Marveltwin: Difference between revisions
Cmloegcmluin (talk | contribs) add unchanged-intervals along with eigenmonzo, per prior agreement on Xenwiki Work Group |
Cleanup (1/) |
||
| Line 1: | Line 1: | ||
__FORCETOC__ | __FORCETOC__ | ||
=Marveltwin and | == Marveltwin and marvel == | ||
The marveltwin comma, [[325/324]], bears a curiously close analogy to the marvel comma, [[225/224]]. 325/324 can be added to the [[11-limit]] version of marvel, which tempers out 225/224 and 385/384, to get [[13-limit]] marvel, aka hecate. But it's also interesting to leave 11 out of it. From 225/224 we get that a 5-limit approximation for 7 is 225/224 * 7 = 225/32. Similarly from 325/324 we get a 5-limit approximation of 13 from 324/325 * 13 = 324/25. If we define the major/minor transformation of the 5-limit as the result of fixing 2 and 3 and replacing 5 by 24/5, then major/minor applied to 225/32 is 162/25, which is (324/25)/2. Similarly, major/minor applied to 324/25 is 225/16 = 2 * (225/32). 225/224 tells us that two 16/15 in a row are an approximate 8/7, and 325/324 tells us two 10/9 in a row are an approximate 16/13. Needless to say, major/minor applied to 16/15 is 10/9, and applied to 10/9 is 16/15. | The marveltwin comma, [[325/324]], bears a curiously close analogy to the marvel comma, [[225/224]]. 325/324 can be added to the [[11-limit]] version of marvel, which tempers out 225/224 and 385/384, to get [[13-limit]] marvel, aka hecate. But it's also interesting to leave 11 out of it. From 225/224 we get that a 5-limit approximation for 7 is 225/224 * 7 = 225/32. Similarly from 325/324 we get a 5-limit approximation of 13 from 324/325 * 13 = 324/25. If we define the major/minor transformation of the 5-limit as the result of fixing 2 and 3 and replacing 5 by 24/5, then major/minor applied to 225/32 is 162/25, which is (324/25)/2. Similarly, major/minor applied to 324/25 is 225/16 = 2 * (225/32). 225/224 tells us that two 16/15 in a row are an approximate 8/7, and 325/324 tells us two 10/9 in a row are an approximate 16/13. Needless to say, major/minor applied to 16/15 is 10/9, and applied to 10/9 is 16/15. | ||
=Rank | == Rank-5 temperaments == | ||
Comma: 325/324 | [[Subgroup]]: 2.3.5.7.11.13 | ||
[[Comma list]]: 325/324 | |||
13 and 15 limit minimax tuning | 13 and 15 limit minimax tuning | ||
| Line 46: | Line 48: | ||
Edos: 7, 12, 15, 19, 26, 34, 41, 46, 53, 72, 87, 121, 140, 159, 193, 212, 299, 333 | Edos: 7, 12, 15, 19, 26, 34, 41, 46, 53, 72, 87, 121, 140, 159, 193, 212, 299, 333 | ||
=Rank | == Rank-4 temperaments == | ||
=== 225/224 === | |||
==225/224== | |||
13-limit eigenmonzo subgroup (unchanged-interval basis): 2.7.11/5.13/5 | 13-limit eigenmonzo subgroup (unchanged-interval basis): 2.7.11/5.13/5 | ||
| Line 72: | Line 73: | ||
10^6 * Badness: 3.668 | 10^6 * Badness: 3.668 | ||
==385/384== | === 385/384 === | ||
Spectrum: 4/3, 6/5, 10/9, 5/4, 9/8, 8/7, 11/8, 7/6, 12/11, 16/15, 18/13, 13/12, 9/7, 7/5, 16/13, 11/9, 11/10, 13/11, 14/13, 15/14, 15/11, 13/10, 15/13, 14/11 | Spectrum: 4/3, 6/5, 10/9, 5/4, 9/8, 8/7, 11/8, 7/6, 12/11, 16/15, 18/13, 13/12, 9/7, 7/5, 16/13, 11/9, 11/10, 13/11, 14/13, 15/14, 15/11, 13/10, 15/13, 14/11 | ||
| Line 80: | Line 81: | ||
10^6 * Badness: 2.206 | 10^6 * Badness: 2.206 | ||
==364/363== | === 364/363 === | ||
Spectrum: 4/3, 6/5, 10/9, 11/8, 5/4, 12/11, 14/11, 9/8, 13/11, 11/9, 8/7, 13/12, 18/13, 16/15, 7/6, 11/10, 7/5, 16/13, 9/7, 15/11, 15/14, 13/10, 15/13, 14/13 | Spectrum: 4/3, 6/5, 10/9, 11/8, 5/4, 12/11, 14/11, 9/8, 13/11, 11/9, 8/7, 13/12, 18/13, 16/15, 7/6, 11/10, 7/5, 16/13, 9/7, 15/11, 15/14, 13/10, 15/13, 14/13 | ||
| Line 88: | Line 89: | ||
10^6 * Badness: 3.011 | 10^6 * Badness: 3.011 | ||
==441/440== | === 441/440 === | ||
Spectrum: 4/3, 6/5, 8/7, 10/9, 14/11, 5/4, 7/6, 9/8, 7/5, 16/15, 13/12, 18/13, 9/7, 11/8, 12/11, 15/14, 16/13, 11/9, 11/10, 13/11, 15/11, 14/13, 13/10, 15/13 | Spectrum: 4/3, 6/5, 8/7, 10/9, 14/11, 5/4, 7/6, 9/8, 7/5, 16/15, 13/12, 18/13, 9/7, 11/8, 12/11, 15/14, 16/13, 11/9, 11/10, 13/11, 15/11, 14/13, 13/10, 15/13 | ||
| Line 96: | Line 97: | ||
10^6 * Badness: 3.037 | 10^6 * Badness: 3.037 | ||
==169/168== | === 169/168 === | ||
Spectrum: 10/9, 6/5, 4/3, 14/13, 13/12, 16/13, 5/4, 18/13, 9/8, 13/10, 8/7, 7/6, 15/13, 16/15, 9/7, 7/5, 11/8, 12/11, 13/11, 11/10, 15/14, 11/9, 14/11, 15/11 | Spectrum: 10/9, 6/5, 4/3, 14/13, 13/12, 16/13, 5/4, 18/13, 9/8, 13/10, 8/7, 7/6, 15/13, 16/15, 9/7, 7/5, 11/8, 12/11, 13/11, 11/10, 15/14, 11/9, 14/11, 15/11 | ||
| Line 104: | Line 105: | ||
10^6 * Badness: 2.975 | 10^6 * Badness: 2.975 | ||
==540/539== | === 540/539 === | ||
Spectrum: 4/3, 6/5, 7/6, 9/7, 10/9, 5/4, 7/5, 9/8, 15/14, 8/7, 16/15, 18/13, 13/12, 12/11, 11/9, 11/10, 11/8, 15/11, 16/13, 14/13, 15/13, 13/10, 13/11, 14/11 | Spectrum: 4/3, 6/5, 7/6, 9/7, 10/9, 5/4, 7/5, 9/8, 15/14, 8/7, 16/15, 18/13, 13/12, 12/11, 11/9, 11/10, 11/8, 15/11, 16/13, 14/13, 15/13, 13/10, 13/11, 14/11 | ||
| Line 112: | Line 113: | ||
10^6 * Badness: 3.281 | 10^6 * Badness: 3.281 | ||
==352/351== | === 352/351 === | ||
Spectrum: 4/3, 10/9, 6/5, 9/8, 5/4, 13/11, 16/13, 11/9, 13/12, 12/11, 16/15, 18/13, 8/7, 7/6, 11/8, 9/7, 15/11, 11/10, 7/5, 14/13, 13/10, 15/14, 14/11, 15/13 | Spectrum: 4/3, 10/9, 6/5, 9/8, 5/4, 13/11, 16/13, 11/9, 13/12, 12/11, 16/15, 18/13, 8/7, 7/6, 11/8, 9/7, 15/11, 11/10, 7/5, 14/13, 13/10, 15/14, 14/11, 15/13 | ||
| Line 120: | Line 121: | ||
10^6 * Badness: 3.434 | 10^6 * Badness: 3.434 | ||
==625/624== | === 625/624 === | ||
Spectrum: 6/5, 18/13, 15/13, 5/4, 4/3, 10/9, 13/12, 13/10, 16/15, 9/8, 16/13, 8/7, 7/5, 7/6, 15/14, 11/8, 9/7, 11/10, 12/11, 14/13, 15/11, 11/9, 13/11, 14/11 | Spectrum: 6/5, 18/13, 15/13, 5/4, 4/3, 10/9, 13/12, 13/10, 16/15, 9/8, 16/13, 8/7, 7/5, 7/6, 15/14, 11/8, 9/7, 11/10, 12/11, 14/13, 15/11, 11/9, 13/11, 14/11 | ||
| Line 128: | Line 129: | ||
10^6 * Badness: 3.563 | 10^6 * Badness: 3.563 | ||
=Rank | == Rank-3 temperaments == | ||
=== Portending === | |||
==Portending== | |||
Commas: 325/324, 364/363, 441/440 | Commas: 325/324, 364/363, 441/440 | ||
| Line 139: | Line 139: | ||
10^5 * Badness: 62.715 | 10^5 * Badness: 62.715 | ||
==Marvel ( | === Marvel (hecate) === | ||
Commas: 225/224, 325/324, 385/384 | Commas: 225/224, 325/324, 385/384 | ||
| Line 148: | Line 148: | ||
10^5 * Badness: 72.113 | 10^5 * Badness: 72.113 | ||
==Sumatra== | === Sumatra === | ||
Commas: 325/324, 385/384, 625/624 | Commas: 325/324, 385/384, 625/624 | ||
Revision as of 08:48, 29 September 2024
Marveltwin and marvel
The marveltwin comma, 325/324, bears a curiously close analogy to the marvel comma, 225/224. 325/324 can be added to the 11-limit version of marvel, which tempers out 225/224 and 385/384, to get 13-limit marvel, aka hecate. But it's also interesting to leave 11 out of it. From 225/224 we get that a 5-limit approximation for 7 is 225/224 * 7 = 225/32. Similarly from 325/324 we get a 5-limit approximation of 13 from 324/325 * 13 = 324/25. If we define the major/minor transformation of the 5-limit as the result of fixing 2 and 3 and replacing 5 by 24/5, then major/minor applied to 225/32 is 162/25, which is (324/25)/2. Similarly, major/minor applied to 324/25 is 225/16 = 2 * (225/32). 225/224 tells us that two 16/15 in a row are an approximate 8/7, and 325/324 tells us two 10/9 in a row are an approximate 16/13. Needless to say, major/minor applied to 16/15 is 10/9, and applied to 10/9 is 16/15.
Rank-5 temperaments
Subgroup: 2.3.5.7.11.13
Comma list: 325/324
13 and 15 limit minimax tuning
| [1 0 0 0 0 0> |
| [0 1 0 0 0 0> |
| [2/3 4/3 1/3 0 0 -1/3> |
| [2/3 4/3 -2/3 1 0 -1/3> |
| [2/3 4/3 -2/3 0 1 -1/3> |
| [2/3 4/3 -2/3 0 0 2/3> |
Fifths are pure; 5, 7, 11 and 13 are all flat by (325/324)^(1/3), which is 1.778 cents.
Eigenmonzo subgroup (unchanged-interval basis): 2.3.7/5.11/5.13/5
Spectrum: 4/3, 6/5, 10/9, 5/4, 9/8, 8/7, 16/15, 18/13, 13/12, 7/6, 16/13, 7/5, 11/8, 9/7, 12/11, 15/14, 11/10, 11/9, 13/10, 14/13, 15/13, 14/11, 15/11, 13/11
Mapping:
| <1 0 0 0 0 2] |
| <0 1 0 0 0 4] |
| <0 0 1 0 0 -2] |
| <0 0 0 1 0 0] |
| <0 0 0 0 1 0] |
Edos: 7, 12, 15, 19, 26, 34, 41, 46, 53, 72, 87, 121, 140, 159, 193, 212, 299, 333
Rank-4 temperaments
225/224
13-limit eigenmonzo subgroup (unchanged-interval basis): 2.7.11/5.13/5
15-limit eigenmonzo subgroup (unchanged-interval basis): 2.7.15/11.15/13
Spectrum: 4/3, 5/4, 6/5, 16/15, 15/14, 9/8, 10/9, 7/5, 9/7, 7/6, 18/13, 8/7, 13/12, 16/13, 11/8, 12/11, 15/13, 11/10, 15/11, 11/9, 13/10, 14/13, 14/11, 13/11
Mapping:
| <1 0 0 -5 0 2] |
| <0 1 0 2 0 4] |
| <0 0 1 2 0 -2] |
| <0 0 0 0 1 0] |
Edos: 12, 19, 41, 53, 72, 166
10^6 * Badness: 3.668
385/384
Spectrum: 4/3, 6/5, 10/9, 5/4, 9/8, 8/7, 11/8, 7/6, 12/11, 16/15, 18/13, 13/12, 9/7, 7/5, 16/13, 11/9, 11/10, 13/11, 14/13, 15/14, 15/11, 13/10, 15/13, 14/11
Edos: 7, 15, 19, 26, 34, 41, 46, 53, 72, 87, 140, 159, 212, 299
10^6 * Badness: 2.206
364/363
Spectrum: 4/3, 6/5, 10/9, 11/8, 5/4, 12/11, 14/11, 9/8, 13/11, 11/9, 8/7, 13/12, 18/13, 16/15, 7/6, 11/10, 7/5, 16/13, 9/7, 15/11, 15/14, 13/10, 15/13, 14/13
Edos: 15, 26, 41, 46, 72, 87, 121, 159, 193, 239, 280
10^6 * Badness: 3.011
441/440
Spectrum: 4/3, 6/5, 8/7, 10/9, 14/11, 5/4, 7/6, 9/8, 7/5, 16/15, 13/12, 18/13, 9/7, 11/8, 12/11, 15/14, 16/13, 11/9, 11/10, 13/11, 15/11, 14/13, 13/10, 15/13
Edos: 12, 15, 26, 41, 46, 72, 87, 159
10^6 * Badness: 3.037
169/168
Spectrum: 10/9, 6/5, 4/3, 14/13, 13/12, 16/13, 5/4, 18/13, 9/8, 13/10, 8/7, 7/6, 15/13, 16/15, 9/7, 7/5, 11/8, 12/11, 13/11, 11/10, 15/14, 11/9, 14/11, 15/11
Edos: 7, 19, 26, 46, 53, 72, 152
10^6 * Badness: 2.975
540/539
Spectrum: 4/3, 6/5, 7/6, 9/7, 10/9, 5/4, 7/5, 9/8, 15/14, 8/7, 16/15, 18/13, 13/12, 12/11, 11/9, 11/10, 11/8, 15/11, 16/13, 14/13, 15/13, 13/10, 13/11, 14/11
Edos: 19, 41, 53, 72, 121, 166, 193
10^6 * Badness: 3.281
352/351
Spectrum: 4/3, 10/9, 6/5, 9/8, 5/4, 13/11, 16/13, 11/9, 13/12, 12/11, 16/15, 18/13, 8/7, 7/6, 11/8, 9/7, 15/11, 11/10, 7/5, 14/13, 13/10, 15/14, 14/11, 15/13
Edos: 7, 34, 41, 46, 53, 80, 87, 121, 140, 261, 358, 401
10^6 * Badness: 3.434
625/624
Spectrum: 6/5, 18/13, 15/13, 5/4, 4/3, 10/9, 13/12, 13/10, 16/15, 9/8, 16/13, 8/7, 7/5, 7/6, 15/14, 11/8, 9/7, 11/10, 12/11, 14/13, 15/11, 11/9, 13/11, 14/11
Edos: 15, 19, 34, 53, 72, 87, 121, 140, 159, 193, 212, 299, 333
10^6 * Badness: 3.563
Rank-3 temperaments
Portending
Commas: 325/324, 364/363, 441/440
Spectrum: 8/7, 4/3, 11/8, 6/5, 14/11, 7/6, 10/9, 12/11, 5/4, 13/11, 9/8, 7/5, 11/9, 9/7, 18/13, 13/12, 16/15, 11/10, 15/14, 16/13, 14/13, 15/11, 13/10, 15/13
Edos: 15, 26, 41, 46, 72, 87, 159
10^5 * Badness: 62.715
Marvel (hecate)
Commas: 225/224, 325/324, 385/384
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
Edos: 19, 41, 53, 72, 166
10^5 * Badness: 72.113
Sumatra
Commas: 325/324, 385/384, 625/624
Spectrum: 6/5, 18/13, 15/13, 5/4, 4/3, 10/9, 13/12, 13/10, 16/15, 9/8, 8/7, 11/8, 11/10, 7/5, 12/11, 7/6, 16/13, 15/11, 15/14, 11/9, 9/7, 13/11, 14/13, 14/11
Edos: 15, 19, 34, 53, 72, 87, 140, 159, 212, 299
Optimal patent val: 299edo
10^5 * Badness: 68.005