Marvel family: Difference between revisions

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[[de:Marvel]]
[[de:Marvel]]


The '''marvel family''' is the set of temperaments that temper out the [[7-limit]] comma [[225/224|225/224]] = {{monzo|-5 2 2 -1}}, the [[septimal_kleisma|septimal kleisma]] or [[marvel_comma|marvel comma]]. 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.
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 named ''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.


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]] 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]] which temper out the marvel comma.
=Marvel=
=Marvel=
The head of the marvel family is marvel, which tempers out [[225/224|225/224]]. Marvel has a [[Normal_lists|normal list basis]] of [2, 3, 5]; hence a [[Harmonic_Limit|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|197edo]].
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 [[Harmonic Limit|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 [[Kleismic_family|catakleismic temperament]]. Another temperament which does little damage to tuning accuracy is [[Pythagorean_family|compton temperament]], for which [[240edo|240edo]] may be used.
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 also [[Marvel_temperaments|Marvel temperaments]].
See also [[Marvel temperaments]].


==Vital statistics==
==Vital statistics==
[[Comma|Comma]] c = 225/224
[[Comma]] c = 225/224


Related linear temperament: [[Kleismic_family|catakleismic temperament]]
Related linear temperament: [[catakleismic temperament]]


7-limit minimax: 3 and 5 1/4c flat, 7 just
7-limit minimax: 3 and 5 1/4c flat, 7 just
Line 21: Line 22:
[|1 0 0 0>, |5/4 1/2 -1/2 1/4>, |5/4 -1/2 1/2 1/4>, |0 0 0 1>]
[|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_subgroup|Eigenmonzo subgroup]]: 2.5/3.7
[[Eigenmonzo subgroup]]: 2.5/3.7


9-limit minimax: 3 1/6c flat, 5 1/3c flat, 7 just
9-limit minimax: 3 1/6c flat, 5 1/3c flat, 7 just
Line 27: Line 28:
[|1 0 0 0>, |5/6 2/3 -1/3 1/6>, |5/3 -2/3 1/3 1/3>, |0 0 0 1>]
[|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_subgroup|Eigenmonzo subgroup]]: 2.9/5.7
[[Eigenmonzo subgroup]]: 2.9/5.7


Lattice basis: secor length 1.256, 3/2 length 1.369
Lattice basis: secor length 1.256, 3/2 length 1.369
Line 39: Line 40:
Generators: 2, 3, 5
Generators: 2, 3, 5


[[EDO|EDOs]]: [[10edo|10]], [[12edo|12]], [[19edo|19]], [[22edo|22]], [[31edo|31]], [[41edo|41]],[[72edo|72]], [[197edo|197]], [[269edo|269c]]
{{EDOs|legend=1| 10, 12, 19, 22, 31, 41, 72, 197, 269c }}


Badness: 0.0000365
Badness: 0.0000365


[[Projection_pair|Projection pair]]s: 7 225/32
[[Projection pair]]s: 7 225/32


[[Spectrum_of_a_temperament|Spectrum]]: 4/3, 5/4, 7/5, 7/6, 8/7, 6/5, 9/8, 9/7, 10/9
[[Spectrum]]: 4/3, 5/4, 7/5, 7/6, 8/7, 6/5, 9/8, 9/7, 10/9


Scales: [[marvel9|marvel9]], [[marvel10|marvel10]], [[marvel11|marvel11]], [[marvel12|marvel12]], [[marvel19|marvel19]], [[marvel22|marvel22]], [[pump12_1|pump12_1]], [[pump12_2|pump12_2]], [[pump13|pump13]], [[pump14|pump14]], [[pump15|pump15]], [[pump16|pump16]], [[pump17|pump17]], [[pump18|pump18]]
Scales: [[marvel9]], [[marvel10]], [[marvel11]], [[marvel12]], [[marvel19]], [[marvel22]], [[pump12_1]], [[pump12_2]], [[pump13]], [[pump14]], [[pump15]], [[pump16]], [[pump17]], [[pump18]]


==[[Minkowski_blocks|Minkowski blocks]]==
==[[Minkowski blocks]]==
{2, 3, 5} subgroup
{2, 3, 5} subgroup


Line 79: Line 80:


==Music==
==Music==
[http://chrisvaisvil.com/?p=314 Semimarvelous Blue Drawf] by [[Chris_Vaisvil|Chris Vaisvil]]
* [http://chrisvaisvil.com/?p=314 Semimarvelous Blue Drawf] by [[Chris Vaisvil]]
 
* [http://micro.soonlabel.com/gene_ward_smith/Others/Stiltner/marvel10_2.mp3 notJoelTaylorsTreeSpirit-Marvel10-2] by [[Billy Stiltner]]
[http://micro.soonlabel.com/gene_ward_smith/Others/Stiltner/marvel10_2.mp3 notJoelTaylorsTreeSpirit-Marvel10-2] by [[Billy_Stiltner|Billy Stiltner]]


==11-limit (Unimarv)==
==11-limit (Unimarv)==
[[Comma|Commas]]: 225/224, 385/384
[[Commas]]: 225/224, 385/384


Related linear temperament: [[Kleismic_family|catakleismic temperament]]
Related linear temperament: [[catakleismic temperament]]


[[Minimax_tuning|Minimax tuning]]:
[[Minimax tuning]]:


[|1 0 0 0 0>, |4/3 8/9 -1/3 0 -1/9>, |8/3 -2/9 1/3 0 -2/9>,
[|1 0 0 0 0>, |4/3 8/9 -1/3 0 -1/9>, |8/3 -2/9 1/3 0 -2/9>,
|3 4/3 0 0 -2/3>, |8/3 -2/9 -2/3 0 7/9>]
|3 4/3 0 0 -2/3>, |8/3 -2/9 -2/3 0 7/9>]


[[Eigenmonzo_subgroup|Eigenmonzo subgroup]]: 2.9/5.11/9
[[Eigenmonzo subgroup]]: 2.9/5.11/9


Lattice basis: secor length 1.0364 5/4 length 1.0759
Lattice basis: secor length 1.0364 5/4 length 1.0759
Line 103: Line 103:
Map: [<1 0 0 -5 12|, <0 1 0 2 -1|, <0 0 1 2 -3|]
Map: [<1 0 0 -5 12|, <0 1 0 2 -1|, <0 0 1 2 -3|]


[[generator|Generators]]: 2, 3, 5
[[Generator]]s: 2, 3, 5


[[EDO|Edos]]: [[10edo|10]], [[12edo|12e]], [[19edo|19]], [[22edo|22]], [[31edo|31]], [[41edo|41]], [[53edo|53]], [[72edo|72]], [[166edo|166]], [[197edo|197e]], [[269edo|269ce]], [[341edo|341ce]]
{{EDOs|legend=1| 10, 12e, 19, 22, 31, 41, 53, 72, 166, 197e, 269ce, 341ce }}


Badness: 0.000255
Badness: 0.000255


[[Projection_pair|Projection pair]]s: 7 225/32 11 4096/375
[[Projection pair]]s: 7 225/32 11 4096/375


[[Spectrum_of_a_temperament|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
[[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


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


===Hobbit bases===
===Hobbit bases===
Line 139: Line 139:
Map: [<1 0 0 -5 12 -4|, <0 1 0 2 -1 -1|, <0 0 1 2 -3 4|]
Map: [<1 0 0 -5 12 -4|, <0 1 0 2 -1 -1|, <0 0 1 2 -3 4|]


EDOs: 19, 22, 31, 50, 53, 72, 103, [[175edo|175f]], [[300edo|300cef]], [[403bcef|403bcef]]
{{EDOs|legend=1| 19, 22, 31, 50, 53, 72, 103, 175f, 300cef, 403bcef }}


Badness: 0.000690
Badness: 0.000690


[[Spectrum_of_a_temperament|Spectrum]]: 5/4, 4/3, 16/15, 15/14, 9/7, 6/5, 7/6, 11/8, 7/5, 9/8, 8/7, 10/9, 12/11, 13/10, 11/10, 15/11, 16/13, 11/9, 15/13, 14/13, 13/12, 14/11, 18/13, 13/11
[[Spectrum]]: 5/4, 4/3, 16/15, 15/14, 9/7, 6/5, 7/6, 11/8, 7/5, 9/8, 8/7, 10/9, 12/11, 13/10, 11/10, 15/11, 16/13, 11/9, 15/13, 14/13, 13/12, 14/11, 18/13, 13/11


=Hecate=
=Hecate=
Line 154: Line 154:
Map: [<1 0 0 -5 12 2|, <0 1 0 2 -1 4|, <0 0 1 2 -3 -2|]
Map: [<1 0 0 -5 12 2|, <0 1 0 2 -1 4|, <0 0 1 2 -3 -2|]


EDOs: 19, 41, 53, 72, 113, 125f, 166, 238cf, 404cef
{{EDOs|legend=1| 19, 41, 53, 72, 113, 125f, 166, 238cf, 404cef }}


Badness: 0.000721
Badness: 0.000721
Line 160: Line 160:
Projection pairs: 7 225/32 11 4096/375 13 324/25
Projection pairs: 7 225/32 11 4096/375 13 324/25


[[Spectrum_of_a_temperament|Spectrum]]: 4/3, 5/4, 16/15, 15/14, 6/5, 9/8, 7/5, 9/7, 7/6, 10/9, 8/7, 18/13, 11/8, 12/11, 13/12, 11/9, 11/10, 15/13, 15/11, 16/13, 13/11, 14/13, 13/10, 14/11
[[Spectrum]]: 4/3, 5/4, 16/15, 15/14, 6/5, 9/8, 7/5, 9/7, 7/6, 10/9, 8/7, 18/13, 11/8, 12/11, 13/12, 11/9, 11/10, 15/13, 15/11, 16/13, 13/11, 14/13, 13/10, 14/11


==17-limit==
==17-limit==
Line 167: Line 167:
Map: [<1 0 0 -5 12 2 18|, <0 1 0 2 -1 4 0|, <0 0 1 2 -3 -2 -6|]
Map: [<1 0 0 -5 12 2 18|, <0 1 0 2 -1 4 0|, <0 0 1 2 -3 -2 -6|]


EDOs: 19, 41, 53g, 72, 113, 166g, 238cfg, 351cfg, 404cefg
{{EDOs|legend=1| 19, 41, 53g, 72, 113, 166g, 238cfg, 351cfg, 404cefg }}


Badness: 0.000869
Badness: 0.000869
Line 176: Line 176:
Map: [<1 0 0 -5 12 2 18|, <0 1 0 2 -1 4 0|, <0 0 1 2 -3 -2 6|]
Map: [<1 0 0 -5 12 2 18|, <0 1 0 2 -1 4 0|, <0 0 1 2 -3 -2 6|]


EDOs: 41g, 53, 72, 125f, 166g, 238cfg, 363cefg, 404cefg
{{EDOs|legend=1| 41g, 53, 72, 125f, 166g, 238cfg, 363cefg, 404cefg }}


Badness: .000917
Badness: .000917
Line 185: Line 185:
Map: [<1 0 0 -5 12 -|, <0 2 0 4 -2 3|, <0 0 1 2 -3 1|]
Map: [<1 0 0 -5 12 -|, <0 2 0 4 -2 3|, <0 0 1 2 -3 1|]


EDOs: 9, 10, 19, 44, 53, 72, 125f, 197ef, 269cef
{{EDOs|legend=1| 9, 10, 19, 44, 53, 72, 125f, 197ef, 269cef }}


Badness: 0.0009997
Badness: 0.0009997
Line 198: Line 198:
Map: [<1 0 0 -5 12 -29|, <0 1 0 2 -1 6|, <0 0 1 2 -3 10|]
Map: [<1 0 0 -5 12 -29|, <0 1 0 2 -1 6|, <0 0 1 2 -3 10|]


EDOs: 9, 31, [[63edo|63]], 72, [[103edo|103]], [[166edo|166]], [[238edo|238cf]], [[269edo|269ce]], 507bcef, 610bcef
{{EDOs|legend=1| 9, 31, 63, 72, 103, 166, 238cf, 269ce, 507bcef, 610bcef }}


Badness: 0.000862
Badness: 0.000862
Line 207: Line 207:
Map: [<1 0 0 -5 12 17|, <0 1 0 2 -1 4|, <0 0 1 2 -3 -3|]
Map: [<1 0 0 -5 12 17|, <0 1 0 2 -1 4|, <0 0 1 2 -3 -3|]


EDOs: 10, 22, 31, 41, 53, 94
{{EDOs|legend=1| 10, 22, 31, 41, 53, 94 }}


Badness: 0.000866
Badness: 0.000866
Line 222: Line 222:
Map: [<1 0 0 -5 12 27|, <0 1 0 2 -1 -3|, <0 0 1 2 -3 -8|]
Map: [<1 0 0 -5 12 27|, <0 1 0 2 -1 -3|, <0 0 1 2 -3 -8|]


EDOs: 9, 22, 41, 63, 72, [[185edo|185cf]], [[257edo|257cf]]
{{EDOs|legend=1| 9, 22, 41, 63, 72, 185cf, 257cf }}


Badness: 0.000920
Badness: 0.000920
Line 233: Line 233:
Map: [<1 1 3 3 2 0|, <0 6 -7 -2 15 0|, <0 0 0 0 0 1|]
Map: [<1 1 3 3 2 0|, <0 6 -7 -2 15 0|, <0 0 0 0 0 1|]


EDOs: 10, 31, 41, 62, 72, 103, 175f, 216c, 288cdf, 391bcdef
{{EDOs|legend=1| 10, 31, 41, 62, 72, 103, 175f, 216c, 288cdf, 391bcdef }}


Badness: 0.000738
Badness: 0.000738


=Minerva=
=Minerva=
[[Comma|Commas]]: 99/98, 176/175
[[Comma]]s: 99/98, 176/175


Related linear temperament: [[Semicomma_family|orwell]]
Related linear temperament: [[Semicomma family#Orwell|orwell]]


[[Eigenmonzo_subgroup|Eigenmonzo subgroup]]: 2.7/5.11/9
[[Eigenmonzo subgroup]]: 2.7/5.11/9


Lattice basis: 16/15 length 0.8997 5/4 length 1.0457
Lattice basis: 16/15 length 0.8997 5/4 length 1.0457
Line 252: Line 252:
Map: [<1 0 0 -5 -9|, <0 1 0 2 2|, <0 0 1 2 4|]
Map: [<1 0 0 -5 -9|, <0 1 0 2 2|, <0 0 1 2 4|]


[[generator|Generators]]: 2, 3, 5
[[Generator]]s: 2, 3, 5


[[EDO|EDOs]]: [[9edo|9]], [[12edo|12]], [[21edo|21]], [[22edo|22]], [[31edo|31]], [[43edo|43]], [[53edo|53]], [[74edo|74]], [[75edo|75]], [[96edo|96]], [[127edo|127]]
{{EDOs|legend=1| 9, 12, 21, 22, 31, 43, 53, 74, 75, 96, 127 }}


Badness: 0.000381
Badness: 0.000381


[[Projection_pair|Projection pair]]s: 7 225/32 11 5625/512
[[Projection pair]]s: 7 225/32 11 5625/512


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


==Athene==
==Athene==
Line 271: Line 271:
Map: [<1 0 0 -5 -9 -4|, <0 1 0 2 2 -1|, <0 0 1 2 4 4|]
Map: [<1 0 0 -5 -9 -4|, <0 1 0 2 2 -1|, <0 0 1 2 4 4|]


EDOs: 22, 31, 53, 84e, 118d, 149df, 171de, 202def
{{EDOs|legend=1| 22, 31, 53, 84e, 118d, 149df, 171de, 202def }}


Badness: 0.000818
Badness: 0.000818


[[Projection_pair|Projection pair]]s: 7 225/32 11 5625/512 13 625/48
[[Projection pair]]s: 7 225/32 11 5625/512 13 625/48


==Other eleven limit marvel children==
==Other eleven limit marvel children==
The second comma of the [[Normal_lists|normal comma list]] defines which 11-limit family member we are looking at. Adding 4125/4096 gives unidecimal marvel, 91125/90112 gives prodigy, 5632/5625 minerva and 243/242 spectacle.
The second comma of the [[Normal lists|normal comma list]] defines which 11-limit family member we are looking at. Adding 4125/4096 gives unidecimal marvel, 91125/90112 gives prodigy, 5632/5625 minerva and 243/242 spectacle.


=Spectacle=
=Spectacle=
[[Comma|Commas]]: 225/224, 243/242
[[Comma]]s: 225/224, 243/242


Related linear temperament: [[Marvel_temperaments|marvo]]
Related linear temperament: [[Marvel temperaments#Marvo|marvo]]


[[Minimax_tuning|Minimax tuning]]:
[[Minimax tuning]]:


[|1 0 0 0 0>, |1/5 0 0 0 2/5>, |2/5 -2 1 0 4/5>, |-19/5 -4 2 0 12/5>, |0 0 0 0 1>]
[|1 0 0 0 0>, |1/5 0 0 0 2/5>, |2/5 -2 1 0 4/5>, |-19/5 -4 2 0 12/5>, |0 0 0 0 1>]


[[Eigenmonzo_subgroup|Eigenmonzo subgroup]]: 2.9/5.11
[[Eigenmonzo subgroup]]: 2.9/5.11


Map: [<1 1 0 -3 2|, <0 2 0 4 5|, <0 0 1 2 0|]
Map: [<1 1 0 -3 2|, <0 2 0 4 5|, <0 0 1 2 0|]


[[generator|Generators]]: 2, 11/9, 5
[[Generator]]s: 2, 11/9, 5


[[EDO|EDOs]]: [[10edo|10]], [[31edo|31]], [[41edo|41]], [[72edo|72]], [[240edo|240]], [[259edo|259b]], [[269edo|269ce]], [[310edo|310c]], [[331edo|331bc]], [[353edo|353c]], [[497edo|497bc]], [[569edo|569bc]]
{{EDOs|legend=1| 10, 31, 41, 72, 240, 259b, 269ce, 310c, 331bc, 353c, 497bc, 569bc }}


Badness: 0.000499
Badness: 0.000499
Line 301: Line 301:
Projection pairs: 3 242/81 7 366025/52488 11 644204/59049 to 2.5.11/9
Projection pairs: 3 242/81 7 366025/52488 11 644204/59049 to 2.5.11/9


Scales: [[spectacle31|spectacle31]]
Scales: [[spectacle31]]


==13-limit==
==13-limit==
Line 308: Line 308:
Map: [<1 1 0 -3 2 -5|, <0 2 0 4 5 -2|, <0 0 1 2 0 4|]
Map: [<1 1 0 -3 2 -5|, <0 2 0 4 5 -2|, <0 0 1 2 0 4|]


EDOs: 31, 72, 103, 175f, 209, 240
{{EDOs|legend=1| 31, 72, 103, 175f, 209, 240 }}


Badness: 0.001009
Badness: 0.001009


=Apollo=
=Apollo=
[[Comma|Commas]]: 100/99, 225/224
[[Comma]]s: 100/99, 225/224


Related linear temperament: [[Magic]]
Related linear temperament: [[Magic]]
Line 321: Line 321:
Map: [<1 0 0 -5 2|, <0 1 0 2 -2|, <0 0 1 2 2|]
Map: [<1 0 0 -5 2|, <0 1 0 2 -2|, <0 0 1 2 2|]


EDOs: 12, 19, 22, 41, [[104edo|104edo]], [[157edo|157ce]], [[198edo|198ce]], [[220edo|220ce]], [[261edo|261ce]]
{{EDOs|legend=1| 12, 19, 22, 41, 104, 157ce, 198ce, 220ce, 261ce }}


Projection pairs: 7 225/32 11 100/9
Projection pairs: 7 225/32 11 100/9


==13-limit==
==13-limit==
[[Comma|Commas]]: 100/99, 225/224, 245/243
[[Comma]]s: 100/99, 225/224, 245/243


Eigenmonzo subgroup: 2.11/9.13/9
Eigenmonzo subgroup: 2.11/9.13/9
Line 332: Line 332:
Map: [<1 0 0 -5 2 7|, <0 1 0 2 -2 -5|, <0 0 1 2 2 2|]
Map: [<1 0 0 -5 2 7|, <0 1 0 2 -2 -5|, <0 0 1 2 2 2|]


EDOs: 22, 29, 41, 63, 104, 179cef, 242cde, 283def, 346bcdef
{{EDOs|legend=1| 22, 29, 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 341: Line 341:
Map: [<1 0 1 -3 2|, <0 1 1 4 1|, <0 0 2 4 1|]
Map: [<1 0 1 -3 2|, <0 1 1 4 1|, <0 0 2 4 1|]


EDOs: {{EDOs|9, 20, 22, 31, 53}}
{{EDOs|legend=1| 9, 20, 22, 31, 53 }}


== 13-limit ==
== 13-limit ==
Line 348: Line 348:
Map: [<1 0 1 -3 2 -5|, <0 1 1 4 1 6|, <0 0 2 4 1 6|]
Map: [<1 0 1 -3 2 -5|, <0 1 1 4 1 6|, <0 0 2 4 1 6|]


EDOs: {{EDOs|9, 20, 29, 31}}
{{EDOs|legend=1| 9, 20, 29, 31 }}


== Diana ==
== Diana ==
Line 355: Line 355:
Map: [<1 0 1 -3 2 7|, <0 1 1 4 1 -2|, <0 0 2 4 1 1|]
Map: [<1 0 1 -3 2 7|, <0 1 1 4 1 -2|, <0 0 2 4 1 1|]


EDOs: {{EDOs|22, 29, 31, 53}}
{{EDOs|legend=1| 22, 29, 31, 53 }}


=Potassium=
=Potassium=
Line 364: Line 364:
Map: [&lt;1 0 0 -5 -2|, &lt;0 1 0 2 2|, &lt;0 0 1 2 1|]
Map: [&lt;1 0 0 -5 -2|, &lt;0 1 0 2 2|, &lt;0 0 1 2 1|]


EDOs: 9, 10, 12, 19, 31e, 50e
{{EDOs|legend=1| 9, 10, 12, 19, 31e, 50e }}


Badness: 0.000464
Badness: 0.000464
Line 379: Line 379:
Map: [&lt;1 0 0 -5 -2 -8|, &lt;0 1 0 2 2 3|, &lt;0 0 1 2 1 3|]
Map: [&lt;1 0 0 -5 -2 -8|, &lt;0 1 0 2 2 3|, &lt;0 0 1 2 1 3|]


EDOs: 9, 10, 19, 31e, 50e
{{EDOs|legend=1| 9, 10, 19, 31e, 50e }}


Badness: 0.000733
Badness: 0.000733
Line 390: Line 390:
Map: [&lt;2 0 0 -10 -7|, &lt;0 1 0 2 0|, &lt;0 0 1 2 3|]
Map: [&lt;2 0 0 -10 -7|, &lt;0 1 0 2 0|, &lt;0 0 1 2 3|]


EDOs: 12, 22, 50, 72, 166, 238c, 310c
{{EDOs|legend=1| 12, 22, 50, 72, 166, 238c, 310c }}


Badness: 0.000743
Badness: 0.000743
Line 399: Line 399:
Map: [&lt;1 0 1 -3 0|, &lt;0 6 5 22 0|, &lt;0 0 0 0 1|]
Map: [&lt;1 0 1 -3 0|, &lt;0 6 5 22 0|, &lt;0 0 0 0 1|]


EDOs: 19, 53, 72, 197e, 269ce
{{EDOs|legend=1| 19, 53, 72, 197e, 269ce }}


Badness: 0.001275
Badness: 0.001275
Line 408: Line 408:
Map: [&lt;1 0 1 -3 0 0|, &lt;0 6 5 22 0 14|, &lt;0 0 0 0 1 0|]
Map: [&lt;1 0 1 -3 0 0|, &lt;0 6 5 22 0 14|, &lt;0 0 0 0 1 0|]


EDOs: 19, 53, 72, 125f, 197ef, 269cef
{{EDOs|legend=1| 19, 53, 72, 125f, 197ef, 269cef }}


Badness: 0.000916
Badness: 0.000916
Line 417: Line 417:
Map: [&lt;1 0 0 -5 9|, &lt;0 2 0 4 -7|, &lt;0 0 1 2 0|]
Map: [&lt;1 0 0 -5 9|, &lt;0 2 0 4 -7|, &lt;0 0 1 2 0|]


EDOs: 19, 29, 43, 53, 72, 197e, 269ce, 341ce, 610bce
{{EDOs|legend=1| 19, 29, 43, 53, 72, 197e, 269ce, 341ce, 610bce }}


Badness: 0.00154
Badness: 0.00154
Line 426: Line 426:
Map: [&lt;1 0 0 -5 9 -1|, &lt;0 2 0 4 -7 3|, &lt;0 0 1 2 0 1|]
Map: [&lt;1 0 0 -5 9 -1|, &lt;0 2 0 4 -7 3|, &lt;0 0 1 2 0 1|]


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


Badness: 0.001062
Badness: 0.001062
Line 435: Line 435:
Map: [&lt;1 0 0 -5 -3|, &lt;0 1 0 2 7|, &lt;0 0 1 2 -2|]
Map: [&lt;1 0 0 -5 -3|, &lt;0 1 0 2 7|, &lt;0 0 1 2 -2|]


EDOs: 41, 53, 94, 229c, 248ce, 289ce, 342ce, 383ce
{{EDOs|legend=1| 41, 53, 94, 229c, 248ce, 289ce, 342ce, 383ce }}


Badness: 0.001250
Badness: 0.001250
Line 444: Line 444:
Map: [&lt;1 0 0 -5 -3 2|, &lt;0 1 0 2 7 4|, &lt;0 0 1 2 -2 -2|]
Map: [&lt;1 0 0 -5 -3 2|, &lt;0 1 0 2 7 4|, &lt;0 0 1 2 -2 -2|]


EDOs: 41, 53, 94, 429cdef, 523cdef
{{EDOs|legend=1| 41, 53, 94, 429cdef, 523cdef }}


Badness: 0.001075
Badness: 0.001075
Line 457: Line 457:
Map: [&lt;1 0 0 -5 12 -8|, &lt;0 1 0 2 -1 3|, &lt;0 0 1 2 -3 3|]
Map: [&lt;1 0 0 -5 12 -8|, &lt;0 1 0 2 -1 3|, &lt;0 0 1 2 -3 3|]


EDOs: 9, 10, 19, 31, 41, 72f, [[81edo|81]], [[91edo|91]], [[122edo|122f]], [[163edo|163df]]
{{EDOs|legend=1| 9, 10, 19, 31, 41, 72f, 81, 91, 122f, 163df }}


Badness: 0.000745
Badness: 0.000745
Line 464: Line 464:


=Prodigy=
=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 [[Gamelismic_clan|miracle temperament]]. 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.
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 temperament]]. 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.


[[Comma|Commas]]: 225/224, 441/440
[[Comma]]s: 225/224, 441/440


Related linear temperament: [[Gamelismic_clan|miracle]]
Related linear temperament: [[Gamelismic clan#Miracle|miracle]]


[[Minimax_tuning|Minimax tuning]]:
[[Minimax tuning]]:


[|1 0 0 0 0&gt;, |13/12 1/2 -1/4 0 1/12&gt;,
[|1 0 0 0 0&gt;, |13/12 1/2 -1/4 0 1/12&gt;,
Line 476: Line 476:
|3/2 -1 1/2 0 1/2&gt;, |0 0 0 0 1&gt;]
|3/2 -1 1/2 0 1/2&gt;, |0 0 0 0 1&gt;]


[[Eigenmonzo_subgroup|Eigenmonzo subgroup]]: 2.9/5.11
[[Eigenmonzo subgroup]]: 2.9/5.11


Lattice basis: secor length 0.9111, 3/2 length 0.9477
Lattice basis: secor length 0.9111, 3/2 length 0.9477
Line 486: Line 486:
Map: [&lt;1 0 0 -5 -13|, &lt;0 1 0 2 6|, &lt;0 0 1 2 3|]
Map: [&lt;1 0 0 -5 -13|, &lt;0 1 0 2 6|, &lt;0 0 1 2 3|]


[[generator|Generators]]: 2, 3, 5
[[Generator]]s: 2, 3, 5


[[EDO|EDOs]]: [[10edo|10]], [[12edo|12]], [[29edo|29]], [[31edo|31]], [[41edo|41]], [[72edo|72]], [[247edo|247c]], [[319edo|319bcde]], [[391edo|391bcde]], [[463edo|463bcde]]
{{EDOs|legend=1| 10, 12, 29, 31, 41, 72, 247c, 319bcde, 391bcde, 463bcde }}


Badness: 0.000344
Badness: 0.000344


[[Projection_pair|Projection pair]]s: 7 225/32 11 91125/8192
[[Projection pair]]s: 7 225/32 11 91125/8192


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


==Hobbit bases==
==Hobbit bases==
Line 508: Line 508:
Map: [&lt;1 0 0 -5 -13 -8|, &lt;0 1 0 2 6 3|, &lt;0 0 1 2 3 3|]
Map: [&lt;1 0 0 -5 -13 -8|, &lt;0 1 0 2 6 3|, &lt;0 0 1 2 3 3|]


EDOs: 10, 29, 31, 41, 60e, 72f, 91e, 101cd, 132def, 233cdef, 274cdef, 305cdef
{{EDOs|legend=1| 10, 29, 31, 41, 60e, 72f, 91e, 101cd, 132def, 233cdef, 274cdef, 305cdef }}


Badness: 0.000736
Badness: 0.000736
Line 517: Line 517:
Map: [&lt;1 0 0 -5 -13 -23|, &lt;0 1 0 2 6 11|, &lt;0 0 1 2 3 4|]
Map: [&lt;1 0 0 -5 -13 -23|, &lt;0 1 0 2 6 11|, &lt;0 0 1 2 3 4|]


EDOs: 29, 41, 72, 113, 185cf, 341cf, 413bcf, 526bcdf
{{EDOs|legend=1| 29, 41, 72, 113, 185cf, 341cf, 413bcf, 526bcdf }}


Badness: 0.000900
Badness: 0.000900
Line 526: Line 526:
Map: [&lt;1 0 0 -5 -13 -4|, &lt;0 1 0 2 6 -1|, &lt;0 0 1 2 3 4|]
Map: [&lt;1 0 0 -5 -13 -4|, &lt;0 1 0 2 6 -1|, &lt;0 0 1 2 3 4|]


EDOs: 31, 72, 103, 175f
{{EDOs|legend=1| 31, 72, 103, 175f }}


Badness: 0.000889
Badness: 0.000889

Revision as of 14:38, 29 November 2020


The marvel family is the set of temperaments that temper out the 7-limit marvel comma (225/224 = [-5 2 2 -1⟩) which is also named 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.

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 which temper out the marvel comma.

Marvel

The head of the marvel family is marvel, which tempers out 225/224. Marvel has a 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.

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 also Marvel temperaments.

Vital statistics

Comma c = 225/224

Related linear temperament: catakleismic temperament

7-limit minimax: 3 and 5 1/4c flat, 7 just

[|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 subgroup: 2.5/3.7

9-limit minimax: 3 1/6c flat, 5 1/3c flat, 7 just

[|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 subgroup: 2.9/5.7

Lattice basis: secor length 1.256, 3/2 length 1.369

Angle(secor, 3/2) = 106.958 degrees

Map to lattice: [<0 0 -1 -2|, <0 1 -1 0|]

Map: [<1 0 0 -5|, <0 1 0 2|, <0 0 1 2|]

Generators: 2, 3, 5

EDOs: 10, 12, 19, 22, 31, 41, 72, 197, 269c

Badness: 0.0000365

Projection pairs: 7 225/32

Spectrum: 4/3, 5/4, 7/5, 7/6, 8/7, 6/5, 9/8, 9/7, 10/9

Scales: marvel9, marvel10, marvel11, marvel12, marvel19, marvel22, pump12_1, pump12_2, pump13, pump14, pump15, pump16, pump17, pump18

Minkowski blocks

{2, 3, 5} subgroup

8: 16/15, 250/243

9: 135/128, 128/125

10: 25/24, 2048/2025

11: 135/128, 2048/1875

12: 2048/2025, 128/125

15: 128/125, 32768/30375

17: 25/24, 2278125/2097152

19: 16875/16384, 81/80

21: 128/125, 273375/262144

22: 2048/2025, 3125/3072

29: 16875/16384, 32805/32768

31: 81/80, 34171875/33554432

41: 34171875/33554432, 3125/3072

Music

11-limit (Unimarv)

Commas: 225/224, 385/384

Related linear temperament: catakleismic temperament

Minimax tuning:

[|1 0 0 0 0>, |4/3 8/9 -1/3 0 -1/9>, |8/3 -2/9 1/3 0 -2/9>, |3 4/3 0 0 -2/3>, |8/3 -2/9 -2/3 0 7/9>]

Eigenmonzo subgroup: 2.9/5.11/9

Lattice basis: secor length 1.0364 5/4 length 1.0759

Angle(secor, 5/4) = 104.028 degrees

Map to lattice: [<0 -1 0 -2 1|, <0 -1 1 0 -2|]

Map: [<1 0 0 -5 12|, <0 1 0 2 -1|, <0 0 1 2 -3|]

Generators: 2, 3, 5

EDOs: 10, 12e, 19, 22, 31, 41, 53, 72, 166, 197e, 269ce, 341ce

Badness: 0.000255

Projection pairs: 7 225/32 11 4096/375

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

Scales: marvel22_11, unimarv19, unimarv22

Hobbit bases

{2, 3, 5} subgroup

12: 128/125, 2048/2025

15: 128/125, 32768/30375

19: 16875/16384, 81/80

22: 2048/2025, 2109375/2097152

31: 2109375/2097152, 81/80

41: 3125/3072, 34171875/33554432

13-limit

Commas: 225/224, 385/384, 351/350

13-limit eigenmonzo subgroup: 2.11/9.13/9

15-limit eigenmonzo subgroup: 2.15/11.15/13

Map: [<1 0 0 -5 12 -4|, <0 1 0 2 -1 -1|, <0 0 1 2 -3 4|]

EDOs: 19, 22, 31, 50, 53, 72, 103, 175f, 300cef, 403bcef

Badness: 0.000690

Spectrum: 5/4, 4/3, 16/15, 15/14, 9/7, 6/5, 7/6, 11/8, 7/5, 9/8, 8/7, 10/9, 12/11, 13/10, 11/10, 15/11, 16/13, 11/9, 15/13, 14/13, 13/12, 14/11, 18/13, 13/11

Hecate

Commas: 225/224, 385/384, 325/324

13-limit eigenmonzo subgroup: 2.7.13/5

15-limit eigenmonzo subgroup: 2.7.15/13

Map: [<1 0 0 -5 12 2|, <0 1 0 2 -1 4|, <0 0 1 2 -3 -2|]

EDOs: 19, 41, 53, 72, 113, 125f, 166, 238cf, 404cef

Badness: 0.000721

Projection pairs: 7 225/32 11 4096/375 13 324/25

Spectrum: 4/3, 5/4, 16/15, 15/14, 6/5, 9/8, 7/5, 9/7, 7/6, 10/9, 8/7, 18/13, 11/8, 12/11, 13/12, 11/9, 11/10, 15/13, 15/11, 16/13, 13/11, 14/13, 13/10, 14/11

17-limit

Commas: 225/224, 385/384, 325/324, 595/594

Map: [<1 0 0 -5 12 2 18|, <0 1 0 2 -1 4 0|, <0 0 1 2 -3 -2 -6|]

EDOs: 19, 41, 53g, 72, 113, 166g, 238cfg, 351cfg, 404cefg

Badness: 0.000869

Enodia

Commas: 225/224, 385/384, 325/324, 375/374

Map: [<1 0 0 -5 12 2 18|, <0 1 0 2 -1 4 0|, <0 0 1 2 -3 -2 6|]

EDOs: 41g, 53, 72, 125f, 166g, 238cfg, 363cefg, 404cefg

Badness: .000917

Marvelcat

Commas: 169/168, 225/224, 385/384

Map: [<1 0 0 -5 12 -|, <0 2 0 4 -2 3|, <0 0 1 2 -3 1|]

EDOs: 9, 10, 19, 44, 53, 72, 125f, 197ef, 269cef

Badness: 0.0009997

Marvell

Commas: 225/224, 385/384, 1573/1568

13-limit eigenmonzo subgroup: 2.9/5.11/9

15-limit eigenmonzo subgroup: 2.7.15/13

Map: [<1 0 0 -5 12 -29|, <0 1 0 2 -1 6|, <0 0 1 2 -3 10|]

EDOs: 9, 31, 63, 72, 103, 166, 238cf, 269ce, 507bcef, 610bcef

Badness: 0.000862

Isis

Commas: 225/224, 385/384, 275/273

Map: [<1 0 0 -5 12 17|, <0 1 0 2 -1 4|, <0 0 1 2 -3 -3|]

EDOs: 10, 22, 31, 41, 53, 94

Badness: 0.000866

Projection pairs: 7 225/32 11 4096/375 13 131072/10125

Deecee

Commas: 225/224, 385/384, 364/363

13-limit eigenmonzo subgroup: 2.9/5.13/9

15-limit eigenmonzo subgroup: 2.3.13/5

Map: [<1 0 0 -5 12 27|, <0 1 0 2 -1 -3|, <0 0 1 2 -3 -8|]

EDOs: 9, 22, 41, 63, 72, 185cf, 257cf

Badness: 0.000920

Projection pairs: 7 225/32 11 4096/375 13 134217728/10546875

Mirage

Commas: 225/224, 243/242, 385/384

Map: [<1 1 3 3 2 0|, <0 6 -7 -2 15 0|, <0 0 0 0 0 1|]

EDOs: 10, 31, 41, 62, 72, 103, 175f, 216c, 288cdf, 391bcdef

Badness: 0.000738

Minerva

Commas: 99/98, 176/175

Related linear temperament: orwell

Eigenmonzo subgroup: 2.7/5.11/9

Lattice basis: 16/15 length 0.8997 5/4 length 1.0457

Angle(16/15, 5/4) = 98.6044 degrees

Map to lattice: [<0 -1 0 -2 -2|, <0 -1 1 0 2|]

Map: [<1 0 0 -5 -9|, <0 1 0 2 2|, <0 0 1 2 4|]

Generators: 2, 3, 5

EDOs: 9, 12, 21, 22, 31, 43, 53, 74, 75, 96, 127

Badness: 0.000381

Projection pairs: 7 225/32 11 5625/512

Scales: minerva12, minerva22x

Athene

Commas: 99/98, 176/175, 275/273

13-limit eigenmonzo subgroup: 2.11/9.13/7

15-limit eigenmonzo subgroup: 2.11/9.13/7

Map: [<1 0 0 -5 -9 -4|, <0 1 0 2 2 -1|, <0 0 1 2 4 4|]

EDOs: 22, 31, 53, 84e, 118d, 149df, 171de, 202def

Badness: 0.000818

Projection pairs: 7 225/32 11 5625/512 13 625/48

Other eleven limit marvel children

The second comma of the normal comma list defines which 11-limit family member we are looking at. Adding 4125/4096 gives unidecimal marvel, 91125/90112 gives prodigy, 5632/5625 minerva and 243/242 spectacle.

Spectacle

Commas: 225/224, 243/242

Related linear temperament: marvo

Minimax tuning:

[|1 0 0 0 0>, |1/5 0 0 0 2/5>, |2/5 -2 1 0 4/5>, |-19/5 -4 2 0 12/5>, |0 0 0 0 1>]

Eigenmonzo subgroup: 2.9/5.11

Map: [<1 1 0 -3 2|, <0 2 0 4 5|, <0 0 1 2 0|]

Generators: 2, 11/9, 5

EDOs: 10, 31, 41, 72, 240, 259b, 269ce, 310c, 331bc, 353c, 497bc, 569bc

Badness: 0.000499

Projection pairs: 3 242/81 7 366025/52488 11 644204/59049 to 2.5.11/9

Scales: spectacle31

13-limit

Commas: 225/224, 243/242, 351/350

Map: [<1 1 0 -3 2 -5|, <0 2 0 4 5 -2|, <0 0 1 2 0 4|]

EDOs: 31, 72, 103, 175f, 209, 240

Badness: 0.001009

Apollo

Commas: 100/99, 225/224

Related linear temperament: Magic

Eigenmonzo subgroup: 2.7/5.11/9

Map: [<1 0 0 -5 2|, <0 1 0 2 -2|, <0 0 1 2 2|]

EDOs: 12, 19, 22, 41, 104, 157ce, 198ce, 220ce, 261ce

Projection pairs: 7 225/32 11 100/9

13-limit

Commas: 100/99, 225/224, 245/243

Eigenmonzo subgroup: 2.11/9.13/9

Map: [<1 0 0 -5 2 7|, <0 1 0 2 -2 -5|, <0 0 1 2 2 2|]

EDOs: 22, 29, 41, 63, 104, 179cef, 242cde, 283def, 346bcdef

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

Artemis

Commas: 121/120, 225/224

Map: [<1 0 1 -3 2|, <0 1 1 4 1|, <0 0 2 4 1|]

EDOs: 9, 20, 22, 31, 53

13-limit

Commas: 105/104, 121/120, 196/195

Map: [<1 0 1 -3 2 -5|, <0 1 1 4 1 6|, <0 0 2 4 1 6|]

EDOs: 9, 20, 29, 31

Diana

Commas: 121/120, 225/224, 275/273

Map: [<1 0 1 -3 2 7|, <0 1 1 4 1 -2|, <0 0 2 4 1 1|]

EDOs: 22, 29, 31, 53

Potassium

Commas: 45/44, 56/55

Eigenmonzo subgroup: 2.9/7.11

Map: [<1 0 0 -5 -2|, <0 1 0 2 2|, <0 0 1 2 1|]

EDOs: 9, 10, 12, 19, 31e, 50e

Badness: 0.000464

Projection pairs: 7 225/32 11 45/4

13-limit

Commas: 45/44, 56/55, 78/77

13-limit eigenmonzo subgroup: 2.9/7.13/9

15-limit eigenmonzo subgroup: 2.9/7.13/9

Map: [<1 0 0 -5 -2 -8|, <0 1 0 2 2 3|, <0 0 1 2 1 3|]

EDOs: 9, 10, 19, 31e, 50e

Badness: 0.000733

Projection pairs: 7 225/32 11 45/4 13 3375/256

Fantastic

Commas: 225/224, 4375/4356

Map: [<2 0 0 -10 -7|, <0 1 0 2 0|, <0 0 1 2 3|]

EDOs: 12, 22, 50, 72, 166, 238c, 310c

Badness: 0.000743

Catakleismoid

Commas: 225/224, 4375/4374

Map: [<1 0 1 -3 0|, <0 6 5 22 0|, <0 0 0 0 1|]

EDOs: 19, 53, 72, 197e, 269ce

Badness: 0.001275

13-limit

Commas: 169/168, 225/224, 325/324

Map: [<1 0 1 -3 0 0|, <0 6 5 22 0 14|, <0 0 0 0 1 0|]

EDOs: 19, 53, 72, 125f, 197ef, 269cef

Badness: 0.000916

Hestia

Commas: 225/224, 125000/124509

Map: [<1 0 0 -5 9|, <0 2 0 4 -7|, <0 0 1 2 0|]

EDOs: 19, 29, 43, 53, 72, 197e, 269ce, 341ce, 610bce

Badness: 0.00154

13-limit

Commas: 169/168, 225/224, 1001/1000

Map: [<1 0 0 -5 9 -1|, <0 2 0 4 -7 3|, <0 0 1 2 0 1|]

EDOs: 19, 29, 43, 53, 72, 125f, 197ef, 269cef

Badness: 0.001062

Malcolm

Commas: 225/224, 2200/2187

Map: [<1 0 0 -5 -3|, <0 1 0 2 7|, <0 0 1 2 -2|]

EDOs: 41, 53, 94, 229c, 248ce, 289ce, 342ce, 383ce

Badness: 0.001250

13-limit

commas: 225/224, 275/273, 325/324

Map: [<1 0 0 -5 -3 2|, <0 1 0 2 7 4|, <0 0 1 2 -2 -2|]

EDOs: 41, 53, 94, 429cdef, 523cdef

Badness: 0.001075

Tripod

Commas: 105/104, 144/143, 196/195

13-limit eigenmonzo subgroup: 2.9/7.13/11

15-limit eigenmonzo subgroup: 2.5/3.13/11

Map: [<1 0 0 -5 12 -8|, <0 1 0 2 -1 3|, <0 0 1 2 -3 3|]

EDOs: 9, 10, 19, 31, 41, 72f, 81, 91, 122f, 163df

Badness: 0.000745

Projection pairs: 7 225/32 11 4096/375 13 3375/256

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 temperament. 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.

Commas: 225/224, 441/440

Related linear temperament: miracle

Minimax tuning:

[|1 0 0 0 0>, |13/12 1/2 -1/4 0 1/12>, |13/6 -1 1/2 0 1/6>, |3/2 -1 1/2 0 1/2>, |0 0 0 0 1>]

Eigenmonzo subgroup: 2.9/5.11

Lattice basis: secor length 0.9111, 3/2 length 0.9477

Angle(secor, 3/2) = 65.933

Map to lattice: [<0 0 -1 -2 -3|, <0 1 -1 0 3|]

Map: [<1 0 0 -5 -13|, <0 1 0 2 6|, <0 0 1 2 3|]

Generators: 2, 3, 5

EDOs: 10, 12, 29, 31, 41, 72, 247c, 319bcde, 391bcde, 463bcde

Badness: 0.000344

Projection pairs: 7 225/32 11 91125/8192

Scales: prodigy11, prodigy12, prodigy29

Hobbit bases

{2, 3, 5} subgroup

31: 81/80, 34171875/33554432

41: 34171875/33554432, 32805/32768

13-limit

Commas: 105/104, 196/195, 352/351

Map: [<1 0 0 -5 -13 -8|, <0 1 0 2 6 3|, <0 0 1 2 3 3|]

EDOs: 10, 29, 31, 41, 60e, 72f, 91e, 101cd, 132def, 233cdef, 274cdef, 305cdef

Badness: 0.000736

Prodigious

Commas: 225/224, 441/440, 364/363

Map: [<1 0 0 -5 -13 -23|, <0 1 0 2 6 11|, <0 0 1 2 3 4|]

EDOs: 29, 41, 72, 113, 185cf, 341cf, 413bcf, 526bcdf

Badness: 0.000900

Prodigal

Commas: 225/224, 441/440, 351/350

Map: [<1 0 0 -5 -13 -4|, <0 1 0 2 6 -1|, <0 0 1 2 3 4|]

EDOs: 31, 72, 103, 175f

Badness: 0.000889

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