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 are with ~2/1 period and ~5/4 generator. There is one-to-one correspondence between 2.5 subgroup and mapped intervals.
  • Didacus is with a ~28/25 generator, two of which give the ~5/4.
  • Ostara, movila and vengeance are with variety expressed generator, three of which give the ~5/2.
  • Insect is with a ~55/32 generator, three of which give the ~5/1.
  • Frostburn is with a ~28/25 generator, four of which give the ~8/5.

Others are more far.

Temperaments discussed elsewhere include

2.5.7 temperaments

Didacus

See also: Hemimean clan § Didacus

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

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 pure 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, ...

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

Llywelyn a.k.a. shoe

See also: Llywelynsmic clan § Llywelyn a.k.a. shoe

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, ...

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, ...

Mercy

See also: Quince clan § Mercy

Two generators make an 8/7; seven generators make an 8/5. Mercy can be thought of as a way to conceptualize the 2.5.7.13.17.19 subgroup of 31edo, and is the no-threes or elevens version of miracle.

Subgroup: 2.5.7

Comma list: 823543/819200

Sval mapping: [⟨1 3 3], ⟨0 -7 -2]]

Gencom: [2 2744/2560; 823543/819200]

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

Optimal tuning (POTE): ~343/320 = 116.291

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

2.5.7.13

Subgroup: 2.5.7.13

Comma list: 343/338, 640/637

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

Gencom: [2 14/13; 343/338 640/637]

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

Optimal tuning (POTE): ~14/13 = 116.094

Optimal ET sequence: 10, 21, 31

2.5.7.13.17

Subgroup: 2.5.7.13.17

Comma list: 170/169, 224/221, 640/637

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

Gencom: [2 14/13; 170/169 224/221 640/637]

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

Optimal tuning (POTE): ~14/13 = 115.769

Optimal ET sequence: 10, 21, 31

2.5.7.13.17.19

Subgroup: 2.5.7.13.17.19

Comma list: 170/169, 343/338, 640/637, 16384/16055

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

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

Gencom: [2 14/13; 170/169 343/338 640/637 16384/16055]

Optimal tuning (POTE): ~14/13 = 115.716

Optimal ET sequence: 10, 21, 31, 52f

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

Pure onzonic

The 2.5.11.13 subgroup primarily contains temperaments developed for 1789edo, since it tempers out the jacobin comma 6656/6655, for which 2.5.11.13 is the subgroup, and the year 1789 is hallmark for the French revolution.

Subgroup: 2.5.11.13

Comma list: 6656/6655, [-119 -46 15 47⟩

Mapping: [⟨1 74 3 74], ⟨0 -156 1 -153]]

Optimal tuning (POTE): ~11/8 = 551.370

Optimal ET sequence: 37, 1789

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

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

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

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

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

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

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

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

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