Garischismic clan: Difference between revisions

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== 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 the double-diminished octave (C–C𝄫), or the minor seventh minus a generic comma step ~~(sometimes called ''gary comma'')~~ 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.

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]].

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==== Subgroup extensions ====

Gary can be naturally extended into the no-5's 11-limit with ~~incredible~~ accuracy by equating (64/63)<sup>2</sup> with 33/32 at the cost of doubling the complexity, acting as a temperament of [[~~Olympic~~ clan#Olympian|olympian]] which accurately represents the entire 2.3.7.11 into the chain of fifths. The major third additionally is extremely close to [[19/15]], and is a natural interpretation in extensions with 5 and 19.

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, acting as a temperament of [[olympic clan #Olympian|olympian]] which accurately represents the entire 2.3.7.11 into the chain of fifths. The major third additionally is extremely close to [[19/15]], and is a natural interpretation in extensions with 5 and 19.

=== 2.3.7.11 subgroup ===

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Optimal tunings:

* WE: ~2 = 1199.9631{{c}}, ~3/2 = 702.2077{{c}}~~Subgroup [[~~error map]]: {{val|-0.037, 0.216, -0.139, -0.173}}

* WE: ~2 = 1199.9631{{c}}, ~3/2 = 702.2077{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2290{{c}}~~Subgroup [[~~error map]]: {{val|-0.000, 0.274, -0.032, -0.051}}

: error map: {{val| -0.037 0.216 -0.139 -0.173 }}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2290{{c}}

: error map: {{val| -0.000 0.274 -0.032 -0.051 }}

{{Optimal ET sequence|legend=0| 12e, 41, 94, 135, 716, 851, 986, 1121, 1256 }}

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 == 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 the double-diminished octave (C–C𝄫), or the minor seventh minus a generic comma step (sometimes called ''gary comma'') 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.
+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]].
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 ==== Subgroup extensions ====
-Gary can be naturally extended into the no-5's 11-limit with incredible accuracy by equating (64/63)<sup>2</sup> with 33/32 at the cost of doubling the complexity, acting as a temperament of [[Olympic clan#Olympian|olympian]] which accurately represents the entire 2.3.7.11 into the chain of fifths. The major third additionally is extremely close to [[19/15]], and is a natural interpretation in extensions with 5 and 19.
+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, acting as a temperament of [[olympic clan #Olympian|olympian]] which accurately represents the entire 2.3.7.11 into the chain of fifths. The major third additionally is extremely close to [[19/15]], and is a natural interpretation in extensions with 5 and 19.
 === 2.3.7.11 subgroup ===
@@ Line 51: / Line 51: @@
 Optimal tunings:
-* WE: ~2 = 1199.9631{{c}}, ~3/2 = 702.2077{{c}}Subgroup [[error map]]: {{val|-0.037, 0.216, -0.139, -0.173}}
+* WE: ~2 = 1199.9631{{c}}, ~3/2 = 702.2077{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2290{{c}}Subgroup [[error map]]: {{val|-0.000, 0.274, -0.032, -0.051}}
+: error map: {{val| -0.037 0.216 -0.139 -0.173 }}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2290{{c}}
+: error map: {{val| -0.000 0.274 -0.032 -0.051 }}
 {{Optimal ET sequence|legend=0| 12e, 41, 94, 135, 716, 851, 986, 1121, 1256 }}