Garischismic clan: Difference between revisions

The '''garischismic clan''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[garischisma]] ({{monzo|legend=1| 25 -14 0 -1 }}, [[ratio]]: 33554432/33480783). The head of this clan is gary, which is generated by a perfect fifth. Two apotomes i.e. 14 fifths octave reduced make a [[8/7|septimal major second (8/7)]]. Equivalently stated, the [[7/4|harmonic seventh (7/4)]] is found at the double-diminished octave (C–Cbb).  

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.

Newt adds [[2401/2400]], slicing the fifth in two. Sextile adds [[250047/250000]] with a 1/3-octave period. 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. 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]] → [[Schismatic family #Garibaldi|Schismatic family]] (+225/224)

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

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

[[Subgroup]]: 2.3.7

[[Comma list]]: 33554432/33480783

{{Mapping|legend=2| 1 0 25 | 0 1 -14 }}

: sval mapping generators: ~2, ~3

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 702.2079

[[Badness]]: 0.0135

=== 2.3.7.11 subgroup ===

Subgroup: 2.3.7.11

Comma list: 19712/19683, 41503/41472

Sval mapping: {{mapping| 1 0 25 -33 | 0 1 -14 23 }}

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.2292

{{Optimal ET sequence|legend=1| 12e, 41, 94, 135, 716, 851, 986, 1121, 1256 }}

Badness: 0.00731

{{Main| Cotoneum }}

The cotoneum temperament tempers out 10976/10935 ([[hemimage comma]]), and 823543/819200 ([[quince comma]]) in addition to the garischisma. This temperament can be described as 41 & 217, and is supported by [[176edo|176-]], [[217edo|217-]], and [[258edo]]. 5/4 is found at the septuple diminished octave (C-Cbbbbbbb) or equivalently at the perfect fourth minus four Pyth. commas. It can be extended to the 11-, 13-, 17-, and 19-limit by adding 441/440, 364/363, 595/594, and 343/342 to the comma list in this order.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 10976/10935, 823543/819200

{{Mapping|legend=1| 1 0 80 25 | 0 1 -49 -14 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 702.317

{{Optimal ET sequence|legend=1| 41, 135c, 176, 217, 258, 475 }}

[[Badness]]: 0.105632

=== 11-limit ===

Subgroup: 2.3.5.7.11

Comma list: 441/440, 10976/10935, 16384/16335

Mapping: {{mapping| 1 0 80 25 -33 | 0 1 -49 -14 23 }}

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.303

Badness: 0.050966

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 441/440, 3584/3575, 10976/10935

Mapping: {{mapping| 1 0 80 25 -33 -93 | 0 1 -49 -14 23 61 }}

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Badness: 0.0378