Rastmic clan: Difference between revisions

[[Tp tuning #T2 tuning|RMS error]]: 0.3021 cents

Namo adds [[144/143]] to the comma list and finds ~[[16/13]] at the same neutral third. With {{nowrap|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 ~16/13 implies that the 11 is extremely sharp because 11/9 must be tuned sharp so that 11 must be sharper than 9, which is thus four times as sharp as however sharp of the (3/2)^1/2 neutral third is, while tuning it much flatter means that you increase the error of 16/13, which in 58edo is already as almost 8{{cent}} off and in [[99edo|99ef]] it is only slightly worse. For these reasons, Godtone is not fond of the recommendations by the various [[optimal tuning]]s to tune flat of 58edo, although it is clear that in an optimal tuning nothing much sharper than 58edo should be used, as making 11 more off than 13 would imply damaging 3 and 11/9 more than necessary. Curiously, [[POTE]] recommends a sharper tuning than both [[CTE]] and [[CWE]] here, but still flat of 58edo.

[[Optimal tuning]]s:

* [[CTE]]: 2 = 1200.000{{c}}, ~11/9 = 350.746{{c}}

* [[CWE]]: 2 = 1200.000{{c}}, ~11/9 = 351.270{{c}}

* [[POTE]]: 2 = 1200.000{{c}}, ~11/9 = 351.488{{c}}

{{Optimal ET sequence|legend=1| 7, 10, 17, 24, 41, 58, 99ef, 239eefff, 338eeeffff }}

[[Tp tuning #T2 tuning|RMS error]]: 0.7038 cents

* [[CTE]]: ~2 = 1200.000{{c}}, ~11/9 = 353.139{{c}}

* [[POTE]]: ~2 = 1200.000{{c}}, ~11/9 = 353.958{{c}}

* CTE: ~2 = 1200.000{{c}}, ~11/9 = 353.219{{c}}

* POTE: ~2 = 1200.000{{c}}, ~11/9 = 353.775{{c}}

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

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 [[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]]).

* [[CTE]]: ~2 = 1200.0000{{c}}, ~11/9 = 348.8296{{c}}

* [[POTE]]: ~2 = 1200.0000{{c}}, ~11/9 = 348.0938{{c}}

* CTE: ~2 = 1200.0000{{c}}, ~11/9 = 348.8794{{c}}

* POTE: ~2 = 1200.0000{{c}}, ~11/9 = 348.9155{{c}}

* CTE: ~2 = 1200.0000{{c}}, ~11/9 = 348.5324{{c}}

* POTE: ~2 = 1200.000{{c}}, ~11/9 = 348.182{{c}}

Badness: 0.025516

* CTE: ~2 = 1200.0000{{c}}, ~11/9 = 348.5444{{c}}

* POTE: ~2 = 1200.000{{c}}, ~11/9 = 348.490{{c}}

Badness: 0.028071

* CTE: ~2 = 1200.000{{c}}, ~11/9 = 346.905{{c}}

* POTE: ~2 = 1200.000{{c}}, ~11/9 = 346.922{{c}}

* CTE: ~2 = 1200.000{{c}}, ~11/9 = 346.754{{c}}

* POTE: ~2 = 1200.000{{c}}, ~11/9 = 346.910{{c}}

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

Badness: 0.034316

* CTE: ~2 = 1200.000{{c}}, ~11/9 = 349.865{{c}}

* POTE: ~2 = 1200.000{{c}}, ~11/9 = 350.934{{c}}

Badness: 0.040240

* CTE: ~2 = 1200.000{{c}}, ~11/9 = 349.904{{c}}

* POTE: ~2 = 1200.000{{c}}, ~11/9 = 350.816{{c}}

Badness: 0.027214

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 {{nowrap|7/6 ~ 8/7}}.

Subgroup: 2.3.7.11

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

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

: mapping generators: ~77/54, ~11/9

Optimal tunings:

* CTE: ~77/54 = 600.000{{c}}, ~11/9 = 351.181{{c}}

* POTE: ~77/54 = 600.000{{c}}, ~11/9 = 348.728{{c}}

{{Optimal ET sequence|legend=0| 10, 14, 24, 62dd }}