Werckismic temperaments
This is a collection of rank-3 temperaments tempering out 441/440
Temperaments discussed elsewhere are:
- Festival (+45/44 or 50/49) → Jubilismic family
- Vulcan (+56/55 or 64/63) → Archytas family
- Euterpe (+81/80 or 99/98) → Didymus rank-3 family
- Nakika (+100/99 or 245/242) → Octagar family
- Aphrodite (+121/120) → Biyatismic clan
- Thrush (+126/125) → Starling family
- Prodigy (+225/224) → Marvel family
- Jove (+243/242) → Breed family
- Sensawer (+245/243) → Sensamagic family
- Portent (+385/384) → Gamelismic family
- Demeter (+686/675) → Sengic family
- Pele (+896/891) → Hemifamity family
- Tiberius (+2048/2025) → Diaschismic rank-3 family
- Belobog (+3136/3125) → Hemimean family
- Turan (+10976/10935) → Hemimage family
- Snape (+19683/19600) → Cataharry family
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
Mapping: [⟨1 0 0 0 -3], ⟨0 1 0 0 2], ⟨0 0 1 0 -1], ⟨0 0 0 1 2]]
- [[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 sequence: 12, 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
- [[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 sequence: 14c, 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 sequence: 14cf, 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 sequence: 12, 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 sequence: 12, 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 sequence: 12f, 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 sequence: 27, 31, 58, 87, 118, 267d, 385d, 412d *
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 sequence: 27e, 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 sequence: 27e, 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 sequence: 29, 54ceeff, 58, 87, 145, 149, 207, 236d, 294
Badness: 1.478 × 10-3