# Rastmic rank three clan

(Redirected from Mirwomo)

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:

## 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

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 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

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]]

## 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

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

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

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

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]]

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

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

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

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

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