The Quartercache

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The Quartercache is a collection of temperaments of different ranks, including subgroup temperaments, that all 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 quartismatic, Altierran, meanquarter, coin, escapismic, dietismic, kleirtismic, doublefour and quarterframe.

Quartismic

Main article: Quartismic family

The 11-limit parent comma for the quartismic family and for the Quartercache is 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.

Subgroup: 2.3.5.7.11

Comma list: 117440512/117406179

Mapping: [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]]]]

POTE generators: ~3/2 = 701.9826, ~5/4 = 386.3427, ~33/32 = 53.3748

Optimal GPV sequence21, 22, 43, 46, 65d, 68, 89, 111, 159, 202, 224, 270, 494, 742, 764, 966, 1236, 1506, 2159, 2653, 3125, 3395, 7060, 7554, 10949e, 14614e, 15850ee, 22168bdee, 23404bcdee, 26799bcdeee, 34353bcdeeee

Badness: 0.274 × 10-6

Quartismatic

There are some temperaments in the Quartercache 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.

Subgroup: 2.3.7.11

Comma list: 117440512/117406179

Sval mapping: [1 0 1 5], 0 1 1 -1], 0 0 5 1]]

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

Optimal GPV sequence21, 22, 24, 43, 46, 89, 135, 359, 494, 629, 742, 877, 1012, 1506, 2248, 2383, 2518, 7419, 8431e, 10949e, 13467e

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

Altierran

In altierran, both the schisma and the quartisma are tempered out.

Subgroup: 2.3.5.7.11

Comma list: 32805/32768, 161280/161051

Mapping: [1 0 15 1 5], 0 1 -8 1 -1], 0 0 0 5 1]]

Wedgie⟨⟨⟨-102 24 -15 75 6 -8 40 1 -5 0]]]

POTE generators: ~3/2 = 701.7299, ~33/32 = 53.3889

Optimal GPV sequence24, 46c, 65d, 89, 135, 159, 224, 383, 472, 696, 1168, 1327, 1551, 2023e

Badness: 4.563 × 10-3

Tenierian

Subgroup: 2.3.5.7.11.13

Comma list: 10985/10976, 32805/32768, 161280/161051

Mapping: [1 2 -1 3 3 5], 0 -3 24 -3 3 -11], 0 0 0 5 1 5]]

POTE generators: ~11/10 = 166.0628, ~33/32 = 53.4151

Badness: 16.903 × 10-3

Meanquarter

In meanquarter, both the meantone comma and the quartisma are tempered out.

Subgroup: 2.3.5.7.11

Comma list: 81/80, 4128768/4026275

Mapping: [1 0 -4 1 5], 0 1 4 1 -1], 0 0 5 1]]

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

Optimal GPV sequence24, 43, 67, 110c

Badness: 15.125 × 10-3

Coin

In coin, both the magic comma and the quartisma are tempered out.

Subgroup: 2.3.5.7.11

Comma list: 3125/3072, 117440512/117406179

Mapping: [1 0 2 1 5], 0 5 1 0 -6], 0 0 0 5 1]]

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

Optimal GPV sequence19d, 22

Badness: 70.470 × 10-3

Escapismic

In escapisimic, both the escapade comma and the quartisma are tempered out, thus, it is essentially an escapade expansion.

Subgroup: 2.3.5.7.11

Comma list: 117440512/117406179, 4294967296/4271484375

Mapping: [1 2 2 3 3], 0 -9 7 -4 10], 0 0 0 5 1]]

POTE generators: ~33/32 = 55.3538

Optimal GPV sequence21, 22, 43, 65d, 521d, 543, 564, 586, 629c, 651

Badness: 64.233 × 10-3

Dietismic

In dietismic, both the diaschisma and the quartisma are tempered out. Dietismic can easily be further tempered to shrutar, and in fact, it is rather unusual to find a different tempering option.

Subgroup: 2.3.5.7.11

Comma list: 2048/2025, 117440512/117406179

POTE generators: ~3/2 = 704.5238, ~33/32 = 53.4408

Mapping: [2 0 11 2 10], 0 1 -2 1 -1], 0 0 0 5 1]]

Optimal GPV sequence22, 46, 68, 114

Badness: 23.250 × 10-3

Scales:

Rank 2 scale (52.6800, 2/1), 13|9

Rank 2 scale (53.3742, 2/1), 13|9

Kleirtismic

In kleirtismic, both the kleisma and the quartisma are tempered out. The "kleir-" in "kleirtismic" is pronounced the same as "Clair".

Subgroup: 2.3.5.7.11

Comma list: 15625/15552, 117440512/117406179

Mapping: [1 0 1 1 5], 0 6 5 1 -7], 0 0 0 5 1]]

POTE generators: ~6/5 = 317.0291, ~68/55 = 370.2940

Optimal GPV sequence68, 91, 159, 246, 337, 405

Badness: 26.882 × 10-3

Doublefour

In doublefour, both the tetracot comma and the quartisma are tempered out.

Subgroup: 2.3.5.7.11

Comma list: 20000/19683, 100656875/99090432

Mapping: [1 1 1 2 4], 0 4 9 4 -4], 0 0 0 5 1]]

POTE generators: ~425/384 = 175.9566, ~33/32 = 52.9708

Optimal GPV sequence48d, 68, 116d, 157c, 225

Badness: 81.083 × 10-3

Quarterframe

This is actually a microtemperament involving the lehmerisma and the frameshift comma. It is also a weak extension of the monzismic temperament.

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 26214400/26198073, 29296875/29218112

Mapping: [1 4 47 130 26], 0 -6 -111 -316 -56]]

POTE generator: ~160/121 = 483.0098

Optimal GPV sequence159, 559d, 718, 877

Badness: 0.154578

Ravine

Initially defined upon the 33/32 generator in 1619edo, producing a 832 & 1619 temperament.

Subgroup: 2.3.5.7.11

Comma list: 514714375/514434888, 117440512/117406179, 1220703125/1219784832

Mapping: [1 26 21 27 -21], 0 -549 -420 -544 550]]

Optimal tuning (CTE): ~33/32 = 53.366

Optimal GPV sequence832, 1619

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196625/196608, 200000/199927, 2912000/2910897, 3764768/3764475

Mapping: [1 26 21 27 -21 21], 0 -549 -420 -544 550 -389]]

Optimal tuning (CTE): ~33/32 = 53.366

Optimal GPV sequence: 832, 1619

Prequartismic

Named because 3125edo was the only one confirmed for tempering out the quartisma before it's discovery as significance of the difference between 5 33/32s and 7/6. Defined upon the 33/32 (or 64/33) generator in 3125edo, in terms of patent vals it can be described as 3125 & 4991 or 3125 & 1866.

Subgroup: 2.3.5.7.11

Comma list: [24, -6, 0, 1, -5, [-1, 4, 11, -11, 0, [-19 -25 14  13 -3

Mapping: [1 1389 890 1395 -1383], 0 -1452 -929 -1457 1451]]

Optimal tuning (CTE): ~64/33 = 1146.624

Vals: 1259e, 1866, 3125, 4384e, 4991, 6250e, 8116d, 7509ee, 9375e, 11241de

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1990656/1990625, 14236560/14235529, 117440512/117406179, 1181640625/1181599328

Mapping: [1 1389 890 1395 -1383 -282], 0 -1452 -929 -1457 1451 299]]

Optimal tuning (CTE): ~64/33 = 1146.624