No-threes subgroup 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 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.
• Trader has 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

• Jubilic (50/49) → Jubilismic clan
• Didacus (3136/3125) → Hemimean clan
• Sidewalk (823543/800000) → 2023/2000 #Sidewalk

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

Mabilic

Main article: Mabilic and trismegistus

See also: Chromatic pairs § Mabilic

Mabilic is the no-3 restriction of armodue (7 & 16), trismegistus (16 & 25), and semabila (25 & 34). It is the 7 & 9 temperament in the 2.5.7 subgroup, and tempers out 1071875/1048576, the mabilisma. Weak extensions of mabilic include sixix (7 & 25), vishnean (16 & 34), septimin (9 & 41) and monkey (7 & 34).

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

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

Frostburn

"Frostburn" redirects here. For the software developer, see Lumi Pakkanen.

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

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

Llywelyn a.k.a. shoe

Llywelyn a.k.a. shoe is the common restriction of gorgo (16 & 21), gorgik (21 & 37), and semaja (16 & 37). It can be described as 16 & 21, and tempers out the llywelynsma. Its generator is a flattened ~8/7 whole tone, seven of which give the interval class of 5. Its ploidacot is alpha-heptaseph. Weak extensions to the full 7-limit include misneb (21 & 32), which splits 8/7 into two for 15/14~16/15 each, ultrapyth (32 & 37), which splits 16/7 into two for a perfect fifth, and buzzard (53 & 58), which splits 16/7 into three, each for 21/16.

Scales

• Shoe5, Shoe6, Shoe11, Shoe16

Subgroup: 2.5.7

Comma list: 4194304/4117715

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

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

mapping generators: ~2, ~8/7

Optimal tunings:

• WE: ~2 = 1199.1974 ¢, ~8/7 = 226.7586 ¢

error map: ⟨-0.803 +0.194 +2.008]

• CWE: ~2 = 1200.0000 ¢, ~8/7 = 226.8388 ¢

error map: ⟨0.000 +1.558 +4.335]

Optimal ET sequence: 5, 11c, 16, 21, 37, 164d

Badness (Sintel): 2.06

2.5.7.11 subgroup

Subgroup: 2.5.7.11

Comma list: 176/175, 1310720/1294139

Subgroup-val mapping: [⟨1 1 3 1], ⟨0 7 -1 13]]

Gencom mapping: [⟨1 0 1 3 1], ⟨0 0 7 -1 13]]

Optimal tunings:

• WE: ~2 = 1199.1784 ¢, ~8/7 = 226.9582 ¢
• CWE: ~2 = 1200.0000 ¢, ~8/7 = 227.0438 ¢

Optimal ET sequence: 16, 21, 37, 206cd, 243cd, …, 354ccdd

Badness (Sintel): 0.651

2.5.7.11.13 subgroup

Subgroup: 2.5.7.11.13

Comma list: 176/175, 640/637, 847/845

Subgroup-val mapping: [⟨1 1 3 1 2], ⟨0 7 -1 13 9]]

Gencom mapping: [⟨1 0 1 3 1 2], ⟨0 0 7 -1 13 9]]

Optimal tunings:

• WE: ~2 = 1199.1643 ¢, ~8/7 = 226.9495 ¢
• CWE: ~2 = 1200.0000 ¢, ~8/7 = 227.0315 ¢

Optimal ET sequence: 16, 21, 37, 206cdf, 243cdf, …, 354ccddff

Badness (Sintel): 0.319

2.5.7.11.13.17 subgroup

Subgroup: 2.5.7.11.13.17

Comma list: 176/175, 221/220, 640/637, 847/845

Subgroup-val mapping: [⟨1 1 3 1 2 2], ⟨0 7 -1 13 9 11]]

Gencom mapping: [⟨1 0 1 3 1 2 2], ⟨0 0 7 -1 13 9 11]]

Optimal tunings:

• WE: ~2 = 1199.3162 ¢, ~8/7 = 227.1125 ¢
• CWE: ~2 = 1200.0000 ¢, ~8/7 = 227.1774 ¢

Optimal ET sequence: 16, 21, 37, 95

Badness (Sintel): 0.458

Mercy

Named by KingHyperio in 2021, mercy is the common restriction of miracle and countermiracle. It tempers out the quince comma. In this temperament, two generators make ~8/7, five make ~7/5, and seven make ~8/5. The ploidacot for this temperament is omega-heptaseph. Weak extensions of this temperament include cassablanca, slendi, quincy, cotoneum, and octowerck.

Subgroup: 2.5.7

Comma list: 823543/819200

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

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

mapping generators: ~2, ~343/320

Optimal tunings:

• WE: ~2 = 1200.2169 ¢, ~343/320 = 116.3116 ¢

error map: ⟨+0.217 +0.156 -0.798]

• CWE: ~2 = 1200.0000 ¢, ~343/320 = 116.2576 ¢

error map: ⟨0.000 -0.117 -1.341]

Optimal ET sequence: 10, 21, 31, 134, 165, 196, 227, 485d, 712d, 1197dd

Badness (Sintel): 0.428

2.5.7.13 subgroup

Presented here is a low-accuracy extension, which may be thought of as a way to conceptualize the 2.5.7.13.17.19 subgroup of 31edo.

Subgroup: 2.5.7.13

Comma list: 343/338, 640/637

Subgroup-val mapping: [⟨1 3 3 4], ⟨0 -7 -2 -3]]

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

Optimal tunings:

• WE: ~2 = 1199.2166 ¢, ~14/13 = 116.0181 ¢
• CWE: ~2 = 1200.0000 ¢, ~14/13 = 116.2099 ¢

Optimal ET sequence: 10, 21, 31

Badness (Sintel): 0.546

2.5.7.13.17 subgroup

Subgroup: 2.5.7.13.17

Comma list: 170/169, 224/221, 343/338

Subgroup-val mapping: [⟨1 3 3 4 4], ⟨0 -7 -2 -3 1]]

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

Optimal tunings:

• WE: ~2 = 1198.5476 ¢, ~14/13 = 115.6294 ¢
• CWE: ~2 = 1200.0000 ¢, ~14/13 = 115.9132 ¢

Optimal ET sequence: 10, 21, 31, 83fg

Badness (Sintel): 0.451

2.5.7.13.17.19 subgroup

Subgroup: 2.5.7.13.17.19

Comma list: 170/169, 224/221, 343/338, 476/475

Subgroup-val mapping: [⟨1 3 3 4 4 3], ⟨0 -7 -2 -3 1 13]]

Gencom mapping: [⟨1 0 3 3 0 4 4 3], ⟨0 0 -7 -2 0 -3 1 13]]

Optimal tunings:

• WE: ~2 = 1198.4839 ¢, ~14/13 = 115.5700 ¢
• CWE: ~2 = 1200.0000 ¢, ~14/13 = 115.7299 ¢

Optimal ET sequence: 10h, 21, 31, 52f, 83fg

Badness (Sintel): 0.759

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

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

No-threes naiad (rank-3)

Todo: review Devise the permanent name for this temperament; determine its status with respect to its extensions linked below.

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:

• Jacobin superfamily (6656/6655) → The Jacobins

Sephiroth

See also: Chromatic pairs § Sephiroth

Sephiroth is the common restriction of muggles and submerged.

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

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

Trader

Subgroup: 2.5.13

Comma list: 26/25

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

mapping generators, ~2, ~5

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

• Orgone → Orgonia
• 21-23-commatic → 21st-octave temperaments
• 31-17/13-commatic → 31st-octave temperaments
• 37-11-commatic (rank-1) → 37th-octave temperaments
• etc.

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, equating 14/11 to a step of 3edo. 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

Scaevola

Scaevola is the 2.7.11-subgroup temperament tempering out 14641/14336, with a generator of 11/8 and mapping +4 generators to 7/4. Two strong extensions are score to 2.7.11.13 and joan to 2.9.7.11.

Scaevola has a structure that is very similar to mavila, but whereas mavila has a "fifth" of 3/2, "major third" of 5/4, and "minor third" of 6/5, scaevola has a "fifth" of 16/11, "major third" of 14/11, and "minor third" of 8/7, substituting mavila's 4:5:6 for 11:14:16. Essentially, Scaevola can be seen as a counterpart of mavila for fifths even flatter than 9edo, such as 11edo's fifth.

It was named by CompactStar in 2026, referring to Scaevola (the genus of flowering plants), which is meant to sound reminiscent of score and mavila (the score and joan temperaments were actually named prior to this temperament).

Subgroup: 2.7.11

Comma list: 14641/14336

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

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

mapping generators: ~2, ~11/8

Optimal tunings:

• WE: ~2 = 1201.2479 ¢, ~11/8 = 542.1162 ¢

error map: ⟨+1.248 +0.887 -5.458]

• CWE: ~2 = 1200.0000 ¢, ~11/8 = 541.9403 ¢

error map: ⟨0.000 -1.065 -9.378]

Optimal ET sequence: 2, 7d, 9, 11, 20, 31, 51, 82e, 113e, 195eee

Badness (Sintel): 0.578

Score

See also: Chromatic pairs § Score

Score is an extension of scaevola that maps 13/8 to +6 generators.

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]]

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, 7^1/4 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 1 0], ⟨0 2 3]]

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

mapping generators: ~2, ~26/11

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

Arsenic

Arsenic was named by Xenllium in 2026 after the 33rd chemical element.

Subgroup: 2.11.13.17.19

Comma list: 2432/2431, 7086244/7054567, 33698267/33554432

Subgroup-val mapping: [⟨33 0 8 249 26], ⟨0 1 1 -1 1]]

mapping generators: ~247/242, ~11

Optimal tunings:

• WE: ~247/242 = 36.3676 ¢, ~11/8 = 550.4246 ¢
• CWE: ~247/242 = 36.3636 ¢, ~11/8 = 550.4187 ¢

Optimal ET sequence: 33, 165, 198, 231, 495h

Badness (Sintel): 2.30

Berylic

Berylic tempers out the berylisma in the 2.11.37 subgroup, representing the fact that 44/37 is a continued fraction convergent to 2^1/4 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