No-threes subgroup temperaments

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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

Berylic

Berylic tempers out the berylisma in the 2.11.37 subgroup, representing the fact that 44/37 is a continued fraction convergent to 21/4 the fourth root of 2. Beryllic is an example of a temperament which has an astronomically low badness, being a very high-accuracy microtemperament with low-to-average complexity for the harmonics in its subgroup. This also makes it simultaneously supported by edo systems as low as 16edo and up into the tens of thousands. The tradeoff with this temperament, not captured within the metric of badness, is that it is defined within an obscure subgroup, 2.11.37.

If one wishes to explore harmony in this temperament, a great way is to use the 8-note 4L 4s mos, and use the 32:37:44 triad and its inversion 1/(44:37:32) as the root chords. However, the consonance of the 37th harmonic is questionable.

Subgroup: 2.11.37

Comma list: 1874161/1874048

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

mapping generators: ~44/37, ~11

Optimal tunings:

  • WE: ~44/37 = 300.0003 ¢, ~11/8 = 551.3211 ¢
error map: ⟨+0.001 +0.007 -0.017]
  • CWE: ~44/37 = 300.0000 ¢, ~11/8 = 551.3237 ¢
error map: ⟨0.000 +0.006 -0.020]

Optimal ET sequence: 4, 16, 20, 24, 76, 100, 124, 148, 616, 764, 912, 1060, 3328, 4388, 5448

Badness (Sintel): 0.00188

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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