Xenharmonic Wiki

No-threes subgroup temperaments

Revision as of 12:06, 2 July 2026 by FloraC (talk | contribs) (→Temperaments with a 2.5.7 gene: move pakkanen to the comma page as this page is mainly for rank-2)
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 is a collection of subgroup temperaments which omit the prime harmonic of 3.

Overview by mapping of 5

Classified by focusing on the mapping of 5th harmonic, similar to Rank-2 temperaments by mapping of 3.

  • For no-fives, see #No-threes no-fives subgroup temperaments.
  • French decimal and trader have a ~2/1 period and ~5/4 generator. There is a one-to-one correspondence between the 2.5 subgroup and mapped intervals.
  • Ostara, movila and vengeance have variantly expressed generators, three of which give the ~5/2.
  • Insect has a ~55/32 generator, three of which give the ~5/1.
  • Frostburn has a ~28/25 generator, four of which give the ~8/5.

Others have a more complex mapping of 5.

Temperaments with a 2.5.7 gene

Temperaments discussed elsewhere include

Rainy

In rainy, three generators make an 8/7; five generators make a 5/4. It is the no-3's restriction of tertiaseptal (and valentine), notable theoretically as it equates (2/1)/(5/4)3 (128/125, the lesser diesis) with (2/1)/(8/7)5 (the 2.7-subgroup cloudy comma, which is similar to the 2.5-subgroup lesser diesis in that tempering it out tunes the 8/7 about 8.8 ¢ sharp, while tempering out 128/125 similarly sharpens the 5/4 by about 13.7 ¢). By tempering out their difference, stacked 5's and stacked 7's become easier to navigate, using the general-purpose diesis to simplify clusters.

A highly notable tuning of rainy not shown here is 311edo, which is 140 + 171 so tuned between them.

Subgroup: 2.5.7

Comma list: 2100875/2097152

Subgroup-val mapping: [⟨1 2 3], ⟨0 5 -3]]

Gencom mapping: [⟨1 0 2 3], ⟨0 0 5 -3]]

mapping generators: ~2, ~256/245

Optimal tunings:

  • WE: ~2 = 1200.0939 ¢, ~256/245 = 77.2107 ¢
error map: ⟨+0.094 -0.072 -0.176]
  • CWE: ~2 = 1200.0000 ¢, ~256/245 = 77.2093 ¢
error map: ⟨0.000 -0.267 -0.454]

Optimal ET sequence: 15, 16, 31, 109, 140, 171, 373, 544, 1259, 1803d

Badness (Sintel): 0.156

Augment

See also: Chromatic pairs § Augment

Augment is related to augmented, but for 2.5.7 instead of 2.3.5.

Subgroup: 2.5.7

Comma list: 128/125

Subgroup-val mapping: [⟨3 7 0], ⟨0 0 1]]

Gencom mapping: [⟨3 0 7 0], ⟨0 0 0 1]]

mapping generators: ~5/4, ~7

Optimal tunings:

  • WE: ~5/4 = 399.0128 ¢, ~7/4 = 974.7085 ¢
error map: ⟨-2.962 +6.776 -0.040]
  • CWE: ~5/4 = 400.0000 ¢, ~7/4 = 974.3418 ¢
error map: ⟨0.000 +13.686 +5.516]

Optimal ET sequence: 3, 6, 15, 21, 27, 102ccd, 129ccd

Badness (Sintel): 0.296

2.5.7.11 subgroup

Subgroup: 2.5.7.11

Comma list: 56/55, 128/125

Subgroup-val mapping: [⟨3 7 0 2], ⟨0 0 1 1]]

Gencom mapping: [⟨3 0 7 0 2], ⟨0 0 0 1 1]]

Optimal tunings:

  • WE: ~5/4 = 398.9239 ¢, ~7/4 = 969.1106 ¢
  • CWE: ~5/4 = 400.0000 ¢, ~7/4 = 968.4397 ¢

Optimal ET sequence: 3, 6, 15, 21

Badness (Sintel): 0.196

Frostburn

Frostburn is the common restriction of quadrimage and baldy.

Subgroup: 2.5.7

Comma list: 78125/76832

Subgroup-val mapping: [⟨1 3 4], ⟨0 -4 -7]]

mapping generators: ~2, ~28/25

Optimal tunings:

  • WE: ~2 = 1200.3462 ¢, ~28/25 = 204.3386 ¢
error map: ⟨+0.346 -2.630 +2.189]
  • CWE: ~2 = 1200.0000 ¢, ~28/25 = 204.2027 ¢
error map: ⟨0.000 -3.125 +1.755]

Optimal ET sequence: 6, 29, 35, 41, 47

Badness (Sintel): 0.886

2.5.7.11 subgroup

Subgroup: 2.5.7.11

Comma list: 245/242, 625/616

Subgroup-val mapping: [⟨1 3 4 5], ⟨0 -4 -7 -9]]

mapping generators: ~2, ~28/25

Optimal tunings:

  • WE: ~2 = 1200.6817 ¢, ~28/25 = 205.0734 ¢
  • CWE: ~2 = 1200.0000 ¢, ~28/25 = 204.8199 ¢

Optimal ET sequence: 6, 23de, 29, 35, 41

Badness (Sintel): 0.463

Mabilic

Main article: Mabilic and trismegistus
See also: Chromatic pairs § Mabilic

Mabilic is the no-3 restriction of armodue, semabila, and trismegistus. It is the 7 & 9 temperament in the 2.5.7 subgroup, and tempers out 1071875/1048576, the mabilisma.

Subgroup: 2.5.7

Comma list: 1071875/1048576

Subgroup-val mapping: [⟨1 1 5], ⟨0 3 -5]]

Gencom mapping: [⟨1 0 1 5], ⟨0 0 3 -5]]

mapping generators: ~2, ~175/128

Optimal tunings:

  • WE: ~2 = 1201.2543 ¢, ~175/128 = 527.7872 ¢
error map: ⟨+1.254 -1.698 -1.491]
  • CWE: ~2 = 1200.0000 ¢, ~175/128 = 527.2041 ¢
error map: ⟨0.000 -4.701 -4.846]

Optimal ET sequence: 7, 9, 16, 25, 41, 66, 305ccdd, 371ccddd

Badness (Sintel): 1.70

Huntington

See also: Chromatic pairs § Huntington

Huntington may be described as the 10 & 37 temperament in the 2.5.7.13 subgroup.

Subgroup: 2.5.7

Comma list: 40960000/40353607

Subgroup-val mapping: [⟨1 -4 0], ⟨0 9 4]]

Gencom mapping: [⟨1 0 -4 0], ⟨0 0 9 4]]

mapping generators: ~2, ~80/49

Optimal tunings:

  • WE: ~2 = 1199.5781 ¢, ~80/49 = 842.6730 ¢
error map: ⟨-0.422 -0.569 +1.866]
  • CWE: ~2 = 1200.0000 ¢, ~80/49 = 842.9136 ¢
error map: ⟨0.000 -0.091 +2.828]

Optimal ET sequence: 7c, 10, 27, 37, 84, 121

Badness (Sintel): 1.87

2.5.7.13 subgroup

Subgroup: 2.5.7.13

Comma list: 640/637, 10985/10976

Subgroup-val mapping: [⟨1 -4 0 3], ⟨0 9 4 1]]

Gencom mapping: [⟨1 0 -4 0 0 3], ⟨0 0 9 4 0 1]]

mapping generators: ~2, ~13/8

Optimal tunings:

  • WE: ~2 = 1199.4788 ¢, ~13/8 = 842.6318 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/8 = 842.9447 ¢

Optimal ET sequence: 7c, 10, 17, 27, 37, 84, 121, 279df, 400ddf

Badness (Sintel): 0.319

Silver

See also: Chromatic pairs § Silver

Silver can be described as the 10 & 37 temperament in the 2.5.7.13.17 subgroup.

Subgroup: 2.5.7.13.17

Comma list: 170/169, 640/637, 5525/5488

Subgroup-val mapping: [⟨1 -4 0 3 9], ⟨0 9 4 1 -7]]

Gencom mapping: [⟨1 0 -4 0 0 3 9], ⟨0 0 9 4 0 1 -7]]

Optimal tunings:

  • WE: ~2 = 1200.0932 ¢, ~13/8 = 842.7764 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/8 = 842.7143 ¢

Optimal ET sequence: 10, 27, 37, 47, 84, 131, 178g

Badness (Sintel): 0.504

Ostara

Ostara is a temperament that is derived from 93 & 524 solar calendar leap rule scale, interpreted in general no-3's 19-limit. It is a weak extension of the unnamed 2.5.7-subgroup 28 & 31 temperament, which tempers out 8589934592/8544921875.

Subgroup: 2.5.7.11

Comma list: 8589934592/8544921875, 30691800524/30517578125

Subgroup-val mapping: [⟨1 1 20 -49], ⟨0 3 -39 119]]

mapping generators: ~2, ~5120/3773

Optimal tunings:

  • WE: ~2 = 1199.9115 ¢, ~5120/3773 = 528.9650 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5120/3773 = 529.0037 ¢

Optimal ET sequence: 93, 245e, 338, 955c, 1386c

Badness (Sintel): 11.7

2.5.7.11.13 subgroup

Subgroup: 2.5.7.11.13

Comma list: 1001/1000, 34420736/34328125, 5670699008/5661858125

Subgroup-val mapping: [⟨1 1 20 -49 35], ⟨0 3 -39 119 -71]]

Optimal tunings:

  • WE: ~2 = 1199.9194 ¢, ~1664/1225 = 528.9681 ¢
  • CWE: ~2 = 1200.0000 ¢, ~1664/1225 = 529.0036 ¢

Optimal ET sequence: 93, 245e, 338, 431, 1386c

Badness (Sintel): 3.42

2.5.7.11.13.17 subgroup

Subgroup: 2.5.7.11.13.17

Comma list: 1001/1000, 32768/32725, 147968/147875, 537824/537251

Subgroup-val mapping: [⟨1 1 20 -49 35 42], ⟨0 3 -39 119 -71 -86]]

Optimal tunings:

  • WE: ~2 = 1199.9054 ¢, ~1664/1225 = 528.9628 ¢
  • CWE: ~2 = 1200.0000 ¢, ~1664/1225 = 529.0046 ¢

Optimal ET sequence: 93, 338, 431, 955c, 1386cg

Badness (Sintel): 1.99

2.5.7.11.13.17.19 subgroup

Subgroup: 2.5.7.11.13.17.19

Comma list: 1001/1000, 2128/2125, 3328/3325, 16807/16796, 147968/147875

Subgroup-val mapping: [⟨1 1 20 -49 35 42 21], ⟨0 3 -39 119 -71 -86 -38]]

Optimal tunings:

  • WE: ~2 = 1199.9081 ¢, ~19/14 = 528.9639 ¢
  • CWE: ~2 = 1200.0000 ¢, ~19/14 = 529.0045 ¢

Optimal ET sequence: 93, 338, 431, 955c, 1386cg

Badness (Sintel): 1.29

French decimal

French decimal is conceived upon the fact that 1789edo has an excellent 5/4, and uses it as the generator. This rings particularly true for the French attempts to decimalize a lot more things than we are used to today. Using the maximal evenness method of finding rank-2 temperaments, a 1525 & 1789 temperament is obtained.

Subgroup: 2.5.7

Comma list: [372 -159 -1⟩

Subgroup-val mapping: [⟨1 0 372], ⟨0 1 -159]]

mapping generators: ~2, ~5

Optimal tunings:

  • WE: ~2 = 1199.9901 ¢, ~5/4 = 386.3562 ¢
error map: ⟨-0.010 +0.023 +0.000]
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 386.3595 ¢
error map: ⟨0.000 +0.046 +0.019]

Optimal ET sequence: 205, 264, 733, 997, 2258, 3255, 7507, 10762

Badness (Sintel): 148

2.5.7.11 subgroup

Subgroup: 2.5.7.11

Comma list: [-49 8 17 -5⟩, [45 -27 10 -3⟩

Subgroup-val mapping: [⟨1 0 372 1255], ⟨0 1 -159 -539]]

Optimal tunings:

  • WE: ~2 = 1200.0130 ¢, ~5/4 = 386.3653 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 386.3611 ¢

Optimal ET sequence: 264, 997e, 1261e, 1525, 1789

Badness (Sintel): 52.2

2.5.7.11.13 subgroup

Subgroup: 2.5.7.11.13

Comma basis: 28824005/28792192, 200126927/200000000, 6106906624/6103515625

Subgroup-val mapping: [⟨1 0 372 1255 -398], ⟨0 1 -159 -539 173]]

Optimal tunings:

  • WE: ~2 = 1200.0137 ¢, ~5/4 = 386.3655 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 386.3611 ¢

Optimal ET sequence: 261, 1261e, 1525, 1789

Badness (Sintel): 10.5

Bastille

Main article: Bastille

Bastille is described as the 2.5.7-subgroup 1407 & 1789 temperament, and named after an eponymous historical event which took place on July 14, 1789 (14/07/1789). Extensions discussed elsewhere include double bastille.

Subgroup: 2.5.7

Comma list: [1426 -596 -15⟩

Subgroup-val mapping: [⟨1 -4 254], ⟨0 15 -596]]

mapping generators: ~2, ~[-380 159 4⟩

Optimal tunings:

  • WE: ~2 = 1199.9911 ¢, ~[-380 159 4⟩ = 505.7532 ¢
error map: ⟨-0.009 +0.020 +0.001]
  • CWE: ~2 = 1200.0000 ¢, ~[-380 159 4⟩ = 505.7570 ¢
error map: ⟨0.000 +0.041 +0.018]

Optimal ET sequence: 382, 1025, 1407, 14452, 15859c, 17266c, …, 27115cd

Badness (Sintel): 7.18 × 103

Tricesimoprimal miracloid

See also: Tricesimoprimal miracloid/Eliora's approach
  Todo: complete section

Add missing data from 2.5.7 to 2.5.7.11.19.29.

Tricesimoprimal miracloid is described as the 52 & 1789 temperament in the 2.5.7.11.19.29.31 subgroup, with harmonics specifically selected for 52edo and 1789edo. Its generator is 31/29, which is also close to the secor. In terms of microtempering, a circle of 52 generators is essentially a barely noticeable well temperament for 52edo.

2.5.7.11.19.29.31 subgroup

Subgroup: 2.5.7.11.19.29.31

Comma list: 10241/10240, 5858783/5856400, 4093705/4090624, 15109493/15089800, 102942875/102834688

Subgroup-val mapping: [⟨1 -42 -2 -15 -12 -61 -61], ⟨0 461 50 192 169 685 686]]

Optimal tunings:

  • WE: ~2 = 1200.0079 ¢, ~31/29 = 115.3723 ¢
  • CWE: ~2 = 1200.0000 ¢, ~31/29 = 115.3716 ¢

Optimal ET sequence: 52, 1737, 1789

Badness (Sintel): 4.51

No-threes naiad (rank-3)

See also: Wizardharry clan § Naiad, and Werckismic temperaments § Seminaiad

This temperament can be described as the 21 & 29 & 37 temperament in no-threes subgroups. It expands tridec and naiadec.

Subgroup: 2.5.7.11

Comma list: 5021863/5000000

Subgroup-val mapping: [⟨1 0 -2 3], ⟨0 1 1 1], ⟨0 0 4 -3]]

mapping generators: ~2, ~5, ~77/50

Optimal tunings:

  • WE: ~2 = 1200.0805 ¢, ~5/4 = 386.6593 ¢, ~77/50 = 745.4622 ¢
error map: ⟨+0.080 +0.507 -1.318 -0.643]
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 386.7404 ¢, ~77/50 = 745.4102 ¢
error map: ⟨0.000 +0.427 -0.445 -0.808]

Optimal ET sequence: 16, 21, 29, 37, 87, 103, 124, 161, 227, 264, 388, 425, 652e, 689e, 1077de

Badness (Sintel): 1.86

2.5.7.11.13 subgroup

Subgroup: 2.5.7.11.13

Comma list: 847/845, 1001/1000

Subgroup-val mapping: [⟨1 0 -2 3 2], ⟨0 1 1 1 1], ⟨0 0 4 -3 -1]]

Optimal tunings:

  • WE: ~2 = 1200.0343 ¢, ~5/4 = 386.6098 ¢, ~20/13 = 745.4658 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 386.6458 ¢, ~20/13 = 745.4431 ¢

Optimal ET sequence: 16, 21, 29, 37, 87, 103, 124, 161, 227, 264, 565e, 689e

Badness (Sintel): 0.179

2.5.7.11.13.17 subgroup

Subgroup: 2.5.7.11.13.17

Comma list: 170/169, 221/220, 847/845

Subgroup-val mapping: [⟨1 0 -2 3 2 3], ⟨0 1 1 1 1 1], ⟨0 0 4 -3 -1 -2]]

Optimal tunings:

  • WE: ~2 = 1200.4068 ¢, ~5/4 = 386.6701 ¢, ~17/11 = 745.3706 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 387.1074 ¢, ~17/11 = 745.0940 ¢

Optimal ET sequence: 16, 21, 29g, 37, 66g, 87g, 124g

Badness (Sintel): 0.438

Temperaments with a higher 2.5.p gene

Temperaments discussed elsewhere include:

Wizz

See also: Chromatic pairs § Wizz

Wizz, the 6 & 16 temperament in the 2.5.11 subgroup, tempers out 15625/15488, and is the common restriction of astrology and wizard.

Subgroup: 2.5.11

Comma list: 15625/15488

Subgroup-val mapping: [⟨2 0 -7], ⟨0 1 3]]

Gencom mapping: [⟨2 0 4 0 5], ⟨0 0 1 0 3]]

mapping generators: ~125/88, ~5/4

Optimal tunings:

  • WE: ~125/88 = 600.1831 ¢, ~5/4 = 383.8848 ¢
error map: ⟨+0.366 -1.697 +1.252]
  • CWE: ~125/88 = 600.0000 ¢, ~5/4 = 383.9977 ¢
error map: ⟨0.000 -2.316 +0.675]

Optimal ET sequence: 6, 16, 22, 28, 50, 122, 172, 222, 394c

Badness (Sintel): 0.266

Insect

Subgroup: 2.5.11

Comma list: 33275/32768

Subgroup-val mapping: [⟨1 0 5], ⟨0 3 -2]]

mapping generators, ~2, ~55/32

Optimal tunings:

  • WE: ~2 = 1201.1238 ¢, ~55/32 = 928.5003 ¢
error map: ⟨+1.124 -0.813 -2.700]
  • CWE: ~2 = 1200.0000 ¢, ~55/32 = 927.7384 ¢
error map: ⟨0.000 -3.099 -6.975]

Optimal ET sequence: 9, 13, 22, 97e, 119e, 141e, 163e, 304ceee

Badness (Sintel): 0.564

Movila

This temperament has a structure very similar to mavila but is somewhat more accurate because the generator is a flat 11/8 rather than a sharp 4/3. The major third is still ~5/4, but the minor third is now ~64/55 instead of ~6/5.

Subgroup: 2.5.11

Comma list: 1331/1280

Subgroup-val mapping: [⟨1 1 3], ⟨0 3 1]]

mapping generators: ~2, ~11/8

Optimal tunings:

  • WE: ~2 = 1203.0339 ¢, ~11/8 = 528.4296 ¢
error map: ⟨+3.034 +2.009 -13.787]
  • CWE: ~2 = 1200.0000 ¢, ~11/8 = 528.1575 ¢
error map: ⟨0.000 -1.841 -23.160]

Optimal ET sequence: 7, 9, 16, 25, 41e, 66ee

Badness (Sintel): 0.718

Sephiroth

See also: Chromatic pairs § Sephiroth

Sephiroth is the no-7 restriction of muggles.

Subgroup: 2.5.11

Comma list: 34375/32768

Subgroup-val mapping: [⟨1 0 15], ⟨0 1 -5]]

Gencom mapping: [⟨1 0 0 0 15], ⟨0 0 1 0 -5]]

mapping generators: ~2, ~5

Optimal tunings:

  • WE: ~2 = 1203.3290 ¢, ~5/4 = 373.6097 ¢
error map: ⟨+3.329 -6.046 -2.722]
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 372.1586 ¢
error map: ⟨0.000 -14.155 -12.111]

Optimal ET sequence: 3, 10, 13, 16, 29, 132cceee

Badness (Sintel): 1.85

2.5.11.13 subgroup

Subgroup: 2.5.11.13

Comma list: 65/64, 6875/6656

Subgroup-val mapping: [⟨1 0 15 6], ⟨0 1 -5 -1]]

Gencom mapping: [⟨1 0 0 0 15 6], ⟨0 0 1 0 -5 -1]]

Optimal tunings:

  • WE: ~2 = 1203.3825 ¢, ~5/4 = 373.6318 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 372.1519 ¢

Optimal ET sequence: 3, 10, 13, 16, 29, 132cceeeff

Badness (Sintel): 0.410

2.5.11.13.17 subgroup

Subgroup: 2.5.11.13.17

Comma list: 65/64, 170/169, 221/220

Subgroup-val mapping: [⟨1 0 15 6 11], ⟨0 1 -5 -1 -3]]

Gencom mapping: [⟨1 0 0 0 15 6 11], ⟨0 0 1 0 -5 -1 -3]]

Optimal tunings:

  • WE: ~2 = 1203.6741 ¢, ~5/4 = 373.3775 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 371.6773 ¢

Optimal ET sequence: 3, 10, 13, 16, 29g, 129ccceeffgggg

Badness (Sintel): 0.299

Trader

Subgroup: 2.5.13

Comma list: 26/25

Subgroup-val mapping: [⟨1 2 3], ⟨0 1 2]]

mapping generators, ~2, ~5/4

Optimal tunings:

  • WE: ~2 = 1198.0216 ¢, ~5/4 = 410.2152 ¢
error map: ⟨-1.978 +19.945 -26.033]
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 408.9029 ¢
error map: ⟨0.000 +22.589 -22.722]

Optimal ET sequence: 3, 20c, 23c, 26c

Badness (Sintel): 0.138

Superquintal

Subgroup: 2.5.13

Comma list: 64000000/62748517

Subgroup-val mapping: [⟨1 -2 0], ⟨0 7 6]]

mapping generators, ~2, ~20/13

Optimal tunings:

  • WE: ~2 = 1199.5925 ¢, ~20/13 = 740.6286 ¢
error map: ⟨-0.408 -1.098 +3.244]
  • CWE: ~2 = 1200.0000 ¢, ~20/13 = 740.8058 ¢
error map: ⟨0.000 -0.673 +4.307]

Optimal ET sequence: 8, 13, 21, 34, 81, 115

Badness (Sintel): 1.93

No-threes no-fives subgroup temperaments

Temperaments discussed elsewhere include

Amaranthine

See also: No-fives subgroup temperaments § Chrysanthemum

Amaranthine is the high-accuracy 2.7.11-subgroup strong restriction of undecimal mothra.

Subgroup: 2.7.11

Comma list: 5767168/5764801

Subgroup-val mapping: [⟨1 0 -19], ⟨0 1 8]]

mapping generators: ~2, ~7

Optimal tunings:

  • WE: ~2 = 1199.9846 ¢, ~7/4 = 968.9078 ¢
error map: ⟨-0.015 +0.051 -0.010]
  • CWE: ~2 = 1200.0000 ¢, ~7/4 = 968.9174 ¢
error map: ⟨0.000 +0.091 +0.021]

Optimal ET sequence: 26, 83, 109, 135, 161, 296, 1641, 1937, 2233, 2529, 2825, 3121, 6538d, 9659d, 12780dd

Badness (Sintel): 0.0309

Argument

Argument tempers out 1372/1331 in the 2.7.11 subgroup. It is the no-3 restriction of augment.

Subgroup: 2.7.11

Comma list: 1372/1331

Subgroup-val mapping: [⟨3 0 2], ⟨0 1 1]]

mapping generators: ~14/11, ~7

Optimal tunings:

  • WE: ~14/11 = 399.8041 ¢, ~7/4 = 963.1666 ¢
error map: ⟨-0.588 -6.835 +10.281]
  • CWE: ~14/11 = 400.0000 ¢, ~4/4 = 962.7466 ¢
error map: ⟨0.000 -6.079 +11.429]

Optimal ET sequence: 6, 9, 15, 36, 51e, 66e

Badness (Sintel): 0.475

Score

See also: Chromatic pairs § Score

Score is a low-accuracy extension of the unnamed 2.7.11-subgroup temperament tempering out 14641/14336.

Subgroup: 2.7.11.13

Comma list: 343/338, 847/832

Subgroup-val mapping: [⟨1 1 3 1], ⟨0 4 1 6]]

Gencom mapping: [⟨1 0 0 1 3 1], ⟨0 0 0 4 1 6]]

mapping generators: ~2, ~11/8

Optimal tunings:

  • WE: ~2 = 1201.5484 ¢, ~11/8 = 540.7963 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/8 = 540.5091 ¢

Optimal ET sequence: 9, 11, 20

Badness (Sintel): 0.368

Bossier

See also: Chromatic pairs § Bossier

Bossier can be described as the 3 & 17 in the 2.7.11.13 subgroup, tempering out 1573/1568 and 15488/15379.

Subgroup: 2.7.11

Comma list: 214358881/210827008

Subgroup-val mapping: [⟨1 0 1], ⟨0 8 7]]

Gencom mapping: [⟨1 0 0 0 1], ⟨0 0 0 8 7]]

mapping generators: ~2, ~14/11

Optimal tunings:

  • WE: ~2 = 1200.1886 ¢, ~14/11 = 421.2661 ¢
error map: ⟨+0.189 +1.303 -2.266]
  • CWE: ~2 = 1200.0000 ¢, ~14/11 = 421.2365 ¢
error map: ⟨0.000 +1.066 -2.662]

Optimal ET sequence: 17, 20, 37, 57, 94, 151

Badness (Sintel): 1.73

2.7.11.13 subgroup

Subgroup: 2.7.11.13

Comma list: 1573/1568, 15488/15379

Subgroup-val mapping: [⟨1 0 1 3], ⟨0 8 7 2]]

Gencom mapping: [⟨], ⟨1 0 0 0 1 3], ⟨0 0 0 8 7 2]]

Optimal tunings:

  • WE: ~2 = 1199.8668 ¢, ~14/11 = 421.2623 ¢
  • CWE: ~2 = 1200.0000 ¢, ~14/11 = 421.2874 ¢

Optimal ET sequence: 17, 20, 37, 57, 94, 225

Badness (Sintel): 0.307

Voltage

Voltage is the 3 & 7 temperament in the 2.7.13 subgroup. Among the notable tunings is pure-7 tuning, 71/4 of 842.2065 ¢, which is also the CTC (constrained Tenney–Chebyshevian) tuning.

Subgroup: 2.7.13

Comma list: 28672/28561

Subgroup-val mapping: [⟨1 0 3], ⟨0 4 1]]

Gencom mapping: [⟨1 0 0 0 0 3], ⟨0 0 0 4 0 1]]

mapping generators: ~2, ~13

Optimal tunings:

  • WE: ~2 = 1199.7827 ¢, ~13/8 = 842.1707 ¢
error map: ⟨-0.217 -0.143 +0.991]
  • CWE: ~2 = 1200.0000 ¢, ~13/8 = 842.2568 ¢
error map: ⟨0.000 +0.201 +1.729]

Optimal ET sequence: 3, 7, 10, 27, 37, 47, 57, 104, 463f, 567f, 671ff, 775ff

Badness (Sintel): 0.115

Ultrakleismic

Subgroup: 2.7.17

Comma list: 4913/4802

Subgroup-val mapping: [⟨1 2 3], ⟨0 3 4]]

mapping generators, ~2, ~17/14

Optimal tunings:

  • WE: ~2 = 1200.1379 ¢, ~17/14 = 324.3440 ¢
error map: ⟨+0.138 +4.482 -7.166]
  • CWE: ~2 = 1200.000 ¢, ~17/14 = 324.3738 ¢
error map: ⟨0.000 +4.295 -7.460]

Optimal ET sequence: 4, 7, 11, 26, 37

Badness (Sintel): 0.460

Counterultrakleismic

Subgroup: 2.7.17

Comma list: 2024782584832/2015993900449

Subgroup-val mapping: [⟨1 0 1], ⟨0 10 11]]

mapping generators, ~2, ~17/14

Optimal tunings:

  • WE: ~2 = 1199.9723 ¢, ~17/14 = 336.8586 ¢
error map: ⟨-0.028 -0.240 +0.462]
  • CWE: ~2 = 1200.000 ¢, ~17/14 = 336.8621 ¢
error map: ⟨0.000 -0.205 +0.528]

Optimal ET sequence: 7, 18dg, 25, 32, 57, 488, 545, 602, 659, 716, 773, 830, 887, 1717g

Badness (Sintel): 0.860

Shipwreck

Subgroup: 2.7.53

Comma list: 1048576/1042139

Subgroup-val mapping: [⟨1 0 6], ⟨0 3 -1]]]

mapping generators, ~2, ~64/53

Optimal tunings:

  • WE: ~2 = 1199.6967 ¢, ~64/53 = 323.1839 ¢
error map: ⟨-0.303 +0.119 +1.491]
  • CWE: ~2 = 1200.0000 ¢, ~64/53 = 323.1959 ¢
error map: ⟨0.000 +0.762 +3.300]

Optimal ET sequence: 4, 7, 11, 15, 26, 141, 167, 193p, 219p, 245p

Badness (Sintel): 0.224

Lovecraft

See also: Chromatic pairs § Lovecraft

Lovecraft, the 4 & 13 temperament in the 2.11.13 subgroup, tempers out 1352/1331, and is generated by ~13/11. Two generator steps give ~11/8 and three generator steps give ~13/8.

Subgroup: 2.11.13

Comma list: 1352/1331

Subgroup-val mapping: [⟨1 3 3], ⟨0 2 3]]

Gencom mapping: [⟨1 0 0 0 3 3], ⟨0 0 0 0 2 3]]

mapping generators, ~2, ~13/11

Optimal tunings:

  • WE: ~2 = 1199.5223 ¢, ~13/11 = 279.2064 ¢
error map: ⟨-0.478 +5.662 -4.341]
  • CWE: ~2 = 1200.0000 ¢, ~13/11 = 278.9918 ¢
error map: ⟨0.000 +6.666 -3.552]

Optimal ET sequence: 4, 9, 13, 30, 43, 73, 116e

Badness (Sintel): 0.175

Bluebirds

Not to be confused with Bluebird.
See also: Chromatic pairs § Bluebirds

Subgroup: 2.11.13

Comma list: 265837/262144

Subgroup-val mapping: [⟨1 0 6], ⟨0 3 -2]]

Gencom mapping: [⟨1 0 0 0 3 4], ⟨0 0 0 0 3 -2]]

mapping generators, ~2, ~143/128

Optimal tunings:

  • WE: ~2 = 1200.8795 ¢, ~143/128 = 182.5017 ¢
error map: ⟨+0.880 -1.174 -2.013]
  • CWE: ~2 = 1200.0000 ¢, ~143/128 = 182.4386 ¢
error map: ⟨0.000 -4.002 -5.405]

Optimal ET sequence: 6, 7, 13, 33, 46, 79, 125f, 204ef, 329eeff

Badness (Sintel): 0.451

Blackbirds

See also: Chromatic pairs § Blackbirds

Blackbirds is a fairly straightforward temperament. It simply equates ~13/11 to 1/4 of the octave with a generator for prime 11 or 13.

Subgroup: 2.11.13

Comma list: 29282/28561

Subgroup-val mapping: [⟨4 0 1], ⟨0 1 1]]

Gencom mapping: [⟨4 0 0 0 12 13], ⟨0 0 0 0 1 1]]

mapping generators, ~13/11, ~11

Optimal tunings:

  • WE: ~13/11 = 299.9728 ¢, ~11/8 = 546.6107 ¢
error map: ⟨-0.109 -5.033 +5.730]
  • CWE: ~13/11 = 300.0000 ¢, ~11/8 = 546.4664 ¢
error map: ⟨0.000 -4.852 +5.939]

Optimal ET sequence: 4, 12e, 16, 20, 24, 44, 68

Badness (Sintel): 0.668

Yamablu

Yamablu, with a generator of ~26/17, is named for its tempering of the yama comma (209/208) and the blume comma (2057/2048), which also implies the blumeyer comma (2432/2431). It extends the 2.11.13-subgroup temperament tempering out 556573090931/549755813888. The 13th Yamablu[13] scale is a linear-temperament version of Gjaeck.

Subgroup: 2.11.13.17.19

Comma list: 209/208, 2057/2048, 83521/83486

Subgroup-val mapping: [⟨1 1 8 9 11], ⟨0 4 -7 -8 -11]]

mapping generators: ~2, ~26/17

Optimal tunings:

  • WE: ~2 = 1200.4661 ¢, ~26/17 = 737.3256 ¢
  • CWE: ~2 = 1200.0000 ¢, ~26/17 = 737.0014 ¢

Optimal ET sequence: 13, 44, 57, 70, 127, 197eh

Badness (Sintel): 0.386

Mavericks

Subgroup: 2.13.19

Comma list: 47525504/47045881

Subgroup-val mapping: [⟨1 1 2], ⟨0 6 5]]

mapping generators: ~2, ~26/19

Optimal tunings:

  • WE: ~2 = 1199.8817 ¢, ~26/19 = 539.9150 ¢
error map: ⟨-0.118 -1.156 +1.825]
  • CWE: ~2 = 1200.0000 ¢, ~26/19 = 539.9280 ¢
error map: ⟨0.000 -0.960 +2.127]

Optimal ET sequence: 9, 11, 20

Badness (Sintel): 0.559

Yer (rank 3)

Subgroup: 2.11.13.17.19

Comma list: 209/208, 2057/2048

Subgroup-val mapping: [⟨1 0 0 11 4], ⟨0 1 0 -2 -1], ⟨0 0 1 0 1]]

mapping generators: ~2, ~11, ~13

Optimal tunings:

  • WE: ~2 = 1200.4447 ¢, ~11/8 = 548.4929 ¢, ~13/8 = 841.3613 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/8 = 548.2193 ¢, ~13/8 = 841.4707 ¢

Optimal ET sequence: 11, 13, 24, 33, 37, 46, 57, 70, 127, 197eh

Badness (Sintel): 0.106

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