Quartismic family
The quartismic family is built up from temperaments that temper out the quartisma- the unnoticeable comma with the ratio 117440512/117406179, and a monzo of [24 -6 0 1 -5⟩. Among the members of this family are Altierran, Meanquarter, Coin, Kleirtismic, and Doublefour.
Quartismic
The 11-limit parent comma for the quartismic family is the the quartisma with a ratio of 117440512/117406179 and a monzo of [24 -6 0 1 -5⟩. As the quartisma is an unnoticeable comma, this rank-4 temperament is a microtemperament.
Comma: 117440512/117406179
POTE generators: ~3/2 = 701.9826, ~5/4 = 386.3427, ~33/32 = 53.3748
Map: [<1 0 0 1 5|, <0 1 0 1 -1|, <0 0 1 0 0|, <0 0 0 5 1|]
Wedgie: <<<<5 1 0 -6 -24||||
EDOs: 21, 22, 24, 25, 43, 45, 46, 67, 68, 89, 90, 91, 92, 110, 111, 113, 114, 132, 134, 135, 138, 156, 157, 159, 178, 179, 180, 181, 202, 224, 270, 359, 494, 629, 653, 742, 877, 1012, 1236, 1506, 2159, 2248, 2383, 2518, 3125, 7419
Badness: 0.274 × 10-6
13-limit
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17-limit
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Quartismatic
There are some temperaments in the quartismic family in which the quartisma is tempered out, but without any sort of five-limit representation. This particular temperament is the parent temperament of all such no-fives children, and is referred to as Saquinlu-azo temperament in color notation.
Comma: 117440512/117406179
POTE generators: ~3/2 = 701.9826, ~33/32 = 53.3748
Map: [<1 0 1 5|, <0 1 1 -1|, <0 0 5 1|]
EDOs: 21, 22, 24, 43, 46, 89, 135, 270, 359, 494, 629, 653, 742, 877, 1012, 1236, 1506, 2159, 2248, 2383, 2518, 7419
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The following scale tree has been found:
The following rank-2 quartismic temperament MOS scales have been found:
13-limit
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17-limit
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Additional Options for 5-limit Representation
Among quartismic temperaments, there are a number of other options for 5-limit representation depending on which among the various 5-limit commas is tempered out. Adding the schisma to the list of tempered-out commas results in some form of Altierran temperament. Adding the meantone comma results in some form of Meanquarter temperament. Adding the magic comma results in some form of Coin temperament. Adding the kleisma results in some form of Kleirtismic temperament- the "kleir-" in "Kleirtismic" is pronounced the same as "Clair". Adding the tetracot comma results in some form of Doublefour temperament. Other possible extensions are listed here.
Shrutar extension
This is the 22&46 temperament. See Shrutar.
Escapade extension
This is the 22&43 temperament. See Escapade.
Altierran
The Altierran clan is the temperament clan consisting of those temperaments in which both the schisma and the quartisma are tempered out.
Commas: 32805/32768, 117440512/117406179
POTE generators: ~3/2 = 701.7299, ~33/32 = 53.3889
Map: [<1 0 15 1 5|, <0 1 -8 1 -1|, <0 0 0 5 1|]
Badness:
13-limit
Commas: 10985/10976, 32805/32768, 117440512/117406179
POTE generators: ~11/10 = 166.0628, ~33/32 = 53.4151
Map: [<1 2 -1 3 3 5|, <0 -3 24 -3 3 -11|, <0 0 0 5 1 5|]
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17-limit
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Meanquarter
The Meanquarter clan is the temperament clan consisting of those temperaments in which both the meantone comma and the quartisma are tempered out. Meanquarter can easily be extended to a form of godzilla, though not all possible tunings for Meanquarter lend themselves to this sort of thing.
Commas: 81/80, 117440512/117406179
POTE generators: ~3/2 = 697.3325, ~33/32 = 54.1064
Map: [<1 0 -4 1 5|, <0 1 4 1 -1|, <0 0 5 1|]
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Coin
The Coin clan is the temperament clan consisting of those temperaments in which both the magic comma and the quartisma are tempered out.
Commas: 3125/3072, 117440512/117406179
POTE generators: ~5/4 = 380.3623, ~9/7 = 433.3120
Map: [<1 0 2 1 5|, <0 5 1 0 -6|, <0 0 0 5 1|]
Badness:
Kleirtismic
The Kleirtismic clan is the temperament clan consisting of those temperaments in which both the kleisma and the quartisma are tempered out.
Commas: 15625/15552, 117440512/117406179
POTE generators: ~6/5 = 317.0291, ~68/55 370.2940
Map: [<1 0 1 1 5|, <0 6 5 1 -7|, <0 0 0 5 1|]
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Doublefour
The Doublefour clan is the temperament clan consisting of those temperaments in which both the tetracot comma and the quartisma are tempered out.
Commas: 20000/19683, 117440512/117406179
POTE generators: ~425/384 = 175.9566, ~33/32 = 52.9708
Map: [<1 1 1 2 4|, <0 4 9 4 -4|, <0 0 0 5 1|]
Badness: