Rastmic clan

From Xenharmonic Wiki
(Redirected from Neutral clan)
Jump to navigation Jump to search

The rastmic clan of temperaments tempers out 243/242.

Neutral

Neutral 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 tuning (POTE): ~2 = 1\1, ~11/9 = 350.525

Optimal ET sequence7, 10, 17, 24, 41, 65, 89, 202, 291, 380

RMS error: 0.3021 cents

Scales: neutral7, neutral10, neutral17

Namo

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 tuning (POTE): ~2 = 1\1, ~11/9 = 351.488

Optimal ET sequence7, 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 sequence7, 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 tuning (POTE): ~11/9 = 353.775

Optimal ET sequence7, 10, 17, 44e, 61de

Scales: suhajira7, suhajira10, suhajira17

Mohaha

See also: No-sevens subgroup temperaments #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.

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 tunings:

  • CTE: ~2 = 1\1, ~11/9 = 348.8296
  • POTE: ~2 = 1\1, ~11/9 = 348.0938

Optimal ET sequence7, 17c, 24, 31, 69e, 100e, 131bee

Scales: mohaha7, mohaha10

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 sequence7, 17c, 24, 31, 55

Scales: mohaha7, mohaha10

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 sequence7d, 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 sequence7d, 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 tuning (POTE): ~2 = 1\1, ~11/9 = 346.922

Optimal ET sequence7, 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 tuning (POTE): ~2 = 1\1, ~11/9 = 346.910

Optimal ET sequence7, 31ddf, 38df, 45ef, 83bcddeeff

Badness: 0.034316

Neutrominant

Deutsch

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 sequence7, 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 sequence7, 17c, 24d, 41cd

Badness: 0.027214