@@ 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 72: / 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 78: / 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 113: / 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 128: / Line 126: @@
 }}
-=== 13-limit ===
+=== Helios ===
 Subgroup: 2.3.5.7.11.13
@@ Line 150: / 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 272: / 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:
@@ Line 341: / Line 341: @@
 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 370: / Line 487: @@
 Scales: [[minerva12]], [[minerva22x]]
-[[Associated temperament]]: [[Würschmidt family #Würschmidt|würschmidt]]
 === Athene ===
@@ Line 396: / 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 416: / Line 537: @@
 [[Projection pair]]s: <code>7 225/32 11 100/9</code>
-[[Associated temperament]]: [[magic]]
 Scales: [[apollo wholetone]], [[indigo17]]
@@ Line 439: / 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 517: / 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|
-.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 707: / Line 739: @@
 [[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 ==

Marvel family: Difference between revisions