Rastmic clan: Difference between revisions
- CTE & POTE tunings |
Another pass on neutrality, explaining what actually happens with CTE/CWE and giving an alternative that bypasses their mechanics |
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=== Namo === | === 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 much sharper to optimize plausibility of | 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)<sup>1/2</sup> neutral third is, while tuning it much flatter means increasing the error of prime 13, which in 58edo is already almost 8{{cent}} off and in [[99edo|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{{c}}), which is still flat of 58edo. | |||
[[Subgroup]]: 2.3.11.13 | [[Subgroup]]: 2.3.11.13 | ||
Latest revision as of 16:07, 24 August 2025
- 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 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 ET sequence: 7, 10, 17, 41, 58, 99ef, 239eefff, 338eeeffff
Badness (Sintel): 0.133
Suhajira
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 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 ET sequence: 7, 17c, 24, 31, 69e, 100e, 131bee
Badness (Sintel): 0.401
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:
Optimal ET sequence: 7, 17c, 24, 31, 55
Badness (Sintel): 0.401
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
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
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 ET sequence: 10, 14, 24, 62dd
Badness (Sintel): 0.776