The Quartercache

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This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

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 quartic, altierran, meanquarter, coin, escapismic, dietismic, kleirtismic, doublefour and quarterframe.

Quartismic

See Catalog of rank-4 temperaments #Quartismic (117440512/117406179).

Quartic

There are some temperaments that temper out the quartisma despite having limited accuracy in their approximations of five-limit intervals. This particular temperament is the parent temperament of all such no-fives children, and is referred to as Saquinlu-azo in color notation.

Subgroup: 2.3.7.11

Comma list: 117440512/117406179

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

mapping generators: ~2, ~3, ~33/32

Optimal tunings:

  • WE: ~2 = 1199.9875 ¢, ~3/2 = 701.9753 ¢, ~33/32 = 53.3743 ¢
error map: ⟨-0.012 +0.008 -0.004 +0.031]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.9802 ¢, ~33/32 = 53.3730 ¢
error map: ⟨0.000 +0.025 +0.019 +0.075]

Optimal ET sequence: 21, 22, 24, 43, 46, 89, 135, 359, 494, 629, 742, 877, 1012, 1506, 2248, 2383, 2518, 7419, 8431e, 10949e, 13467e

Badness (Sintel): 0.416

The following rank-2 mos scales of quartismic have been found:

Direct quartismic

Instead of 11-limit, direct quartismic is defined directly in the 2.7/3.33 subgroup, as the quartisma itself was discovered via a representation where the generator is 33/32 and five of them stack to 7/6. 45edo is an excellent tuning. It was named by Eliora in 2023.

Subgroup: 2.7/3.33

Comma list: 117440512/117406179

Subgroup-val mapping: [⟨1 1 5], ⟨0 5 1]]

mapping generators: ~2, ~33/32

Optimal tunings:

  • Subgroup WE: ~2 = 1199.9875 ¢, ~33/32 = 53.3743 ¢
error map: ⟨-0.012 -0.012 +0.039]
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~33/32 = 53.3730 ¢
error map: ⟨0.000 -0.006 +0.100]

Optimal ET sequence: 21, 22, 45, 337, 382, 427, 472, 517, 562, 607, 652, 1911, 2563, 3215, 5778, 8993*

* wart for 33

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]]

Optimal tuning (POTE): ~2 = 1200.0000 ¢, ~3/2 = 701.7299 ¢, ~33/32 = 53.3889 ¢

Optimal ET sequence: 24, 46c, 65d, 89, 135, 159, 224, 383, 472, 696, 1168, 1327, 1551, 2023e

Badness (Smith): 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]]

mapping generators: ~2, ~11/10, ~33/32

Optimal tuning (POTE): ~2 = 1200.0000 ¢, ~11/10 = 166.0628 ¢, ~33/32 = 53.4151 ¢

Badness (Smith): 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]]

mapping generators: ~2, ~3, ~33/32

Optimal tuning (POTE): ~2 = 1200.0000 ¢, ~3/2 = 697.3325 ¢, ~33/32 = 54.1064 ¢

Optimal ET sequence: 24, 43, 67, 110c

Badness (Smith): 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]]

mapping generators: ~2, ~5/4, ~9/7

Optimal tuning (POTE): ~2 = 1200.0000 ¢, ~5/4 = 380.3623 ¢, ~9/7 = 433.3120 ¢

Optimal ET sequence: 19d, 22

Badness (Smith): 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]]

mapping generators: ~2, ~?, ~33/32

Optimal tuning (POTE): ~2 = 1200.0000 ¢, ~33/32 = 55.3538 ¢

Optimal ET sequence: 21, 22, 43, 65d, 521d, 543, 564, 586, 629c, 651

Badness (Smith): 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

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

mapping generators: ~45/32, ~3, ~33/32

Optimal tuning (POTE): ~45/32 = 600.0000 ¢, ~3/2 = 704.5238 ¢, ~33/32 = 53.4408 ¢

Optimal ET sequence: 22, 46, 68, 114

Badness (Smith): 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 meant to be 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]]

mapping generators: ~2, ~6/5, ~68/55

Optimal tuning (POTE): ~2 = 1200.0000 ¢, ~6/5 = 317.0291 ¢, ~68/55 = 370.2940 ¢

Optimal ET sequence: 68, 91, 159, 246, 337, 405

Badness (Smith): 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]]

mapping generators: ~2, ~425/384, ~33/32

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~425/384 = 175.9566 ¢, ~33/32 = 52.9708 ¢

Optimal ET sequence: 48d, 68, 116d, 157c, 225

Badness (Smith): 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]]

mapping generators: ~2, ~160/121

Optimal tuning (POTE): ~2 = 1200.0000 ¢, ~160/121 = 483.0098 ¢

Optimal ET sequence: 159, 559d, 718, 877

Badness (Smith): 0.154578

Ravine

This temperament was 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]]

mapping generators: ~2, ~33/32

Optimal tuning (CTE): ~2 = 1200.000 ¢, ~33/32 = 53.366 ¢

Optimal ET sequence: 832, 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): ~2 = 1200.000 ¢, ~33/32 = 53.366

Optimal ET sequence: 832, 1619

Prequartismic

This temperament was named because 3125edo was the only one confirmed for tempering out the quartisma before its discovery as significance of the difference between five 33/32's 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]]

mapping generators: ~2, ~64/33

Optimal tuning (CTE): ~2 = 1200.000 ¢, ~64/33 = 1146.624{{c}

Optimal ET sequence: 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): ~2 = 1200.000 ¢, ~64/33 = 1146.624 ¢

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