Werckismic temperaments

This is a collection of rank-3 temperaments tempering out 441/440, but we can begin by looking at the rank-4 temperament.

Temperaments discussed elsewhere are:

• festival, {45/44, 50/49} → Jubilismic family #Festival
• vulcan, {56/55, 64/63} → Archytas family #Vulcan
• euterpe, {81/80, 99/98} → Didymus rank three family #Euterpe
• nakika, {100/99, 245/242} → Octagar family #Nakika
• thrush, {126/125, 441/440} → Starling family #Thrush
• prodigy, {225/224, 441/440} → Marvel family #Prodigy
• jove, {243/242, 441/440} → Breed family #Jove
• sensawer, {245/243, 441/440} → Sensamagic family #Sensawer
• portent, {385/384, 441/440} → Gamelismic family #Portent
• demeter, {441/440, 686/675} → Sengic family #Demeter
• pele, {441/440, 896/891} → Hemifamity family #Pele
• tiberius, {441/440, 2048/2025} → Diaschismic rank three family #Tiberius
• belobog, {441/440, 3136/3125} → Hemimean family #Belobog
• turan, {441/440, 10976/10935} → Hemimage family #Turan
• snape, {441/440, 19683/19600} → Cataharry family #Snape

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:

• 11-odd-limit

[[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⟩]

Eigenmonzos: 2, 5, 11, 9/7

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:

• 11-odd-limit

[[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⟩]

Eigenmonzos: 2, 11, 9/7

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⟩]

Eigenmonzos: 2, 5, 13/7

Vals: Template:Val list

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

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]]

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]]

Vals: Template:Val list

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

Template:Val list *

* 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]]

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]]

Vals: Template:Val list

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

Vals: Template:Val list

Badness: 1.478 × 10^-3

Aphrodite

Aphrodite tempers out the squalentine comma, 64827/64000, in the 7-limit. Its generators can be taken to be 2, 3, and 21/20, and it equates (21/20)³ with 8/7.

7-limit (squalentine)

Subgroup: 2.3.5.7

Comma list: 64827/64000

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

Mapping generators: ~2, ~3, ~21/20

Badness: 0.943 × 10^-3

Projection pairs: 5 320000/64827 7 64000/9261 to 2.3.7/5

11-limit

Subgroup: 2.3.5.7.11

Comma list: 121/120, 441/440

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

Mapping generators: ~2, ~3, ~22/21

Badness: 0.583 × 10^-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 351/350, 441/440

Mapping: [⟨1 0 1 3 2 6], ⟨0 1 1 0 1 -1], ⟨0 0 4 3 2 11]]

Vals: Template:Val list

Badness: 1.456 × 10^-3

Eros

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 196/195, 352/351

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

Vals: Template:Val list

Badness: 1.150 × 10^-3

Inanna

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 121/120, 275/273

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

Vals: Template:Val list

Badness: 1.077 × 10^-3

Ishtar

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 121/120, 441/440

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

Vals: Template:Val list

Badness: 1.151 × 10^-3