No-threes subgroup temperaments

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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-or-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.

2.5.7 temperaments

Temperaments discussed elsewhere include

Frostburn

See also: Magic family § Quadrimage

Subgroup: 2.5.7

Comma list: 78125/76832

Sval mapping: [⟨1 3 4], ⟨0 -4 -7]]

Sval mapping generators: ~2, ~28/25

Optimal tuning (TE): ~2/1 = 1200.3479, ~28/25 = 204.3389

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

2.5.7.11

Subgroup: 2.5.7.11

Comma list: 245/242, 625/616

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

Sval mapping generators: ~2, ~28/25

Optimal tuning (TE): ~2/1 = 1200.6817, ~28/25 = 205.0745

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

Mabilic

See also: Chromatic pairs § Mabilic

Given below is the no-three version of semabila, or equivalently the no-threes version of 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

Sval mapping: [⟨1 1 5], ⟨0 3 -5]]

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

gencom: [2 175/128; 1071875/1048576]

Optimal tuning (POTE): ~2 = 1\1, ~175/128 = 527.236

Optimal ET sequence: 7, 9, 16, 25, 41, 66, 305bc

RMS error: 0.7729 cents

Rainy

Three generators make an 8/7; five generators make a 5/4. This is the no-threes version of tertiaseptal (and valentine). Rainy is 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 5s and stacked 7s become easier to navigate, using the general-purpose diesis to simplify clusters. (Note that this analysis assumes a lattice-based conceptualization of JI which is often called "stacking-based"; see taxonomies of xen approaches.)

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

Sval mapping: [⟨1 2 3], ⟨0 5 -3]]

Gencom: [2 256/245; 2100875/2097152]

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

Optimal tuning (POTE): ~256/245 = 77.205

Optimal ET sequence: 31, 47, 78, 109, 140, 171, 202, 233

RMS error: 0.0586 cents

French decimal

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 basis: [372 -159 -1⟩

Sval mapping: [⟨1 2 54], ⟨0 1 -159]]

Optimal tuning (CTE): ~5/4 = 386.360

Optimal ET sequence: 205, 264, 469, 733, 997, 1261, 1525, 1789, ...

2.5.7.11 subgroup

Subgroup: 2.5.7.11

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

Sval mapping: [⟨1 2 54 -177], ⟨0 1 -159 -539]]

Optimal tuning (CTE): ~5/4 = 386.361

Optimal ET sequence: 264, 733, ...

2.5.7.11.13 subgroup

Subgroup: 2.5.7.11.13

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

Sval mapping: [⟨1 2 54 -177 52], ⟨0 1 -159 -539 173]]

Optimal tuning (CTE): ~5/4 = 386.361

Optimal ET sequence: 1525, 1789, ...

Bastille

Main article: Bastille

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⟩

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

Optimal tuning (CTE): ~[381 0 -159 -4⟩ = 694.243

Optimal ET sequence: 382, 1025, 1407, 1789, 3196, ...

Augment

See also: Chromatic pairs § Augment

Augment is related to augmented.

Subgroup: 2.5.7.11

Comma list: 56/55, 128/125

Sval mapping: [⟨3 7 0 2], ⟨0 0 1 1]]

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

gencom: [5/4 8/7; 56/55 128/125]

Optimal tuning (POTE): ~5/4 = 1\3, ~8/7 = 228.275

Optimal ET sequence: 3, 6, 9, 15, 21

RMS error: 2.422 cents

Ostara

Ostara is a temperament that is derived from 93 & 524 solar calendar leap rule scale. It was initially defined by taking the 2.7.13.17.19 subgroup, but it can also be intepreted in general no-threes 19-limit.

Ostara can also refer to a collection of temperaments which temper out 16807/16796.

Subgroup: 2.5.7.11

Comma list: 8589934592/8544921875, 53710650917/53687091200

Mapping: [⟨1 1 20 -49], ⟨0 3 -39 119]]

Optimal tuning (POTE): ~5120/3773 = 529.003¢

Optimal ET sequence: 93, 431, 338, 524

2.5.7.11.13 subgroup

Subgroup: 2.5.7.11.13

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

Sval Mapping: [⟨1 1 20 -49 35], ⟨0 3 -39 119 -71]]

Optimal tuning (POTE): ~1664/1225 = 529.003¢

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

2.5.7.11.13.17 subgroup

Subgroup: 2.5.7.11.13.17

Sval Mapping: [⟨1 1 20 -49 35 42], ⟨0 3 -39 119 -71 -86]]

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

Optimal tuning (POTE): ~1664/1225 = 529.003¢

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

2.5.7.11.13.17.19 subgroup

Subgroup: 2.5.7.11.13.17.19

Sval Mapping: [⟨1 1 20 -49 35 42], ⟨0 3 -39 119 -71 -86]]

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

Optimal tuning (POTE): ~19/14 = 529.003¢

Tricesimoprimal miracloid

See also: Eliora's approach to tricesimoprimal miracloid

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. Since it is conceived as the temperament in the above specific subgroup, it makes no sense to name it for smaller subgroups. In terms of microtempering, a circle of 52 generators is essentially a barely noticeable well temperament for 52edo.

Subgroup: 2.5.7.11.19.29.31

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

Sval Mapping: [⟨1 419 48 177 157 624 625], ⟨0 -461 -50 -192 -169 -685 -686]]

Optimal tuning (CTE): ~58/31 = 1084.628

Optimal ET sequence: 52, 1737, 1789, ...

Huntington

See also: Chromatic pairs § Huntington

Huntington may be described as the 10 & 27 temperament in the 2.5.7.13 subgroup.

Subgroup: 2.5.7.13

Comma list: 640/637, 10985/10976

Sval mapping: [⟨1 5 4 4], ⟨0 -9 -4 -1]]

Gencom mapping: [⟨1 0 5 4 0 4], ⟨0 0 -9 -4 0 -1]]

gencom: [2 16/13; 640/637 10985/10976]

Optimal tuning (POTE): ~2 = 1\1, ~16/13 = 357.002

Optimal ET sequence: 7, 10, 17, 27, 37, 84, 121, 279cd, 400cd

RMS error: 0.3452 cents

Silver

See also: Chromatic pairs § Silver

Silver can be described as the 10 & 27 temperament in the 2.5.7.13.17 subgroup.

Subgroup: 2.5.7.13.17

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

Sval mapping: [⟨1 5 4 4 2], ⟨0 -9 -4 -1 7]]

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

gencom: [2 13/8; 170/169 640/637 5525/5488]

Optimal tuning (POTE): ~2 = 1\1, ~13/8 = 842.711

Optimal ET sequence: 7, 10, 17, 27, 37, 47, 84, 131, 178e, 309cde, 487bcdee

RMS error: 0.5886 cents

Pakkanen (rank 3)

Subgroup: 2.5.7.11

Comma list: 625/616

Sval mapping: [⟨1 0 0 -3], ⟨0 1 0 4], ⟨0 0 1 -1]]

mapping generators: ~2, ~5, ~11

Optimal tuning (TE): ~2/1 = 1200.6544, ~5/4 = 380.3004, ~11/8 = 551.9653

Optimal ET sequence: 6, 16, 22, 28, 29, 35, 41, 57, 63, 98c

Higher 2.5 temperaments

Temperaments discussed elsewhere include:

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

Mapping: [⟨1 1 3], ⟨0 3 1]]

Optimal tuning (CTE): ~2 = 1/1, ~11/8 = 529.846

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

Wizz

See also: Chromatic pairs § Wizz

Wizz, the 6 & 16 temperament in the 2.5.11 subgroup, is related to wizard.

Subgroup: 2.5.11

Comma list: 15625/15488

Sval mapping: [⟨2 0 -7], ⟨0 1 3]]

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

gencom: [125/88 5/4; 15625/15488]

Optimal tuning (POTE): ~125/88 = 1\2, ~5/4 = 383.768

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

RMS error: 0.3997

Insect

Subgroup: 2.5.11

Comma list: 33275/32768

Sval mapping: [⟨1 0 5], ⟨0 3 -2]]

Mapping generators, ~2, ~55/32

Optimal tuning (CTE): ~2 = 1\1, ~55/32 = 928.032

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

Sephiroth

See also: Chromatic pairs § Sephiroth

Sephiroth is the no-7 restriction of muggles.

Subgroup: 2.5.11.13.17

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

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

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

gencom: [2 5/4; 65/64 170/169 221/220]

Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 372.236

Optimal ET sequence: 10, 13, 16, 29

RMS error: 1.774 cents

Trader

Subgroup: 2.5.13

Comma list: 26/25

Sval mapping: [⟨1 2 3], ⟨0 1 2]]

Mapping generators, ~2, ~5/4

Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 407.079

Optimal ET sequence: 3, 20c, 23c, 26c

Superquintal

Subgroup: 2.5.13

Comma list: 64000000/62748517

Sval mapping: [⟨1 5 6], ⟨0 -7 -6]]

Mapping generators, ~2, ~13/10

Optimal tuning (CTE): ~2 = 1\1, ~13/10 = 459.281

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

Vengeance

Main article: Vengeance

Another lower-error replica of mavila, with the fifth being ~25/17 instead of ~3/2.

Subgroup: 2.5.17

Comma list: 78608/78125

Sval mapping: [⟨1 1 1], ⟨0 3 7]]

Optimal tuning (CTE): ~2 = 1\1, ~34/25 = 529.095

Optimal ET sequence: 7g, 9, 25, 34, 93, 127, 288, 415

No-threes-or-fives subgroup temperaments

Temperaments discussed elsewhere include

Score

See also: Chromatic pairs § Score

Subgroup: 2.7.11.13

Comma list: 343/338, 847/832

Sval mapping: [⟨1 1 3 1], ⟨0 4 1 6]]

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

gencom: [2 11/8; 343/338 847/832]

Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 540.099

Optimal ET sequence: 5, 7, 9, 11, 20

RMS error: 1.282 cents

Bossier

See also: Chromatic pairs § Bossier

Bossier can be described as the 3 & 17 in the 2.7.11.13 subgroup.

Subgroup: 2.7.11.13

Comma list: 1573/1568, 15488/15379

Sval mapping: [⟨1 0 1 3], ⟨0 8 7 2]]

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

gencom: [2 14/11; 1573/1568 15488/15379]

Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 421.309

Optimal ET sequence: 17, 20, 37, 57, 94, 225, 319cd, 413bcd

RMS error: 0.4043 cents

Voltage

Voltage is the 3 & 7 temperament in the 2.7.13 subgroup.

Subgroup: 2.7.13

Comma list: 28672/28561

Sval mapping: [⟨1 4 4], ⟨0 -4 -1]]

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

gencom: [2, 16/13; 28672/28561]

Optimal tuning:

  • POTE: ~2 = 1\1, ~16/13 = 357.677
  • POTT: ~2 = 1\1, ~16/13 = 357.794 (1200 - 300 log2(7))

Optimal ET sequence: 3, 7, 10, 27, 37, 47, 57, 104

RMS error: 0.1423 cents

Ultrakleismic

Subgroup: 2.7.17

Comma list: 4913/4802

Sval mapping: [⟨1 2 3], ⟨0 3 4]]

Mapping generators, ~2, ~17/14

Optimal tuning (CTE): ~2 = 1\1, ~17/14 = 324.446

Optimal ET sequence: 4, 7, 11, 26, 37

Counterultrakleismic

Subgroup: 2.7.17

Comma list: 2024782584832/2015993900449

Sval mapping: [⟨1 0 1], ⟨0 10 11]]

Mapping generators, ~2, ~17/14

Optimal tuning (CTE): ~2 = 1\1, ~17/14 = 336.858

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

Shipwreck

Subgroup: 2.7.53

Comma list: 1048576/1042139

Gencom: [2 64/53; 1048576/1042139]

Mapping: [⟨1 0 6], ⟨0 3 -1]]]

POTE generator: ~64/53 = 323.034

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

Lovecraft

See also: Chromatic pairs § Lovecraft

Lovecraft, the 4 & 13 temperament in the 2.11.13 subgroup, 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

Sval mapping: [⟨1 3 3], ⟨0 2 3]]

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

gencom: [2 13/11; 1352/1331]

Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 279.318

Optimal ET sequence: 13, 30, 43, 73, 116

RMS error: 0.8449 cents

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

Sval mapping: [⟨4 0 1], ⟨0 1 1]]

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

gencom: [13/11 11/8; 29282/28561]

Optimal tuning (POTE): ~13/11 = 1\4, ~11/8 = 546.660

Optimal ET sequence: 4, 16, 20, 24, 44, 68, 112c, 180bc

RMS error: 0.8685 cents

Bluebirds

See also: Chromatic pairs § Bluebirds

Subgroup: 2.11.13

Comma list: 265837/262144

Sval mapping: [⟨1 0 6], ⟨0 3 -2]]

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

gencom: [2 143/128; 265837/262144]

Optimal tuning (POTE): ~2 = 1\1, ~143/128 = 182.368

Optimal ET sequence: 6, 7, 13, 33, 46, 79, 125c, 204bc, 329bc

RMS error: 0.4444 cents

Yamablu

Yamablu, with a generator of ~17/13, 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). 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

Sval mapping: [⟨1 5 1 1 0], ⟨0 -4 7 8 11]]

Optimal tuning (POTE): ~17/13 = 462.9606

Optimal ET sequence: 13, 44, 57, 70

RMS error: 0.4898 cents

Mavericks

Subgroup: 2.13.19

Comma list: 47525504/47045881

Mapping: [⟨1 1 2], ⟨0 6 5]]

Optimal tuning (CTE): ~2 = 1\1, ~26/19 = 539.886

Optimal ET sequence: 7fh, 9, 11, 20

Yer (rank 3)

Subgroup: 2.11.13.17.19

Comma list: 209/208, 2057/2048

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

Optimal tuning (TE): ~2/1 = 1200.4457, ~11/8 = 548.4934, ~16/13 = 358.638

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

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