Garischismic clan: Difference between revisions

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The '''garischismic clan''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[garischisma]] ({{monzo|legend=1| 25 -14 0 -1 }}, [[ratio]]: 33554432/33480783), the amount by which the [[Pythagorean comma]] falls short of the [[septimal comma]].

The '''garischismic clan''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[garischisma]] ({{monzo|legend=1| 25 -14 0 -1 }}, [[ratio]]: 33554432/33480783), the amount by which the [[Pythagorean comma]] falls short of the [[septimal comma]], thus equating the two.

== 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 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 ~~generic~~ 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 a minor seventh minus a Pyth-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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[[Badness]] (Sintel): 0.463

=== ~~Overview to extensions~~ ===

=== 2.3.7.11 subgroup ===

~~The second comma of the comma list determines which full 7~~-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.~~

Gary can be very naturally extended into no-5s 11-limit with incredible accuracy by equating (64/63)^2 = 33/32. Equivalently, {{Nowrap|1=(64/63)^3 = 22/21}}. 135edo provides a basically perfect tuning.

~~Gariwizmic adds the [[wizma]] with a 1~~/2~~-octave period. Newt adds [[2401/2400]], slicing the fifth in two. Alphatrident adds [[6144~~/~~6125]], slicing the twelfth in three~~. ~~Satin adds [[2100875/2097152]]~~, ~~slicing the fourth in three. Vulture adds [[4375~~/~~4374]], slicing the twelfth in four. Sextile adds [[250047~~/~~250000]] with a 1/6-octave period~~. ~~World calendar adds [[390625/388962]] with~~ a ~~1/4-octave period as well as a bisect generator. Quintagar adds [[3136/3125]], slicing the fourth in five. Paramity adds [[65625/65536]], slicing the eleventh in five~~.

~~Temperaments discussed elsewhere are:~~

* [[Garibaldi]] (+225/224) → [[Schismatic family #Garibaldi|Schismatic family]]

* ''[[Newt]]'' (+2401/2400) → [[Breedsmic temperaments #Newt|Breedsmic temperaments]]

* ''[[Alphatrident]]'' (+6144/6125) → [[Alphatricot family #Alphatrident|Alphatricot family]]

* ''[[Vulture]]'' (+4375/4374) → [[Vulture family #Vulture|Vulture family]]

* ''[[Quintagar]]'' (+3136/3125) → [[Quindromeda family #Quintagar|Quindromeda family]]

* ''[[Paramity]]'' (+65625/65536) → [[Amity family #Paramity|Amity family]]

* ''[[Garistearn]]'' (+118098/117649) → [[Stearnsmic clan #Garistearn|Stearnsmic clan]]

~~Considered below are cotoneum, gariwizmic, satin, sextile, and world calendar.~~

~~=== 2.3.7.11 subgroup ===~~

Subgroup: 2.3.7.11

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Badness (Sintel): 0.276

=== Overview to 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.

Gariwizmic adds the [[wizma]] with a 1/2-octave period. Newt adds [[2401/2400]], halving the fifth. Alphatrident adds [[6144/6125]], slicing the twelfth in three. Satin adds [[2100875/2097152]], slicing the fourth in three. Vulture adds [[4375/4374]], slicing the twelfth in four. Sextile adds [[250047/250000]] with a 1/6-octave period. World calendar adds [[390625/388962]] with a 1/4-octave period as well as a halved fifth. Quintagar adds [[3136/3125]], slicing the fourth in five. Paramity adds [[65625/65536]], slicing the eleventh in five.

Temperaments discussed elsewhere are: