Marvel family: Difference between revisions

(12 intermediate revisions by 3 users not shown)

Line 1:

~~{{interwiki~~

~~| en = Marvel family~~

~~| de = Marvel~~

~~| es =~~

~~| ja =~~

}}

The '''marvel family''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[7-limit]] [[marvel comma]] ([[ratio]]: [[225/224]], {{monzo|legend=1| -5 2 2 -1 }}), also known as ''septimal kleisma''. These temperaments hence equate [[16/15]] and [[15/14]], or equivalently they equate two [[5/4]]'s and one [[14/9]]. The marvel comma is noteworthy in that it is tempered out by many common [[equal temperament|equal]] and [[rank-2 temperament]]s.

Line 13:

Line 7:

The head of the marvel family is marvel, which tempers out [[225/224]]. Marvel has a [[normal ~~lists~~|normal generator list]] of {2, 3, 5}; hence a [[5-limit]] scale can be converted to marvel simply by tempering it. One way to do that, and an excellent marvel tuning, is given by [[197edo]].

The head of the marvel family is marvel, which tempers out [[225/224]]. Marvel has a [[normal forms|normal generator list]] of {2, 3, 5}; hence a [[5-limit]] scale can be converted to marvel simply by tempering it. One way to do that, and an excellent marvel tuning, is given by [[197edo]].

Little is gained in tuning accuracy by not tempering out 4375/4374 as well as 225/224, leading to [[catakleismic]], with which it shares the [[optimal patent val]]. Another temperament which does little damage to tuning accuracy is [[compton]], for which [[240edo]] may be used. See [[~~marvel~~ temperaments]] for some other rank-2 temperaments.

Little is gained in tuning accuracy by not tempering out 4375/4374 as well as 225/224, leading to [[catakleismic]], with which it shares the [[optimal patent val]]. Another temperament which does little damage to tuning accuracy is [[compton]], for which [[240edo]] may be used. See [[Marvel temperaments]] for some other rank-2 temperaments.

[[Subgroup]]: 2.3.5.7

Line 35:

Line 29:

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 700.6222{{c}}, ~5/4 = 383.8540{{c}}

: error map: {{val| 0.000 -1.333 -2.460 +0.127 }}

~~~~

[[Minimax tuning]]:

Line 73:

Line 66:

=== Overview to extensions ===

The second comma of the [[~~Normal lists~~ #Normal ~~interval list~~|normal comma list]] defines which 11-limit family member we are looking at. 4125/4096 gives unidecimal marvel; 91125/90112 gives prodigy; 5632/5625 gives minerva. These and many others considered below use the same generators as marvel.

The second comma of the [[normal forms #Normal forms for commas|normal comma list]] defines which 11-limit family member we are looking at. 4125/4096 gives unidecimal marvel; 91125/90112 gives prodigy; 5632/5625 gives minerva. These and many others considered below use the same generators as marvel.

Temperaments discussed elsewhere include

Line 79:

Line 72:

* ''[[Artemis]]'' (+121/120) → [[Biyatismic clan #Artemis|Biyatismic clan]]

* ''[[Spectacle]]'' (+243/242) → [[Rastmic rank-3 clan #Spectacle|Rastmic rank-3 clan]]

* ''[[Marvelpine]]'' (+4000/3993) → [[Wizardharry clan #Marvelpine|Wizardharry clan]]

* ''[[Mirage]]'' (+243/242, +385/384) → [[Rastmic rank-3 clan #Spectacle|Rastmic rank-3 clan]]

* ''[[Catakleismoid]]'' (+4375/4374) → [[Kleismic rank-3 family #Catakleismoid|Kleismic rank-3 family]]

== Undecimal marvel ~~(unimarv)~~ ==

== Undecimal marvel ==

Undecimal marvel tempers out [[385/384]] as well as [[540/539]], and is loosely [[associated temperament|associated]] with [[wizard]]. This extension is natural because of the factorization 225/224 = (385/384)⋅(540/539). [[197edo]] remains useful as a tuning, with the 197e val, but [[166edo]], which among other things has a virtually pure 7, works as well.

In the 13-limit, 225/224 factors as ([[351/350]])⋅([[625/624]]) or ([[325/324]])⋅([[729/728]]). Tempering out 351/350 and 625/624 leads to helios, tempering out 325/324 and 729/728 leads to hecate. Tempering out all of them leads to the 13-limit version of [[catakleismic]].

[[Subgroup]]: 2.3.5.7.11

Line 101:

Line 100:

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 700.6048{{c}}, ~5/4 = 383.4538{{c}}

: error map: {{val| 0.000 -1.350 -2.860 -0.709 -2.284 }}

~~~~

[[Minimax tuning]]:

Line 115:

Line 113:

[[Complexity spectrum]]: 5/4, 4/3, 7/6, 8/7, 7/5, 6/5, 9/7, 12/11, 9/8, 11/8, 11/9, 10/9, 11/10, 14/11

~~[[Associated temperament]]: [[catakleismic]]~~

Scales: [[marvel22_11]], [[unimarv19]], [[unimarv22]]

Line 130:

Line 126:

}}

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

=== Helios ===

Subgroup: 2.3.5.7.11.13

Line 140:

Line 136:

* WE: ~2 = 1200.8043{{c}}, ~3/2 = 700.2057{{c}}, ~5/4 = 384.3188{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 699.8109{{c}}, ~5/4 = 384.1177{{c}}

~~~~

Minimax tuning:

Line 153:

Line 148:

=== Hecate ===

Hecate tempers out [[325/324]], the marveltwin comma, such that [[16/13]] is found by a stack of two ~[[10/9]]'s, similar to how [[8/7]] is found by a stack of two [[15/14]]~[[16/15]]'s. Hecate has a natural extension to include [[prime interval|prime]] [[19/1|19]], where it further tempers out [[400/399]] and [[513/512]], taking advantage of the factorization 225/224 = (400/399)⋅(513/512). For both of these cases [[166edo]] remains an excellent tuning.

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

Line 163:

Line 160:

* WE: ~2 = 1200.5788{{c}}, ~3/2 = 701.3161{{c}}, ~5/4 = 383.3471{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.1003{{c}}, ~5/4 = 383.1315{{c}}

~~~~

Minimax tuning:

Line 187:

Line 183:

* WE: ~2 = 1200.4716{{c}}, ~3/2 = 701.4395{{c}}, ~5/4 = 383.2779{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.2002{{c}}, ~5/4 = 383.1136{{c}}

~~~~

Line 203:

Line 198:

* WE: ~2 = 1200.6057{{c}}, ~3/2 = 701.3157{{c}}, ~5/4 = 383.2243{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.0877{{c}}, ~5/4 = 382.9709{{c}}

~~~~

Line 219:

Line 213:

* WE: ~2 = 1200.4927{{c}}, ~3/2 = 701.4506{{c}}, ~5/4 = 383.1291{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.2007{{c}}, ~5/4 = 382.9404{{c}}

~~~~

Line 235:

Line 228:

* WE: ~2 = 1200.6165{{c}}, ~3/2 = 701.3259{{c}}, ~5/4 = 383.5032{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.0949{{c}}, ~5/4 = 383.3140{{c}}

~~~~

Line 251:

Line 243:

* WE: ~2 = 1200.5038{{c}}, ~3/2 = 701.4654{{c}}, ~5/4 = 383.4549{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.2113{{c}}, ~5/4 = 383.3052{{c}}

~~~~

Line 267:

Line 258:

* WE: ~2 = 1200.6404{{c}}, ~3/2 = 700.7675{{c}}, ~5/4 = 383.7772{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.6113{{c}}, ~5/4 = 383.4925{{c}}

~~~~

Minimax tuning:

Line 282:

Line 272:

Comma list: 225/224, 275/273, 385/384

Mapping: {{mapping| 1 0 0 -5 12 17 | 0 1 0 2 -1 4 | 0 0 1 2 -3 -3 }}

Mapping: {{mapping| 1 0 0 -5 12 17 | 0 1 0 2 -1 -4 | 0 0 1 2 -3 -3 }}

Optimal tunings:

* WE: ~2 = 1200.1824{{c}}, ~3/2 = 702.0223{{c}}, ~5/4 = 383.3028{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.9276{{c}}, ~5/4 = 383.2283{{c}}

~~~~

{{Optimal ET sequence|legend=0| 10, 19f, 22, 31, 41, 53, 84e, 94 }}

Line 305:

Line 294:

* WE: ~2 = 1200.7533{{c}}, ~3/2 = 700.8957{{c}}, ~5/4 = 383.0580{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.7184{{c}}, ~5/4 = 382.6611{{c}}

~~~~

Minimax tuning:

Line 327:

Line 315:

* WE: ~2 = 1200.4789{{c}}, ~3/2 = 699.5125{{c}}, ~5/4 = 383.1304{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 699.4041{{c}}, ~5/4 = 382.9166{{c}}

~~~~

Minimax tuning:

Line 350:

Line 337:

* WE: ~2 = 1200.7720{{c}}, ~3/2 = 950.8976{{c}}, ~5/4 = 383.8283{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 950.4265{{c}}, ~5/4 = 383.4790{{c}}

~~~~

{{Optimal ET sequence|legend=0| 9, 10, 19, 43e, 44, 53, 72, 125f, 197ef }}

Badness (Sintel): 0.935

== Prodigy ==

Prodigy tempers out [[441/440]] and shrinks [[243/242]], [[385/384|384/385]], [[1029/1024|1024/1029]] and [[2401/2400|2400/2401]] down to the same tiny interval. Hence in practice it probably makes the most sense to temper this out as well, leading to [[miracle]]. This, however, does not render it pointless to consider prodigy; for one thing, scales in prodigy such as [[hobbit]] scales translate into interesting scales for miracle.

[[Subgroup]]: 2.3.5.7.11

[[Comma list]]: 225/224, 441/440

Map to lattice: [{{val| 0 0 -1 -2 -3 }}, {{val| 0 1 -1 0 3 }}]

Lattice basis:

: ~15/14 length = 0.9111, ~3/2 length = 0.9477

: angle (~15/14, ~3/2) = 65.933°

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.7854{{c}}, ~3/2 = 700.2562{{c}}, ~5/4 = 383.7624{{c}}

: [[error map]]: {{val| +0.785 -0.913 -0.980 -0.003 +0.721 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 699.8610{{c}}, ~5/4 = 383.7724{{c}}

: error map: {{val| 0.000 -2.094 -2.541 -1.559 -0.835 }}

[[Minimax tuning]]:

* [[11-odd-limit]]

: [{{monzo| 1 0 0 0 0 }}, {{monzo| 13/12 1/2 -1/4 0 1/12 }}, {{monzo| 13/6 -1 1/2 0 1/6 }}, {{monzo| 3/2 -1 1/2 0 1/2 }}, {{monzo| 0 0 0 0 1 }}]

: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5.11

{{Optimal ET sequence|legend=1| 10, 12, 19e, 29, 31, 41, 60e, 72, 247c, 319bcde, 391bcde, 463bccde }}

[[Badness]] (Sintel): 0.402

[[Projection pair]]s: <code>7 225/32 11 91125/8192</code>

Scales: [[prodigy11]], [[prodigy12]], [[prodigy29]]

{{Databox|Hobbit bases|

2.3.5 subgroup

* 31: 81/80, 34171875/33554432

* 41: 34171875/33554432, 32805/32768

}}

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 196/195, 352/351

Mapping: {{mapping| 1 0 0 -5 -13 -8 | 0 1 0 2 6 3 | 0 0 1 2 3 3 }}

Optimal tunings:

* WE: ~2 = 1200.8252{{c}}, ~3/2 = 700.8823{{c}}, ~5/4 = 381.6647{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.4689{{c}}, ~5/4 = 381.6687{{c}}

{{Optimal ET sequence|legend=0| 10, 12f, 19e, 29, 31, 41, 60e, 72f, 101cd }}

Badness (Sintel): 0.689

=== Prodigious ===

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 364/363, 441/440

Mapping: {{mapping| 1 0 0 -5 -13 -23 | 0 1 0 2 6 11 | 0 0 1 2 3 4 }}

Optimal tunings:

* WE: ~2 = 1200.6284{{c}}, ~3/2 = 700.7075{{c}}, ~5/4 = 383.4599{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.3302{{c}}, ~5/4 = 383.5030{{c}}

{{Optimal ET sequence|legend=0| 12f, 29, 31f, 41, 72, 185cf, 257cff }}

Badness (Sintel): 0.841

=== Prodigal ===

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 351/350, 441/440

Mapping: {{mapping| 1 0 0 -5 -13 -4 | 0 1 0 2 6 -1 | 0 0 1 2 3 4 }}

Optimal tunings:

* WE: ~2 = 1200.7798{{c}}, ~3/2 = 699.9410{{c}}, ~5/4 = 384.3494{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 699.5538{{c}}, ~5/4 = 384.3496{{c}}

{{Optimal ET sequence|legend=0| 12f, 19e, 31, 53e, 60eff, 72, 103, 175f }}

Badness (Sintel): 0.831

=== Protannic ===

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 441/440, 1001/1000

Mapping: {{mapping| 1 0 0 -5 -13 21 | 0 1 0 2 6 -8 | 0 0 1 2 3 -2 }}

Optimal tunings:

* WE: ~2 = 1200.9450{{c}}, ~3/2 = 700.1045{{c}}, ~5/4 = 383.8716{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 699.4224{{c}}, ~5/4 = 383.9828{{c}}

{{Optimal ET sequence|legend=0| 29, 31, 43, 60e, 72, 103, 175f, 482bccddeefff, 554bbccddeeeffff }}

Badness (Sintel): 0.891

==== 17-limit ====

Subgroup: 2.3.5.7.11.13.17

Comma list: 225/224, 273/272, 375/374, 441/440

Mapping: {{mapping| 1 0 0 -5 -13 21 12 | 0 1 0 2 6 -8 -5 | 0 0 1 2 3 -2 0 }}

Optimal tunings:

* WE: ~2 = 1200.9342{{c}}, ~3/2 = 700.1708{{c}}, ~5/4 = 383.7444{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 699.4849{{c}}, ~5/4 = 383.8744{{c}}

{{Optimal ET sequence|legend=0| 29g, 31, 43, 60e, 72, 103, 175f, 307bcdeeffg, 379bccdeeffgg, 482bccddeefffgg, 554bbccddeeeffffgg }}

Badness (Sintel): 0.734

== Minerva ==

Minerva tempers out [[99/98]] as well as [[176/175]]. It may be described as 12 & 22 & 31, and is loosely [[associated temperament|associated]] with [[würschmidt]].

[[Subgroup]]: 2.3.5.7.11

Line 374:

Line 477:

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 700.3006{{c}}, ~5/4 = 386.5785{{c}}

: error map: {{val| 0.000 -1.654 +0.265 +4.932 -4.403 }}

~~~~

[[Minimax tuning]]: [[11-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5.11/9

Line 385:

Line 487:

Scales: [[minerva12]], [[minerva22x]]

~~[[Associated temperament]]: [[Würschmidt family #Würschmidt|würschmidt]]~~

=== Athene ===

Line 398:

Line 498:

* WE: ~2 = 1199.9127{{c}}, ~3/2 = 701.1832{{c}}, ~5/4 = 385.9313{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.2143{{c}}, ~5/4 = 385.9336{{c}}

~~~~

Minimax tuning:

Line 412:

Line 511:

== Apollo ==

[[File:Lattice Apollo.png|thumb|Lattice for apollo.]]

Apollo tempers out not only [[100/99]] but [[896/891]]. Note that marvel tempers together [[25/24]] and [[28/27]], and apollo further equates it with [[33/32]] via the vanishing of 100/99. This makes it a weak extension of [[parapyth]], and [[associated temperament|associates]] it with [[magic]]. The lattice structure is very compact, comparable to that of [[ares]], from which apollo only differs in the mapping of [[prime interval|prime]] [[7/1|7]].

The canonical [[13-limit]] extension is implied by parapyth, tempering out [[352/351]] and [[364/363]], but there are a number of other extenions to consider, these being called phoebus and musagetes, after epithets of Apollo. Phoebus tempers out [[105/104]] and finds ~[[16/13]] as a stack of three [[secor]]s. Musagetes tempers out [[144/143]] and conflates 16/13 and [[11/9]], which in this case is simply a stack of two ~[[10/9]]'s. These extensions are [[support]]ed by 13-limit [[magic]], unlike the canonical one.

[[Subgroup]]: 2.3.5.7.11

Line 424:

Line 529:

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.4612{{c}}, ~5/4 = 381.5071{{c}}

: error map: {{val| 0.000 +1.506 -4.807 +1.111 +4.774 }}

~~~~

[[Minimax tuning]]: [[11-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5.11/9

Line 433:

Line 537:

[[Projection pair]]s: <code>7 225/32 11 100/9</code>

~~[[Associated temperament]]: [[magic]]~~

Scales: [[apollo wholetone]], [[indigo17]]

Line 448:

Line 550:

* WE: ~2 = 1199.6919{{c}}, ~3/2 = 703.8176{{c}}, ~5/4 = 381.4372{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.9853{{c}}, ~5/4 = 381.3579{{c}}

~~~~

Minimax tuning: 13-odd-limit unchanged-interval (eigenmonzo) basis: 2.11/9.13/9

Line 457:

Line 558:

Projection pairs: <code>7 225/32 11 100/9 13 3200/243</code>

=== Phoebus ===

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 105/104, 196/195

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

Optimal tunings:

* WE: ~2 = 1200.3724{{c}}, ~3/2 = 702.5274{{c}}, ~5/4 = 379.6345{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.3357{{c}}, ~5/4 = 379.6830{{c}}

Badness (Sintel): 0.886

=== Musagetes ===

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 144/143, 225/224

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

Optimal tunings:

* WE: ~2 = 1199.2695{{c}}, ~3/2 = 702.5589{{c}}, ~5/4 = 382.7715{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.7998{{c}}, ~5/4 = 382.7740{{c}}

{{Optimal ET sequence|legend=0| 19, 22f, 34d, 41, 75e, 94e, 116ef }}

Badness (Sintel): 1.14

== Potassium ==

Line 470:

Line 601:

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 696.0586{{c}}, ~5/4 = 384.9472{{c}}

: error map: {{val| 0.000 -5.896 -1.367 -6.814 +25.746 }}

~~~~

[[Minimax tuning]]: [[11-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7.11

Line 490:

Line 620:

* WE: ~2 = 1199.8192{{c}}, ~3/2 = 695.9054{{c}}, ~5/4 = 384.6205{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 695.9480{{c}}, ~5/4 = 384.6372{{c}}

~~~~

Minimax tuning:

Line 516:

Line 645:

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.0535{{c}}, ~5/4 = 382.6222{{c}}

: error map: {{val| 0.000 +0.098 -3.692 +0.525 -2.188 }}

~~~~

{{Optimal ET sequence|legend=1| 12e, 19e, 34d, 41, 53, 60e, 94, 229c, 248ce, 289cce }}

Line 532:

Line 660:

* WE: ~2 = 1200.2427{{c}}, ~3/2 = 702.1575{{c}}, ~5/4 = 383.0704{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0882{{c}}, ~5/4 = 383.0630{{c}}

~~~~

Line 539:

Line 666:

Scales: [[malco]]

~~== Prodigy ==~~

Prodigy shrinks 1024/1029, 243/242, 384/385 and 2400/2401 down to the same tiny interval. Hence in practice it probably makes the most sense to temper this out as well, leading to [[miracle]]. This, however, does not render it pointless to consider prodigy; for one thing, scales in prodigy such as hobbit scales translate into interesting scales for miracle.

~~[[Subgroup]]: 2.3.5.7.11~~

~~[[Comma list]]: 225/224, 441/440~~

~~{{Mapping|legend=1| 1 0 0 -5 -13 | 0 1 0 2 6 | 0 0 1 2 3 }}~~

~~Map to lattice: [{{val| 0 0 -1 -2 -3 }}, {{val| 0 1 -1 0 3 }}]~~

~~Lattice basis:~~

~~: ~15/14 length = 0.9111, ~3/2 length = 0.9477~~

~~: angle (~15/14, ~3/2) = 65.933°~~

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

* [[WE]]: ~2 = 1200.7854{{c}}, ~3/2 = 700.2562{{c}}, ~5/4 = 383.7624{{c}}

~~: [[error map]]: {{val| +0.785 -0.913 -0.980 -0.003 +0.721 }}~~

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 699.8610{{c}}, ~5/4 = 383.7724{{c}}

~~: error map: {{val| 0.000 -2.094 -2.541 -1.559 -0.835 }}~~

~~~~

~~[[Minimax tuning]]:~~

* [[11-odd-limit]]

~~: [{{monzo| 1 0 0 0 0 }}, {{monzo| 13/12 1/2 -1/4 0 1/12 }}, {{monzo| 13/6 -1 1/2 0 1/6 }}, {{monzo| 3/2 -1 1/2 0 1/2 }}, {{monzo| 0 0 0 0 1 }}]~~

~~: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5.11~~

~~{{Optimal ET sequence|legend=1| 10, 12, 19e, 29, 31, 41, 60e, 72, 247c, 319bcde, 391bcde, 463bccde }}~~

~~[[Badness]] (Sintel): 0.402~~

~~[[Projection pair]]s: <code>7 225/32 11 91125/8192</code>~~

~~Scales: [[prodigy11]], [[prodigy12]], [[prodigy29]]~~

~~[[Associated temperament]]: [[miracle]]~~

~~{{Databox|Hobbit bases|~~

~~2.3.5 subgroup~~

* 31: 81/80, 34171875/33554432

* 41: 34171875/33554432, 32805/32768

}}

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

~~Subgroup: 2.3.5.7.11.13~~

~~Comma list: 105/104, 196/195, 352/351~~

~~Mapping: {{mapping| 1 0 0 -5 -13 -8 | 0 1 0 2 6 3 | 0 0 1 2 3 3 }}~~

~~Optimal tunings:~~

* WE: ~2 = 1200.8252{{c}}, ~3/2 = 700.8823{{c}}, ~5/4 = 381.6647{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.4689{{c}}, ~5/4 = 381.6687{{c}}

~~~~

~~{{Optimal ET sequence|legend=0| 10, 12f, 19e, 29, 31, 41, 60e, 72f, 101cd }}~~

~~Badness (Sintel): 0.689~~

~~=== Prodigious ===~~

~~Subgroup: 2.3.5.7.11.13~~

~~Comma list: 225/224, 364/363, 441/440~~

~~Mapping: {{mapping| 1 0 0 -5 -13 -23 | 0 1 0 2 6 11 | 0 0 1 2 3 4 }}~~

~~Optimal tunings:~~

* WE: ~2 = 1200.6284{{c}}, ~3/2 = 700.7075{{c}}, ~5/4 = 383.4599{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.3302{{c}}, ~5/4 = 383.5030{{c}}

~~~~

~~{{Optimal ET sequence|legend=0| 12f, 29, 31f, 41, 72, 185cf, 257cff }}~~

~~Badness (Sintel): 0.841~~

~~=== Prodigal ===~~

~~Subgroup: 2.3.5.7.11.13~~

~~Comma list: 225/224, 351/350, 441/440~~

~~Mapping: {{mapping| 1 0 0 -5 -13 -4 | 0 1 0 2 6 -1 | 0 0 1 2 3 4 }}~~

~~Optimal tunings:~~

* WE: ~2 = 1200.7798{{c}}, ~3/2 = 699.9410{{c}}, ~5/4 = 384.3494{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 699.5538{{c}}, ~5/4 = 384.3496{{c}}

~~~~

~~{{Optimal ET sequence|legend=0| 12f, 19e, 31, 53e, 60eff, 72, 103, 175f }}~~

~~Badness (Sintel): 0.831~~

~~=== Protannic ===~~

~~Subgroup: 2.3.5.7.11.13~~

~~Comma list: 225/224, 441/440, 1001/1000~~

~~Mapping: {{mapping| 1 0 0 -5 -13 21 | 0 1 0 2 6 -8 | 0 0 1 2 3 -2 }}~~

~~Optimal tunings:~~

* WE: ~2 = 1200.9450{{c}}, ~3/2 = 700.1045{{c}}, ~5/4 = 383.8716{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 699.4224{{c}}, ~5/4 = 383.9828{{c}}

~~~~

~~{{Optimal ET sequence|legend=0| 29, 31, 43, 60e, 72, 103, 175f, 482bccddeefff, 554bbccddeeeffff }}~~

~~Badness (Sintel): 0.891~~

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

~~Subgroup: 2.3.5.7.11.13.17~~

~~Comma list: 225/224, 273/272, 375/374, 441/440~~

~~Mapping: {{mapping| 1 0 0 -5 -13 21 12 | 0 1 0 2 6 -8 -5 | 0 0 1 2 3 -2 0 }}~~

~~Optimal tunings:~~

* WE: ~2 = 1200.9342{{c}}, ~3/2 = 700.1708{{c}}, ~5/4 = 383.7444{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 699.4849{{c}}, ~5/4 = 383.8744{{c}}

~~~~

~~{{Optimal ET sequence|legend=0| 29g, 31, 43, 60e, 72, 103, 175f, 307bcdeeffg, 379bccdeeffgg, 482bccddeefffgg, 554bbccddeeeffffgg }}~~

~~Badness (Sintel): 0.734~~

== Fantastic ==

Line 678:

Line 682:

* [[CWE]]: ~99/70 = 600.0000{{c}}, ~3/2 = 700.8160{{c}}, ~5/4 = 383.5350{{c}}

: error map: {{val| 0.000 -1.139 -2.779 -0.124 -0.713 }}

~~~~

{{Optimal ET sequence|legend=1| 12, 22, 34d, 50, 60e, 72, 166, 238c, 310c }}

Line 699:

Line 702:

* [[CWE]]: ~2 = 1200.0000{{c}}, ~400/231 = 950.1211{{c}}, ~5/4 = 383.9530{{c}}

: error map: {{val| 0.000 -1.712 -2.361 -0.436 -2.165 }}

~~~~

{{Optimal ET sequence|legend=1| 19, 29, 43, 53, 72, 197e, 269ce, 341ce }}

Line 715:

Line 717:

* WE: ~2 = 1200.8238{{c}}, ~26/15 = 950.8873{{c}}, ~5/4 = 383.8191{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 950.2108{{c}}, ~5/4 = 383.9486{{c}}

~~~~

{{Optimal ET sequence|legend=0| 19, 29, 43, 53, 72, 125f, 197ef }}

Line 734:

Line 735:

* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 701.1108{{c}}, ~55/42 = 460.5488{{c}}

: error map: {{val| 0.000 -0.844 -2.446 +1.131 -3.569 }}

~~~~

[[Badness]] (Sintel): 1.38

~~== Catakleismoid ==~~

~~Catakleismoid is the same as catakleismic in the 2.3.5.7.13 subgroup but with an independent generator for prime 11.~~

~~[[Subgroup]]: 2.3.5.7.11~~

~~[[Comma list]]: 225/224, 4375/4374~~

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

~~: mapping generators: ~2, ~6/5, ~11~~

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

* [[WE]]: ~2 = 1200.5965{{c}}, ~6/5 = 316.8893{{c}}, ~11/8 = 549.5258{{c}}

~~: [[error map]]: {{val| +0.596 -0.619 -1.271 +0.948 -0.003 }}~~

* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 316.7705{{c}}, ~11/8 = 550.0343{{c}}

~~: error map: {{val| 0.000 -1.332 -2.461 +0.126 -1.284 }}~~

~~~~

~~{{Optimal ET sequence|legend=1| 19, 34d, 53, 72, 197e, 269ce, 341ce }}~~

~~[[Badness]] (Sintel): 1.53~~

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

~~Subgroup: 2.3.5.7.11.13~~

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

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

~~Optimal tunings:~~

* WE: ~2 = 1200.8238{{c}}, ~6/5 = 316.9478{{c}}, ~11/8 = 548.9611{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.7939{{c}}, ~11/8 = 549.5903{{c}}

~~~~

~~Optimal ET sequence: {{Optimal ET sequence| 19, 34d, 53, 72, 125f, 197ef }}~~

~~Badness (Sintel): 0.853~~

== Subgroup extensions ==

Line 788:

Line 751:

* WE: ~2 = 1199.3681{{c}}, ~3/2 = 701.3669{{c}}, ~5/4 = 384.9641{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.6286{{c}}, ~5/4 = 385.0841{{c}}

~~~~

Line 804:

Line 766:

* WE: ~2 = 1200.3454{{c}}, ~3/2 = 701.1728{{c}}, ~5/4 = 383.7499{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.9546{{c}}, ~5/4 = 383.7912{{c}}

~~~~

{{Optimal ET sequence|legend=0| 10, 12, 19, 22, 31, 41, 53, 72, 94, 113, 125, 291c }}

@@ Line 1: / Line 1: @@
-{{interwiki
-| en = Marvel family
-| de = Marvel
-| es =
-| ja =
-}}
 {{Technical data page}}
 The '''marvel family''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[7-limit]] [[marvel comma]] ([[ratio]]: [[225/224]], {{monzo|legend=1| -5 2 2 -1 }}), also known as ''septimal kleisma''. These temperaments hence equate [[16/15]] and [[15/14]], or equivalently they equate two [[5/4]]'s and one [[14/9]]. The marvel comma is noteworthy in that it is tempered out by many common [[equal temperament|equal]] and [[rank-2 temperament]]s.
@@ Line 13: / Line 7: @@
 {{Main| Marvel }}
-The head of the marvel family is marvel, which tempers out [[225/224]]. Marvel has a [[normal lists|normal generator list]] of {2, 3, 5}; hence a [[5-limit]] scale can be converted to marvel simply by tempering it. One way to do that, and an excellent marvel tuning, is given by [[197edo]].
+The head of the marvel family is marvel, which tempers out [[225/224]]. Marvel has a [[normal forms|normal generator list]] of {2, 3, 5}; hence a [[5-limit]] scale can be converted to marvel simply by tempering it. One way to do that, and an excellent marvel tuning, is given by [[197edo]].
-Little is gained in tuning accuracy by not tempering out 4375/4374 as well as 225/224, leading to [[catakleismic]], with which it shares the [[optimal patent val]]. Another temperament which does little damage to tuning accuracy is [[compton]], for which [[240edo]] may be used. See [[marvel temperaments]] for some other rank-2 temperaments.
+Little is gained in tuning accuracy by not tempering out 4375/4374 as well as 225/224, leading to [[catakleismic]], with which it shares the [[optimal patent val]]. Another temperament which does little damage to tuning accuracy is [[compton]], for which [[240edo]] may be used. See [[Marvel temperaments]] for some other rank-2 temperaments.
 [[Subgroup]]: 2.3.5.7
@@ Line 35: / Line 29: @@
 * [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 700.6222{{c}}, ~5/4 = 383.8540{{c}}
 : error map: {{val| 0.000 -1.333 -2.460 +0.127 }}
-<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 700.974{{c}}, ~5/4 = 384.208{{c}} -->
 [[Minimax tuning]]:
@@ Line 73: / Line 66: @@
 === Overview to extensions ===
-The second comma of the [[Normal lists #Normal interval list|normal comma list]] defines which 11-limit family member we are looking at. 4125/4096 gives unidecimal marvel;  91125/90112 gives prodigy; 5632/5625 gives minerva. These and many others considered below use the same generators as marvel.
+The second comma of the [[normal forms #Normal forms for commas|normal comma list]] defines which 11-limit family member we are looking at. 4125/4096 gives unidecimal marvel; 91125/90112 gives prodigy; 5632/5625 gives minerva. These and many others considered below use the same generators as marvel.
 Temperaments discussed elsewhere include
@@ Line 79: / Line 72: @@
 * ''[[Artemis]]'' (+121/120) → [[Biyatismic clan #Artemis|Biyatismic clan]]
 * ''[[Spectacle]]'' (+243/242) → [[Rastmic rank-3 clan #Spectacle|Rastmic rank-3 clan]]
+* ''[[Marvelpine]]'' (+4000/3993) → [[Wizardharry clan #Marvelpine|Wizardharry clan]]
 * ''[[Mirage]]'' (+243/242, +385/384) → [[Rastmic rank-3 clan #Spectacle|Rastmic rank-3 clan]]
+* ''[[Catakleismoid]]'' (+4375/4374) → [[Kleismic rank-3 family #Catakleismoid|Kleismic rank-3 family]]
-== Undecimal marvel (unimarv) ==
+== Undecimal marvel ==
 {{Main| Marvel }}
+Undecimal marvel tempers out [[385/384]] as well as [[540/539]], and is loosely [[associated temperament|associated]] with [[wizard]]. This extension is natural because of the factorization 225/224 = (385/384)⋅(540/539). [[197edo]] remains useful as a tuning, with the 197e val, but [[166edo]], which among other things has a virtually pure 7, works as well.
+In the 13-limit, 225/224 factors as ([[351/350]])⋅([[625/624]]) or ([[325/324]])⋅([[729/728]]). Tempering out 351/350 and 625/624 leads to helios, tempering out 325/324 and 729/728 leads to hecate. Tempering out all of them leads to the 13-limit version of [[catakleismic]].
 [[Subgroup]]: 2.3.5.7.11
@@ Line 101: / Line 100: @@
 * [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 700.6048{{c}}, ~5/4 = 383.4538{{c}}
 : error map: {{val| 0.000 -1.350 -2.860 -0.709 -2.284 }}
-<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 701.373{{c}}, ~5/4 = 383.146{{c}} -->
 [[Minimax tuning]]:
@@ Line 115: / Line 113: @@
 [[Complexity spectrum]]: 5/4, 4/3, 7/6, 8/7, 7/5, 6/5, 9/7, 12/11, 9/8, 11/8, 11/9, 10/9, 11/10, 14/11
-[[Associated temperament]]: [[catakleismic]]
 Scales: [[marvel22_11]], [[unimarv19]], [[unimarv22]]
@@ Line 130: / Line 126: @@
 }}
-=== 13-limit ===
+=== Helios ===
 Subgroup: 2.3.5.7.11.13
@@ Line 140: / Line 136: @@
 * WE: ~2 = 1200.8043{{c}}, ~3/2 = 700.2057{{c}}, ~5/4 = 384.3188{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 699.8109{{c}}, ~5/4 = 384.1177{{c}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 700.182{{c}}, ~5/4 = 384.400{{c}} -->
 Minimax tuning:
@@ Line 153: / Line 148: @@
 === Hecate ===
+Hecate tempers out [[325/324]], the marveltwin comma, such that [[16/13]] is found by a stack of two ~[[10/9]]'s, similar to how [[8/7]] is found by a stack of two [[15/14]]~[[16/15]]'s. Hecate has a natural extension to include [[prime interval|prime]] [[19/1|19]], where it further tempers out [[400/399]] and [[513/512]], taking advantage of the factorization 225/224 = (400/399)⋅(513/512). For both of these cases [[166edo]] remains an excellent tuning.
 ==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
@@ Line 163: / Line 160: @@
 * WE: ~2 = 1200.5788{{c}}, ~3/2 = 701.3161{{c}}, ~5/4 = 383.3471{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.1003{{c}}, ~5/4 = 383.1315{{c}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 701.563{{c}}, ~5/4 = 383.015{{c}} -->
 Minimax tuning:
@@ Line 187: / Line 183: @@
 * WE: ~2 = 1200.4716{{c}}, ~3/2 = 701.4395{{c}}, ~5/4 = 383.2779{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.2002{{c}}, ~5/4 = 383.1136{{c}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 701.424{{c}}, ~5/4 = 383.029{{c}} -->
 {{Optimal ET sequence|legend=0| 41, 53, 72, 94, 113, 166 }}
@@ Line 203: / Line 198: @@
 * WE: ~2 = 1200.6057{{c}}, ~3/2 = 701.3157{{c}}, ~5/4 = 383.2243{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.0877{{c}}, ~5/4 = 382.9709{{c}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 701.586{{c}}, ~5/4 = 382.733{{c}} -->
 {{Optimal ET sequence|legend=0| 41, 53g, 72, 166g, 238cfg }}
@@ Line 219: / Line 213: @@
 * WE: ~2 = 1200.4927{{c}}, ~3/2 = 701.4506{{c}}, ~5/4 = 383.1291{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.2007{{c}}, ~5/4 = 382.9404{{c}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 701.443{{c}}, ~5/4 = 382.739{{c}} -->
 {{Optimal ET sequence|legend=0| 41, 53g, 72, 94, 113, 166g }}
@@ Line 235: / Line 228: @@
 * WE: ~2 = 1200.6165{{c}}, ~3/2 = 701.3259{{c}}, ~5/4 = 383.5032{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.0949{{c}}, ~5/4 = 383.3140{{c}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 701.627{{c}}, ~5/4 = 383.346{{c}} -->
 {{Optimal ET sequence|legend=0| 41g, 53, 72, 166g, 238cfg }}
@@ Line 251: / Line 243: @@
 * WE: ~2 = 1200.5038{{c}}, ~3/2 = 701.4654{{c}}, ~5/4 = 383.4549{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.2113{{c}}, ~5/4 = 383.3052{{c}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 701.479{{c}}, ~5/4 = 383.379{{c}} -->
 {{Optimal ET sequence|legend=0| 41g, 53, 72, 94, 125f, 166g }}
@@ Line 267: / Line 258: @@
 * WE: ~2 = 1200.6404{{c}}, ~3/2 = 700.7675{{c}}, ~5/4 = 383.7772{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.6113{{c}}, ~5/4 = 383.4925{{c}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 701.385{{c}}, ~5/4 = 383.208{{c}} -->
 Minimax tuning:
@@ Line 282: / Line 272: @@
 Comma list: 225/224, 275/273, 385/384
-Mapping: {{mapping| 1 0 0 -5 12 17 | 0 1 0 2 -1 4 | 0 0 1 2 -3 -3 }}
+Mapping: {{mapping| 1 0 0 -5 12 17 | 0 1 0 2 -1 -4 | 0 0 1 2 -3 -3 }}
 Optimal tunings:
 * WE: ~2 = 1200.1824{{c}}, ~3/2 = 702.0223{{c}}, ~5/4 = 383.3028{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.9276{{c}}, ~5/4 = 383.2283{{c}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 701.997{{c}}, ~5/4 = 383.134{{c}} -->
 {{Optimal ET sequence|legend=0| 10, 19f, 22, 31, 41, 53, 84e, 94 }}
@@ Line 305: / Line 294: @@
 * WE: ~2 = 1200.7533{{c}}, ~3/2 = 700.8957{{c}}, ~5/4 = 383.0580{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.7184{{c}}, ~5/4 = 382.6611{{c}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 701.702{{c}}, ~5/4 = 382.074{{c}} -->
 Minimax tuning:
@@ Line 327: / Line 315: @@
 * WE: ~2 = 1200.4789{{c}}, ~3/2 = 699.5125{{c}}, ~5/4 = 383.1304{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 699.4041{{c}}, ~5/4 = 382.9166{{c}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 700.028{{c}}, ~5/4 = 382.694{{c}} -->
 Minimax tuning:
@@ Line 350: / Line 337: @@
 * WE: ~2 = 1200.7720{{c}}, ~3/2 = 950.8976{{c}}, ~5/4 = 383.8283{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~26/15 = 950.4265{{c}}, ~5/4 = 383.4790{{c}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~26/15 = 950.961{{c}}, ~5/4 = 383.088{{c}} -->
 {{Optimal ET sequence|legend=0| 9, 10, 19, 43e, 44, 53, 72, 125f, 197ef }}
 Badness (Sintel): 0.935
+== Prodigy ==
+Prodigy tempers out [[441/440]] and shrinks [[243/242]], [[385/384|384/385]], [[1029/1024|1024/1029]] and [[2401/2400|2400/2401]] down to the same tiny interval. Hence in practice it probably makes the most sense to temper this out as well, leading to [[miracle]]. This, however, does not render it pointless to consider prodigy; for one thing, scales in prodigy such as [[hobbit]] scales translate into interesting scales for miracle.
+[[Subgroup]]: 2.3.5.7.11
+[[Comma list]]: 225/224, 441/440
+{{Mapping|legend=1| 1 0 0 -5 -13 | 0 1 0 2 6 | 0 0 1 2 3 }}
+Map to lattice: [{{val| 0 0 -1 -2 -3 }}, {{val| 0 1 -1 0 3 }}]
+Lattice basis:
+: ~15/14 length = 0.9111, ~3/2 length = 0.9477
+: angle (~15/14, ~3/2) = 65.933°
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.7854{{c}}, ~3/2 = 700.2562{{c}}, ~5/4 = 383.7624{{c}}
+: [[error map]]: {{val| +0.785 -0.913 -0.980 -0.003 +0.721 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 699.8610{{c}}, ~5/4 = 383.7724{{c}}
+: error map: {{val| 0.000 -2.094 -2.541 -1.559 -0.835 }}
+[[Minimax tuning]]:
+* [[11-odd-limit]]
+: [{{monzo| 1 0 0 0 0 }}, {{monzo| 13/12 1/2 -1/4 0 1/12 }}, {{monzo| 13/6 -1 1/2 0 1/6 }}, {{monzo| 3/2 -1 1/2 0 1/2 }}, {{monzo| 0 0 0 0 1 }}]
+: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5.11
+{{Optimal ET sequence|legend=1| 10, 12, 19e, 29, 31, 41, 60e, 72, 247c, 319bcde, 391bcde, 463bccde }}
+[[Badness]] (Sintel): 0.402
+[[Projection pair]]s: <code>7 225/32 11 91125/8192</code>
+Scales: [[prodigy11]], [[prodigy12]], [[prodigy29]]
+{{Databox|Hobbit bases|
+.3.5 subgroup
+* 31: 81/80, 34171875/33554432
+* 41: 34171875/33554432, 32805/32768
+}}
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
+Comma list: 105/104, 196/195, 352/351
+Mapping: {{mapping| 1 0 0 -5 -13 -8 | 0 1 0 2 6 3 | 0 0 1 2 3 3 }}
+Optimal tunings:
+* WE: ~2 = 1200.8252{{c}}, ~3/2 = 700.8823{{c}}, ~5/4 = 381.6647{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.4689{{c}}, ~5/4 = 381.6687{{c}}
+{{Optimal ET sequence|legend=0| 10, 12f, 19e, 29, 31, 41, 60e, 72f, 101cd }}
+Badness (Sintel): 0.689
+=== Prodigious ===
+Subgroup: 2.3.5.7.11.13
+Comma list: 225/224, 364/363, 441/440
+Mapping: {{mapping| 1 0 0 -5 -13 -23 | 0 1 0 2 6 11 | 0 0 1 2 3 4 }}
+Optimal tunings:
+* WE: ~2 = 1200.6284{{c}}, ~3/2 = 700.7075{{c}}, ~5/4 = 383.4599{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.3302{{c}}, ~5/4 = 383.5030{{c}}
+{{Optimal ET sequence|legend=0| 12f, 29, 31f, 41, 72, 185cf, 257cff }}
+Badness (Sintel): 0.841
+=== Prodigal ===
+Subgroup: 2.3.5.7.11.13
+Comma list: 225/224, 351/350, 441/440
+Mapping: {{mapping| 1 0 0 -5 -13 -4 | 0 1 0 2 6 -1 | 0 0 1 2 3 4 }}
+Optimal tunings:
+* WE: ~2 = 1200.7798{{c}}, ~3/2 = 699.9410{{c}}, ~5/4 = 384.3494{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 699.5538{{c}}, ~5/4 = 384.3496{{c}}
+{{Optimal ET sequence|legend=0| 12f, 19e, 31, 53e, 60eff, 72, 103, 175f }}
+Badness (Sintel): 0.831
+=== Protannic ===
+Subgroup: 2.3.5.7.11.13
+Comma list: 225/224, 441/440, 1001/1000
+Mapping: {{mapping| 1 0 0 -5 -13 21 | 0 1 0 2 6 -8 | 0 0 1 2 3 -2 }}
+Optimal tunings:
+* WE: ~2 = 1200.9450{{c}}, ~3/2 = 700.1045{{c}}, ~5/4 = 383.8716{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 699.4224{{c}}, ~5/4 = 383.9828{{c}}
+{{Optimal ET sequence|legend=0| 29, 31, 43, 60e, 72, 103, 175f, 482bccddeefff, 554bbccddeeeffff }}
+Badness (Sintel): 0.891
+==== 17-limit ====
+Subgroup: 2.3.5.7.11.13.17
+Comma list: 225/224, 273/272, 375/374, 441/440
+Mapping: {{mapping| 1 0 0 -5 -13 21 12 | 0 1 0 2 6 -8 -5 | 0 0 1 2 3 -2 0 }}
+Optimal tunings:
+* WE: ~2 = 1200.9342{{c}}, ~3/2 = 700.1708{{c}}, ~5/4 = 383.7444{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 699.4849{{c}}, ~5/4 = 383.8744{{c}}
+{{Optimal ET sequence|legend=0| 29g, 31, 43, 60e, 72, 103, 175f, 307bcdeeffg, 379bccdeeffgg, 482bccddeefffgg, 554bbccddeeeffffgg }}
+Badness (Sintel): 0.734
 == Minerva ==
+Minerva tempers out [[99/98]] as well as [[176/175]]. It may be described as 12 & 22 & 31, and is loosely [[associated temperament|associated]] with [[würschmidt]].
 [[Subgroup]]: 2.3.5.7.11
@@ Line 374: / Line 477: @@
 * [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 700.3006{{c}}, ~5/4 = 386.5785{{c}}
 : error map: {{val| 0.000 -1.654 +0.265 +4.932 -4.403 }}
-<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 700.379{{c}}, ~5/4 = 386.617{{c}} -->
 [[Minimax tuning]]: [[11-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5.11/9
@@ Line 385: / Line 487: @@
 Scales: [[minerva12]], [[minerva22x]]
-[[Associated temperament]]: [[Würschmidt family #Würschmidt|würschmidt]]
 === Athene ===
@@ Line 398: / Line 498: @@
 * WE: ~2 = 1199.9127{{c}}, ~3/2 = 701.1832{{c}}, ~5/4 = 385.9313{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.2143{{c}}, ~5/4 = 385.9336{{c}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 701.179{{c}}, ~5/4 = 385.888{{c}} -->
 Minimax tuning:
@@ Line 412: / Line 511: @@
 == Apollo ==
 {{See also| Ptolemismic clan #Apollo }}
+[[File:Lattice Apollo.png|thumb|Lattice for apollo.]]
+Apollo tempers out not only [[100/99]] but [[896/891]]. Note that marvel tempers together [[25/24]] and [[28/27]], and apollo further equates it with [[33/32]] via the vanishing of 100/99. This makes it a weak extension of [[parapyth]], and [[associated temperament|associates]] it with [[magic]]. The lattice structure is very compact, comparable to that of [[ares]], from which apollo only differs in the mapping of [[prime interval|prime]] [[7/1|7]].
+The canonical [[13-limit]] extension is implied by parapyth, tempering out [[352/351]] and [[364/363]], but there are a number of other extenions to consider, these being called phoebus and musagetes, after epithets of Apollo. Phoebus tempers out [[105/104]] and finds ~[[16/13]] as a stack of three [[secor]]s. Musagetes tempers out [[144/143]] and conflates 16/13 and [[11/9]], which in this case is simply a stack of two ~[[10/9]]'s. These extensions are [[support]]ed by 13-limit [[magic]], unlike the canonical one.
 [[Subgroup]]: 2.3.5.7.11
@@ Line 424: / Line 529: @@
 * [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.4612{{c}}, ~5/4 = 381.5071{{c}}
 : error map: {{val| 0.000 +1.506 -4.807 +1.111 +4.774 }}
-<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 703.418{{c}}, ~5/4 = 381.331{{c}} -->
 [[Minimax tuning]]: [[11-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5.11/9
@@ Line 433: / Line 537: @@
 [[Projection pair]]s: <code>7 225/32 11 100/9</code>
-[[Associated temperament]]: [[magic]]
 Scales: [[apollo wholetone]], [[indigo17]]
@@ Line 448: / Line 550: @@
 * WE: ~2 = 1199.6919{{c}}, ~3/2 = 703.8176{{c}}, ~5/4 = 381.4372{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.9853{{c}}, ~5/4 = 381.3579{{c}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 703.959{{c}}, ~5/4 = 381.001{{c}} -->
 Minimax tuning: 13-odd-limit unchanged-interval (eigenmonzo) basis: 2.11/9.13/9
@@ Line 457: / Line 558: @@
 Projection pairs: <code>7 225/32 11 100/9 13 3200/243</code>
+=== Phoebus ===
+Subgroup: 2.3.5.7.11.13
+Comma list: 100/99, 105/104, 196/195
+Mapping: {{mapping| 1 0 0 -5 2 1 | 0 1 0 2 -2 3 | 0 0 1 2 2 3 }}
+Optimal tunings:
+* WE: ~2 = 1200.3724{{c}}, ~3/2 = 702.5274{{c}}, ~5/4 = 379.6345{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.3357{{c}}, ~5/4 = 379.6830{{c}}
+{{Optimal ET sequence|legend=0| 12f, 19, 22f, 29, 41 }}
+Badness (Sintel): 0.886
+=== Musagetes ===
+Subgroup: 2.3.5.7.11.13
+Comma list: 100/99, 144/143, 225/224
+Mapping: {{mapping| 1 0 0 -5 2 2 | 0 1 0 2 -2 4 | 0 0 1 2 2 -2 }}
+Optimal tunings:
+* WE: ~2 = 1199.2695{{c}}, ~3/2 = 702.5589{{c}}, ~5/4 = 382.7715{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.7998{{c}}, ~5/4 = 382.7740{{c}}
+{{Optimal ET sequence|legend=0| 19, 22f, 34d, 41, 75e, 94e, 116ef }}
+Badness (Sintel): 1.14
 == Potassium ==
@@ Line 470: / Line 601: @@
 * [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 696.0586{{c}}, ~5/4 = 384.9472{{c}}
 : error map: {{val| 0.000 -5.896 -1.367 -6.814 +25.746 }}
-<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 695.937{{c}}, ~5/4 = 384.836{{c}} -->
 [[Minimax tuning]]: [[11-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7.11
@@ Line 490: / Line 620: @@
 * WE: ~2 = 1199.8192{{c}}, ~3/2 = 695.9054{{c}}, ~5/4 = 384.6205{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 695.9480{{c}}, ~5/4 = 384.6372{{c}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 695.894{{c}}, ~5/4 = 384.602{{c}} -->
 Minimax tuning:
@@ Line 516: / Line 645: @@
 * [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.0535{{c}}, ~5/4 = 382.6222{{c}}
 : error map: {{val| 0.000 +0.098 -3.692 +0.525 -2.188 }}
-<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 702.322{{c}}, ~5/4 = 382.976{{c}} -->
 {{Optimal ET sequence|legend=1| 12e, 19e, 34d, 41, 53, 60e, 94, 229c, 248ce, 289cce }}
@@ Line 532: / Line 660: @@
 * WE: ~2 = 1200.2427{{c}}, ~3/2 = 702.1575{{c}}, ~5/4 = 383.0704{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0882{{c}}, ~5/4 = 383.0630{{c}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 702.235{{c}}, ~5/4 = 383.205{{c}} -->
 {{Optimal ET sequence|legend=0| 12e, 19e, 34d, 41, 53, 94 }}
@@ Line 539: / Line 666: @@
 Scales: [[malco]]
-== Prodigy ==
-Prodigy shrinks 1024/1029, 243/242, 384/385 and 2400/2401 down to the same tiny interval. Hence in practice it probably makes the most sense to temper this out as well, leading to [[miracle]]. This, however, does not render it pointless to consider prodigy; for one thing, scales in prodigy such as hobbit scales translate into interesting scales for miracle.
-[[Subgroup]]: 2.3.5.7.11
-[[Comma list]]: 225/224, 441/440
-{{Mapping|legend=1| 1 0 0 -5 -13 | 0 1 0 2 6 | 0 0 1 2 3 }}
-Map to lattice: [{{val| 0 0 -1 -2 -3 }}, {{val| 0 1 -1 0 3 }}]
-Lattice basis:
-: ~15/14 length = 0.9111, ~3/2 length = 0.9477
-: angle (~15/14, ~3/2) = 65.933°
-[[Optimal tuning]]s:
-* [[WE]]: ~2 = 1200.7854{{c}}, ~3/2 = 700.2562{{c}}, ~5/4 = 383.7624{{c}}
-: [[error map]]: {{val| +0.785 -0.913 -0.980 -0.003 +0.721 }}
-* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 699.8610{{c}}, ~5/4 = 383.7724{{c}}
-: error map: {{val| 0.000 -2.094 -2.541 -1.559 -0.835 }}
-<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 699.995{{c}}, ~5/4 = 384.330{{c}} -->
-[[Minimax tuning]]:
-* [[11-odd-limit]]
-: [{{monzo| 1 0 0 0 0 }}, {{monzo| 13/12 1/2 -1/4 0 1/12 }}, {{monzo| 13/6 -1 1/2 0 1/6 }}, {{monzo| 3/2 -1 1/2 0 1/2 }}, {{monzo| 0 0 0 0 1 }}]
-: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5.11
-{{Optimal ET sequence|legend=1| 10, 12, 19e, 29, 31, 41, 60e, 72, 247c, 319bcde, 391bcde, 463bccde }}
-[[Badness]] (Sintel): 0.402
-[[Projection pair]]s: <code>7 225/32 11 91125/8192</code>
-Scales: [[prodigy11]], [[prodigy12]], [[prodigy29]]
-[[Associated temperament]]: [[miracle]]
-{{Databox|Hobbit bases|
-.3.5 subgroup
-* 31: 81/80, 34171875/33554432
-* 41: 34171875/33554432, 32805/32768
-}}
-=== 13-limit ===
-Subgroup: 2.3.5.7.11.13
-Comma list: 105/104, 196/195, 352/351
-Mapping: {{mapping| 1 0 0 -5 -13 -8 | 0 1 0 2 6 3 | 0 0 1 2 3 3 }}
-Optimal tunings:
-* WE: ~2 = 1200.8252{{c}}, ~3/2 = 700.8823{{c}}, ~5/4 = 381.6647{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.4689{{c}}, ~5/4 = 381.6687{{c}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 700.617{{c}}, ~5/4 = 382.244{{c}} -->
-{{Optimal ET sequence|legend=0| 10, 12f, 19e, 29, 31, 41, 60e, 72f, 101cd }}
-Badness (Sintel): 0.689
-=== Prodigious ===
-Subgroup: 2.3.5.7.11.13
-Comma list: 225/224, 364/363, 441/440
-Mapping: {{mapping| 1 0 0 -5 -13 -23 | 0 1 0 2 6 11 | 0 0 1 2 3 4 }}
-Optimal tunings:
-* WE: ~2 = 1200.6284{{c}}, ~3/2 = 700.7075{{c}}, ~5/4 = 383.4599{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.3302{{c}}, ~5/4 = 383.5030{{c}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 700.305{{c}}, ~5/4 = 384.075{{c}} -->
-{{Optimal ET sequence|legend=0| 12f, 29, 31f, 41, 72, 185cf, 257cff }}
-Badness (Sintel): 0.841
-=== Prodigal ===
-Subgroup: 2.3.5.7.11.13
-Comma list: 225/224, 351/350, 441/440
-Mapping: {{mapping| 1 0 0 -5 -13 -4 | 0 1 0 2 6 -1 | 0 0 1 2 3 4 }}
-Optimal tunings:
-* WE: ~2 = 1200.7798{{c}}, ~3/2 = 699.9410{{c}}, ~5/4 = 384.3494{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 699.5538{{c}}, ~5/4 = 384.3496{{c}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 699.698{{c}}, ~5/4 = 384.884{{c}} -->
-{{Optimal ET sequence|legend=0| 12f, 19e, 31, 53e, 60eff, 72, 103, 175f }}
-Badness (Sintel): 0.831
-=== Protannic ===
-Subgroup: 2.3.5.7.11.13
-Comma list: 225/224, 441/440, 1001/1000
-Mapping: {{mapping| 1 0 0 -5 -13 21 | 0 1 0 2 6 -8 | 0 0 1 2 3 -2 }}
-Optimal tunings:
-* WE: ~2 = 1200.9450{{c}}, ~3/2 = 700.1045{{c}}, ~5/4 = 383.8716{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 699.4224{{c}}, ~5/4 = 383.9828{{c}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 698.947{{c}}, ~5/4 = 385.480{{c}} -->
-{{Optimal ET sequence|legend=0| 29, 31, 43, 60e, 72, 103, 175f, 482bccddeefff, 554bbccddeeeffff }}
-Badness (Sintel): 0.891
-==== 17-limit ====
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 225/224, 273/272, 375/374, 441/440
-Mapping: {{mapping| 1 0 0 -5 -13 21 12 | 0 1 0 2 6 -8 -5 | 0 0 1 2 3 -2 0 }}
-Optimal tunings:
-* WE: ~2 = 1200.9342{{c}}, ~3/2 = 700.1708{{c}}, ~5/4 = 383.7444{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 699.4849{{c}}, ~5/4 = 383.8744{{c}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 698.969{{c}}, ~5/4 = 385.440{{c}} -->
-{{Optimal ET sequence|legend=0| 29g, 31, 43, 60e, 72, 103, 175f, 307bcdeeffg, 379bccdeeffgg, 482bccddeefffgg, 554bbccddeeeffffgg }}
-Badness (Sintel): 0.734
 == Fantastic ==
@@ Line 678: / Line 682: @@
 * [[CWE]]: ~99/70 = 600.0000{{c}}, ~3/2 = 700.8160{{c}}, ~5/4 = 383.5350{{c}}
 : error map: {{val| 0.000 -1.139 -2.779 -0.124 -0.713 }}
-<!-- * [[CTE]]: ~99/70 = 600.000{{c}}, ~3/2 = 701.132{{c}}, ~5/4 = 383.926{{c}} -->
 {{Optimal ET sequence|legend=1| 12, 22, 34d, 50, 60e, 72, 166, 238c, 310c }}
@@ Line 699: / Line 702: @@
 * [[CWE]]: ~2 = 1200.0000{{c}}, ~400/231 = 950.1211{{c}}, ~5/4 = 383.9530{{c}}
 : error map: {{val| 0.000 -1.712 -2.361 -0.436 -2.165 }}
-<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~400/231 = 950.043{{c}}, ~5/4 = 384.858{{c}} -->
 {{Optimal ET sequence|legend=1| 19, 29, 43, 53, 72, 197e, 269ce, 341ce }}
@@ Line 715: / Line 717: @@
 * WE: ~2 = 1200.8238{{c}}, ~26/15 = 950.8873{{c}}, ~5/4 = 383.8191{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~26/15 = 950.2108{{c}}, ~5/4 = 383.9486{{c}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~26/15 = 950.131{{c}}, ~5/4 = 385.245{{c}} -->
 {{Optimal ET sequence|legend=0| 19, 29, 43, 53, 72, 125f, 197ef }}
@@ Line 734: / Line 735: @@
 * [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 701.1108{{c}}, ~55/42 = 460.5488{{c}}
 : error map: {{val| 0.000 -0.844 -2.446 +1.131 -3.569 }}
-<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 701.459{{c}}, ~84/55 = 739.747{{c}} -->
 {{Optimal ET sequence|legend=1| 29, 31, 60e, 91e, 94, 125 }}
 [[Badness]] (Sintel): 1.38
-== Catakleismoid ==
-Catakleismoid is the same as catakleismic in the 2.3.5.7.13 subgroup but with an independent generator for prime 11.
-[[Subgroup]]: 2.3.5.7.11
-[[Comma list]]: 225/224, 4375/4374
-{{Mapping|legend=1| 1 0 1 -3 0 | 0 6 5 22 0 | 0 0 0 0 1 }}
-: mapping generators: ~2, ~6/5, ~11
-[[Optimal tuning]]s:
-* [[WE]]: ~2 = 1200.5965{{c}}, ~6/5 = 316.8893{{c}}, ~11/8 = 549.5258{{c}}
-: [[error map]]: {{val| +0.596 -0.619 -1.271 +0.948 -0.003 }}
-* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 316.7705{{c}}, ~11/8 = 550.0343{{c}}
-: error map: {{val| 0.000 -1.332 -2.461 +0.126 -1.284 }}
-<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~6/5 = 316.834{{c}}, ~11/8 = 551.318{{c}} -->
-{{Optimal ET sequence|legend=1| 19, 34d, 53, 72, 197e, 269ce, 341ce }}
-[[Badness]] (Sintel): 1.53
-=== 13-limit ===
-Subgroup: 2.3.5.7.11.13
-Comma list: 169/168, 225/224, 325/324
-Mapping: {{mapping| 1 0 1 -3 0 0 | 0 6 5 22 0 14 | 0 0 0 0 1 0 }}
-Optimal tunings:
-* WE: ~2 = 1200.8238{{c}}, ~6/5 = 316.9478{{c}}, ~11/8 = 548.9611{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.7939{{c}}, ~11/8 = 549.5903{{c}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~6/5 = 316.886{{c}}, ~11/8 = 551.318{{c}} -->
-Optimal ET sequence: {{Optimal ET sequence| 19, 34d, 53, 72, 125f, 197ef }}
-Badness (Sintel): 0.853
 == Subgroup extensions ==
@@ Line 788: / Line 751: @@
 * WE: ~2 = 1199.3681{{c}}, ~3/2 = 701.3669{{c}}, ~5/4 = 384.9641{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.6286{{c}}, ~5/4 = 385.0841{{c}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 701.244{{c}}, ~5/4 = 384.789{{c}} -->
 {{Optimal ET sequence|legend=0| 9, 10, 12, 19, 22, 31, 41, 53 }}
@@ Line 804: / Line 766: @@
 * WE: ~2 = 1200.3454{{c}}, ~3/2 = 701.1728{{c}}, ~5/4 = 383.7499{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.9546{{c}}, ~5/4 = 383.7912{{c}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 701.904{{c}}, ~5/4 = 384.260{{c}} -->
 {{Optimal ET sequence|legend=0| 10, 12, 19, 22, 31, 41, 53, 72, 94, 113, 125, 291c }}