Rastmic clan

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This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

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

Subgroup-val mapping: [⟨1 1 2], ⟨0 2 5]]

Gencom mapping: [⟨1 1 0 0 2], ⟨0 2 0 0 5]]

mapping generators: ~2, ~11/9

Optimal tunings:

  • WE: ~2 = 1200.0635 ¢, ~11/9 = 350.5439 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 350.5486 ¢

Optimal ET sequence: 7, 17, 24, 65, 89, 202, 291, 380

Badness (Sintel): 0.0700

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, 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 tuned much sharper to optimize the plausibility of prime 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 four times as sharp as however sharp of just the (3/2)1/2 neutral third is, while tuning it much flatter means increasing the error of prime 13, which in 58edo is already almost 8 ¢ off and in 99ef-edo it is only slightly worse.

The CWE tuning given below takes account of how optimizational resource is best used. As such, the optimum is not raised much in the introduction of prime 13, as each unit increment in the generator implies four and five times more error in the harmonics 9 and 11, respectively, damaging them more than necessary. A sharper pure-octave tuning would be given by CTOP (~351.7142 ¢), which is still flat of 58edo.

Subgroup: 2.3.11.13

Comma list: 144/143, 243/242

Subgroup-val mapping: [⟨1 1 2 4], ⟨0 2 5 -1]]

Gencom mapping: [⟨1 1 0 0 2 4], ⟨0 2 0 0 5 -1]]

Optimal tunings:

  • WE: ~2 = 1198.8553 ¢, ~11/9 = 351.1523 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 351.2700 ¢

Optimal ET sequence: 7, 10, 17, 41, 58, 99ef, 239eefff, 338eeeffff

Badness (Sintel): 0.133

Suhajira

Suhajira can be thought of as "superpyth with quartertones", adding neutral intervals onto the existing pythagorean and septimal ones. Due to the sharper generator, it is now much more natural to add prime 13, with the neutral intervals generally closer to simple intervals of 13 than simple intervals of 11.

Subgroup: 2.3.7.11

Comma list: 64/63, 243/242

Subgroup-val mapping: [⟨1 1 4 2], ⟨0 2 -4 5]]

Gencom mapping: [⟨1 1 0 4 2], ⟨0 2 0 -4 5]]

Optimal tunings:

  • WE: ~2 = 1196.3496 ¢, ~11/9 = 352.8809 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 353.7832 ¢

Optimal ET sequence: 7, 10, 17, 44e, 61de, 78ddee, 139bdddeeee

Badness (Sintel): 0.602

Scales: suhajira7, suhajira10, suhajira17

2.3.7.11.13 subgroup

Subgroup: 2.3.7.11.13

Comma list: 64/63, 78/77, 144/143

Subgroup-val 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]]

Optimal tunings:

  • WE: ~2 = 1196.8114 ¢, ~11/9 = 352.8351 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 353.6791 ¢

Optimal ET sequence: 7, 10, 17, 44e, 61de, 78ddee

Badness (Sintel): 0.425

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 quartertones (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

Subgroup-val mapping: [⟨1 1 0 2], ⟨0 2 8 5]]

Gencom mapping: [⟨1 1 0 0 2], ⟨0 2 8 0 5]]

Optimal tunings:

  • WE: ~2 = 1201.8548 ¢, ~11/9 = 348.6318 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 348.3793 ¢

Optimal ET sequence: 7, 17c, 24, 31, 69e, 100e, 131bee

Badness (Sintel): 0.401

Scales: mohaha7, mohaha10

Mohoho

Subgroup: 2.3.5.11.13

Comma list: 66/65, 81/80, 121/120

Subgroup-val mapping: [⟨1 1 0 2 4], ⟨0 2 8 5 -1]]

Gencom mapping: [⟨1 1 0 0 2 4], ⟨0 2 8 0 5 -1]]

Optimal tunings:

  • WE: ~2 = 1199.8846 ¢, ~11/9 = 348.8820 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 348.9040 ¢

Optimal ET sequence: 7, 17c, 24, 31, 55

Badness (Sintel): 0.545

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]]

Optimal tunings:

  • WE: ~2 = 1201.7423 ¢, ~11/9 = 348.6879 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 348.2867 ¢

Optimal ET sequence: 7d, 24d, 31, 100de, 131bdee, 162bdee

Badness (Sintel): 0.844

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:

  • WE: ~2 = 1200.2590 ¢, ~11/9 = 348.5648 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 348.5009 ¢

Optimal ET sequence: 7d, 24d, 31

Badness (Sintel): 1.16

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]]

Optimal tunings:

  • WE: ~2 = 1203.8812 ¢, ~11/9 = 348.0444 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 346.9194 ¢

Optimal ET sequence: 7, 31dd, 38d, 45e, 83bcddee

Badness (Sintel): 1.94

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:

  • WE: ~2 = 1203.9456 ¢, ~11/9 = 348.0510 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 346.8878 ¢

Optimal ET sequence: 7, 31ddf, 38df, 45ef, 83bcddeeff, 128bccddeeefff

Badness (Sintel): 1.42

Neutrominant

Main article: 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]]

Optimal tunings:

  • WE: ~2 = 1195.9529 ¢, ~11/9 = 349.7505 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 350.7031 ¢

Optimal ET sequence: 7, 17c, 24d, 41cd

Badness (Sintel): 1.33

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:

  • WE: ~2 = 1196.3143 ¢, ~11/9 = 349.7387 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 350.6573 ¢

Optimal ET sequence: 7, 17c, 24d, 41cd

Badness (Sintel): 1.12


Mohajira

Main article: Mohajira

Mohajira can be viewed as derived from mohaha which maps the interval half a chroma flat of the minor seventh to ~7/4 so that 7/4 is mapped to a semidiminished seventh (C–Bdb), although mohajira really makes more sense as an 11-limit temperament. It tempers out 6144/6125, the porwell comma. It can be described as 24 & 31; its ploidacot is dicot. 31edo makes for an excellent mohajira tuning, with generator 9\31. Note that while 24 + 31 = 55edo doesn't apear in the optimal ET sequence, it is a patent val tuning and recommendable if you prefer a light meantone tempering.

Subgroup: 2.3.5.7

Comma list: 81/80, 6144/6125

Mapping: [⟨1 1 0 6], ⟨0 2 8 -11]]

mapping generators: ~2, ~128/105

Optimal tunings:

  • WE: ~2 = 1200.8160 ¢, ~128/105 = 348.6518 ¢
error map: ⟨+0.816 -3.835 +2.901 +0.900]
  • CWE: ~2 = 1200.0000 ¢, ~128/105 = 348.4194 ¢
error map: ⟨0.000 -5.116 +1.041 -1.439]

Minimax tuning:

projection map: [[1 0 0 0⟩, [1 0 1/4 0⟩, [0 0 1 0⟩, [6 0 -11/8 0⟩]
Unchanged-interval (eigenmonzo) basis: 2.5

Tuning ranges:

  • 7- and 9-odd-limit diamond monotone: ~128/105 = [347.368, 350.000] (11\38 to 7\24)
  • 7-odd-limit diamond tradeoff: ~128/105 = [347.393, 350.978]
  • 9-odd-limit diamond tradeoff: ~128/105 = [345.601, 350.978]

Algebraic generator: Mohabis, real root of 3x3 - 3x2 - 1, 348.6067 cents. Corresponding recurrence converges quickly.

Optimal ET sequence: 7, 24, 31

Badness (Sintel): 1.41

Scales: mohaha7, mohaha10

11-limit

Subgroup: 2.3.5.7.11

Comma list: 81/80, 121/120, 176/175

Mapping: [⟨1 1 0 6 2], ⟨0 2 8 -11 5]]

Optimal tunings:

  • WE: ~2 = 1201.1562 ¢, ~11/9 = 348.8124 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 348.4910 ¢

Minimax tuning:

  • 11-odd-limit: ~11/9 = [0 0 1/8⟩
projection map: [[1 0 0 0 0⟩, [1 0 1/4 0 0⟩, [0 0 1 0 0⟩, [6 0 -11/8 0 0⟩, [2 0 5/8 0 0⟩]
unchanged-interval (eigenmonzo) basis: 2.5

Tuning ranges:

  • 11-odd-limit diamond monotone: ~11/9 = [348.387, 350.000] (9\31 to 7\24)
  • 11-odd-limit diamond tradeoff: ~11/9 = [344.999, 350.978]

Optimal ET sequence: 7, 24, 31

Badness (Sintel): 0.862

Scales: mohaha7, mohaha10

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 81/80, 105/104, 121/120

Mapping: [⟨1 1 0 6 2 4], ⟨0 2 8 -11 5 -1]]

Optimal tunings:

  • WE: ~2 = 1200.4256 ¢, ~11/9 = 348.6819 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 348.5622 ¢

Optimal ET sequence: 7, 24, 31

Badness (Sintel): 0.966

Scales: mohaha7, mohaha10

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 66/65, 81/80, 105/104, 121/120, 154/153

Mapping: [⟨1 1 0 6 2 4 7], ⟨0 2 8 -11 5 -1 -10]]

Optimal tunings:

  • WE: ~2 = 1200.0382 ¢, ~11/9 = 348.7471 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 348.7360 ¢

Optimal ET sequence: 7, 24, 31

Badness (Sintel): 1.05

Scales: mohaha7, mohaha10

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 66/65, 77/76, 81/80, 96/95, 105/104, 153/152

Mapping: [⟨1 1 0 6 2 4 7 6], ⟨0 2 8 -11 5 -1 -10 -6]]

Optimal tunings:

  • WE: ~2 = 1199.7469 ¢, ~11/9 = 348.7367 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 348.8117 ¢

Optimal ET sequence: 7, 24, 31, 55

Badness (Sintel): 1.05

Scales: mohaha7, mohaha10

Mohamaq

Mohamaq is a lower-accuracy alternative to mohajira that favors tunings sharp of 24edo. It may be described as 17c & 24; its ploidacot is dicot, the same as mohajira. 7/4 is mapped to the semi-augmented sixth (C–A⁠ ⁠), extending hemif.

Subgroup: 2.3.5.7

Comma list: 81/80, 392/375

Mapping: [⟨1 1 0 -1], ⟨0 2 8 13]]

mapping generators: ~2, ~25/21

Optimal tunings:

  • WE: ~2 = 1199.0661 ¢, ~25/21 = 350.3127 ¢
error map: ⟨-0.934 -2.264 +16.188 -13.827]
  • CWE: ~2 = 1200.0000 ¢, ~25/21 = 350.4856 ¢
error map: ⟨0.000 -0.984 +17.571 -12.513]

Optimal ET sequence: 7d, 17c, 24

Badness (Sintel): 1.97

Scales: mohaha7, mohaha10

11-limit

Subgroup: 2.3.5.7.11

Comma list: 56/55, 77/75, 243/242

Mapping: [⟨1 1 0 -1 2], ⟨0 2 8 13 5]]

Optimal tunings:

  • WE: ~2 = 1199.1924 ¢, ~11/9 = 350.3286 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 350.4821 ¢

Optimal ET sequence: 7d, 17c, 24

Badness (Sintel): 1.20

Scales: mohaha7, mohaha10

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 66/65, 77/75, 243/242

Mapping: [⟨1 1 0 -1 2 4], ⟨0 2 8 13 5 -1]]

Optimal tunings:

  • WE: ~2 = 1198.5986 ¢, ~11/9 = 350.3353 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 350.6459 ¢

Optimal ET sequence: 7d, 17c, 24, 41c

Badness (Sintel): 1.19

Scales: mohaha7, mohaha10

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

Subgroup-val mapping: [⟨2 0 4 -1], ⟨0 2 1 5]]

Gencom mapping: [⟨2 0 0 4 -1], ⟨0 2 0 1 5]]

mapping generators: ~77/54, ~7/4

Optimal tunings:

  • WE: ~2 = 601.5167 ¢, ~7/4 = 951.1267 ¢
  • CWE: ~2 = 600.0000 ¢, ~7/4 = 949.6883 ¢

Optimal ET sequence: 10, 14, 24, 62dd

Badness (Sintel): 0.776

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