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

Llywelyn aka shoe

See also: Chromatic pairs #Shoe
See also: Llywelyn clan #Llywelyn aka shoe

Subgroup: 2.5.7

Comma list: 4194304/4117715

Sval mapping: [⟨1 1 3], ⟨0 7 -1]]

Mapping generators: 2, ~8/7

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

Gencom: [2 8/7; 4194304/4117715]

Optimal tuning (POTE): ~8/7 = 226.910

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

2.5.7.11 subgroup

Subgroup: 2.5.7.11

Comma list: 176/175, 1310720/1294139

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

Gencom: [2 8/7; 176/175 1310720/1294139]

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

Optimal tuning (POTE): ~8/7 = 227.114

Optimal ET sequence: 16, 21, 37

2.5.7.11.13 subgroup

Subgroup: 2.5.7.11.13

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

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

Gencom: [2 8/7; 176/175 640/637, 1304576/1294139]

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

Optimal tuning (POTE): ~8/7 = 227.108

Optimal ET sequence: 16, 21, 37

2.5.7.11.13.17 subgroup

Subgroup: 2.5.7.11.13.17

Comma list: 176/175, 221/200, 640/637, 833/832

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

Gencom: [2 8/7; 176/175 221/200, 640/637, 833/832]

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

Optimal tuning (POTE): ~8/7 = 227.242

Optimal ET sequence: 16, 21, 37

Didacus

See also: Hemimean clan #Didacus

Related temperaments: roulette, hemithirds

Subgroup: 2.5.7

Comma list: 3136/3125

Sval mapping: [⟨1 2 2], ⟨0 2 5]]

Gencom: [2 28/25; 3136/3125]

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

Optimal tuning (POTE): ~28/25 = 93.772

Optimal ET sequence: 6, 19, 25, 31, 37, 99, 130, 161, 353

RMS error: 0.2138 cents

Rainy

Three generators make an 8/7; five generators make a 5/4. This is the no-threes version of tertiaseptal. Rainy is notable theoretically as it equates (2/1)/(5/4)3 = 128/125 (the 2.3.5 diesis) with (2/1)/(8/7)5 (the cloudy comma), which has a similar size and role to the 2.3.5 diesis so might be considered the 2.3.7 diesis, in that if you temper it, it imparts significant error on the tuning of 8/7 (so that it is now ~8.8 ¢ sharp), similar to how tempering 128/125 imparts significant error on the tuning of 5/4 (so that it is ~13.7 ¢ sharp). Combined with the small size of these "dieses", it implies that the locations of 5/4 and 8/7 are in some sense awkward to conceptualize due to the small but not insignificant size of the remnant with the octave, so it is might be favorable to equate 128/125 with the cloudy comma to create a general-purpose diesis. (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

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

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

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¢

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

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, ...

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 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, ...

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

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

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

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

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

No-threes-or-fives subgroup temperaments

Temperaments discussed elsewhere include

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

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

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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