Rastmic rank three clan
The rastmic rank-3 clan of temperaments tempers out the rastma, 243/242. Both no-5 rastmic and no-7 rastmic can be the head of this clan. These temperaments divide the fifth in half and use it as an 11/9 neutral third.
Temperaments discussed elsewhere include:
- Jove (+441/440 or 540/539) → Breed family
- Hagrid (+9801/9800) → Cataharry family
- Rabic (+131769/131072) → Alphaxenic rank-3 clan
Spectacle
Spectacle, named by Gene Ward Smith in 2010[1], can be described as the 31 & 34 & 41 temperament. It tempers out 225/224, making it a sort of marvel infested with neutral thirds. It is therefore generated by octaves, major thirds, and neutral thirds. 3/2 is equated with two a stack of two 11/9's as a corollary of 243/242 being tempered out, 7/4 is equated with a stack of four 11/9's and two 5/4's, 11/8 is equated with a stack of five 11/9's, 13/8 is equated with a stack of two 18/11's and four 5/4's, and 17/16 is equated with three 18/11's and three 5/4's. Every harmonic is reached with help of other intervals at most with three 5/4's.
Subgroup: 2.3.5.7.11
Comma list: 225/224, 243/242
Mapping: [⟨1 1 0 -3 2], ⟨0 2 0 4 5], ⟨0 0 1 2 0]]
- mapping generators: ~2, ~11/9, ~5
Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 350.0570, ~5/4 = 383.9323
- [[1 0 0 0 0⟩, [1/5 0 0 0 2/5⟩, [2/5 -2 1 0 4/5⟩, [-19/5 -4 2 0 12/5⟩, [0 0 0 0 1⟩]
- eigenmonzo (unchanged-interval) basis: 2.9/5.11
Optimal ET sequence: 31, 41, 72, 247c, 281, 353c, 425bc, 497bc
Badness: 0.499 × 10-3
Projection pairs: 3 242/81 7 366025/52488 11 644204/59049 to 2.5.11/9
Scales: spectacle31
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 225/224, 243/242, 351/350
Mapping: [⟨1 1 0 -3 2 -5], ⟨0 2 0 4 5 -2], ⟨0 0 1 2 0 4]]
Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 349.9247, ~5/4 = 384.3505
Optimal ET sequence: 31, 72, 103, 175f *
* optimal patent val: 240
Badness: 1.009 × 10-3
Cuckoo
Subgroup: 2.3.5.7.11
Comma list: 126/125, 243/242
Mapping: [⟨1 1 0 -3 2], ⟨0 2 0 -4 5], ⟨0 0 1 3 0]]
- mapping generators: ~2, ~11/9, ~5
Optimal ET sequence: 24d, 27e, 31, 58, 89, 154, 185
Badness: 0.933 × 10-3
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 126/125, 196/195, 243/242
Mapping: [⟨1 1 0 -3 2 -5], ⟨0 2 0 -4 5 -10], ⟨0 0 1 3 0 5]]
Optimal ET sequence: 27e, 31, 58, 96d, 154
Mirwomo
The mirwomo temperament tunes ~36/35 to exactly half of an apotome ~2187/2048, and tunes ~128/105 to exactly half of a perfect fifth ~3/2.
7-limit
Subgroup: 2.3.5.7
Comma list: 33075/32768
Mapping: [⟨1 1 0 6], ⟨0 2 0 -3], ⟨0 0 1 -1]]
- mapping generators: ~2, ~128/105, ~5
Optimal tuning (POTE): ~2 = 1\1, ~128/105 = 350.1375, ~5/4 = 383.8665
Optimal ET sequence: 17, 21, 24, 31, 41, 72, 281d
Badness: 0.770 × 10-3
11-limit
Subgroup: 2.3.5.7.11
Comma list: 243/242, 385/384
Mapping: [⟨1 1 0 6 2], ⟨0 2 0 -3 5], ⟨0 0 1 -1 0]]
Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 349.9554, ~5/4 = 384.1048
Optimal ET sequence: 17, 24, 31, 41, 72, 312bd, 384bcd, 456bcde, 528bcde, 631bcde
Badness: 0.641 × 10-3
Mirage
Subgroup: 2.3.5.7.11.13
Comma list: 225/224, 243/242, 385/384
Mapping: [⟨1 1 3 3 2 0], ⟨0 6 -7 -2 15 0], ⟨0 0 0 0 0 1]]
- mapping generators: ~2, ~15/14, ~13
Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 116.6327, ~13/8 = 837.7040
Optimal ET sequence: 10, 31, 41, 62, 72, 103, 175f, 216c, 288cdf, 391bcdef
Badness: 0.738 × 10-3
Mandos
Subgroup: 2.3.5.7.11
Comma list: 176/175, 243/242
Mapping: [⟨1 1 0 6 2], ⟨0 2 0 5 5], ⟨0 0 1 -2 0]]
- mapping generators: ~2, ~11/9, ~5
Optimal ET sequence: 7, 24, 31, 58, 89
Badness: 0.751 × 10-3
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 144/143, 176/175, 243/242
Mapping: [⟨1 1 0 6 2 4], ⟨0 2 0 5 5 -1], ⟨0 0 1 -2 0 0]]
Optimal ET sequence: 7, 24, 31, 58
Badness: 0.923 × 10-3
Parahemif
7-limit
Subgroup: 2.3.5.7
Comma list: 1605632/1594323
Mapping: [⟨1 1 0 -1], ⟨0 2 0 13], ⟨0 0 1 0]]
- mapping generators: ~2, ~896/729, ~5
Optimal tuning (POTE): ~2 = 1\1, ~896/729 = 351.4846, ~5/4 = 386.9396
Optimal ET sequence: 17c, 24, 34d, 41, 58, 99, 239, 338
Badness: 0.607 × 10-3
11-limit
Subgroup: 2.3.5.7.11
Comma list: 243/242, 896/891
Mapping: [⟨1 1 0 -1 2], ⟨0 2 0 13 5], ⟨0 0 1 0 0]]
Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 351.5347, ~5/4 = 388.0217
Optimal ET sequence: 17c, 24, 34d, 41, 58, 99e *
Badness: 1.34547 × 10-3
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 144/143, 243/242, 364/363
Mapping: [⟨1 1 0 -1 2 4], ⟨0 2 0 13 5 -1], ⟨0 0 1 0 0 0]]
Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 351.6908, ~5/4 = 389.1892
Optimal ET sequence: 17c, 24, 34d, 41, 58, 99ef, 157eff, 290cdeeefff
Badness: 1.19366 × 10-3
Urania
Subgroup: 2.3.5.7.11
Comma list: 81/80, 121/120
Mapping: [⟨1 1 0 0 2], ⟨0 2 8 0 5], ⟨0 0 0 1 0]]
Mapping to lattice: [⟨0 2 8 0 5], ⟨0 0 0 -1 0]]
Lattice basis:
- 11/9 length = 0.2536, 8/7 length = 2.807
- Angle (11/9, 8/7) = 90 degrees
Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 348.0938, ~7/4 = 963.6042
Optimal ET sequence: 7, 14c, 17c, 24, 31, 100de, 131bde, 162bde
Badness: 0.842 × 10-3
Complexity spectrum: 11/9, 4/3, 12/11, 11/10, 10/9, 9/8, 11/8, 6/5, 5/4, 8/7, 7/6, 9/7, 14/11, 7/5
Scales: urania24