Rastmic clan
The rastmic clan of temperaments tempers out the rastma, 243/242 = [-1 5 0 0 -2⟩.
Neutral
Neutral is the 2.3.11-subgroup temperament with a generator of a neutral third which can be taken to represent 11/9~27/22, two of which make up a perfect fifth of 3/2. It can be thought of as the 2.3.11 version of either mohajira or neutrominant, as well as suhajira and ringo. Among other things, it is the temperament optimizing the neutral tetrad.
Subgroup: 2.3.11
Comma list: 243/242
Sval mapping: [⟨1 1 2], ⟨0 2 5]]
- mapping generators: ~2, ~11/9
Gencom mapping: [⟨1 1 0 0 2], ⟨0 2 0 0 5]]
- gencom: [2 11/9; 243/242]
Optimal ET sequence: 7, 10, 17, 24, 41, 65, 89, 202, 291, 380
RMS error: 0.3021 cents
Scales: neutral7, neutral10, neutral17
Namo
Namo adds 144/143 to the comma list and finds ~16/13 at the same neutral third. With 11/9~16/13, an equivalence which Godtone considers harmonically challenging, it requires a slightly flat ~27/22 as the tuning of the neutral third. 58edo is the largest patent val tuning for it in the optimal ET sequence, with a tuning between that of 17edo and 41edo, so that ~11 and ~13 are practically equally sharp, given that 29edo forms a consistent circle of 13/11's with a closing error of 31.2%. It might be recommended as a tuning for this reason, as having the neutral third much sharper to optimize plausibility of ~16/13 implies that the 11 is extremely sharp because 11/9 must be tuned sharp so that 11 must be sharper than 9, which is thus four times as sharp as however sharp of the (3/2)1/2 neutral third is, while tuning it much flatter means that you increase the error of 16/13, which in 58edo is already as almost 8 ¢ off and in 99ef it is only slightly worse. For these reasons, Godtone is not fond of the recommendations by the various optimal tunings to tune flat of 58edo, although it is clear that in an optimal tuning nothing much sharper than 58edo should be used, as making 11 more off than 13 would imply damaging 3 and 11/9 more than necessary. Curiously, POTE recommends a sharper tuning than both CTE and CWE here, but still flat of 58edo.
Subgroup: 2.3.11.13
Comma list: 144/143, 243/242
Sval mapping: [⟨1 1 2 4], ⟨0 2 5 -1]]
Gencom mapping: [⟨1 1 0 0 2 4], ⟨0 2 0 0 5 -1]]
- gencom: [2 11/9; 144/143 243/242]
Optimal ET sequence: 7, 10, 17, 24, 41, 58, 99ef, 239eefff, 338eeeffff
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 2 0 -4 5]]
- gencom: [2 11/9; 64/63 243/242]
Optimal ET sequence: 7, 10, 17, 44e, 61de
Scales: suhajira7, suhajira10, suhajira17
2.3.7.11.13 subgroup
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 2 0 -4 5 -1]]
- gencom: [2 11/9; 64/63 78/77 144/143]
Optimal tunings:
- CTE: ~2 = 1\1, ~11/9 = 353.219
- POTE: ~2 = 1\1, ~11/9 = 353.775
Optimal ET sequence: 7, 10, 17, 44e, 61de
Scales: suhajira7, suhajira10, suhajira17
Mohaha
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, while tempering out 176/175 gives mohajira (shown at Meantone family).
2.3.5.11 subgroup
The S-expression-based comma list of this temperament is {S6/S8 = S9, S11}.
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 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 tunings:
- CTE: ~2 = 1\1, ~11/9 = 348.8794
- POTE: ~2 = 1\1, ~11/9 = 348.9155
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 tunings:
- CTE: ~2 = 1\1, ~11/9 = 348.5324
- POTE: ~2 = 1\1, ~11/9 = 348.182
Optimal ET sequence: 7d, 24d, 31, 100de, 131bdee, 162bdee
Badness: 0.025516
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 66/65, 81/80, 121/120, 126/125
Mapping: [⟨1 1 0 -3 2 4], ⟨0 2 8 20 5 -1]]
Optimal tunings:
- CTE: ~2 = 1\1, ~11/9 = 348.5444
- POTE: ~2 = 1\1, ~11/9 = 348.490
Optimal ET sequence: 7d, 24d, 31
Badness: 0.028071
Ptolemy
- "Ptolemy" redirects here. For the Ancient Greek polymath, see Wikipedia: 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 tunings:
- CTE: ~2 = 1\1, ~11/9 = 346.905
- POTE: ~2 = 1\1, ~11/9 = 346.922
Optimal ET sequence: 7, 31dd, 38d, 45e, 83bcddee
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 65/64, 81/80, 105/104, 121/120
Mapping: [⟨1 1 0 8 2 6], ⟨0 2 8 -18 5 -8]]
Optimal tunings:
- CTE: ~2 = 1\1, ~11/9 = 346.754
- POTE: ~2 = 1\1, ~11/9 = 346.910
Optimal ET sequence: 7, 31ddf, 38df, 45ef, 83bcddeeff
Badness: 0.034316
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
Wedgie: ⟨⟨ 2 8 -4 5 8 -12 1 -32 -16 28 ]]
Optimal tunings:
- CTE: ~2 = 1\1, ~11/9 = 349.865
- 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 tunings:
- CTE: ~2 = 1\1, ~11/9 = 349.904
- POTE: ~2 = 1\1, ~11/9 = 350.816
Optimal ET sequence: 7, 17c, 24d, 41cd
Badness: 0.027214
Semisema
In addition to dividing the perfect fifth into two equal parts of 11/9~27/22, semisema, being an extension of semaphore, also divides the perfect fourth into two equal parts of 7/6~8/7.
Subgroup: 2.3.7.11
Comma list: 49/48, 243/242
Sval mapping: [⟨2 2 5 4], ⟨0 2 1 5]]
- sval mapping generators: ~77/54, ~11/9
Gencom mapping: [⟨2 2 0 5 4], ⟨0 2 0 1 5]]
- gencom: [77/54 11/9; 49/48 243/242]
Optimal tunings:
- CTE: ~77/54 = 1\2, ~11/9 = 351.181
- POTE: ~77/54 = 1\2, ~11/9 = 348.728