Rastmic clan
The neutral clan of temperaments tempers out 243/242.
Neutral
Neutral can be thought of as the 2.3.11 version of either mohajira or maqamic, as well as suhajira and ringo. Among other things, it is the temperament optimizing the neutral tetrad.
Subgroup: 2.3.11
Comma: 243/242
Gencom: [2 11/9; 243/242]
Gencom mapping: [⟨1 1 0 0 2], ⟨0 2 0 0 5]]
Sval mapping: [⟨1 1 2], ⟨0 2 5]]
POL2 generator: ~11/9 = 350.525
Optimal ET sequence: 7, 10, 17, 24, 41, 65, 89, 202, 291, 380
RMS error: 0.3021 cents
- Scales
Namo
Subgroup: 2.3.11.13
Comma list: 144/143, 243/242
Gencom: [2 11/9; 144/143 243/242]
Gencom mapping: [⟨1 1 0 0 2 4], ⟨0 2 0 0 5 -1]]
Sval mapping: [⟨1 1 2 4], ⟨0 2 5 -1]]
POL2 generator: ~11/9 = 351.488
Optimal ET sequence: 7, 10, 17, 24, 41
RMS error: 0.7038 cents
Suhajira
Subgroup: 2.3.7.11
Comma list: 64/63, 243/242
Sval mapping: [⟨1 1 4 2], ⟨0 2 -4 5]]
- sval mapping generators: ~2, ~11/9
Gencom mapping: [⟨1 1 0 4 2], ⟨0 1 0 -4 5]]
- gencom: [2 3/2; 64/63 99/98]
Optimal tuning (POTE): ~11/9 = 353.958
Optimal ET sequence: 7, 10, 17, 44e, 61de
Scales: suhajira7, suhajira10, suhajira17
2.3.7.11.13
Subgroup: 2.3.7.11.13
Comma list: 64/63, 78/77, 144/143
Sval mapping: [⟨1 1 4 2 4], ⟨0 2 -4 5 -1]]
Gencom mapping: [⟨1 1 0 4 2 4], ⟨0 1 0 -4 5 -1]]
- gencom: [2 3/2; 64/63 78/77 99/98]
Optimal tuning (POTE): ~11/9 = 353.775
Optimal ET sequence: 7, 10, 17, 44e, 61de
Scales: suhajira7, suhajira10, suhajira17
Mohaha
Mohaha is the 2.3.5.11 subgroup temperament with a generator of a neutral third, two of which make up a fifth, and which can be taken to represent 11/9. Mohaha can be thought of, intuitively, as "meantone with quartertones"; as is the 3/2 generator subdivided in half, so is the ~25/24 chromatic semitone divided into two equal ~33/32 quarter tones (in the 2.3.5.11 subgroup). Within this paradigm, mohaha is the temperament that splits the 3/2 into two equal 11/9's, that splits the 6/5 into two equal 11/10~12/11's, and that maps four 3/2's to 5/1. It has a heptatonic mos with three larger steps and four smaller ones, going sLsLsLs. Taking septimal meantone mapping of 7 leads to #Migration, flattone mapping of 7 leads to #Ptolemy, and dominant mapping of 7 leads to #Neutrominant.
Subgroup: 2.3.5.11
Comma list: 81/80, 121/120
Sval mapping: [⟨1 1 0 2], ⟨0 2 8 5]]
- sval mapping generators: ~2, ~11/9
Gencom mapping: [⟨1 1 0 0 2], ⟨0 2 8 0 5]]
- gencom: [2 11/9; 81/80 121/120]
Optimal tuning (CTE): ~2 = 1\1, ~11/9 = 348.8296
Optimal ET sequence: 7, 17c, 24, 31, 69e, 100e, 131bee
Mohoho
Subgroup: 2.3.5.11.13
Comma list: 66/65, 81/80, 121/120
Sval mapping: [⟨1 1 0 2 4], ⟨0 2 8 5 -1]]
- sval mapping generators: ~2, ~11/9
Gencom mapping: [⟨1 1 0 0 2 4], ⟨0 2 8 0 5 -1]]
- gencom: [2 11/9; 66/65 81/80 121/120]
Optimal tuning (CTE): ~2 = 1\1, ~11/9 = 348.8794
Optimal ET sequence: 7, 17c, 24, 31, 55
Migration
Subgroup: 2.3.5.7.11
Comma list: 81/80, 121/120, 126/125
Mapping: [⟨1 1 0 -3 2], ⟨0 2 8 20 5]]
- mapping generators: ~2, ~11/9
Optimal tuning (CTE): ~2 = 1\1, ~11/9 = 348.5324
Optimal ET sequence: 7d, 24d, 31, 100de, 131bdee, 162bdee
Badness: 0.025516
Ptolemy
Subgroup: 2.3.5.7.11
Comma list: 81/80, 121/120, 525/512
Mapping: [⟨1 1 0 8 2], ⟨0 2 8 -18 5]]
- mapping generators: ~2, ~11/9
Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 346.922
Optimal ET sequence: 7, 31dd, 38d, 45e, 83bcddee
Neutrominant
The neutrominant temperament (formerly maqamic temperament) has a hemififth generator (~11/9) and tempers out 36/35 and 121/120. It makes the most sense if viewed as an adaptive temperament, whereby 7/4 and 9/5 simply share an equivalence class in the resulting scales, but don't need to share a particular tempered "middle-of-the-road" intonation.
Subgroup: 2.3.5.7.11
Comma list: 36/35, 64/63, 121/120
Mapping: [⟨1 1 0 4 2], ⟨0 2 8 -4 5]]
- mapping generators: ~2, ~11/9
Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 350.934
Optimal ET sequence: 7, 17c, 24d, 41cd
Badness: 0.040240
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 36/35, 64/63, 66/65, 121/120
Mapping: [⟨1 1 0 4 2 4], ⟨0 2 8 -4 5 -1]]
Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 350.816
Optimal ET sequence: 7, 17c, 24d, 41cd
Badness: 0.027214