Swetismic temperaments

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

This page collects rank-3 swetismic temperaments, which temper out the swetisma (monzo: [2 3 1 -2 -1⟩, ratio: 540/539). For the rank-4 temperament, see Catalog of rank-4 temperaments #Swetismic (540/539).

Temperaments discussed elsewhere are:

• Terpsichore (+81/80) → Didymus rank-3 family
• Big brother (+99/98 or 121/120) → Biyatismic clan
• Octarod (+100/99 or 245/243) → Sensamagic family
• Aplonis (+126/125) → Starling family
• Guanyin (+176/175) → Orwellismic family
• Marvel (+225/224 or 385/384) → Marvel family
• Jove (+243/242 or 441/440) → Breed family
• Uni (+896/891) → Hemimage family
• Gamel (+1029/1024) → Gamelismic family
• Claudius (+2048/2025) → Diaschismic rank-3 family
• Octasand (+2200/2187) → Octagar family
• Gariboh (+3125/3087) → Gariboh family
• Siebog (+3136/3125) → Hemimean family
• Laka (+5120/5103) → Aberschismic family
• Jupiter (+5632/5625) → Porwell family
• Albus (+8019/8000) → Cataharry family
• Orthoschismic (+32805/32768) → Schismic rank-3 family

Considered below are indra, hades, minos, persephone and hypnos, in the order of increasing badness.

Indra

For the 7-limit version, see Miscellaneous 7-limit temperaments #Canopic.

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372

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

Mapping generators: ~2, ~3, ~10/7

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

Lattice basis:

15/14 length = 0.6184, 55/36 length = 0.7585

Angle (15/14, 55/36) = 68.1048 deg

Optimal tunings:

• WE: ~2 = 1199.9693 ¢, ~3/2 = 701.7444 ¢, ~10/7 = 616.1152 ¢

error map: ⟨-0.031 -0.241 -0.443 +0.899 +0.183]

• CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7342 ¢, ~10/7 = 616.1422 ¢

error map: ⟨0.000 -0.221 -0.400 +0.946 +0.256]

Minimax tuning:

• 11-odd-limit

[[1 0 0 0 0⟩, [10 10 -3 0 -5⟩, [0 0 1 0 0⟩, [6 6 -1 0 -3⟩, [20 18 -6 0 -9⟩]

eigenmonzo (unchanged-interval) basis: 2.5.11/9

Optimal ET sequence: 31, 41, 72, 152, 224, 631d, 855d

Badness (Sintel): 0.320

Projection pairs: 5 84375/16807 7 16875/2401 11 1372/125 to 2.3.7/5

Scales: indra31

Shibi

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 729/728, 1375/1372

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

Optimal tunings:

• WE: ~2 = 1199.9762 ¢, ~3/2 = 701.7232 ¢, ~10/7 = 616.1317 ¢
• CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7261 ¢, ~10/7 = 616.1467 ¢

Optimal ET sequence: 31f, 39df, 41, 72, 111, 152f, 183, 224, 631d, 855d

Badness (Sintel): 0.674

Yajna

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 625/624, 1375/1372

Mapping: [⟨1 0 -5 -4 5 -24], ⟨0 1 3 3 0 11], ⟨0 0 5 4 -3 20]]

Optimal tunings:

• WE: ~2 = 1200.0073 ¢, ~3/2 = 701.7067 ¢, ~10/7 = 616.1052 ¢
• CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7089 ¢, ~10/7 = 616.0986 ¢

Minimax tuning:

• 13-odd-limit

[[1 0 0 0 0 0⟩, [4/5 6/5 -4/5 0 0 1/5⟩, [8/5 2/5 -3/5 0 0 2/5⟩, [44/25 26/25 -24/25 0 0 11/25⟩, [62/25 48/25 -27/25 0 0 3/25⟩, [8/5 2/5 -8/5 0 0 7/5⟩]

eigenmonzo (unchanged-interval) basis: 2.9/5.13/5

• 15-odd-limit

[[1 0 0 0 0 0⟩, [0 1 0 0 0 0⟩, [5/4 1/2 0 -5/16 0 5/16⟩, [1 1 0 -1/4 0 1/4⟩, [5/4 3/2 0 3/16 0 -3/16⟩, [1 1 0 -5/4 0 5/4⟩]

eigenmonzo (unchanged-interval) basis: 2.3.13/7

Optimal ET sequence: 31, 49f, 72, 103, 121, 152f, 193, 224

Badness (Sintel): 0.655

Vibhu

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 1375/1372

Mapping: [⟨1 0 -5 -4 5 -13], ⟨0 1 3 3 0 6], ⟨0 0 5 4 -3 14]]

Optimal tunings:

• WE: ~2 = 1200.1587 ¢, ~3/2 = 701.2394 ¢, ~10/7 = 616.6507 ¢
• CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.2689 ¢, ~10/7 = 616.5306 ¢

Optimal ET sequence: 31, 64cdf, 72, 103, 111, 152, 175f, 183, 469f

Badness (Sintel): 0.772

Vish

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 364/363, 540/539

Mapping: [⟨1 0 0 0 2 2], ⟨0 1 3 3 0 -2], ⟨0 0 5 4 -3 -10]]

Optimal tunings:

• WE: ~2 = 1199.9463 ¢, ~3/2 = 702.3912 ¢, ~10/7 = 615.5733 ¢
• CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.3742 ¢, ~10/7 = 615.6198 ¢

Optimal ET sequence: 31f, 41, 72, 121, 193

Badness (Sintel): 0.870

Hades

For the 7-limit version, see Miscellaneous 7-limit temperaments #Stearnsmic.

Subgroup: 2.3.5.7.11

Comma list: 540/539, 4000/3993

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

mapping generators: ~99/70, ~9/7, ~5

Optimal tunings:

• WE: ~99/70 = 599.9486 ¢, ~9/7 = 433.9084 ¢, ~5/4 = 386.2290 ¢

error map: ⟨-0.103 -0.281 -0.290 +0.613 +0.643]

• CWE: ~99/70 = 600.0000 ¢, ~9/7 = 433.9204 ¢, ~5/4 = 386.1272 ¢

error map: ⟨0.000 -0.194 -0.187 +0.776 +0.889]

Optimal ET sequence: 22, 50, 58, 72, 130, 152, 224, 354, 506e, 578, 730e, 1084de, 1308dee

Badness (Sintel): 0.434

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 729/728, 1575/1573

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

Optimal tunings:

• WE: ~99/70 = 599.9598 ¢, ~9/7 = 433.8891 ¢, ~5/4 = 386.1680 ¢
• CWE: ~99/70 = 600.0000 ¢, ~9/7 = 433.9059 ¢, ~5/4 = 386.1014 ¢

Optimal ET sequence: 22f, 36ce, 58, 72, 130, 224, 354, 578

Badness (Sintel): 0.501

Cerberus

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 364/363, 540/539

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

Optimal tunings:

• WE: ~55/39 = 599.9565 ¢, ~9/7 = 433.8372 ¢, ~5/4 = 387.1905 ¢
• CWE: ~55/39 = 600.0000 ¢, ~9/7 = 433.8475 ¢, ~5/4 = 387.1030 ¢

Optimal ET sequence: 8d, 14cf, 22, 36f, 44ceff, 50, 58, 72, 130

Badness (Sintel): 0.716

Minos

Subgroup: 2.3.5.7.11

Comma list: 540/539, 12005/11979

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

mapping generators: ~2, ~121/70, ~66/49

Optimal tunings:

• WE: ~2 = 1199.8923 ¢, ~121/70 = 950.8327 ¢, ~66/49 = 516.4492 ¢

error map: ⟨-0.108 -0.290 -0.414 +0.121 +1.468]

• CWE: ~2 = 1200.0000 ¢, ~121/70 = 950.8712 ¢, ~66/49 = 516.5067 ¢

error map: ⟨0.000 -0.213 -0.295 +0.294 +1.687]

Optimal ET sequence: 9, …, 49, 58, 72, 130, 193, 202, 323e, 395e, 525e *

* optimal patent val: 467

Badness (Sintel): 1.06

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 540/539, 676/675

Mapping: [⟨1 0 -3 0 -1 -4], ⟨0 2 4 3 4 7], ⟨0 0 5 1 3 5]]

Optimal tunings:

• WE: ~2 = 1199.7669 ¢, ~26/15 = 950.9360 ¢, ~66/49 = 516.3652 ¢
• CWE: ~2 = 1200.0000 ¢, ~26/15 = 951.0324 ¢, ~66/49 = 516.4883 ¢

Optimal ET sequence: 9, …, 49f, 58, 72, 121, 130, 193, 251e, 323e

Badness (Sintel): 0.677

Persephone

Subgroup: 2.3.5.7.11

Comma list: 540/539, 88209/87500

Mapping: [⟨1 0 -6 4 -12], ⟨0 1 4 0 7], ⟨0 0 5 -3 11]]

mapping generators: ~2, ~3, ~33/25

Optimal tunings:

• WE: ~2 = 1200.0185 ¢, ~3/2 = 700.9179 ¢, ~33/25 = 476.7538 ¢

error map: ⟨+0.019 -1.019 +1.090 +0.987 -0.694]

• CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.9220 ¢, ~33/25 = 476.7393 ¢

error map: ⟨0.000 -1.033 +1.071 +0.956 -0.732]

Optimal ET sequence: 8d, …, 45e, 50, 53, 58, 103, 161, 214, 272, 375

Badness (Sintel): 2.42

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 847/845

Mapping: [⟨1 0 -6 4 -12 -7], ⟨0 1 4 0 7 5], ⟨0 0 5 -3 11 7]]

Optimal tunings:

• WE: ~2 = 1199.9553 ¢, ~3/2 = 700.9477 ¢, ~33/25 = 476.6709 ¢
• CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.9378 ¢, ~33/25 = 476.7054 ¢

Optimal ET sequence: 8d, …, 45ef, 50, 53, 58, 103, 111, 161, 214, 375

Badness (Sintel): 0.933

Hypnos

Subgroup: 2.3.5.7.11

Comma list: 540/539, 72171/71680

Mapping: [⟨1 0 0 -3 8], ⟨0 3 0 11 -13], ⟨0 0 1 0 1]]

mapping generators: ~2, ~81/56, ~5

Optimal tunings:

• WE: ~2 = 1200.3393 ¢, ~81/56 = 633.6837 ¢, ~5/4 = 385.6892 ¢

error map: ⟨+0.339 -0.904 +0.054 +0.677 -0.124]

• CWE: ~2 = 1200.0000 ¢, ~81/56 = 633.5228 ¢, ~5/4 = 385.9359 ¢

error map: ⟨0.000 -1.387 -0.378 -0.075 -1.178]

Optimal ET sequence: 17c, 19, …, 36, 53, 72, 161, 214, 233, 286, 358, 591be

Badness (Sintel): 1.96

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 540/539, 729/728

Mapping: [⟨1 0 0 -3 8 0], ⟨0 3 0 11 -13 7], ⟨0 0 1 0 1 0]]

Optimal tunings:

• WE: ~2 = 1200.6306 ¢, ~81/56 = 633.8505 ¢, ~5/4 = 385.0539 ¢
• CWE: ~2 = 1200.0000 ¢, ~81/56 = 633.5546 ¢, ~5/4 = 385.4668 ¢

Optimal ET sequence: 17c, 19, 36ce, 53, 72, 125f, 197ef

Badness (Sintel): 1.35