Quartismic family: Difference between revisions

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{{todo|inline=1| discuss title | comment= This doesn't follow the definition of temperament family }}
{{Technical data page}}
 
The '''quartismic family''' is a family of [[rank-4]] temperaments tempers out the [[quartisma]] – the unnoticeable comma with the ratio 117440512/117406179, and a monzo of {{monzo|24 -6 0 1 -5}}, however, most of the members of this rank-4 family currently have yet to be explored. For other families that are defined by the tempering of this comma, see [[the Quartercache]].
The '''quartismic family''' is built up from temperaments of various ranks that 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 quartismatic, altierran, meanquarter, coin, escapismic, dietismic, kleirtismic, and doublefour.


== Quartismic ==
== Quartismic ==
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The 11-limit parent comma for the quartismic family is the the quartisma with a ratio of 117440512/117406179 and a monzo of {{monzo| 24 -6 0 1 -5 }}. As the quartisma is an unnoticeable comma, this rank-4 temperament is a [[microtemperament]].
The 11-limit parent comma for the quartismic family is the the quartisma with a ratio of 117440512/117406179 and a monzo of {{monzo| 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
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 117440512/117406179
[[Comma list]]: 117440512/117406179
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[[Mapping]]: [{{val| 1 0 0 1 5 }}, {{val| 0 1 0 1 -1 }}, {{val| 0 0 1 0 0 }}, {{val| 0 0 0 5 1 }}]
[[Mapping]]: [{{val| 1 0 0 1 5 }}, {{val| 0 1 0 1 -1 }}, {{val| 0 0 1 0 0 }}, {{val| 0 0 0 5 1 }}]


{{Multival|legend=1|rank=4| 5 1 0 -6 -24 }}
Mapping generators: ~2, ~3, ~5, ~33/32


[[POTE generator]]s: ~3/2 = 701.9826, ~5/4 = 386.3427, ~33/32 = 53.3748
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.9742, ~5/4 = 386.3137, ~33/32 = 53.3683


{{Val list|legend=1| 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, 313, 359, 494, 629, 653, 742, 877, 1012, 1236, 1506, 2159, 2248, 2383, 2518, 3125, 7419 }}
{{Optimal ET sequence|legend=1| 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<sup>-6</sup>
[[Badness]]: 0.274 × 10<sup>-6</sup>


== Quartismatic ==
== Tridecimal quartismic ==
 
[[Subgroup]]: 2.3.5.7.11.13
There are some temperaments in the quartismic family 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]]: [{{val| 1 0 1 5 }}, {{val| 0 1 1 -1 }}, {{val| 0 0 5 1 }}]
 
[[POTE generator]]s: ~3/2 = 701.9826, ~33/32 = 53.3748
 
{{Val list|legend=1| 21, 22, 24, 43, 46, 89, 135, 270, 359, 494, 629, 653, 742, 877, 1012, 1236, 1506, 2159, 2248, 2383, 2518, 7419 }}
 
The following unnamed rank-2 quartismic temperament MOS scales 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]
 
== 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]]: [{{val| 1 0 15 1 5 }}, {{val| 0 1 -8 1 -1 }}, {{val| 0 0 0 5 1 }}]  
 
{{Multival|legend=1|rank=3| -102 24 -15 75 6 -8 40 1 -5 0 }}
 
[[POTE generator]]s: ~3/2 = 701.7299, ~33/32 = 53.3889
 
{{Val list|legend=1| 135, 159, 224, 248, 313, 472 }}
 
=== 13-limit ===
 
{{todo|inline=1| rename | comment = Not an immediate extension, must be renamed }}
 
Subgroup: 2.3.5.7.11.13
 
[[Comma list]]: 10985/10976, 32805/32768, 161280/161051
 
[[Mapping]]: [{{val| 1 2 -1 3 3 5 }}, {{val| 0 -3 24 -3 3 -11 }}, {{val| 0 0 0 5 1 5 }}]
 
[[POTE generator]]s: ~11/10 = 166.0628, ~33/32 = 53.4151
 
== 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]]: [{{val| 1 0 -4 1 5 }}, {{val| 0 1 4 1 -1 }}, {{val| 0 0 5 1 }}]
 
[[POTE generator]]s: ~3/2 = 697.3325, ~33/32 = 54.1064
 
{{Val list|legend=1| 24, 43, 45, 67 }}
 
== 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]]: [{{val| 1 0 2 1 5 }}, {{val| 0 5 1 0 -6 }}, {{val| 0 0 0 5 1 }}]
 
[[POTE generator]]s: ~5/4 = 380.3623, ~9/7 = 433.3120
 
{{Val list|legend=1| 22, 25, 139cdd }}
 
== Escapismic ==
 
In escapisimic, both the escapade comma and the quartisma are tempered out, thus, it is essentially an [[Escapade family|Escapade expansion]].
 
Subgroup: 2.3.5.7.11
 
[[Comma list]]: 117440512/117406179, 4294967296/4271484375
 
[[Mapping]]: [{{val| 1 2 2 3 3 }}, {{val| 0 -9 7 -4 10 }}, {{val| 0 0 0 5 1 }}]
 
[[POTE generator]]s: ~33/32 = 55.3538
 
{{Val list|legend=1| 21, 22, 43 }}
 
== Dietismic ==
 
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
 
[[Comma list]]: 2048/2025, 117440512/117406179
 
[[POTE generator]]s: ~3/2 = 704.5238, ~33/32 = 53.4408
 
[[Mapping]]: [{{val| 2 3 5 5 7 }}, {{val| 0 2 -4 7 -1 }}]
 
{{Val list|legend=1| 22, 24, 38cdde, 46, 68, 114 }}
 
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 ==
 
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]]: [{{val| 1 0 1 1 5 }}, {{val| 0 6 5 1 -7 }}, {{val| 0 0 0 5 1 }}]
 
[[POTE generator]]s: ~6/5 = 317.0291, ~68/55 = 370.2940
 
{{Val list|legend=1| 159, 178, 246 }}
 
== Doublefour ==
 
In doublefour, both the tetracot comma and the quartisma are tempered out.


Subgroup: 2.3.5.7.11
[[Comma list]]: 6656/6655, 123201/123200


[[Comma list]]: 20000/19683, 100656875/99090432
[[Mapping]]: [{{val| 1 0 0 1 5 6 }}, {{val| 0 1 0 1 -1 -3 }}, {{val| 0 0 1 0 0 1 }}, {{val| 0 0 0 5 1 3 }}]


[[Mapping]]: [{{val| 1 1 1 2 4 }}, {{val| 0 4 9 4 -4 }}, {{val| 0 0 0 5 1 }}]
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.9695, ~5/4 = 386.3174, ~33/32 = 53.3698


[[POTE generator]]s: ~425/384 = 175.9566, ~33/32 = 52.9708
{{Optimal ET sequence|legend=1| 22, 43f, 46, 65d, 89f, 111, 159, 224, 270, 494, 764, 1012, 1236, 1506, 2901, 3125, 3395, 8026e, 8296e, 11421e, 11691e, 12927e, 13421e, 16322ee, 16816dee }}


{{Val list|legend=1| 48d, 68, 89c }}
[[Badness]]: 1.739 × 10<sup>-6</sup>


[[Category:Regular temperament theory]]
[[Category:Temperament families]]
[[Category:Temperament family]]
[[Category:Pages with mostly numerical content]]
[[Category:Microtemperament]]
[[Category:Microtemperaments]]
[[Category:Quartismic]]
[[Category:Quartismic]]
[[Category:Rank 4]]
[[Category:Rank 4]]

Latest revision as of 00:26, 24 June 2025

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 quartismic family is a family of rank-4 temperaments tempers out the quartisma – the unnoticeable comma with the ratio 117440512/117406179, and a monzo of [24 -6 0 1 -5⟩, however, most of the members of this rank-4 family currently have yet to be explored. For other families that are defined by the tempering of this comma, see the Quartercache.

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.

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

Tridecimal quartismic

Subgroup: 2.3.5.7.11.13

Comma list: 6656/6655, 123201/123200

Mapping: [⟨1 0 0 1 5 6], ⟨0 1 0 1 -1 -3], ⟨0 0 1 0 0 1], ⟨0 0 0 5 1 3]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.9695, ~5/4 = 386.3174, ~33/32 = 53.3698

Optimal ET sequence: 22, 43f, 46, 65d, 89f, 111, 159, 224, 270, 494, 764, 1012, 1236, 1506, 2901, 3125, 3395, 8026e, 8296e, 11421e, 11691e, 12927e, 13421e, 16322ee, 16816dee

Badness: 1.739 × 10-6

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