Quartismic family: Difference between revisions

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Made the rest of the chart consistent, and added approximations for a few of the generators
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Removed "mapping generators" thing
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POTE generators: ~3/2 = 701.9826, ~5/4 = 386.3427, ~33/32 = 53.3748
POTE generators: ~3/2 = 701.9826, ~5/4 = 386.3427, ~33/32 = 53.3748
Mapping generator:


Map: [<1 0 0 1 5|, <0 1 0 1 -1|, <0 0 1 0 0|, <0 0 0 5 1|]
Map: [<1 0 0 1 5|, <0 1 0 1 -1|, <0 0 1 0 0|, <0 0 0 5 1|]
Line 24: Line 22:


POTE generators: ~3/2 = 701.9826, ~33/32 = 53.3748
POTE generators: ~3/2 = 701.9826, ~33/32 = 53.3748
Mapping generator:


Map: [<1 0 1 5|, <0 1 1 -1|, <0 0 5 1|]
Map: [<1 0 1 5|, <0 1 1 -1|, <0 0 5 1|]
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POTE generators: ~3/2 = 701.7299, ~33/32 = 53.3889  
POTE generators: ~3/2 = 701.7299, ~33/32 = 53.3889  
Mapping generators:


Map: [<1 0 15 1 5|, <0 1 -8 1 -1|, <0 0 0 5 1|]  
Map: [<1 0 15 1 5|, <0 1 -8 1 -1|, <0 0 0 5 1|]  
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POTE generators: ~11/10 = 166.0628, ~33/32 = 53.4151  
POTE generators: ~11/10 = 166.0628, ~33/32 = 53.4151  
Mapping generators:


Map: [<1 2 -1 3 3 5|, <0 -3 24 -3 3 -11|, <0 0 0 5 1 5|]  
Map: [<1 2 -1 3 3 5|, <0 -3 24 -3 3 -11|, <0 0 0 5 1 5|]  
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POTE generator:  
POTE generator:  
Mapping generator:


Map:  
Map:  
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POTE generators: ~3/2 = 697.3325, ~33/32 = 54.1064
POTE generators: ~3/2 = 697.3325, ~33/32 = 54.1064
Mapping generators:


Map: [<1 0 -4 1 5|, <0 1 4 1 -1|, <0 0 5 1|]
Map: [<1 0 -4 1 5|, <0 1 4 1 -1|, <0 0 5 1|]
Line 109: Line 97:


POTE generators: ~5/4 = 380.3623, ~9/7 = 433.3120
POTE generators: ~5/4 = 380.3623, ~9/7 = 433.3120
Mapping generators:


Map: [<1 0 2 1 5|, <0 5 1 0 -6|, <0 0 0 5 1|]
Map: [<1 0 2 1 5|, <0 5 1 0 -6|, <0 0 0 5 1|]
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POTE generators: ~6/5 = 317.0291, ~68/55 370.2940
POTE generators: ~6/5 = 317.0291, ~68/55 370.2940
Mapping generators:


Map: [<1 0 1 1 5|, <0 6 5 1 -7|, <0 0 0 5 1|]
Map: [<1 0 1 1 5|, <0 6 5 1 -7|, <0 0 0 5 1|]
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POTE generators: ~425/384 = 175.9566, ~33/32 = 52.9708
POTE generators: ~425/384 = 175.9566, ~33/32 = 52.9708
Mapping generators:


Map: [<1 1 1 2 4|, <0 4 9 4 -4|, <0 0 0 5 1|]
Map: [<1 1 1 2 4|, <0 4 9 4 -4|, <0 0 0 5 1|]

Revision as of 14:29, 14 September 2020

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

No-five Children

There are some temperaments in the quartismic family in which only the quartisma is tempered out, but without any regard to the five-limit.

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

Badness:

The following scale tree has been found:

The following rank-2 quartismic temperament MOS scales have been found:

5-limit Representation

Among quartismic temperaments, there are several options for 5-limit representation depending 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|]

EDOs: 135, 159, 224, 472

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|]

EDOs:

Badness:

17-limit

Commas:

POTE generator:

Map:

EDOs:

Badness:

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|]

EDOs: 24, 43, 45, 67

Badness:

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|]

EDOs: 22, 25, 139cdd

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|]

EDOs: 159, 178, 246

Badness:

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|]

EDOs: 48d, 68, 89c

Badness:

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