# The Quartercache

**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

*For extensions, see 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]]

Mapping generators: ~2, ~3, ~5, ~33/32

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.9742, ~5/4 = 386.3137, ~33/32 = 53.3683

Optimal ET sequence: 21, 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 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 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 ET sequence: 21, 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

- Rank 2 scale (106.71461627796054, 1200.0), 5|5
- The following scale tree has been found: 1200-106.71461627796054-12-11 Scale Tree

## Direct quartismic

Instead of 11-limit, defined directly in the 2.33/2.7/6 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.

Subgroup: 2.33/32.7/6

Comma list: 117440512/117406179

Sval mapping: [⟨1 0 0], ⟨0 1 5]]

Sval mapping generators: ~2 = 1\1, ~33/32 = 53.368

## 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 ET sequence: 24, 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 ET sequence: 24, 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

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 ET sequence: 21, 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 ET sequence: 22, 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 ET sequence: 68, 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 ET sequence: 48d, 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 ET sequence: 159, 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 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): ~33/32 = 53.366

Optimal ET 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

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): ~64/33 = 1146.624