Ragismic family
The ragismic family of temperaments are rank-3 microtemperaments which temper out 4375/4374.
Temperaments discussed elsewhere include:
Considered below are beyla, ulmo, thor, heimdall, bragi, and vidar.
Ragismic
Subgroup: 2.3.5.7
Mapping: [⟨1 0 0 1], ⟨0 1 0 7], ⟨0 0 1 -4]]
- mapping generators: ~2, ~3, ~5
Mapping to lattice: [⟨0 -1 -1 -3], ⟨0 -1 -2 1]]
Lattice basis:
- 10/9 length = 0.789, 6/5 length = 0.921
- Angle (10/9, 6/5) = 105.299°
- 7-odd-limit eigenmonzo (unchanged-interval) basis: 2.5.7
- 9-odd-limit eigenmonzo (unchanged-interval) basis: 2.3.7/5
Optimal ET sequence: 19, 27, 45, 46, 53, 72, 99, 171, 441, 612, 935, 1106, 1277, 1547, 1718, 4983, 6701, 8419, 17279c
Projection pair: ~7 = ~4374/625
Scales: Ragismic19, Ragismic chromatic
Beyla
Subgroup: 2.3.5.7.11
Comma list: 385/384, 4375/4374
Mapping: [⟨1 0 0 1 6], ⟨0 1 0 7 -6], ⟨0 0 1 -4 3]]
Optimal ET sequence: 7, 19, 26, 45, 46, 53, 72, 118, 190, 315e, 361e, 433de, 623cdee
Badness: 0.721 × 10-3
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 169/168, 325/324, 385/384
Mapping: [⟨1 0 0 1 6 2], ⟨0 1 0 7 -6 4], ⟨0 0 1 -4 3 -2]]
Optimal ET sequence: 7, 19, 26, 46, 53, 72, 118f, 125f, 171ef, 190ff, 197ef, 243eff
Badness: 0.840 × 10-3
Ulmo
Subgroup: 2.3.5.7.11
Comma list: 176/175, 2200/2187
Mapping: [⟨1 0 0 1 -3], ⟨0 1 0 7 7], ⟨0 0 1 -4 -2]]
Optimal ET sequence: 7, 39, 46, 53, 80, 126, 179, 259cd, 385bcd
Badness: 1.23 × 10-3
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 169/168, 176/175, 325/324
Mapping: [⟨1 0 0 1 -3 2], ⟨0 1 0 7 7 4], ⟨0 0 1 -4 -2 -2]]
Optimal ET sequence: 7, 39, 46, 53, 80, 133d, 179, 312cdf
Badness: 0.932 × 10-3
Thor
Subgroup: 2.3.5.7.11
Comma list: 3025/3024, 4375/4374
Mapping: [⟨2 0 0 2 5], ⟨0 1 0 7 5], ⟨0 0 1 -4 -3]]
- mapping generators: ~99/70, ~3, ~5
Optimal ET sequence: 26, 34d, 46, 72, 118, 152, 224, 270, 342, 494, 612, 836, 1106, 1448, 2554, 4002e, 5720e, 7168cee
Badness: 0.0888 × 10-3
Projection pairs: 2 = ~9801/4900, ~7 = ~21434787/3062500, ~11 = ~2310905821257/210087500000 to 3.5.14/11
Thunor
Subgroup: 2.3.5.7.11.13
Comma list: 1716/1715, 2080/2079, 3025/3024
Mapping: [⟨2 0 0 2 5 -3], ⟨0 1 0 7 5 15], ⟨0 0 1 -4 -3 -8]]
Mapping to lattice: [⟨0 -1 -2 -3 -2 -7], ⟨0 -1 -2 1 1 1]]
Lattice basis:
- 10/9 length = 0.4234, 6/5 length = 0.8426
- Angle (10/9, 6/5) = 84.126°
Minimax tuning:
- 13- and 15-odd-limit eigenmonzo (unchanged-interval) basis: 2.9/7.13
Optimal ET sequence: 26, 46, 72, 152f, 198, 224, 270, 494, 764, 1258
Badness: 0.341 × 10-3
Rym
Subgroup: 2.3.5.7.11.13
Comma list: 3025/3024, 4096/4095, 4375/4374
Mapping: [⟨2 0 0 2 5 22], ⟨0 1 0 7 5 -9], ⟨0 0 1 -4 -3 3]]
Optimal ET sequence: 46, 72f, 106, 118, 152, 224, 270, 494, 764, 1106, 1376, 1870
Badness: 0.549 × 10-3
Donar
Subgroup: 2.3.5.7.11.13
Comma list: 3025/3024, 4225/4224, 4375/4374
Mapping: [⟨2 0 1 -2 2 7], ⟨0 1 0 7 5 3], ⟨0 0 2 -8 -6 -5]]
- mapping generators: ~99/70, ~3, ~220/117
Mapping to lattice: [⟨0 2 2 6 4 1], ⟨0 -1 -2 1 1 2]]
Lattice basis:
- 44/39 length = 0.3480, 6/5 length = 0.7828
- Angle (44/39, 6/5) = 85.3977°
Minimax tuning:
- 13-odd-limit eigenmonzo (unchanged-interval) basis: 2.7.11/5
- 15-odd-limit eigenmonzo (unchanged-interval) basis: 2.7.15/11
Optimal ET sequence: 34d, 46, 80, 144, 178, 190, 224, 494, 684, 764, 954, 1178, 1448, 6970ceeff, 8418ceeff
Badness: 0.322 × 10-3
Scales: Donar46
Heimdall
Subgroup: 2.3.5.7.11
Comma list: 4375/4374, 117649/117612
Mapping: [⟨1 0 0 1 2], ⟨0 2 0 14 37], ⟨0 0 1 -4 -12]]
- mapping generators: ~2, ~343/198, ~5
Optimal ET sequence: 72, 198, 270, 342, 612, 954, 1205, 1475, 1547, 1817, 2159, 4660, 4930, 6819, 7089, 9590d, 11749d
Bragi
Subgroup: 2.3.5.7.11
Comma list: 4375/4374, 2097152/2096325
Mapping: [⟨1 0 2 -7 8], ⟨0 1 0 7 -3], ⟨0 0 3 -12 2]]
- mapping generators: ~2, ~3, ~320/297
Optimal ET sequence: 46, 111d, 159d, 178, 205, 224, 270, 494, 764, 1053, 1277, 1323, 1547, 1817, 4128, 10073bdee, 14201bdeee
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 2080/2079, 4375/4374, 10985/10976
Mapping: [⟨1 0 2 -7 8 -6], ⟨0 1 0 7 -3 7], ⟨0 0 3 -12 2 -13]]
Optimal ET sequence: 46, 111df, 159df, 178, 205, 224, 270, 494, 764, 1258, 1771, 2041f, 2265, 2535f
Vidar
Subgroup: 2.3.5.7.11
Comma list: 4375/4374, 100663296/100656875
Mapping: [⟨1 0 0 1 5], ⟨0 1 4 -9 -3], ⟨0 0 -5 20 4]]
- mapping generators: ~2, ~3, ~96/55
Optimal ET sequence: 46, 91, 133d, 137d, 178, 224, 270, 494, 764, 1395, 1619, 1665, 1889, 2159, 2653, 4812, 7735, 9894
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 4225/4224, 4375/4374, 6656/6655
Mapping: [⟨1 0 0 1 5 6], ⟨0 1 4 -9 -3 -5], ⟨0 0 -5 20 4 7]]
Optimal ET sequence: 46, 87d, 91, 133d, 137d, 178, 224, 270, 494, 764, 1125, 1395, 1619, 1665, 1889, 4048, 4542, 6431f