Quartismic family: Difference between revisions

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The '''quartismic family''' is built up from temperaments 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 Altierran, Meanquarter, Coin, Kleirtismic, and Doublefour.
{{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]].


= Quartismic =
== 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 [[Microtempering|microtemperament]].


Comma: 117440512/117406179
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]].


POTE generators: ~3/2 = 701.9826, ~5/4 = 386.3427, ~33/32 = 53.3748
[[Subgroup]]: 2.3.5.7.11


Mapping generator:
[[Comma list]]: 117440512/117406179


Map: [<1 0 0 1 5|, <0 1 0 1 -1|, <0 0 1 0 0|, <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 }}]


Wedgie: <<<<5 1 0 -6 -24||||
Mapping generators: ~2, ~3, ~5, ~33/32


EDOs: {{EDOs|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, 359, 494, 629, 653, 742, 877, 1012, 1236, 1506, 2159, 2248, 2383, 2518, 3125, 7419}}
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.9742, ~5/4 = 386.3137, ~33/32 = 53.3683


Badness: 0.274 × 10<sup>-6</sup>
{{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 }}


== No-five Children ==
[[Badness]]: 0.274 × 10<sup>-6</sup>
There are some temperaments in the quartismic family in which only the quartisma is tempered out, but without any regard to the five-limit.


Comma: 117440512/117406179
== Tridecimal quartismic ==
[[Subgroup]]: 2.3.5.7.11.13


POTE generators: ~3/2 = 701.9826, ~33/32 = 53.3748
[[Comma list]]: 6656/6655, 123201/123200


Mapping generator:
[[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 }}]


Map: [<1 0 1 5|, <0 1 1 -1|, <0 0 5 1|]
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.9695, ~5/4 = 386.3174, ~33/32 = 53.3698


EDOs: {{EDOs|21, 22, 24, 43, 46, 89, 135, 270, 359, 494, 629, 653, 742, 877, 1012, 1236, 1506, 2159, 2248, 2383, 2518, 7419}}
{{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 }}


Badness:
[[Badness]]: 1.739 × 10<sup>-6</sup>
 
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]
The following rank-2 quartismic temperament MOS scales have been found:
* [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=&waveform=triangle&ampenv=organ Rank 2 scale (53.37418112074753, 2/1), 13|9]
* [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]
 
== 5-limit Representation ==
Among quartismic temperaments, there are several options for 5-limit representation depending which among the various 5-limit commas is tempered out.  Adding the [[schisma]] to the list of tempered-out commas results in some form of Altierran temperament.  Adding the [[81/80|meantone comma]] results in some form of Meanquarter temperament.  Adding the [[Magic_comma|magic comma]] results in some form of Coin temperament.  Adding the [[15625/15552|kleisma]] results in some form of Kleirtismic temperament- the "kleir-" in "Kleirtismic" is pronounced the same as "Clair".  Adding the [[Tetracot_comma|tetracot comma]] results in some form of Doublefour temperament.  Other possible extensions are listed here.
 
===Shrutar extension===
This is the 22&46 temperament.  See [[Diaschismic_family#Shrutar|Shrutar]].
===Escapade extension===
This is the 22&43 temperament.  See [[Escapade_family|Escapade]].
 
= Altierran =
The Altierran clan is the temperament clan consisting of those temperaments in which both the schisma and the quartisma are tempered out.
 
Commas: 32805/32768, 117440512/117406179
 
POTE generators: ~3/2 = 701.7299, ~33/32 = 53.3889
 
Mapping generators:
 
Map: [<1 0 15 1 5|, <0 1 -8 1 -1|, <0 0 0 5 1|]
 
EDOs:  {{EDOs|135, 159, 224, 472}}
 
Badness:
 
== 13-limit ==
Commas: 10985/10976, 32805/32768, 117440512/117406179
 
POTE generators: ~11/10 = 166.0628, ~33/32 = 53.4151
 
Mapping generators:
 
Map: [<1 2 -1 3 3 5|, <0 -3 24 -3 3 -11|, <0 0 0 5 1 5|]
 
EDOs: {{EDOs}}
 
Badness:
 
== 17-limit ==
Commas:
 
POTE generator:
 
Mapping generator:
 
Map:
 
EDOs:
 
Badness:
 
= Meanquarter =
The Meanquarter clan is the temperament clan consisting of those temperaments in which both the meantone comma and the quartisma are tempered out.  Meanquarter can easily be extended to a form of [[Semaphore_and_Godzilla|godzilla]], though not all possible tunings for Meanquarter lend themselves to this sort of thing.
 
Commas: 81/80, 117440512/117406179
 
POTE generators: ~3/2 = 697.3325, ~33/32 = 54.1064
 
Mapping generators:
 
Map: [<1 0 -4 1 5|, <0 1 4 1 -1|, <0 0 5 1|]
 
EDOs: {{EDOs|24, 43, 45, 67}}
 
Badness:
 
= Coin =
The Coin clan is the temperament clan consisting of those temperaments in which both the magic comma and the quartisma are tempered out.
 
Commas: 3125/3072, 117440512/117406179
 
POTE generators: ~5/4 = 380.3623, ~9/7 = 433.3120
 
Mapping generators:
 
Map: [<1 0 2 1 5|, <0 5 1 0 -6|, <0 0 0 5 1|]
 
EDOs: {{EDOs|22, 25, 139cdd}}
 
Badness:
 
= Kleirtismic =
The Kleirtismic clan is the temperament clan consisting of those temperaments in which both the kleisma and the quartisma are tempered out.
 
Commas: 15625/15552, 117440512/117406179
 
POTE generators: ~6/5 = 317.0291, ~68/55 370.2940
 
Mapping generators:
 
Map: [<1 0 1 1 5|, <0 6 5 1 -7|, <0 0 0 5 1|]
 
EDOs: {{EDOs|159, 178, 246}}
 
Badness:
 
= Doublefour =
The Doublefour clan is the temperament clan consisting of those temperaments in which both the tetracot comma and the quartisma are tempered out.
 
Commas: 20000/19683, 117440512/117406179
 
POTE generators: ~425/384 = 175.9566, ~33/32 = 52.9708
 
Mapping generators:
 
Map: [<1 1 1 2 4|, <0 4 9 4 -4|, <0 0 0 5 1|]
 
EDOs: {{EDOs|48d, 68, 89c}}
 
Badness:


[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Microtemperaments]]
[[Category:Quartismic]]
[[Category:Quartismic]]
[[Category:Microtemperament]]
[[Category:Rank 4]]
[[Category:Rank 2]]
[[Category:Temperament]]

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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