Rastmic clan: Difference between revisions

The '''rastmic clan''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[rastma]], 243/242 ({{monzo| -1 5 0 0 -2 }}).

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]].

{{Mapping|legend=3| 1 1 2 | 0 2 5 }}

 

{{Mapping|legend=5| 1 1 0 0 2 | 0 2 0 0 5 }}

* [[WE]]: ~2 = 1200.0635{{c}}, ~11/9 = 350.5439{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~11/9 = 350.5486{{c}}

[[Badness]] (Sintel): 0.0700

Scales: [[neutral7]], [[neutral10]], [[neutral17]]

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{{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.

{{Mapping|legend=3| 1 1 2 4 | 0 2 5 -1 }}

{{Mapping|legend=5| 1 1 0 0 2 4 | 0 2 0 0 5 -1 }}

[[Optimal tuning]]s:  

* [[WE]]: ~2 = 1198.8553{{c}}, ~11/9 = 351.1523{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~11/9 = 351.2700{{c}}

{{Optimal ET sequence|legend=1| 7, 10, 17, 41, 58, 99ef, 239eefff, 338eeeffff }}

[[Badness]] (Sintel): 0.133

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

{{Mapping|legend=3| 1 1 4 2 | 0 2 -4 5 }}

{{Mapping|legend=5| 1 1 0 4 2 | 0 2 0 -4 5 }}

[[Optimal tuning]]s:  

* [[WE]]: ~2 = 1196.3496{{c}}, ~11/9 = 352.8809{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~11/9 = 353.7832{{c}}

{{Optimal ET sequence|legend=1| 7, 10, 17, 44e, 61de, 78ddee, 139bdddeeee }}

{{Mapping|legend=2| 1 1 4 2 4 | 0 2 -4 5 -1 }}

{{Mapping|legend=4| 1 1 0 4 2 4 | 0 2 0 -4 5 -1 }}

Optimal tunings:  

* WE: ~2 = 1196.8114{{c}}, ~11/9 = 352.8351{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 353.6791{{c}}

{{Optimal ET sequence|legend=0| 7, 10, 17, 44e, 61de, 78ddee }}

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 [[3L 4s|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 #Mohajira|Meantone family]]).

{{Mapping|legend=3| 1 1 0 2 | 0 2 8 5 }}

{{Mapping|legend=5| 1 1 0 0 2 | 0 2 8 0 5 }}

* [[WE]]: ~2 = 1201.8548{{c}}, ~11/9 = 348.6318{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~11/9 = 348.3793{{c}}

{{Mapping|legend=2| 1 1 0 2 4| 0 2 8 5 -1 }}

{{Mapping|legend=4| 1 1 0 0 2 4 | 0 2 8 0 5 -1 }}

Optimal tunings:  

* [[WE]]: ~2 = 1199.8846{{c}}, ~11/9 = 348.8820{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~11/9 = 348.9040{{c}}

{{Optimal ET sequence|legend=0| 7, 17c, 24, 31, 55 }}

{{Mapping|legend=0| 1 1 0 -3 2 | 0 2 8 20 5 }}

* WE: ~2 = 1201.7423{{c}}, ~11/9 = 348.6879{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 348.2867{{c}}

{{Optimal ET sequence|legend=0| 7d, 24d, 31, 100de, 131bdee, 162bdee }}

Badness (Sintel): 0.844

{{Mapping|legend=0| 1 1 0 -3 2 4 | 0 2 8 20 5 -1 }}

* WE: ~2 = 1200.2590{{c}}, ~11/9 = 348.5648{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 348.5009{{c}}

{{Optimal ET sequence|legend=0| 7d, 24d, 31 }}

Badness (Sintel): 1.16

{{Mapping|legend=0| 1 1 0 8 2 | 0 2 8 -18 5 }}

Optimal tunings:  

* WE: ~2 = 1203.8812{{c}}, ~11/9 = 348.0444{{c}}

{{Optimal ET sequence|legend=0| 7, 31dd, 38d, 45e, 83bcddee }}

Badness (Sintel): 1.94

{{Mapping|legend=0| 1 1 0 8 2 6 | 0 2 8 -18 5 -8 }}

Optimal tunings:

* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 346.8878{{c}}

{{Optimal ET sequence|legend=0| 7, 31ddf, 38df, 45ef, 83bcddeeff, 128bccddeeefff }}

Badness (Sintel): 1.42

=== Neutrominant ===

{{Mapping|legend=0| 1 1 0 4 2 | 0 2 8 -4 5 }}

Optimal tunings:  

* WE: ~2 = 1195.9529{{c}}, ~11/9 = 349.7505{{c}}

{{Optimal ET sequence|legend=0| 7, 17c, 24d, 41cd }}

Badness (Sintel): 1.33

==== 13-limit ====

{{Mapping|legend=0| 1 1 0 4 2 4 | 0 2 8 -4 5 -1 }}

Optimal tunings:

* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 350.6573{{c}}

{{Optimal ET sequence|legend=0| 7, 17c, 24d, 41cd }}

Badness (Sintel): 1.12

[[Category:Catalogs of rank-2 temperaments]]