Werckismic temperaments

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This is a collection of rank-3 temperaments tempering out 441/440

Temperaments discussed elsewhere are:

Considered below are history, varuna, and trimyna, but we can begin by looking at the rank-4 temperament.

Werckismic

Subgroup: 2.3.5.7.11

Comma list: 441/440

Mapping[1 0 0 0 -3], 0 1 0 0 2], 0 0 1 0 -1], 0 0 0 1 2]]

Minimax tuning:

[[1 0 0 0 0, [1/2 2/3 1/6 -1/3 1/6, [0 0 1 0 0, [1 -2/3 1/3 1/3 1/3, [0 0 0 0 1]
eigenmonzo (unchanged-interval) basis: 2.5.9/7.11

Optimal ET sequence12, 15, 19e, 26, 27e, 31, 41, 58, 72, 118, 130, 190, 248, 289, 320, 609d

History

Subgroup: 2.3.5.7.11

Comma list: 441/440, 4000/3993

Mapping[1 2 0 0 1], 0 -6 0 7 2], 0 0 1 1 1]]

mapping generators: ~2, ~21/20, ~5

Minimax tuning:

[[1 0 0 0 0, [4/17 12/17 0 -6/17 6/17, [-27/17 4/17 0 -2/17 19/17, [8/17 -10/17 0 5/17 12/17, [0 0 0 0 1]
eigenmonzo (unchanged-interval) basis: 2.9/7.11

Optimal ET sequence14c, 15, 29, 43, 57, 58, 72, 130, 202, 217, 289, 491

Badness: 0.604 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 441/440, 676/675

Mapping: [1 2 0 0 1 2], 0 -6 0 7 2 -9], 0 0 1 1 1 1]]

mapping generators: ~2, ~21/20, ~5

Minimax tuning:

  • 13-odd-limit
[[1 0 0 0 0 0, [5/4 0 0 -3/8 0 3/8, [0 0 1 0 0 0, [7/8 0 1 7/16 0 -7/16, [5/4 0 1 1/8 0 -1/8, [7/8 0 1 -9/16 0 9/16]
eigenmonzo (unchanged-interval) basis: 2.5.13/7

Optimal ET sequence14cf, 15, 29, 43, 58, 72, 87, 130, 217, 289

Badness: 0.540 × 10-3

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

Varuna

7-limit

Subgroup: 2.3.5.7

Comma list: 321489/320000

Mapping[2 0 0 9], 0 1 0 -4], 0 0 1 2]]

mapping generators: ~567/400, ~3, ~5

Optimal ET sequence12, 26, 34, 46, 58, 72, 118, 130, 202, 320, 450, 522, 972bd, 1174bd

Badness: 0.501 × 10-3

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 8019/8000

Mapping[2 0 0 9 12], 0 1 0 -4 -6], 0 0 1 2 3]]

Optimal ET sequence12, 26, 44ce, 46, 58, 72, 118, 130, 190, 248, 320

Badness: 0.419 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 364/363, 441/440

Mapping: [2 0 0 9 12 11], 0 1 0 -4 -6 -7], 0 0 1 2 3 4]]

Optimal ET sequence12f, 14cf, 26, 46, 58, 72, 130

Badness: 0.636 × 10-3

Trimyna

7-limit

Subgroup: 2.3.5.7

Comma list: 50421/50000

Mapping[1 4 0 0], 0 -5 0 1], 0 0 1 1]]

mapping generators: ~2, ~7/5, ~5

Optimal ET sequence27, 31, 58, 87, 118, 267d, 385d, 412d *

* optimal patent val: 294

Badness: 0.870 × 10-3

Projection pair: 3 50000/16807 to 2.5.7

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 3388/3375

Mapping[1 4 0 0 5], 0 -5 0 1 -8], 0 0 1 1 1]]

Optimal ET sequence27e, 31, 58, 87, 118, 294, 412d

Badness: 0.512 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 352/351, 1001/1000

Mapping: [1 4 0 0 5 -2], 0 -5 0 1 -8 7], 0 0 1 1 1 1]]

Optimal ET sequence27e, 29, 31, 56, 58, 87, 118, 145, 232

Badness: 0.814 × 10-3

Trimy

Subgroup: 2.3.5.7.11.13

Comma list: 441/440, 676/675, 847/845

Mapping: [1 4 0 0 5 5], 0 -10 0 2 -16 -15], 0 0 1 1 1 1]]

mapping generators: ~2, ~13/11, ~5

Optimal ET sequence29, 54ceeff, 58, 87, 145, 149, 207, 236d, 294

Badness: 1.478 × 10-3