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 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 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.


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 34b 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 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]].
Line 20: Line 20:
[[Comma list]]: 225/224
[[Comma list]]: 225/224


[[Mapping]]: [{{val| 1 0 0 -5 }}, {{val| 0 1 0 2 }}, {{val| 0 0 1 2 }}]
{{Mapping|legend=1| 1 0 0 -5 | 0 1 0 2 | 0 0 1 2 }}


Mapping generators: ~2, ~3, ~5
: mapping generators: ~2, ~3, ~5


Map to lattice: [{{val| 0 0 -1 -2 }}, {{val| 0 1 -1 0 }}]
Map to lattice: [{{val| 0 0 -1 -2 }}, {{val| 0 1 -1 0 }}]
Line 30: Line 30:
: Angle (secor, 3/2) = 106.958 degrees
: Angle (secor, 3/2) = 106.958 degrees


[[Optimal tuning]] ([[POTE]]): ~3/2 = 700.4075, ~5/4 = 383.6376
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 700.4075, ~5/4 = 383.6376


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[7-odd-limit]]: 3 and 5 1/4c flat, 7 just
* [[7-odd-limit]]: 3 and 5 1/4-comma flat, 7 just
: [{{monzo| 1 0 0 0 }}, {{monzo| 5/4 1/2 -1/2 1/4 }}, {{monzo| 5/4 -1/2 1/2 1/4 }}, {{monzo| 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 subgroup]] ([[unchanged-interval basis]]): 2.5/3.7
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5/3.7
* [[9-odd-limit]]: 3 1/6c flat, 5 1/3c flat, 7 just
* [[9-odd-limit]]: 3 1/6-comma flat, 5 1/3-comma flat, 7 just
: [{{monzo| 1 0 0 0 }}, {{monzo| 5/6 2/3 -1/3 1/6 }}, {{monzo| 5/3 -2/3 1/3 1/3 }}, {{monzo| 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 subgroup]] ([[unchanged-interval basis]]): 2.9/5.7
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) 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 53: Line 53:


{{Databox|[[Minkowski blocks]]|
{{Databox|[[Minkowski blocks]]|
{2, 3, 5} subgroup
2.3.5 subgroup
* 8: 16/15, 250/243
* 8: 16/15, 250/243
* 9: 135/128, 128/125
* 9: 135/128, 128/125
Line 83: Line 83:
[[Comma list]]: 225/224, 385/384
[[Comma list]]: 225/224, 385/384


[[Mapping]]: [{{val| 1 0 0 -5 12 }}, {{val| 0 1 0 2 -1 }}, {{val| 0 0 1 2 -3 }}]
{{Mapping|legend=1| 1 0 0 -5 12 | 0 1 0 2 -1 | 0 0 1 2 -3 }}


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


Lattice basis:  
Lattice basis:  
Line 91: Line 91:
: Angle (secor, 5/4) = 104.028 degrees
: Angle (secor, 5/4) = 104.028 degrees


[[Optimal tuning]] ([[POTE]]): ~3/2 = 700.3887, ~5/4 = 383.5403
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 700.3887, ~5/4 = 383.5403


[[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 subgroup]] ([[unchanged-interval basis]]): 2.9/5.11/9
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) 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 111: Line 111:


{{Databox|Hobbit bases|
{{Databox|Hobbit bases|
{2, 3, 5} subgroup
2.3.5 subgroup
* 12: 128/125, 2048/2025
* 12: 128/125, 2048/2025
* 15: 128/125, 32768/30375
* 15: 128/125, 32768/30375
Line 125: Line 125:
Comma list: 225/224, 351/350, 385/384
Comma list: 225/224, 351/350, 385/384


Mapping: [{{val| 1 0 0 -5 12 -4 }}, {{val| 0 1 0 2 -1 -1 }}, {{val| 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): ~3/2 = 699.7367, ~5/4 = 384.0613
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 699.7367, ~5/4 = 384.0613


Minimax tuning:  
Minimax tuning:  
* 13-odd-limit eigenmonzo subgroup (unchanged-interval basis): 2.11/9.13/9
* 13-odd-limit eigenmonzo (unchanged-interval) basis: 2.11/9.13/9
* 15-odd-limit eigenmonzo subgroup (unchanged-interval basis): 2.15/11.15/13
* 15-odd-limit eigenmonzo (unchanged-interval) 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|legend=1| 19, 22, 31, 50, 53, 72, 103, 175f, 300ceff, 403bceeff, 578bbccdeeefff }}
Line 144: Line 144:
Comma list: 225/224, 325/324, 385/384
Comma list: 225/224, 325/324, 385/384


Mapping: [{{val| 1 0 0 -5 12 2 }}, {{val| 0 1 0 2 -1 4 }}, {{val| 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): ~3/2 = 700.9779, ~5/4 = 383.1622
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.9779, ~5/4 = 383.1622


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


{{Optimal ET sequence|legend=1| 19, 22f, 31f, 41, 53, 72, 125f, 166, 238cf }}
{{Optimal ET sequence|legend=1| 19, 22f, 31f, 41, 53, 72, 125f, 166, 238cf }}
Line 165: Line 165:
Comma list: 225/224, 325/324, 385/384, 595/594
Comma list: 225/224, 325/324, 385/384, 595/594


Mapping: [{{val| 1 0 0 -5 12 2 18 }}, {{val| 0 1 0 2 -1 4 0 }}, {{val| 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 (POTE): ~3/2 = 700.9619, ~5/4 = 383.0310
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.9619, ~5/4 = 383.0310


{{Optimal ET sequence|legend=1| 19, 22f, 31fg, 41, 53g, 72, 166g, 238cfg, 404ccefgg }}
{{Optimal ET sequence|legend=1| 19, 22f, 31fg, 41, 53g, 72, 166g, 238cfg, 404ccefgg }}
Line 178: Line 178:
Comma list: 225/224, 325/324, 375/374, 385/384
Comma list: 225/224, 325/324, 375/374, 385/384


Mapping: [{{val| 1 0 0 -5 12 2 18 }}, {{val| 0 1 0 2 -1 4 0 }}, {{val| 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 (POTE): ~3/2 = 700.9658, ~5/4 = 383.3063
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.9658, ~5/4 = 383.3063


{{Optimal ET sequence|legend=1| 19g, 22f, 31f, 41g, 53, 72, 166g, 238cfg, 404ccefgg }}
{{Optimal ET sequence|legend=1| 19g, 22f, 31f, 41g, 53, 72, 166g, 238cfg, 404ccefgg }}
Line 191: Line 191:
Comma list: 225/224, 385/384, 1573/1568
Comma list: 225/224, 385/384, 1573/1568


Mapping: [{{val| 1 0 0 -5 12 -29 }}, {{val| 0 1 0 2 -1 6 }}, {{val| 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): ~3/2 = 700.3937, ~5/4 = 383.5725
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.3937, ~5/4 = 383.5725


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


{{Optimal ET sequence|legend=1| 9, 22f, 31, 63, 72, 103, 166, 238cf, 269ce, 507bcceff, 610bcceeff }}
{{Optimal ET sequence|legend=1| 9, 22f, 31, 63, 72, 103, 166, 238cf, 269ce, 507bcceff, 610bcceeff }}
Line 208: Line 208:
Comma list: 225/224, 275/273, 385/384
Comma list: 225/224, 275/273, 385/384


Mapping: [{{val| 1 0 0 -5 12 17 }}, {{val| 0 1 0 2 -1 4 }}, {{val| 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): ~3/2 = 701.9156, ~5/4 = 383.2445
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.9156, ~5/4 = 383.2445


{{Optimal ET sequence|legend=1| 10, 19f, 22, 31, 41, 53, 94 }}
{{Optimal ET sequence|legend=1| 10, 19f, 22, 31, 41, 53, 94 }}
Line 223: Line 223:
Comma list: 225/224, 364/363, 385/384
Comma list: 225/224, 364/363, 385/384


Mapping: [{{val| 1 0 0 -5 12 27 }}, {{val| 0 1 0 2 -1 -3 }}, {{val| 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): ~3/2 = 700.4560, ~5/4 = 382.8177
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.4560, ~5/4 = 382.8177


Minimax tuning:  
Minimax tuning:  
Line 242: Line 242:
Comma list: 105/104, 144/143, 196/195
Comma list: 105/104, 144/143, 196/195


Mapping: [{{val| 1 0 0 -5 12 -8 }}, {{val| 0 1 0 2 -1 3 }}, {{val| 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): ~3/2 = 699.2335, ~5/4 = 382.9775
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 699.2335, ~5/4 = 382.9775


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


{{Optimal ET sequence|legend=1| 9, 10, 19, 22f, 31, 41, 72f, 91, 122f, 163df }}
{{Optimal ET sequence|legend=1| 9, 10, 19, 22f, 31, 41, 72f, 91, 122f, 163df }}
Line 261: Line 261:
Comma list: 169/168, 225/224, 385/384
Comma list: 169/168, 225/224, 385/384


Mapping: [{{val| 1 0 0 -5 12 -1 }}, {{val| 0 2 0 4 -2 3 }}, {{val| 0 0 1 2 -3 1 }}]
Mapping: {{mapping| 1 0 0 -5 12 -1 | 0 2 0 4 -2 3 | 0 0 1 2 -3 1 }}


Mapping generators: ~2, ~26/15, ~5
: mapping generators: ~2, ~26/15, ~5


Optimal tuning (POTE): ~15/13 = 249.7138, ~5/4 = 383.5816
Optimal tuning (POTE): ~2 = 1\1, ~15/13 = 249.7138, ~5/4 = 383.5816


{{Optimal ET sequence|legend=1| 9, 10, 19, 44, 53, 72, 125f, 197ef, 269ceff }}
{{Optimal ET sequence|legend=1| 9, 10, 19, 44, 53, 72, 125f, 197ef, 269ceff }}
Line 276: Line 276:
[[Comma list]]: 99/98, 176/175
[[Comma list]]: 99/98, 176/175


[[Mapping]]: [{{val| 1 0 0 -5 -9 }}, {{val| 0 1 0 2 2 }}, {{val| 0 0 1 2 4 }}]
{{Mapping|legend=1| 1 0 0 -5 -9 | 0 1 0 2 2 | 0 0 1 2 4 }}


Map to lattice: [{{val| 0 -1 0 -2 -2 }}, {{val| 0 -1 1 0 2 }}]
Map to lattice: [{{val| 0 -1 0 -2 -2 }}, {{val| 0 -1 1 0 2 }}]
Line 284: Line 284:
: Angle (16/15, 5/4) = 98.6044 degrees
: Angle (16/15, 5/4) = 98.6044 degrees


[[Optimal tuning]] ([[POTE]]): ~3/2 = 700.2593, ~5/4 = 386.5581
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 700.2593, ~5/4 = 386.5581


[[Minimax tuning]]: [[11-odd-limit]] [[Eigenmonzo subgroup]] ([[unchanged-interval basis]]): 2.7/5.11/9
[[Minimax tuning]]: [[11-odd-limit]] [[Eigenmonzo subgroup]] ([[unchanged-interval basis]]): 2.7/5.11/9
Line 294: Line 294:
[[Projection pair]]s: 7 225/32 11 5625/512
[[Projection pair]]s: 7 225/32 11 5625/512


Scales (Scala files): [[minerva12]], [[minerva22x]]
Scales: [[minerva12]], [[minerva22x]]


[[Associated temperament]]: [[Würschmidt family #Würschmidt|würschmidt]]
[[Associated temperament]]: [[Würschmidt family #Würschmidt|würschmidt]]
Line 303: Line 303:
Comma list: 99/98, 176/175, 275/273
Comma list: 99/98, 176/175, 275/273


Mapping: [{{val| 1 0 0 -5 -9 -4 }}, {{val| 0 1 0 2 2 -1 }}, {{val| 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): ~3/2 = 701.2342, ~5/4 = 385.9594
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.2342, ~5/4 = 385.9594


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


{{Optimal ET sequence|legend=1| 12f, 19e, 22, 31, 53, 84e, 118d, 171de, 202def }}
{{Optimal ET sequence|legend=1| 12f, 19e, 22, 31, 53, 84e, 118d, 171de, 202def }}
Line 322: Line 322:
[[Comma list]]: 100/99, 225/224
[[Comma list]]: 100/99, 225/224


[[Mapping]]: [{{val| 1 0 0 -5 2 }}, {{val| 0 1 0 2 -2 }}, {{val| 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]]): ~3/2 = 703.4846, ~5/4 = 381.6033
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 703.4846, ~5/4 = 381.6033


[[Minimax tuning]]: [[11-odd-limit]] [[Eigenmonzo subgroup]] ([[unchanged-interval basis]]): 2.7/5.11/9
[[Minimax tuning]]: [[11-odd-limit]] [[Eigenmonzo basis|eigenmonzo (unchanged-interval) 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 332: Line 332:
[[Projection pair]]s: 7 225/32 11 100/9
[[Projection pair]]s: 7 225/32 11 100/9


[[Associated temperament]]: [[Magic family #magic|magic]]
[[Associated temperament]]: [[magic]]


=== 13-limit ===
=== 13-limit ===
Line 339: Line 339:
Comma list: 100/99, 225/224, 275/273
Comma list: 100/99, 225/224, 275/273


Mapping: [{{val| 1 0 0 -5 2 7 }}, {{val| 0 1 0 2 -2 -5 }}, {{val| 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): ~3/2 = 703.9984, ~5/4 = 381.5352
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 703.9984, ~5/4 = 381.5352


Minimax tuning: 13-odd-limit eigenmonzo subgroup (unchanged-interval basis): 2.11/9.13/9
Minimax tuning: 13-odd-limit eigenmonzo (unchanged-interval) 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|legend=1| 12f, 19f, 22, 29, 34d, 41, 63, 104, 179cef, 242cde, 283def, 346bcdef }}
Line 354: Line 354:
[[Comma list]]: 45/44, 56/55
[[Comma list]]: 45/44, 56/55


[[Mapping]]: [{{val| 1 0 0 -5 -2 }}, {{val| 0 1 0 2 2 }}, {{val| 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]]): ~3/2 = 696.1714, ~5/4 = 385.0500
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 696.1714, ~5/4 = 385.0500


[[Minimax tuning]]: [[11-odd-limit]] [[Eigenmonzo subgroup]] ([[unchanged-interval basis]]): 2.9/7.11
[[Minimax tuning]]: [[11-odd-limit]] [[Eigenmonzo basis|eigenmonzo (unchanged-interval) 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 371: Line 371:
Comma list: 45/44, 56/55, 78/77
Comma list: 45/44, 56/55, 78/77


Mapping: [{{val| 1 0 0 -5 -2 -8 }}, {{val| 0 1 0 2 2 3 }}, {{val| 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): ~3/2 = 696.0103, ~5/4 = 384.6785
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 696.0103, ~5/4 = 384.6785


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


{{Optimal ET sequence|legend=1| 9, 10, 12f, 19, 31e, 50e }}
{{Optimal ET sequence|legend=1| 9, 10, 12f, 19, 31e, 50e }}
Line 390: Line 390:
[[Comma list]]: 225/224, 2200/2187
[[Comma list]]: 225/224, 2200/2187


[[Mapping]]: [{{val| 1 0 0 -5 -3 }}, {{val| 0 1 0 2 7 }}, {{val| 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]]): ~3/2 = 701.8913, ~5/4 = 382.4083
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 701.8913, ~5/4 = 382.4083


{{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 403: Line 403:
Comma list: 225/224, 275/273, 325/324
Comma list: 225/224, 275/273, 325/324


Mapping: [{{val| 1 0 0 -5 -3 2 }}, {{val| 0 1 0 2 7 4 }}, {{val| 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): ~3/2 = 701.8913, ~5/4 = 382.4083
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.8913, ~5/4 = 382.4083


{{Optimal ET sequence|legend=1| 12e, 19e, 34d, 41, 53, 94, 429ccdeef, 523ccdeef }}
{{Optimal ET sequence|legend=1| 12e, 19e, 34d, 41, 53, 94, 429ccdeef, 523ccdeef }}
Line 412: Line 412:


== 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 [[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]]. 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
[[Subgroup]]: 2.3.5.7.11
Line 418: Line 418:
[[Comma list]]: 225/224, 441/440
[[Comma list]]: 225/224, 441/440


[[Mapping]]: [{{val| 1 0 0 -5 -13 }}, {{val| 0 1 0 2 6 }}, {{val| 0 0 1 2 3 }}]
{{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 }}]
Map to lattice: [{{val| 0 0 -1 -2 -3 }}, {{val| 0 1 -1 0 3 }}]
Line 426: Line 426:
: Angle (secor, 3/2) = 65.933
: Angle (secor, 3/2) = 65.933


[[Optimal tuning]] ([[POTE]]): ~3/2 = 699.7981, ~5/4 = 383.5114
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 699.7981, ~5/4 = 383.5114


[[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 subgroup]] ([[unchanged-interval basis]]): 2.9/5.11
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) 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 441: Line 441:
Scales: [[prodigy11]], [[prodigy12]], [[prodigy29]]
Scales: [[prodigy11]], [[prodigy12]], [[prodigy29]]


[[Associated temperament]]: [[Gamelismic clan #Miracle|miracle]]
[[Associated temperament]]: [[miracle]]


{{Databox|Hobbit bases|
{{Databox|Hobbit bases|
{2, 3, 5} subgroup
2.3.5 subgroup
* 31: 81/80, 34171875/33554432
* 31: 81/80, 34171875/33554432
* 41: 34171875/33554432, 32805/32768
* 41: 34171875/33554432, 32805/32768
Line 454: Line 454:
Comma list: 105/104, 196/195, 352/351
Comma list: 105/104, 196/195, 352/351


Mapping: [{{val| 1 0 0 -5 -13 -8 }}, {{val| 0 1 0 2 6 3 }}, {{val| 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): ~3/2 = 700.4006, ~5/4 = 381.4025
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.4006, ~5/4 = 381.4025


{{Optimal ET sequence|legend=1| 10, 12f, 19e, 29, 31, 41, 60e, 72f, 91e, 101cd, 132def, 233ccddeef, 274ccddeff, 305ccddeeff }}
{{Optimal ET sequence|legend=1| 10, 12f, 19e, 29, 31, 41, 60e, 72f, 91e, 101cd, 132def, 233ccddeef, 274ccddeff, 305ccddeeff }}
Line 467: Line 467:
Comma list: 225/224, 364/363, 441/440
Comma list: 225/224, 364/363, 441/440


Mapping: [{{val| 1 0 0 -5 -13 -23 }}, {{val| 0 1 0 2 6 11 }}, {{val| 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): ~3/2 = 700.3407, ~5/4 = 383.2592
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.3407, ~5/4 = 383.2592


{{Optimal ET sequence|legend=1| 12f, 29, 31f, 41, 72, 185cf, 341cf, 413bcff, 526bccdff }}
{{Optimal ET sequence|legend=1| 12f, 29, 31f, 41, 72, 185cf, 341cf, 413bcff, 526bccdff }}
Line 480: Line 480:
Comma list: 225/224, 351/350, 441/440
Comma list: 225/224, 351/350, 441/440


Mapping: [{{val| 1 0 0 -5 -13 -4 }}, {{val| 0 1 0 2 6 -1 }}, {{val| 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): ~3/2 = 699.4864, ~5/4 = 384.0998
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 699.4864, ~5/4 = 384.0998


{{Optimal ET sequence|legend=1| 12f, 19e, 31, 53e, 60eff, 72, 103, 175f }}
{{Optimal ET sequence|legend=1| 12f, 19e, 31, 53e, 60eff, 72, 103, 175f }}
Line 493: Line 493:
Comma list: 225/224, 441/440, 1001/1000
Comma list: 225/224, 441/440, 1001/1000


Mapping: [{{val| 1 0 0 -5 -13 21 }}, {{val| 0 1 0 2 6 -8 }}, {{val| 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): ~3/2 = 699.5536, ~5/4 = 383.5696
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 699.5536, ~5/4 = 383.5696


{{Optimal ET sequence|legend=1| 29, 31, 43, 60e, 72, 103, 175f, 482bccddeefff, 554bbccddeeeffff }}
{{Optimal ET sequence|legend=1| 29, 31, 43, 60e, 72, 103, 175f, 482bccddeefff, 554bbccddeeeffff }}
Line 506: Line 506:
Comma list: 225/224, 273/272, 375/374, 441/440
Comma list: 225/224, 273/272, 375/374, 441/440


Mapping: [{{val| 1 0 0 -5 -13 21 12 }}, {{val| 0 1 0 2 6 -8 -5 }}, {{val| 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): ~3/2 = 699.6262, ~5/4 = 383.4458
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 699.6262, ~5/4 = 383.4458


{{Optimal ET sequence|legend=1| 29g, 31, 43, 60e, 72, 103, 175f, 307bcdeeffg, 379bccdeeffgg, 482bccddeefffgg, 554bbccddeeeffffgg }}
{{Optimal ET sequence|legend=1| 29g, 31, 43, 60e, 72, 103, 175f, 307bcdeeffg, 379bccdeeffgg, 482bccddeefffgg, 554bbccddeeeffffgg }}
Line 519: Line 519:
[[Comma list]]: 225/224, 4375/4356
[[Comma list]]: 225/224, 4375/4356


[[Mapping]]: [{{val| 2 0 0 -10 -7 }}, {{val| 0 1 0 2 0 }}, {{val| 0 0 1 2 3 }}]
{{Mapping|legend=1| 2 0 0 -10 -7 | 0 1 0 2 0 | 0 0 1 2 3 }}


Mapping generators: ~99/70, ~3, ~5
: mapping generators: ~99/70, ~3, ~5


[[Optimal tuning]] ([[POTE]]): ~3/2 = 700.6242, ~5/4 = 383.2978
[[Optimal tuning]] ([[POTE]]): ~99/70 = 1\2, ~3/2 = 700.6242, ~5/4 = 383.2978


{{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 534: Line 534:
[[Comma list]]: 225/224, 243/242
[[Comma list]]: 225/224, 243/242


[[Mapping]]: [{{val| 1 1 0 -3 2 }}, {{val| 0 2 0 4 5 }}, {{val| 0 0 1 2 0 }}]
{{Mapping|legend=1| 1 1 0 -3 2 | 0 2 0 4 5 | 0 0 1 2 0 }}


Mapping generators: ~2, ~11/9, ~5
: mapping generators: ~2, ~11/9, ~5


[[Optimal tuning]] ([[POTE]]): ~11/9 = 350.0570, ~5/4 = 383.9323
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~11/9 = 350.0570, ~5/4 = 383.9323


[[Minimax tuning]]:
[[Minimax tuning]]:
* [[11-odd-limit]]
* [[11-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 1/5 0 0 0 2/5 }}, {{monzo| 2/5 -2 1 0 4/5 }}, {{monzo| -19/5 -4 2 0 12/5 }}, {{monzo| 0 0 0 0 1 }}]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 1/5 0 0 0 2/5 }}, {{monzo| 2/5 -2 1 0 4/5 }}, {{monzo| -19/5 -4 2 0 12/5 }}, {{monzo| 0 0 0 0 1 }}]
: [[Eigenmonzo subgroup]] ([[unchanged-interval basis]]): 2.9/5.11
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.9/5.11


{{Optimal ET sequence|legend=1| 31, 41, 72, 247c, 281, 353c, 425bc, 497bc }}
{{Optimal ET sequence|legend=1| 31, 41, 72, 247c, 281, 353c, 425bc, 497bc }}
Line 551: Line 551:
[[Projection pair]]s: 3 242/81 7 366025/52488 11 644204/59049 to 2.5.11/9
[[Projection pair]]s: 3 242/81 7 366025/52488 11 644204/59049 to 2.5.11/9


Scales (Scala files): [[spectacle31]]
Scales: [[spectacle31]]


[[Associated temperament]]: [[Marvel temperaments #Marvo|marvo]]
[[Associated temperament]]: [[marvo]]


=== 13-limit ===
=== 13-limit ===
Line 560: Line 560:
Comma list: 225/224, 243/242, 351/350
Comma list: 225/224, 243/242, 351/350


Mapping: [{{val| 1 1 0 -3 2 -5 }}, {{val| 0 2 0 4 5 -2 }}, {{val| 0 0 1 2 0 4 }}]
Mapping: {{mapping| 1 1 0 -3 2 -5 | 0 2 0 4 5 -2 | 0 0 1 2 0 4 }}
 
Mapping generators: ~2, ~11/9, ~5


Optimal tuning (POTE): ~11/9 = 349.9247, ~5/4 = 384.3505
Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 349.9247, ~5/4 = 384.3505


{{Optimal ET sequence|legend=1| 31, 72, 103, 175f }} <nowiki>*</nowiki>
{{Optimal ET sequence|legend=1| 31, 72, 103, 175f }} <nowiki>*</nowiki>
Line 577: Line 575:
[[Comma list]]: 225/224, 125000/124509
[[Comma list]]: 225/224, 125000/124509


[[Mapping]]: [{{val| 1 0 0 -5 9 }}, {{val| 0 2 0 4 -7 }}, {{val| 0 0 1 2 0 }}]
{{Mapping|legend=1| 1 0 0 -5 9 | 0 2 0 4 -7 | 0 0 1 2 0 }}


Mapping generators: ~2, ~400/231, ~5
: mapping generators: ~2, ~400/231, ~5


[[Optimal tuning]] ([[POTE]]): ~400/231 = 950.1474, ~5/4 = 383.6467
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~400/231 = 950.1474, ~5/4 = 383.6467


{{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 592: Line 590:
Comma list: 169/168, 225/224, 1001/1000
Comma list: 169/168, 225/224, 1001/1000


Mapping: [{{val| 1 0 0 -5 9 -1 }}, {{val| 0 2 0 4 -7 3 }}, {{val| 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): ~26/15 = 950.2349, ~5/4 = 383.5558
Optimal tuning (POTE): ~2 = 1\1, ~26/15 = 950.2349, ~5/4 = 383.5558


{{Optimal ET sequence|legend=1| 19, 29, 43, 53, 72, 125f, 197ef, 269cef }}
{{Optimal ET sequence|legend=1| 19, 29, 43, 53, 72, 125f, 197ef, 269cef }}
Line 605: Line 603:
[[Comma list]]: 121/120, 225/224
[[Comma list]]: 121/120, 225/224


[[Mapping]]: [{{val| 1 0 1 -3 2 }}, {{val| 0 1 1 4 1 }}, {{val| 0 0 -2 -4 -1 }}]
{{Mapping|legend=1| 1 0 1 -3 2 | 0 1 1 4 1 | 0 0 -2 -4 -1 }}


Mapping generators: ~2, ~3, ~11/10
: mapping generators: ~2, ~3, ~11/10


[[Optimal tuning]] ([[POTE]]): ~3/2 = 699.8719, ~11/10 = 158.3232
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 699.8719, ~11/10 = 158.3232


{{Optimal ET sequence|legend=1| 9, 15d, 16d, 20, 22, 31, 53, 82e, 84e, 113e, 144ee }}
{{Optimal ET sequence|legend=1| 9, 15d, 16d, 20, 22, 31, 53, 82e, 84e, 113e, 144ee }}
Line 618: Line 616:
Comma list: 105/104, 121/120, 196/195
Comma list: 105/104, 121/120, 196/195


Mapping: [{{val| 1 0 1 -3 2 -5 }}, {{val| 0 1 1 4 1 6 }}, {{val| 0 0 -2 -4 -1 -6 }}]
Mapping: {{mapping| 1 0 1 -3 2 -5 | 0 1 1 4 1 6 | 0 0 -2 -4 -1 -6 }}
 
Mapping generators: ~2, ~3, ~11/10


Optimal tuning (POTE): ~3/2 = 698.7090, ~11/10 = 158.7117
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 698.7090, ~11/10 = 158.7117


{{Optimal ET sequence|legend=1| 9, 20, 22f, 29, 31 }}
{{Optimal ET sequence|legend=1| 9, 20, 22f, 29, 31 }}
Line 631: Line 627:
Comma list: 121/120, 225/224, 275/273
Comma list: 121/120, 225/224, 275/273


Mapping: [{{val| 1 0 1 -3 2 7 }}, {{val| 0 1 1 4 1 -2 }}, {{val| 0 0 -2 -4 -1 -1 }}]
Mapping: {{mapping| 1 0 1 -3 2 7 | 0 1 1 4 1 -2 | 0 0 -2 -4 -1 -1 }}


Mapping generators: ~2, ~3, ~11/10
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.9789, ~11/10 = 159.0048
 
Optimal tuning (POTE): ~3/2 = 700.9789, ~11/10 = 159.0048


{{Optimal ET sequence|legend=1| 22, 29, 31, 53, 82e, 84e, 113e, 166ee }}
{{Optimal ET sequence|legend=1| 22, 29, 31, 53, 82e, 84e, 113e, 166ee }}
Line 644: Line 638:
[[Comma list]]: 225/224, 1331/1323
[[Comma list]]: 225/224, 1331/1323


[[Mapping]]: [{{val| 1 0 1 -3 -2 }}, {{val| 0 1 2 6 5 }}, {{val| 0 0 -3 -6 -4 }}]
{{Mapping|legend=1| 1 0 1 -3 -2 | 0 1 2 6 5 | 0 0 -3 -6 -4 }}


Mapping generators: ~2, ~3, ~84/55
: mapping generators: ~2, ~3, ~84/55


[[Optimal tuning]] ([[POTE]]): ~3/2 = 700.8983, ~84/55 = 739.3812
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 700.8983, ~84/55 = 739.3812


{{Optimal ET sequence|legend=1| 29, 31, 60e, 91e, 94, 125 }}
{{Optimal ET sequence|legend=1| 29, 31, 60e, 91e, 94, 125 }}
Line 659: Line 653:
[[Comma list]]: 225/224, 4375/4374
[[Comma list]]: 225/224, 4375/4374


[[Mapping]]: [{{val| 1 0 1 -3 0 }}, {{val| 0 6 5 22 0 }}, {{val| 0 0 0 0 1 }}]
{{Mapping|legend=1| 1 0 1 -3 0 | 0 6 5 22 0 | 0 0 0 0 1 }}


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


[[Optimal tuning]] ([[POTE]]): ~6/5 = 316.7318, ~11/8 = 549.2528
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 316.7318, ~11/8 = 549.2528


{{Optimal ET sequence|legend=1| 19, 34d, 53, 72, 197e, 269ce }}
{{Optimal ET sequence|legend=1| 19, 34d, 53, 72, 197e, 269ce }}
Line 674: Line 668:
Comma list: 169/168, 225/224, 325/324
Comma list: 169/168, 225/224, 325/324


Mapping: [{{val| 1 0 1 -3 0 0 }}, {{val| 0 6 5 22 0 14 }}, {{val| 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 }}
 
Mapping generators: ~2, ~6/5, ~11


Optimal tuning (POTE): ~6/5 = 316.7410, ~11/8 = 548.6028
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.7410, ~11/8 = 548.6028


{{Optimal ET sequence|legend=1| 19, 34d, 53, 72, 125f, 197ef, 269cef }}
{{Optimal ET sequence|legend=1| 19, 34d, 53, 72, 125f, 197ef, 269cef }}
Line 689: Line 681:
Comma list: 225/224, 243/242, 385/384
Comma list: 225/224, 243/242, 385/384


Mapping: [{{val| 1 1 3 3 2 0 }}, {{val| 0 6 -7 -2 15 0 }}, {{val| 0 0 0 0 0 1 }}]
Mapping: {{mapping| 1 1 3 3 2 0 | 0 6 -7 -2 15 0 | 0 0 0 0 0 1 }}


Mapping generators: ~2, ~15/14, ~13
: mapping generators: ~2, ~15/14, ~13


Optimal tuning (POTE): ~15/14 = 116.6327, ~13/8 = 837.7040
Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 116.6327, ~13/8 = 837.7040


{{Optimal ET sequence|legend=1| 10, 31, 41, 62, 72, 103, 175f, 216c, 288cdf, 391bcdef }}
{{Optimal ET sequence|legend=1| 10, 31, 41, 62, 72, 103, 175f, 216c, 288cdf, 391bcdef }}
Line 701: Line 693:
[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Marvel family| ]] <!-- main article -->
[[Category:Marvel family| ]] <!-- main article -->
[[Category:Marvel]]
[[Category:Marvel| ]] <!-- key article -->
[[Category:Rank 3]]
[[Category:Rank 3]]

Revision as of 14:43, 10 August 2023

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 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.

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.

Marvel

Main article: 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 marvel temperaments for some other rank-2 temperaments.

Subgroup: 2.3.5.7

Comma list: 225/224

Mapping: [⟨1 0 0 -5], ⟨0 1 0 2], ⟨0 0 1 2]]

mapping generators: ~2, ~3, ~5

Map to lattice: [⟨0 0 -1 -2], ⟨0 1 -1 0]]

Lattice basis:

secor length = 1.256, 3/2 length = 1.369
Angle (secor, 3/2) = 106.958 degrees

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.4075, ~5/4 = 383.6376

Minimax tuning:

[[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 (unchanged-interval) basis: 2.5/3.7
  • 9-odd-limit: 3 1/6-comma flat, 5 1/3-comma 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 (unchanged-interval) basis: 2.9/5.7

Optimal ET sequence: 9, 10, 12, 19, 31, 41, 53, 72, 197, 269c

Badness: 0.0365 × 10-3

Projection pairs: 7 225/32

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

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

Overview to extensions

The second comma of the 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,
  • 243/242 gives spectacle,

as well as others considered below. Temperaments discussed elsewhere include supernatural (→ Keemic family).

Undecimal marvel (unimarv)

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384

Mapping: [⟨1 0 0 -5 12], ⟨0 1 0 2 -1], ⟨0 0 1 2 -3]]

Map to lattice: [⟨0 -1 0 -2 1], ⟨0 -1 1 0 -2]]

Lattice basis:

secor length = 1.0364, 5/4 length = 1.0759
Angle (secor, 5/4) = 104.028 degrees

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.3887, ~5/4 = 383.5403

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 (unchanged-interval) basis: 2.9/5.11/9

Optimal ET sequence: 9, 10, 12e, 19, 22, 31, 41, 53, 72, 166, 197e, 238c, 269ce, 341ce

Badness: 0.255 × 10-3

Projection pairs: 7 225/32 11 4096/375

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

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

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 351/350, 385/384

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

Minimax tuning:

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

Optimal ET sequence: 19, 22, 31, 50, 53, 72, 103, 175f, 300ceff, 403bceeff, 578bbccdeeefff

Badness: 0.690 × 10-3

Complexity 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

Subgroup: 2.3.5.7.11.13

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

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

Minimax tuning:

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

Optimal ET sequence: 19, 22f, 31f, 41, 53, 72, 125f, 166, 238cf

Badness: 0.721 × 10-3

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

Complexity 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

Subgroup: 2.3.5.7.11.13.17

Comma list: 225/224, 325/324, 385/384, 595/594

Mapping: [⟨1 0 0 -5 12 2 18], ⟨0 1 0 2 -1 4 0], ⟨0 0 1 2 -3 -2 -6]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.9619, ~5/4 = 383.0310

Optimal ET sequence: 19, 22f, 31fg, 41, 53g, 72, 166g, 238cfg, 404ccefgg

Badness: 0.869 × 10-3

Enodia

Subgroup: 2.3.5.7.11.13.17

Comma list: 225/224, 325/324, 375/374, 385/384

Mapping: [⟨1 0 0 -5 12 2 18], ⟨0 1 0 2 -1 4 0], ⟨0 0 1 2 -3 -2 6]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.9658, ~5/4 = 383.3063

Optimal ET sequence: 19g, 22f, 31f, 41g, 53, 72, 166g, 238cfg, 404ccefgg

Badness: 0.917 × 10-3

Marvell

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 385/384, 1573/1568

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

Minimax tuning:

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

Optimal ET sequence: 9, 22f, 31, 63, 72, 103, 166, 238cf, 269ce, 507bcceff, 610bcceeff

Badness: 0.862 × 10-3

Isis

Subgroup: 2.3.5.7.11.13

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

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 ET sequence: 10, 19f, 22, 31, 41, 53, 94

Badness: 0.866 × 10-3

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

Deecee

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 364/363, 385/384

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

Minimax tuning:

  • 13-odd-limit eigenmonzo subgroup (unchanged-interval basis): 2.9/5.13/9
  • 15-odd-limit eigenmonzo subgroup (unchanged-interval basis): 2.3.13/5

Optimal ET sequence: 9, 19f, 22, 31f, 41, 63, 72, 185cf, 257cff

Badness: 0.920 × 10-3

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

Tripod

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 144/143, 196/195

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

Minimax tuning:

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

Optimal ET sequence: 9, 10, 19, 22f, 31, 41, 72f, 91, 122f, 163df

Badness: 0.745 × 10-3

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

Marvelcat

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 0 0 -5 12 -1], ⟨0 2 0 4 -2 3], ⟨0 0 1 2 -3 1]]

mapping generators: ~2, ~26/15, ~5

Optimal tuning (POTE): ~2 = 1\1, ~15/13 = 249.7138, ~5/4 = 383.5816

Optimal ET sequence: 9, 10, 19, 44, 53, 72, 125f, 197ef, 269ceff

Badness: 0.9997 × 10-3

Minerva

Subgroup: 2.3.5.7.11

Comma list: 99/98, 176/175

Mapping: [⟨1 0 0 -5 -9], ⟨0 1 0 2 2], ⟨0 0 1 2 4]]

Map to lattice: [⟨0 -1 0 -2 -2], ⟨0 -1 1 0 2]]

Lattice basis:

16/15 length = 0.8997, 5/4 length = 1.0457
Angle (16/15, 5/4) = 98.6044 degrees

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.2593, ~5/4 = 386.5581

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

Optimal ET sequence: 9, 12, 19e, 22, 31, 53, 84e, 96, 127

Badness: 0.381 × 10-3

Projection pairs: 7 225/32 11 5625/512

Scales: minerva12, minerva22x

Associated temperament: würschmidt

Athene

Subgroup: 2.3.5.7.11.13

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

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

Minimax tuning:

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

Optimal ET sequence: 12f, 19e, 22, 31, 53, 84e, 118d, 171de, 202def

Badness: 0.818 × 10-3

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

Apollo

Subgroup: 2.3.5.7.11

Comma list: 100/99, 225/224

Mapping: [⟨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

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

Optimal ET sequence: 12, 19, 22, 34d, 41, 104, 157ce, 198ce, 220ce, 261ce

Projection pairs: 7 225/32 11 100/9

Associated temperament: magic

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 225/224, 275/273

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

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

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

Potassium

Subgroup: 2.3.5.7.11

Comma list: 45/44, 56/55

Mapping: [⟨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

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

Optimal ET sequence: 7d, 9, 10, 12, 19, 31e, 50e

Badness: 0.464 × 10-3

Projection pairs: 7 225/32 11 45/4

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 56/55, 78/77

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

Minimax tuning:

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

Optimal ET sequence: 9, 10, 12f, 19, 31e, 50e

Badness: 0.733 × 10-3

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

Malcolm

Subgroup: 2.3.5.7.11

Comma list: 225/224, 2200/2187

Mapping: [⟨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 ET sequence: 12e, 19e, 34d, 41, 53, 60e, 94, 229c, 248ce, 289cce, 342ccee, 383cce

Badness: 1.250 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 275/273, 325/324

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 ET sequence: 12e, 19e, 34d, 41, 53, 94, 429ccdeef, 523ccdeef

Badness: 1.075 × 10-3

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: [⟨1 0 0 -5 -13], ⟨0 1 0 2 6], ⟨0 0 1 2 3]]

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

Lattice basis:

secor length = 0.9111, 3/2 length = 0.9477
Angle (secor, 3/2) = 65.933

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 699.7981, ~5/4 = 383.5114

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 (unchanged-interval) basis: 2.9/5.11

Optimal ET sequence: 10, 12, 19e, 29, 31, 41, 60e, 72, 247c, 319bcde, 391bcde, 463bccde

Badness: 0.344 × 10-3

Projection pairs: 7 225/32 11 91125/8192

Scales: prodigy11, prodigy12, prodigy29

Associated temperament: miracle

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: [⟨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 ET sequence: 10, 12f, 19e, 29, 31, 41, 60e, 72f, 91e, 101cd, 132def, 233ccddeef, 274ccddeff, 305ccddeeff

Badness: 0.736 × 10-3

Prodigious

Subgroup: 2.3.5.7.11.13

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

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 ET sequence: 12f, 29, 31f, 41, 72, 185cf, 341cf, 413bcff, 526bccdff

Badness: 0.900 × 10-3

Prodigal

Subgroup: 2.3.5.7.11.13

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

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 ET sequence: 12f, 19e, 31, 53e, 60eff, 72, 103, 175f

Badness: 0.889 × 10-3

Protannic

Subgroup: 2.3.5.7.11.13

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

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 ET sequence: 29, 31, 43, 60e, 72, 103, 175f, 482bccddeefff, 554bbccddeeeffff

Badness: 0.953 × 10-3

17-limit

Subgroup: 2.3.5.7.11.13.17

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

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 ET sequence: 29g, 31, 43, 60e, 72, 103, 175f, 307bcdeeffg, 379bccdeeffgg, 482bccddeefffgg, 554bbccddeeeffffgg

Badness: 0.772 × 10-3

Fantastic

Subgroup: 2.3.5.7.11

Comma list: 225/224, 4375/4356

Mapping: [⟨2 0 0 -10 -7], ⟨0 1 0 2 0], ⟨0 0 1 2 3]]

mapping generators: ~99/70, ~3, ~5

Optimal tuning (POTE): ~99/70 = 1\2, ~3/2 = 700.6242, ~5/4 = 383.2978

Optimal ET sequence: 12, 22, 34d, 50, 60e, 72, 166, 238c, 310c

Badness: 0.743 × 10-3

Spectacle

Subgroup: 2.3.5.7.11

Comma list: 225/224, 243/242

Mapping: [⟨1 1 0 -3 2], ⟨0 2 0 4 5], ⟨0 0 1 2 0]]

mapping generators: ~2, ~11/9, ~5

Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 350.0570, ~5/4 = 383.9323

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 (unchanged-interval) basis: 2.9/5.11

Optimal ET sequence: 31, 41, 72, 247c, 281, 353c, 425bc, 497bc

Badness: 0.499 × 10-3

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

Scales: spectacle31

Associated temperament: marvo

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 243/242, 351/350

Mapping: [⟨1 1 0 -3 2 -5], ⟨0 2 0 4 5 -2], ⟨0 0 1 2 0 4]]

Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 349.9247, ~5/4 = 384.3505

Optimal ET sequence: 31, 72, 103, 175f *

* optimal patent val: 240

Badness: 1.009 × 10-3

Hestia

Subgroup: 2.3.5.7.11

Comma list: 225/224, 125000/124509

Mapping: [⟨1 0 0 -5 9], ⟨0 2 0 4 -7], ⟨0 0 1 2 0]]

mapping generators: ~2, ~400/231, ~5

Optimal tuning (POTE): ~2 = 1\1, ~400/231 = 950.1474, ~5/4 = 383.6467

Optimal ET sequence: 19, 29, 43, 53, 72, 197e, 269ce, 341ce, 610bce

Badness: 1.54 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 225/224, 1001/1000

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

Badness: 1.062 × 10-3

Artemis

Subgroup: 2.3.5.7.11

Comma list: 121/120, 225/224

Mapping: [⟨1 0 1 -3 2], ⟨0 1 1 4 1], ⟨0 0 -2 -4 -1]]

mapping generators: ~2, ~3, ~11/10

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 699.8719, ~11/10 = 158.3232

Optimal ET sequence: 9, 15d, 16d, 20, 22, 31, 53, 82e, 84e, 113e, 144ee

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 121/120, 196/195

Mapping: [⟨1 0 1 -3 2 -5], ⟨0 1 1 4 1 6], ⟨0 0 -2 -4 -1 -6]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 698.7090, ~11/10 = 158.7117

Optimal ET sequence: 9, 20, 22f, 29, 31

Diana

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 225/224, 275/273

Mapping: [⟨1 0 1 -3 2 7], ⟨0 1 1 4 1 -2], ⟨0 0 -2 -4 -1 -1]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.9789, ~11/10 = 159.0048

Optimal ET sequence: 22, 29, 31, 53, 82e, 84e, 113e, 166ee

Morfil

Subgroup: 2.3.5.7.11

Comma list: 225/224, 1331/1323

Mapping: [⟨1 0 1 -3 -2], ⟨0 1 2 6 5], ⟨0 0 -3 -6 -4]]

mapping generators: ~2, ~3, ~84/55

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.8983, ~84/55 = 739.3812

Optimal ET sequence: 29, 31, 60e, 91e, 94, 125

Badness: 1.152 × 10-3

Catakleismoid

Subgroup: 2.3.5.7.11

Comma list: 225/224, 4375/4374

Mapping: [⟨1 0 1 -3 0], ⟨0 6 5 22 0], ⟨0 0 0 0 1]]

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

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.7318, ~11/8 = 549.2528

Optimal ET sequence: 19, 34d, 53, 72, 197e, 269ce

Badness: 1.275 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

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

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 ET sequence: 19, 34d, 53, 72, 125f, 197ef, 269cef

Badness: 0.916 × 10-3

Mirage

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 243/242, 385/384

Mapping: [⟨1 1 3 3 2 0], ⟨0 6 -7 -2 15 0], ⟨0 0 0 0 0 1]]

mapping generators: ~2, ~15/14, ~13

Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 116.6327, ~13/8 = 837.7040

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

Badness: 0.738 × 10-3

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