Garischismic clan: Difference between revisions
Reorganized things a bit and acknowledged gary as basically 2.3.7.11 |
Partial reversal (last edit removed where 7/4 is mapped on the generator chain). Review on the subgroup extension |
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== Gary == | == Gary == | ||
Gary, the head of this clan, may be viewed as the [[2.3.7 subgroup|2.3.7-subgroup]] counterpart of [[schismic]]. It is generated by a [[3/2|perfect fifth]], and 7/4 is found at | Gary, the head of this clan, may be viewed as the [[2.3.7 subgroup|2.3.7-subgroup]] counterpart of [[schismic]]. It is generated by a [[3/2|perfect fifth]], and 7/4 is found at the double-diminished octave (C–C𝄫), or the minor seventh minus a generic comma step which stands in for both the Pythagorean comma and the septimal comma. Gary can therefore use [[chain-of-fifths notation]] with an additional set of accidentals such as arrows to represent the comma step. | ||
Just as there is the 1/8-schisma tuning for schismic, there is the 1/14-schisma tuning for gary, which tunes 7/4 pure by sharpening the perfect fifth by about 0.272 cents. Similarly, the 1/15-schisma tuning tunes [[7/6]] pure, and the 2/29-schisma tuning splits their difference, tuning the septimal diesis of [[49/48]] pure. [[135edo]] is close to the 1/14-schisma tuning, whereas [[634edo]] gives a tuning practically identical to 1/15-schisma. Other notable tunings not appearing in the optimal ET sequence include [[311edo]] and [[323edo]]. | Just as there is the 1/8-schisma tuning for schismic, there is the 1/14-schisma tuning for gary, which tunes 7/4 pure by sharpening the perfect fifth by about 0.272 cents. Similarly, the 1/15-schisma tuning tunes [[7/6]] pure, and the 2/29-schisma tuning splits their difference, tuning the septimal diesis of [[49/48]] pure. [[135edo]] is close to the 1/14-schisma tuning, whereas [[634edo]] gives a tuning practically identical to 1/15-schisma. Other notable tunings not appearing in the optimal ET sequence include [[311edo]] and [[323edo]]. | ||
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[[Badness]] (Sintel): 0.463 | [[Badness]] (Sintel): 0.463 | ||
=== Overview to extensions === | === Overview to extensions === | ||
==== Full 11-limit extensions ==== | |||
The second comma of the comma list determines which full 7-limit or 11-limit family member we are looking at. Garibaldi adds the [[schisma]], or equivalently [[225/224]] and finds 5/4 at the diminished fourth. Cotoneum adds [[10976/10935]] and finds 5/4 at the septuple-diminished octave. These are generated by the fifth as is gary. | The second comma of the comma list determines which full 7-limit or 11-limit family member we are looking at. Garibaldi adds the [[schisma]], or equivalently [[225/224]] and finds 5/4 at the diminished fourth. Cotoneum adds [[10976/10935]] and finds 5/4 at the septuple-diminished octave. These are generated by the fifth as is gary. | ||
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Considered below are cotoneum, gariwizmic, satin, sextile, and world calendar. | Considered below are cotoneum, gariwizmic, satin, sextile, and world calendar. | ||
==== Subgroup extensions ==== | |||
Gary can be naturally extended into the no-5's 11-limit with good accuracy by equating (64/63)<sup>2</sup> with 33/32, at the cost of doubling the complexity. | |||
=== 2.3.7.11 subgroup === | |||
Subgroup: 2.3.7.11 | |||
Comma list: 19712/19683, 41503/41472 | |||
Subgroup-val mapping: {{mapping| 1 0 25 -33 | 0 1 -14 23 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1199.9631{{c}}, ~3/2 = 702.2077{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2290{{c}} | |||
{{Optimal ET sequence|legend=0| 12e, 41, 94, 135, 716, 851, 986, 1121, 1256 }} | |||
Badness (Sintel): 0.276 | |||
== Cotoneum == | == Cotoneum == | ||