The Quartercache: Difference between revisions

'''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 {{monzo| 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)]].

== Rank-3 temperaments ==

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

{{Mapping|legend=1| 1 0 1 5 | 0 1 1 -1 | 0 0 5 1 }}

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

* [[WE]]: ~2 = 1199.9875{{c}}, ~3/2 = 701.9753{{c}}, ~33/32 = 53.3743{{c}}

: [[error map]]: {{val| -0.012 +0.008 -0.004 +0.031 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9802{{c}}, ~33/32 = 53.3730{{c}}

: error map: {{val| 0.000 +0.025 +0.019 +0.075 }}

{{Optimal ET sequence|legend=1| 21, 22, 24, 43, 46, 89, 135, 359, 494, 629, 742, 877, 1012, 1506, 2248, 2383, 2518, 7419, 8431e, 10949e, 13467e }}

The following rank-2 [[mos scale]]s of quartismic have been found:  

* [https://sevish.com/scaleworkshop/?name=Rank%202%20scale%20(106.71461627796054%2C%201200.0)%2C%205%7C5&data=106.714616%0A213.429233%0A320.143849%0A426.858465%0A533.573081%0A666.426919%0A773.141535%0A879.856151%0A986.570767%0A1093.285384%0A1200.000000&freq=440&midi=69&vert=9&horiz=1&colors=&waveform=triangle&ampenv=organ Rank 2 scale (106.71461627796054, 1200.0), 5|5] 

* The following scale tree has been found: [http://www.microtonalsoftware.com/scale-tree.html?left=12&right=11&rr=1200&ioi=106.71461627796054 1200-106.71461627796054-12-11 Scale Tree]

=== Meanquarter ===

[[Subgroup]]: 2.3.5.7.11

[[Comma list]]: 81/80, 4128768/4026275

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

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 697.3325{{c}}, ~33/32 = 54.1064{{c}}

{{Optimal ET sequence|legend=1| 24, 43, 67, 110c }}

=== Coin ===

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

[[Comma list]]: 3125/3072, 117440512/117406179

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

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~5/4 = 380.3623{{c}}, ~9/7 = 433.3120{{c}}

{{Optimal ET sequence|legend=1| 19d, 22 }}

[[Badness]] (Smith): 70.470 × 10^-3

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

[[Comma list]]: 117440512/117406179, 4294967296/4271484375

{{Mapping|legend=1| 1 2 2 3 3 | 0 -9 7 -4 10 | 0 0 0 5 1 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~33/32 = 55.3538{{c}}

{{Optimal ET sequence|legend=1| 21, 22, 43, 65d, 521d, 543, 564, 586, 629c, 651 }}

[[Badness]] (Smith): 64.233 × 10^-3

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

[[Subgroup]]: 2.3.5.7.11

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

[[Optimal tuning]] ([[POTE]]): ~45/32 = 600.0000{{c}}, ~3/2 = 704.5238{{c}}, ~33/32 = 53.4408{{c}}

{{Optimal ET sequence|legend=1| 22, 46, 68, 114 }}

[[Badness]] (Smith): 23.250 × 10^-3

Scales:  

[https://sevish.com/scaleworkshop/?name=Rank%202%20scale%20(52.6800%2C%202%2F1)&data=52.680000%0A105.360000%0A158.040000%0A210.720000%0A263.400000%0A316.080000%0A368.760000%0A421.440000%0A474.120000%0A526.800000%0A579.480000%0A632.160000%0A684.840000%0A725.880000%0A778.560000%0A831.240000%0A883.920000%0A936.600000%0A989.280000%0A1041.960000%0A1094.640000%0A1147.320000%0A1200.000000&freq=440&midi=69&vert=9&horiz=1&colors=&waveform=triangle&ampenv=organ Rank 2 scale (52.6800, 2/1), 13|9]

[https://sevish.com/scaleworkshop/?name=Rank%202%20scale%20(53.37418112074753%2C%202%2F1)%2C%2013%7C9&data=53.374181%0A106.748362%0A160.122543%0A213.496724%0A266.870906%0A320.245087%0A373.619268%0A426.993449%0A480.367630%0A533.741811%0A587.115992%0A640.490173%0A693.864355%0A719.632370%0A773.006551%0A826.380732%0A879.754913%0A933.129094%0A986.503276%0A1039.877457%0A1093.251638%0A1146.625819%0A1200.000000&freq=440&midi=69&vert=9&horiz=1&colors=white%20black%20white%20white%20black%20white%20black%20white%20white%20black%20white%20black&waveform=triangle&ampenv=organ Rank 2 scale (53.3742, 2/1), 13|9]

=== Kleirtismic ===

[[Subgroup]]: 2.3.5.7.11

[[Comma list]]: 15625/15552, 117440512/117406179

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

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~6/5 = 317.0291{{c}}, ~68/55 = 370.2940{{c}}

{{Optimal ET sequence|legend=1| 68, 91, 159, 246, 337, 405 }}

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

[[Comma list]]: 20000/19683, 100656875/99090432

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

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~425/384 = 175.9566{{c}}, ~33/32 = 52.9708{{c}}

{{Optimal ET sequence|legend=1| 48d, 68, 116d, 157c, 225 }}

[[Badness]] (Smith): 81.083 × 10^-3

=== 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

{{Mapping|legend=1| 1 1389 890 1395 -1383 | 0 -1452 -929 -1457 1451 }}

: mapping generators: ~2, ~64/33

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

{{Optimal ET sequence|legend=1| 1259e, 1866, 3125, 4384e, 4991, 6250e, 8116d, 7509ee, 9375e, 11241de }}

Subgroup: 2.3.5.7.11.13

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

{{Mapping|legend=0| 1 1389 890 1395 -1383 -282 | 0 -1452 -929 -1457 1451 299 }}

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

== Rank-2 temperaments ==

=== 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

: mapping generators: ~2, ~33/32

* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~33/32 = 53.3730{{c}}

: error map: {{val| 0.000 -0.006 +0.100 }}

{{Optimal ET sequence|legend=1| 21, 22, 45, 337, 382, 427, 472, 517, 562, 607, 652, 1911, 2563, 3215, 5778, 8993* }}

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

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

{{Mapping|legend=1| 1 4 47 130 26 | 0 -6 -111 -316 -56 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~160/121 = 483.0098{{c}}

{{Optimal ET sequence|legend=1| 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.

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

{{Mapping|legend=1| 1 26 21 27 -21 | 0 -549 -420 -544 550 }}

[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~33/32 = 53.366{{c}}

Subgroup: 2.3.5.7.11.13

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

{{Mapping|legend=0| 1 26 21 27 -21 21 | 0 -549 -420 -544 550 -389 }}

Optimal tuning (CTE): ~2 = 1200.000{{c}}, ~33/32 = 53.366

[[Category:Commatic realms]]

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