No-threes subgroup 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-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
- Jubilic → Jubilismic clan #Jubilic
2.5.7 temperaments
Didacus
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
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
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
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
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
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
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
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
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
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
- Orgone → Orgonia #Orgone
- Berylic → 4th-octave temperaments #Berylic
- 21st-octave temperaments #21-23-commatic
- 31st-octave temperaments #31-17/13-commatic
- 37th-octave temperaments #37-11-commatic (rank-1)
- etc.
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