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Marvel family: Difference between revisions

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The '''marvel family''' is the set of temperaments that temper out the [[7-limit]] marvel comma ([[225/224]] = {{monzo| -5 2 2 -1 }}) which is 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 edos and rank-2 temperaments.
{{Technical data page}}
The '''marvel family''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] tempers out the [[7-limit]] marvel comma, [[225/224]] = {{monzo| -5 2 2 -1 }}, which is 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.


The marvel comma can also be viewed as a comma of the 2.9.25.7 subgroup. Hence it is tempered out by any subset edos of marvel-supporting edos that have this subgroup, such as [[11edo]] and [[17edo]] which are subsets of [[22edo]] and [[34edo]] (when using the 34b val) which temper out the marvel comma.
The marvel comma can also be viewed as a comma of the 2.9.25.7 [[subgroup]]. Hence it is tempered out by any subset edos of marvel-supporting edos that have this subgroup, such as [[11edo]] and [[17edo]] which are subsets of [[22edo]] and [[34edo]] (when using the 34d val) which temper out the marvel comma.


== Marvel ==
== Marvel ==
{{Main| Marvel }}
{{Main| Marvel }}


The head of the marvel family is marvel, which tempers out [[225/224]]. Marvel has a [[Normal lists|normal list basis]] 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 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]].


Little is gained in tuning accuracy by not tempering out 4375/4374 as well as 225/224, leading to [[catakleismic temperament]]. Another temperament which does little damage to tuning accuracy is [[compton temperament]], 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
[[Subgroup]]: 2.3.5.7
Line 30: Line 31:
: Angle (secor, 3/2) = 106.958 degrees
: Angle (secor, 3/2) = 106.958 degrees


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 700.4075, ~5/4 = 383.6376
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~3/2 = 700.974, ~5/4 = 384.208
* [[CWE]]: ~2 = 1\1, ~3/2 = 700.622, ~5/4 = 383.854


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[7-odd-limit]]: 3 and 5 1/4-comma flat, 7 just
* [[7-odd-limit]]: 3 and 5 1/4-comma flat, 7 just
: {{monzo list| 1 0 0 0 | 5/4 1/2 -1/2 1/4 | 5/4 -1/2 1/2 1/4  | 0 0 0 1 }}
: {{monzo list| 1 0 0 0 | 5/4 1/2 -1/2 1/4 | 5/4 -1/2 1/2 1/4  | 0 0 0 1 }}
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5/3.7
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5/3.7
* [[9-odd-limit]]: 3 1/6-comma flat, 5 1/3-comma flat, 7 just
* [[9-odd-limit]]: 3 1/6-comma flat, 5 1/3-comma flat, 7 just
: {{monzo list| 1 0 0 0 | 5/6 2/3 -1/3 1/6 | 5/3 -2/3 1/3 1/3 | 0 0 0 1 }}
: {{monzo list| 1 0 0 0 | 5/6 2/3 -1/3 1/6 | 5/3 -2/3 1/3 1/3 | 0 0 0 1 }}
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.9/5.7
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5.7


{{Optimal ET sequence|legend=1| 9, 10, 12, 19, 31, 41, 53, 72, 197, 269c }}
{{Optimal ET sequence|legend=1| 9, 10, 12, 19, 31, 41, 53, 72, 197, 269c }}
Line 48: Line 51:
[[Complexity spectrum]]: 4/3, 5/4, 7/5, 7/6, 8/7, 6/5, 9/8, 9/7, 10/9
[[Complexity spectrum]]: 4/3, 5/4, 7/5, 7/6, 8/7, 6/5, 9/8, 9/7, 10/9


[[Associated temperament]]: [[catakleismic]]
Scales: [[marvel9]], [[marvel10]], [[marvel11]], [[marvel12]], [[marvel19]], [[marvel22]], [[pump12_1]], [[pump12_2]], [[pump13]], [[pump14]], [[pump15]], [[pump16]], [[pump17]], [[pump18]], [[marvel wholetone]]
 
Scales: [[marvel9]], [[marvel10]], [[marvel11]], [[marvel12]], [[marvel19]], [[marvel22]], [[pump12_1]], [[pump12_2]], [[pump13]], [[pump14]], [[pump15]], [[pump16]], [[pump17]], [[pump18]]


{{Databox|[[Minkowski blocks]]|
{{Databox|[[Minkowski blocks]]|
Line 75: Line 76:
* ''[[Supernatural]]'' (+245/243) → [[Keemic family #Supernatural|Keemic family]]
* ''[[Supernatural]]'' (+245/243) → [[Keemic family #Supernatural|Keemic family]]
* ''[[Artemis]]'' (+121/120) → [[Biyatismic clan #Artemis|Biyatismic clan]]
* ''[[Artemis]]'' (+121/120) → [[Biyatismic clan #Artemis|Biyatismic clan]]
* ''[[Spectacle]]'' (+243/242) → [[Rastmic rank three clan #Spectacle|Rastmic rank-3 clan]]
* ''[[Spectacle]]'' (+243/242) → [[Rastmic rank-3 clan #Spectacle|Rastmic rank-3 clan]]
* ''[[Mirage]]'' (+243/242, +385/384) → [[Rastmic rank three clan #Spectacle|Rastmic rank-3 clan]]
* ''[[Mirage]]'' (+243/242, +385/384) → [[Rastmic rank-3 clan #Spectacle|Rastmic rank-3 clan]]


== Undecimal marvel (unimarv) ==
== Undecimal marvel (unimarv) ==
Line 93: Line 94:
: Angle (secor, 5/4) = 104.028 degrees
: Angle (secor, 5/4) = 104.028 degrees


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 700.3887, ~5/4 = 383.5403
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~3/2 = 701.373, ~5/4 = 383.146
* [[CWE]]: ~2 = 1\1, ~3/2 = 700.605, ~5/4 = 383.454


[[Minimax tuning]]:
[[Minimax tuning]]:
* [[11-odd-limit]]
* [[11-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 4/3 8/9 -1/3 0 -1/9 }}, {{monzo| 8/3 -2/9 1/3 0 -2/9 }}, {{monzo| 3 4/3 0 0 -2/3 }}, {{monzo| 8/3 -2/9 -2/3 0 7/9 }}]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 4/3 8/9 -1/3 0 -1/9 }}, {{monzo| 8/3 -2/9 1/3 0 -2/9 }}, {{monzo| 3 4/3 0 0 -2/3 }}, {{monzo| 8/3 -2/9 -2/3 0 7/9 }}]
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.9/5.11/9
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5.11/9


{{Optimal ET sequence|legend=1| 9, 10, 12e, 19, 22, 31, 41, 53, 72, 166, 197e, 238c, 269ce, 341ce }}
{{Optimal ET sequence|legend=1| 9, 10, 12e, 19, 22, 31, 41, 53, 72, 166, 197e, 238c, 269ce, 341ce }}
Line 112: Line 115:
Scales: [[marvel22_11]], [[unimarv19]], [[unimarv22]]
Scales: [[marvel22_11]], [[unimarv19]], [[unimarv22]]


{{Databox|Hobbit bases|
{{Databox|[[Hobbit]] bases|
2.3.5 subgroup
2.3.5 subgroup
* 12: 128/125, 2048/2025
* 12: 128/125, 2048/2025
Line 129: Line 132:
Mapping: {{mapping| 1 0 0 -5 12 -4 | 0 1 0 2 -1 -1 | 0 0 1 2 -3 4 }}
Mapping: {{mapping| 1 0 0 -5 12 -4 | 0 1 0 2 -1 -1 | 0 0 1 2 -3 4 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 699.7367, ~5/4 = 384.0613
Optimal tunings:
* CTE: ~2 = 1\1, ~3/2 = 700.182, ~5/4 = 384.400
* CWE: ~2 = 1\1, ~3/2 = 699.811, ~5/4 = 384.118


Minimax tuning:  
Minimax tuning:  
* 13-odd-limit eigenmonzo (unchanged-interval) basis: 2.11/9.13/9
* 13-odd-limit unchanged-interval (eigenmonzo) basis: 2.11/9.13/9
* 15-odd-limit eigenmonzo (unchanged-interval) basis: 2.15/11.15/13
* 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.15/11.15/13


{{Optimal ET sequence|legend=1| 19, 22, 31, 50, 53, 72, 103, 175f, 300ceff, 403bceeff, 578bbccdeeefff }}
Optimal ET sequence: {{Optimal ET sequence| 19, 22, 31, 50, 53, 72, 103, 175f, 300ceff, 403bceeff, 578bbccdeeefff }}


Badness: 0.690 × 10<sup>-3</sup>
Badness: 0.690 × 10<sup>-3</sup>
Line 148: Line 153:
Mapping: {{mapping| 1 0 0 -5 12 2 | 0 1 0 2 -1 4 | 0 0 1 2 -3 -2 }}
Mapping: {{mapping| 1 0 0 -5 12 2 | 0 1 0 2 -1 4 | 0 0 1 2 -3 -2 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.9779, ~5/4 = 383.1622
Optimal tunings:
* CTE: ~2 = 1\1, ~3/2 = 701.563, ~5/4 = 383.015
* CWE: ~2 = 1\1, ~3/2 = 701.100, ~5/4 = 383.132


Minimax tuning:  
Minimax tuning:  
* 13-odd-limit eigenmonzo (unchanged-interval) basis: 2.7.13/5
* 13-odd-limit unchanged-interval (eigenmonzo) basis: 2.7.13/5
* 15-odd-limit eigenmonzo (unchanged-interval) basis: 2.7.15/13
* 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.7.15/13


{{Optimal ET sequence|legend=1| 19, 22f, 31f, 41, 53, 72, 125f, 166, 238cf }}
Optimal ET sequence: {{Optimal ET sequence| 19, 22f, 31f, 41, 53, 72, 125f, 166, 238cf }}


Badness: 0.721 × 10<sup>-3</sup>
Badness: 0.721 × 10<sup>-3</sup>
Line 169: Line 176:
Sval mapping: {{mapping| 1 0 0 -5 12 2 9 | 0 1 0 2 -1 4 -3 | 0 0 1 2 -3 -2 0 }}
Sval mapping: {{mapping| 1 0 0 -5 12 2 9 | 0 1 0 2 -1 4 -3 | 0 0 1 2 -3 -2 0 }}


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.4245, ~5/4 = 383.0293
Optimal tunings:
* CTE: ~2 = 1\1, ~3/2 = 701.424, ~5/4 = 383.029
* CWE: ~2 = 1\1, ~3/2 = 701.200, ~5/4 = 383.114


{{Optimal ET sequence|legend=1| 41, 53, 72, 94, 113, 166 }}
Optimal ET sequence: {{Optimal ET sequence| 41, 53, 72, 94, 113, 166 }}


Badness: 0.773 × 10<sup>-3</sup>
Badness: 0.773 × 10<sup>-3</sup>
Line 182: Line 191:
Mapping: {{mapping| 1 0 0 -5 12 2 18 | 0 1 0 2 -1 4 0 | 0 0 1 2 -3 -2 -6 }}
Mapping: {{mapping| 1 0 0 -5 12 2 18 | 0 1 0 2 -1 4 0 | 0 0 1 2 -3 -2 -6 }}


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.5860, ~5/4 = 382.7331
Optimal tunings:
* CTE: ~2 = 1\1, ~3/2 = 701.586, ~5/4 = 382.733
* CWE: ~2 = 1\1, ~3/2 = 701.088, ~5/4 = 382.971


{{Optimal ET sequence|legend=1| 41, 53g, 72, 166g, 238cfg }}
Optimal ET sequence: {{Optimal ET sequence| 41, 53g, 72, 166g, 238cfg }}


Badness: 0.869 × 10<sup>-3</sup>
Badness: 0.869 × 10<sup>-3</sup>
Line 195: Line 206:
Mapping: {{mapping| 1 0 0 -5 12 2 18 9 | 0 1 0 2 -1 4 0 -3 | 0 0 1 2 -3 -2 -6 0 }}
Mapping: {{mapping| 1 0 0 -5 12 2 18 9 | 0 1 0 2 -1 4 0 -3 | 0 0 1 2 -3 -2 -6 0 }}


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.4435, ~5/4 = 382.7395
Optimal tunings:
* CTE: ~2 = 1\1, ~3/2 = 701.443, ~5/4 = 382.739
* CWE: ~2 = 1\1, ~3/2 = 701.201, ~5/4 = 382.940


{{Optimal ET sequence|legend=1| 41, 53g, 72, 94, 113, 166g }}
Optimal ET sequence: {{Optimal ET sequence| 41, 53g, 72, 94, 113, 166g }}


Badness: 0.978 × 10<sup>-3</sup>
Badness: 0.978 × 10<sup>-3</sup>
Line 208: Line 221:
Mapping: {{mapping| 1 0 0 -5 12 2 -13 | 0 1 0 2 -1 4 2 | 0 0 1 2 -3 -2 6 }}
Mapping: {{mapping| 1 0 0 -5 12 2 -13 | 0 1 0 2 -1 4 2 | 0 0 1 2 -3 -2 6 }}


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.6270, ~5/4 = 383.3456
Optimal tunings:
* CTE: ~2 = 1\1, ~3/2 = 701.627, ~5/4 = 383.346
* CWE: ~2 = 1\1, ~3/2 = 701.095, ~5/4 = 383.314


{{Optimal ET sequence|legend=1| 41g, 53, 72, 166g, 238cfg }}
Optimal ET sequence: {{Optimal ET sequence| 41g, 53, 72, 166g, 238cfg }}


Badness: 0.917 × 10<sup>-3</sup>
Badness: 0.917 × 10<sup>-3</sup>
Line 221: Line 236:
Mapping: {{mapping| 1 0 0 -5 12 2 -13 9 | 0 1 0 2 -1 4 2 -3 | 0 0 1 2 -3 -2 6 0 }}
Mapping: {{mapping| 1 0 0 -5 12 2 -13 9 | 0 1 0 2 -1 4 2 -3 | 0 0 1 2 -3 -2 6 0 }}


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.4791, ~5/4 = 383.3795
Optimal tunings:
* CTE: ~2 = 1\1, ~3/2 = 701.479, ~5/4 = 383.379
* CWE: ~2 = 1\1, ~3/2 = 701.211, ~5/4 = 383.305


{{Optimal ET sequence|legend=1| 41g, 53, 72, 94, 125f, 166g }}
Optimal ET sequence: {{Optimal ET sequence| 41g, 53, 72, 94, 125f, 166g }}


Badness: 1.03 × 10<sup>-3</sup>
Badness: 1.03 × 10<sup>-3</sup>
Line 234: Line 251:
Mapping: {{mapping| 1 0 0 -5 12 -29 | 0 1 0 2 -1 6 | 0 0 1 2 -3 10 }}
Mapping: {{mapping| 1 0 0 -5 12 -29 | 0 1 0 2 -1 6 | 0 0 1 2 -3 10 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.3937, ~5/4 = 383.5725
Optimal tunings:
* CTE: ~2 = 1\1, ~3/2 = 701.385, ~5/4 = 383.208
* CWE: ~2 = 1\1, ~3/2 = 700.611, ~5/4 = 383.493


Minimax tuning:  
Minimax tuning:  
* 13-odd-limit eigenmonzo (unchanged-interval) basis: 2.9/5.11/9
* 13-odd-limit unchanged-interval (eigenmonzo) basis: 2.9/5.11/9
* 15-odd-limit eigenmonzo (unchanged-interval) basis: 2.7.15/13
* 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.7.15/13


{{Optimal ET sequence|legend=1| 9, 22f, 31, 63, 72, 103, 166, 238cf, 269ce, 507bcceff, 610bcceeff }}
Optimal ET sequence: {{Optimal ET sequence| 9, 22f, 31, 63, 72, 103, 166, 238cf, 269ce, 507bcceff, 610bcceeff }}


Badness: 0.862 × 10<sup>-3</sup>
Badness: 0.862 × 10<sup>-3</sup>
Line 251: Line 270:
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 tuning (POTE): ~2 = 1\1, ~3/2 = 701.9156, ~5/4 = 383.2445
Optimal tunings:
* CTE: ~2 = 1\1, ~3/2 = 701.997, ~5/4 = 383.134
* CWE: ~2 = 1\1, ~3/2 = 701.928, ~5/4 = 383.228


{{Optimal ET sequence|legend=1| 10, 19f, 22, 31, 41, 53, 94 }}
Optimal ET sequence: {{Optimal ET sequence| 10, 19f, 22, 31, 41, 53, 94 }}


Badness: 0.866 × 10<sup>-3</sup>
Badness: 0.866 × 10<sup>-3</sup>
Line 266: Line 287:
Mapping: {{mapping| 1 0 0 -5 12 27 | 0 1 0 2 -1 -3 | 0 0 1 2 -3 -8 }}
Mapping: {{mapping| 1 0 0 -5 12 27 | 0 1 0 2 -1 -3 | 0 0 1 2 -3 -8 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.4560, ~5/4 = 382.8177
Optimal tunings:
* CTE: ~2 = 1\1, ~3/2 = 701.702, ~5/4 = 382.074
* CWE: ~2 = 1\1, ~3/2 = 700.718, ~5/4 = 382.661


Minimax tuning:  
Minimax tuning:  
Line 272: Line 295:
* 15-odd-limit eigenmonzo subgroup (unchanged-interval basis): 2.3.13/5
* 15-odd-limit eigenmonzo subgroup (unchanged-interval basis): 2.3.13/5


{{Optimal ET sequence|legend=1| 9, 19f, 22, 31f, 41, 63, 72, 185cf, 257cff }}
Optimal ET sequence: {{Optimal ET sequence| 9, 19f, 22, 31f, 41, 63, 72, 185cf, 257cff }}


Badness: 0.920 × 10<sup>-3</sup>
Badness: 0.920 × 10<sup>-3</sup>
Line 285: Line 308:
Mapping: {{mapping| 1 0 0 -5 12 -8 | 0 1 0 2 -1 3 | 0 0 1 2 -3 3 }}
Mapping: {{mapping| 1 0 0 -5 12 -8 | 0 1 0 2 -1 3 | 0 0 1 2 -3 3 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 699.2335, ~5/4 = 382.9775
Optimal tunings:
* CTE: ~2 = 1\1, ~3/2 = 700.028, ~5/4 = 382.694
* CWE: ~2 = 1\1, ~3/2 = 699.404, ~5/4 = 382.917


Minimax tuning:  
Minimax tuning:  
* 13-odd-limit eigenmonzo (unchanged-interval) basis: 2.9/7.13/11
* 13-odd-limit unchanged-interval (eigenmonzo) basis: 2.9/7.13/11
* 15-odd-limit eigenmonzo (unchanged-interval) basis: 2.5/3.13/11
* 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.5/3.13/11


{{Optimal ET sequence|legend=1| 9, 10, 19, 22f, 31, 41, 72f, 91, 122f, 163df }}
Optimal ET sequence: {{Optimal ET sequence| 9, 10, 19, 22f, 31, 41, 72f, 91, 122f, 163df }}


Badness: 0.745 × 10<sup>-3</sup>
Badness: 0.745 × 10<sup>-3</sup>
Line 306: Line 331:
: mapping generators: ~2, ~26/15, ~5
: mapping generators: ~2, ~26/15, ~5


Optimal tuning (POTE): ~2 = 1\1, ~15/13 = 249.7138, ~5/4 = 383.5816
Optimal tunings:
* CTE: ~2 = 1\1, ~26/15 = 950.961, ~5/4 = 383.088
* CWE: ~2 = 1\1, ~26/15 = 950.426, ~5/4 = 383.479


{{Optimal ET sequence|legend=1| 9, 10, 19, 44, 53, 72, 125f, 197ef, 269ceff }}
Optimal ET sequence: {{Optimal ET sequence| 9, 10, 19, 44, 53, 72, 125f, 197ef, 269ceff }}


Badness: 0.9997 × 10<sup>-3</sup>
Badness: 0.9997 × 10<sup>-3</sup>
Line 325: Line 352:
: Angle (16/15, 5/4) = 98.6044 degrees
: Angle (16/15, 5/4) = 98.6044 degrees


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 700.2593, ~5/4 = 386.5581
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~3/2 = 700.379, ~5/4 = 386.617
* [[CWE]]: ~2 = 1\1, ~3/2 = 700.301, ~5/4 = 386.579


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


{{Optimal ET sequence|legend=1| 9, 12, 19e, 22, 31, 53, 84e, 96, 127 }}
{{Optimal ET sequence|legend=1| 9, 12, 19e, 22, 31, 53, 84e, 96, 127 }}
Line 346: Line 375:
Mapping: {{mapping| 1 0 0 -5 -9 -4 | 0 1 0 2 2 -1 | 0 0 1 2 4 4 }}
Mapping: {{mapping| 1 0 0 -5 -9 -4 | 0 1 0 2 2 -1 | 0 0 1 2 4 4 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.2342, ~5/4 = 385.9594
Optimal tunings:
* CTE: ~2 = 1\1, ~3/2 = 701.179, ~5/4 = 385.888
* CWE: ~2 = 1\1, ~3/2 = 701.214, ~5/4 = 385.934


Minimax tuning:  
Minimax tuning:  
* 13-odd-limit eigenmonzo (unchanged-interval) basis: 2.11/9.13/7
* 13-odd-limit unchanged-interval (eigenmonzo) basis: 2.11/9.13/7
* 15-odd-limit eigenmonzo (unchanged-interval) basis: 2.11/9.13/7
* 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.11/9.13/7


{{Optimal ET sequence|legend=1| 12f, 19e, 22, 31, 53, 84e, 118d, 171de, 202def }}
Optimal ET sequence: {{Optimal ET sequence| 12f, 19e, 22, 31, 53, 84e, 118d, 171de, 202def }}


Badness: 0.818 × 10<sup>-3</sup>
Badness: 0.818 × 10<sup>-3</sup>
Line 367: Line 398:
{{Mapping|legend=1| 1 0 0 -5 2 | 0 1 0 2 -2 | 0 0 1 2 2 }}
{{Mapping|legend=1| 1 0 0 -5 2 | 0 1 0 2 -2 | 0 0 1 2 2 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 703.4846, ~5/4 = 381.6033
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~3/2 = 703.418, ~5/4 = 381.331
* [[CWE]]: ~2 = 1\1, ~3/2 = 703.461, ~5/4 = 381.507


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


{{Optimal ET sequence|legend=1| 12, 19, 22, 34d, 41, 104, 157ce, 198ce, 220ce, 261ce }}
{{Optimal ET sequence|legend=1| 12, 19, 22, 34d, 41, 104, 157ce, 198ce, 220ce, 261ce }}
Line 376: Line 409:


[[Associated temperament]]: [[magic]]
[[Associated temperament]]: [[magic]]
Scales: [[apollo wholetone]], [[indigo17]]


=== 13-limit ===
=== 13-limit ===
Line 384: Line 419:
Mapping: {{mapping| 1 0 0 -5 2 7 | 0 1 0 2 -2 -5 | 0 0 1 2 2 2 }}
Mapping: {{mapping| 1 0 0 -5 2 7 | 0 1 0 2 -2 -5 | 0 0 1 2 2 2 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 703.9984, ~5/4 = 381.5352
Optimal tunings:
* CTE: ~2 = 1\1, ~3/2 = 703.959, ~5/4 = 381.001
* CWE: ~2 = 1\1, ~3/2 = 703.985, ~5/4 = 381.358


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


{{Optimal ET sequence|legend=1| 12f, 19f, 22, 29, 34d, 41, 63, 104, 179cef, 242cde, 283def, 346bcdef }}
Optimal ET sequence: {{Optimal ET sequence| 12f, 19f, 22, 29, 34d, 41, 63, 104, 179cef, 242cde, 283def, 346bcdef }}


Projection pairs: 7 225/32 11 100/9 13 3200/243
Projection pairs: 7 225/32 11 100/9 13 3200/243
Line 399: Line 436:
{{Mapping|legend=1| 1 0 0 -5 -2 | 0 1 0 2 2 | 0 0 1 2 1 }}
{{Mapping|legend=1| 1 0 0 -5 -2 | 0 1 0 2 2 | 0 0 1 2 1 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 696.1714, ~5/4 = 385.0500
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~3/2 = 695.937, ~5/4 = 384.836
* [[CWE]]: ~2 = 1\1, ~3/2 = 696.059, ~5/4 = 384.947


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


{{Optimal ET sequence|legend=1| 7d, 9, 10, 12, 19, 31e, 50e }}
{{Optimal ET sequence|legend=1| 7d, 9, 10, 12, 19, 31e, 50e }}
Line 416: Line 455:
Mapping: {{mapping| 1 0 0 -5 -2 -8 | 0 1 0 2 2 3 | 0 0 1 2 1 3 }}
Mapping: {{mapping| 1 0 0 -5 -2 -8 | 0 1 0 2 2 3 | 0 0 1 2 1 3 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 696.0103, ~5/4 = 384.6785
Optimal tunings:
* CTE: ~2 = 1\1, ~3/2 = 695.894, ~5/4 = 384.602
* CWE: ~2 = 1\1, ~3/2 = 695.948, ~5/4 = 384.637


Minimax tuning:  
Minimax tuning:  
* 13-odd-limit eigenmonzo (unchanged-interval) basis: 2.9/7.13/9
* 13-odd-limit unchanged-interval (eigenmonzo) basis: 2.9/7.13/9
* 15-odd-limit eigenmonzo (unchanged-interval) basis: 2.9/7.13/9
* 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.9/7.13/9


{{Optimal ET sequence|legend=1| 9, 10, 12f, 19, 31e, 50e }}
Optimal ET sequence: {{Optimal ET sequence| 9, 10, 12f, 19, 31e, 50e }}


Badness: 0.733 × 10<sup>-3</sup>
Badness: 0.733 × 10<sup>-3</sup>
Line 437: Line 478:
{{Mapping|legend=1| 1 0 0 -5 -3 | 0 1 0 2 7 | 0 0 1 2 -2 }}
{{Mapping|legend=1| 1 0 0 -5 -3 | 0 1 0 2 7 | 0 0 1 2 -2 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 701.8913, ~5/4 = 382.4083
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~3/2 = 702.322, ~5/4 = 382.976
* [[CWE]]: ~2 = 1\1, ~3/2 = 702.054, ~5/4 = 382.622


{{Optimal ET sequence|legend=1| 12e, 19e, 34d, 41, 53, 60e, 94, 229c, 248ce, 289cce, 342ccee, 383cce }}
{{Optimal ET sequence|legend=1| 12e, 19e, 34d, 41, 53, 60e, 94, 229c, 248ce, 289cce, 342ccee, 383cce }}
Line 450: Line 493:
Mapping: {{mapping| 1 0 0 -5 -3 2 | 0 1 0 2 7 4 | 0 0 1 2 -2 -2 }}
Mapping: {{mapping| 1 0 0 -5 -3 2 | 0 1 0 2 7 4 | 0 0 1 2 -2 -2 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.8913, ~5/4 = 382.4083
Optimal tunings:
* CTE: ~2 = 1\1, ~3/2 = 702.235, ~5/4 = 383.205
* CWE: ~2 = 1\1, ~3/2 = 702.088, ~5/4 = 383.063


{{Optimal ET sequence|legend=1| 12e, 19e, 34d, 41, 53, 94, 429ccdeef, 523ccdeef }}
Optimal ET sequence: {{Optimal ET sequence| 12e, 19e, 34d, 41, 53, 94, 429ccdeef, 523ccdeef }}


Badness: 1.075 × 10<sup>-3</sup>
Badness: 1.075 × 10<sup>-3</sup>
Line 473: Line 518:
: Angle (secor, 3/2) = 65.933
: Angle (secor, 3/2) = 65.933


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 699.7981, ~5/4 = 383.5114
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~3/2 = 699.995, ~5/4 = 384.330
* [[CWE]]: ~2 = 1\1, ~3/2 = 699.861, ~5/4 = 383.772


[[Minimax tuning]]:
[[Minimax tuning]]:
* [[11-odd-limit]]
* [[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 }}]
: [{{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|eigenmonzo (unchanged-interval) basis]]: 2.9/5.11
: [[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 }}
{{Optimal ET sequence|legend=1| 10, 12, 19e, 29, 31, 41, 60e, 72, 247c, 319bcde, 391bcde, 463bccde }}
Line 503: Line 550:
Mapping: {{mapping| 1 0 0 -5 -13 -8 | 0 1 0 2 6 3 | 0 0 1 2 3 3 }}
Mapping: {{mapping| 1 0 0 -5 -13 -8 | 0 1 0 2 6 3 | 0 0 1 2 3 3 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.4006, ~5/4 = 381.4025
Optimal tunings:
* CTE: ~2 = 1\1, ~3/2 = 700.617, ~5/4 = 382.244
* CWE: ~2 = 1\1, ~3/2 = 700.469, ~5/4 = 381.669


{{Optimal ET sequence|legend=1| 10, 12f, 19e, 29, 31, 41, 60e, 72f, 91e, 101cd, 132def, 233ccddeef, 274ccddeff, 305ccddeeff }}
Optimal ET sequence: {{Optimal ET sequence| 10, 12f, 19e, 29, 31, 41, 60e, 72f, 91e, 101cd, 132def, 233ccddeef, 274ccddeff, 305ccddeeff }}


Badness: 0.736 × 10<sup>-3</sup>
Badness: 0.736 × 10<sup>-3</sup>
Line 516: Line 565:
Mapping: {{mapping| 1 0 0 -5 -13 -23 | 0 1 0 2 6 11 | 0 0 1 2 3 4 }}
Mapping: {{mapping| 1 0 0 -5 -13 -23 | 0 1 0 2 6 11 | 0 0 1 2 3 4 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.3407, ~5/4 = 383.2592
Optimal tunings:
* CTE: ~2 = 1\1, ~3/2 = 700.305, ~5/4 = 384.075
* CWE: ~2 = 1\1, ~3/2 = 700.330, ~5/4 = 383.503


{{Optimal ET sequence|legend=1| 12f, 29, 31f, 41, 72, 185cf, 341cf, 413bcff, 526bccdff }}
Optimal ET sequence: {{Optimal ET sequence| 12f, 29, 31f, 41, 72, 185cf, 341cf, 413bcff, 526bccdff }}


Badness: 0.900 × 10<sup>-3</sup>
Badness: 0.900 × 10<sup>-3</sup>
Line 529: Line 580:
Mapping: {{mapping| 1 0 0 -5 -13 -4 | 0 1 0 2 6 -1 | 0 0 1 2 3 4 }}
Mapping: {{mapping| 1 0 0 -5 -13 -4 | 0 1 0 2 6 -1 | 0 0 1 2 3 4 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 699.4864, ~5/4 = 384.0998
Optimal tunings:
* CTE: ~2 = 1\1, ~3/2 = 699.698, ~5/4 = 384.884
* CWE: ~2 = 1\1, ~3/2 = 699.554, ~5/4 = 384.350


{{Optimal ET sequence|legend=1| 12f, 19e, 31, 53e, 60eff, 72, 103, 175f }}
Optimal ET sequence: {{Optimal ET sequence| 12f, 19e, 31, 53e, 60eff, 72, 103, 175f }}


Badness: 0.889 × 10<sup>-3</sup>
Badness: 0.889 × 10<sup>-3</sup>
Line 542: Line 595:
Mapping: {{mapping| 1 0 0 -5 -13 21 | 0 1 0 2 6 -8 | 0 0 1 2 3 -2 }}
Mapping: {{mapping| 1 0 0 -5 -13 21 | 0 1 0 2 6 -8 | 0 0 1 2 3 -2 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 699.5536, ~5/4 = 383.5696
Optimal tunings:
* CTE: ~2 = 1\1, ~3/2 = 698.947, ~5/4 = 385.480
* CWE: ~2 = 1\1, ~3/2 = 699.422, ~5/4 = 383.983


{{Optimal ET sequence|legend=1| 29, 31, 43, 60e, 72, 103, 175f, 482bccddeefff, 554bbccddeeeffff }}
Optimal ET sequence: {{Optimal ET sequence| 29, 31, 43, 60e, 72, 103, 175f, 482bccddeefff, 554bbccddeeeffff }}


Badness: 0.953 × 10<sup>-3</sup>
Badness: 0.953 × 10<sup>-3</sup>
Line 555: Line 610:
Mapping: {{mapping| 1 0 0 -5 -13 21 12 | 0 1 0 2 6 -8 -5 | 0 0 1 2 3 -2 0 }}
Mapping: {{mapping| 1 0 0 -5 -13 21 12 | 0 1 0 2 6 -8 -5 | 0 0 1 2 3 -2 0 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 699.6262, ~5/4 = 383.4458
Optimal tunings:
* CTE: ~2 = 1\1, ~3/2 = 698.969, ~5/4 = 385.440
* CWE: ~2 = 1\1, ~3/2 = 699.485, ~5/4 = 383.874


{{Optimal ET sequence|legend=1| 29g, 31, 43, 60e, 72, 103, 175f, 307bcdeeffg, 379bccdeeffgg, 482bccddeefffgg, 554bbccddeeeffffgg }}
Optimal ET sequence: {{Optimal ET sequence| 29g, 31, 43, 60e, 72, 103, 175f, 307bcdeeffg, 379bccdeeffgg, 482bccddeefffgg, 554bbccddeeeffffgg }}


Badness: 0.772 × 10<sup>-3</sup>
Badness: 0.772 × 10<sup>-3</sup>


== Fantastic ==
== Fantastic ==
Besides 4375/4356, fantastic also tempers out [[9801/9800]] and splits the octave in two.
[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


Line 570: Line 629:
: mapping generators: ~99/70, ~3, ~5
: mapping generators: ~99/70, ~3, ~5


[[Optimal tuning]] ([[POTE]]): ~99/70 = 1\2, ~3/2 = 700.6242, ~5/4 = 383.2978
[[Optimal tuning]]s:
* [[CTE]]: ~99/70 = 1\2, ~3/2 = 701.132, ~5/4 = 383.926
* [[CWE]]: ~99/70 = 1\2, ~3/2 = 700.816, ~5/4 = 383.535


{{Optimal ET sequence|legend=1| 12, 22, 34d, 50, 60e, 72, 166, 238c, 310c }}
{{Optimal ET sequence|legend=1| 12, 22, 34d, 50, 60e, 72, 166, 238c, 310c }}
Line 587: Line 648:
: mapping generators: ~2, ~400/231, ~5
: mapping generators: ~2, ~400/231, ~5


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~400/231 = 950.1474, ~5/4 = 383.6467
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~400/231 = 950.043, ~5/4 = 384.858
* [[CWE]]: ~2 = 1\1, ~400/231 = 950.121, ~5/4 = 383.953


{{Optimal ET sequence|legend=1| 19, 29, 43, 53, 72, 197e, 269ce, 341ce, 610bce }}
{{Optimal ET sequence|legend=1| 19, 29, 43, 53, 72, 197e, 269ce, 341ce, 610bce }}
Line 600: Line 663:
Mapping: {{mapping| 1 0 0 -5 9 -1 | 0 2 0 4 -7 3 | 0 0 1 2 0 1 }}
Mapping: {{mapping| 1 0 0 -5 9 -1 | 0 2 0 4 -7 3 | 0 0 1 2 0 1 }}


Optimal tuning (POTE): ~2 = 1\1, ~26/15 = 950.2349, ~5/4 = 383.5558
Optimal tunings:
* CTE: ~2 = 1\1, ~26/15 = 950.131, ~5/4 = 385.245
* CWE: ~2 = 1\1, ~26/15 = 950.211, ~5/4 = 383.949


{{Optimal ET sequence|legend=1| 19, 29, 43, 53, 72, 125f, 197ef, 269cef }}
Optimal ET sequence: {{Optimal ET sequence| 19, 29, 43, 53, 72, 125f, 197ef, 269cef }}


Badness: 1.062 × 10<sup>-3</sup>
Badness: 1.062 × 10<sup>-3</sup>
Line 615: Line 680:
: mapping generators: ~2, ~3, ~84/55
: mapping generators: ~2, ~3, ~84/55


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 700.8983, ~84/55 = 739.3812
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~3/2 = 701.459, ~84/55 = 739.747
* [[CWE]]: ~2 = 1\1, ~3/2 = 701.111, ~84/55 = 739.451


{{Optimal ET sequence|legend=1| 29, 31, 60e, 91e, 94, 125 }}
{{Optimal ET sequence|legend=1| 29, 31, 60e, 91e, 94, 125 }}
Line 622: Line 689:


== Catakleismoid ==
== 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
[[Subgroup]]: 2.3.5.7.11


Line 630: Line 699:
: mapping generators: ~2, ~6/5, ~11
: mapping generators: ~2, ~6/5, ~11


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 316.7318, ~11/8 = 549.2528
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~6/5 = 316.834, ~11/8 = 551.318
* [[CWE]]: ~2 = 1\1, ~6/5 = 316.771, ~11/8 = 550.034


{{Optimal ET sequence|legend=1| 19, 34d, 53, 72, 197e, 269ce }}
{{Optimal ET sequence|legend=1| 19, 34d, 53, 72, 197e, 269ce }}
Line 643: Line 714:
Mapping: {{mapping| 1 0 1 -3 0 0 | 0 6 5 22 0 14 | 0 0 0 0 1 0 }}
Mapping: {{mapping| 1 0 1 -3 0 0 | 0 6 5 22 0 14 | 0 0 0 0 1 0 }}


Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.7410, ~11/8 = 548.6028
Optimal tunings:
* CTE: ~2 = 1\1, ~6/5 = 316.886, ~11/8 = 551.318
* CWE: ~2 = 1\1, ~6/5 = 316.794, ~11/8 = 549.590


{{Optimal ET sequence|legend=1| 19, 34d, 53, 72, 125f, 197ef, 269cef }}
Optimal ET sequence: {{Optimal ET sequence| 19, 34d, 53, 72, 125f, 197ef, 269cef }}


Badness: 0.916 × 10<sup>-3</sup>
Badness: 0.916 × 10<sup>-3</sup>
== Subgroup extensions ==
=== Char ===
Subgroup: 2.3.5.7.17
Comma list: 120/119, 225/224
Mapping: {{mapping| 1 0 0 -5 8 | 0 1 0 2 -1 | 0 0 1 2 -1 }}
Optimal tuning:
* CTE: ~2 = 1\1, ~3/2 = 701.244, ~5/4 = 384.789
{{Optimal ET sequence| legend=1 | 9, 10, 12, 19, 22, 31, 41, 53 }}
=== Devimarvel ===
Subgroup: 2.3.5.7.19
Comma list: 225/224, 400/399
Mapping: {{mapping| 1 0 0 -5 9 | 0 1 0 2 -3 | 0 0 1 2 0 }}
Optimal tuning:
* CTE: ~2 = 1\1, ~3/2 = 701.904, ~5/4 = 384.260
{{Optimal ET sequence| legend=1 | 10, 12, 19, 22, 31, 41, 53, 72, 94 }}


== Notes ==
== Notes ==


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Marvel family| ]] <!-- main article -->
[[Category:Marvel family| ]] <!-- main article -->
[[Category:Marvel| ]] <!-- key article -->
[[Category:Marvel| ]] <!-- key article -->
[[Category:Rank 3]]
[[Category:Rank 3]]
<references />
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