# 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 1 0 -4 5]]

- gencom: [2 3/2; 64/63 99/98]

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 1 0 -4 5 -1]]

- gencom: [2 3/2; 64/63 78/77 99/98]

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

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