372, 6250/6237
-Mapping: {{mapping| 4 0 4 7 17 | 0 6 5 4 -3 }}
+{{Mapping|legend=0| 4 0 4 7 17 | 0 6 5 4 -3 }}
 Optimal tunings:
@@ Line 976: / Line 995: @@
 Comma list: 325/324, 385/384, 625/624, 1375/1372
-Mapping: {{mapping| 4 0 4 7 17 0 | 0 6 5 4 -3 14 }}
+{{Mapping|legend=0| 4 0 4 7 17 0 | 0 6 5 4 -3 14 }}
 Optimal tunings:
@@ Line 991: / Line 1,010: @@
 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 }}
+{{Mapping|legend=0| 4 0 4 7 17 0 10 | 0 6 5 4 -3 14 6 }}
 Optimal tunings:
@@ Line 1,026: / Line 1,045: @@
 Comma list: 1375/1372, 15625/15552, 19712/19683
-Mapping: {{mapping| 1 -6 -4 -19 -43 | 0 24 20 69 147 }}
+{{Mapping|legend=0| 1 -6 -4 -19 -43 | 0 24 20 69 147 }}
 Optimal tunings:
@@ Line 1,041: / Line 1,060: @@
 Comma list: 325/324, 625/624, 1375/1372, 19712/19683
-Mapping: {{mapping| 1 -6 -4 -19 -43 -14 | 0 24 20 69 147 56 }}
+{{Mapping|legend=0| 1 -6 -4 -19 -43 -14 | 0 24 20 69 147 56 }}
 Optimal tunings:
@@ Line 1,080: / Line 1,099: @@
 Comma list: 540/539, 1375/1372, 4375/4356
-Mapping: {{mapping| 1 -18 -14 -22 -22 | 0 30 25 38 39 }}
+{{Mapping|legend=0| 1 -18 -14 -22 -22 | 0 30 25 38 39 }}
 Optimal tunings:
@@ Line 1,095: / Line 1,114: @@
 Comma list: 325/324, 364/363, 625/624, 1375/1372
-Mapping: {{mapping| 1 -18 -14 -22 -22 -42 | 0 30 25 38 39 70 }}
+{{Mapping|legend=0| 1 -18 -14 -22 -22 -42 | 0 30 25 38 39 70 }}
 Optimal tunings:
@@ Line 1,110: / Line 1,129: @@
 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|legend=0| 1 -18 -14 -22 -22 -42 -39 | 0 30 25 38 39 70 66 }}
 Optimal tunings:
@@ Line 1,125: / Line 1,144: @@
 Comma list: 325/324, 364/363, 375/374, 400/399, 442/441, 595/594
-Mapping: {{mapping| 1 -18 -14 -22 -22 -42 -39 16 | 0 30 25 38 39 70 66 -18 }}
+{{Mapping|legend=0| 1 -18 -14 -22 -22 -42 -39 16 | 0 30 25 38 39 70 66 -18 }}
 Optimal tunings:
@@ Line 1,160: / Line 1,179: @@
 Comma list: 385/384, 6250/6237, 67228/66825
-Mapping: {{mapping| 1 0 1 1 5 | 0 36 30 41 -35 }}
+{{Mapping|legend=0| 1 0 1 1 5 | 0 36 30 41 -35 }}
 Optimal tunings:
@@ Line 1,175: / Line 1,194: @@
 Comma list: 325/324, 385/384, 625/624, 16807/16731
-Mapping: {{mapping| 1 0 1 1 5 0 | 0 36 30 41 -35 84 }}
+{{Mapping|legend=0| 1 0 1 1 5 0 | 0 36 30 41 -35 84 }}
 Optimal tunings:
@@ Line 1,208: / Line 1,227: @@
 Comma list: 1375/1372, 4000/3993, 15625/15552
-Mapping: {{mapping| 9 0 9 11 24 | 0 6 5 6 3 }}
+{{Mapping|legend=0| 9 0 9 11 24 | 0 6 5 6 3 }}
 Optimal tunings:
@@ Line 1,223: / Line 1,242: @@
 Comma list: 325/324, 625/624, 1375/1372, 4000/3993
-Mapping: {{mapping| 9 0 9 11 24 0 | 0 6 5 6 3 14 }}
+{{Mapping|legend=0| 9 0 9 11 24 0 | 0 6 5 6 3 14 }}
 Optimal tunings:
@@ Line 1,233: / Line 1,252: @@
 Badness (Sintel): 1.61
-== Subgroup extensions ==
+[[Category:Kleismic family| ]] <!-- main article -->
-=== Kleismic (2.3.5.13) a.k.a. cata ===
+[[Category:Kleismic]]
-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
-Comma list: 325/324, 625/624
-Subgroup-val mapping: {{mapping| 1 0 1 0 | 0 6 5 14 }}
-Optimal tunings:
-* WE: ~2 = 1200.1210{{c}}, ~6/5 = 317.1076{{c}}
-* CWE: ~2 = 1200.1210{{c}}, ~6/5 = 317.0920{{c}}
-{{Optimal ET sequence|legend=0| 15, 19, 34, 53, 140, 193, 246 }}
-Badness (Sintel): 0.131
-==== 2.3.5.13.37 subgroup ====
-Hanson can be extended even further to the 2.3.5.13.37.41 subgroup while maintaining a rather low complexity and high accuracy.
-Subgroup: 2.3.5.13.37.41
-Comma list: 325/324, 481/480, 625/624
-Subgroup-val mapping: {{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 = 1200.000{{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, 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 = 1200.1651{{c}}, ~6/5 = 317.1126{{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:Temperament families]]
-[[Category:Kleismic family| ]] <!-- main article -->
+[[Category:Catalogs of rank-2 temperaments]]
-[[Category:Rank 2]]
-[[Category:Listen]]