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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}}).  Despite that fact that the quartisma is an unnoticeable comma in JI, a number of reasonably well known EDOs (such as [[17edo]], [[26edo]] and [[34edo]]) actually fail to temper it out, rendering them unsuitable for temperaments in this family. In fact, there are even some EDOs such as [[23edo]] and [[70edo]] that seem to temper out the comma when one merely examines the patent vals, yet, upon closer examination, actually fail to temper out the comma, and are thus unsuitable for temperaments in this family as [https://www.wolframalpha.com/input/?i=dot+product+of+%2823%2C+round%28log%283%29%2Flog%282%29*23%29%2C+round%28log%285%29%2Flog%282%29*23%29%2C+round%28log%287%29%2Flog%282%29*23%29%2C+round%28log%2811%29%2Flog%282%29*23%29%29++and+%2824%2C+-6%2C+0%2C+1%2C+-5%29 these] [https://www.wolframalpha.com/input/?i=dot+product+of+%2870%2C+round%28log%283%29%2Flog%282%29*70%29%2C+round%28log%285%29%2Flog%282%29*70%29%2C+round%28log%287%29%2Flog%282%29*70%29%2C+round%28log%2811%29%2Flog%282%29*70%29%29++and+%2824%2C+-6%2C+0%2C+1%2C+-5%29 calculations] prove.  Examples of edos that actually ''do'' temper out the quartisma are [[22edo]], [[24edo]], [[46edo]], [[68edo]], [[159edo]], [[224edo]] and [[3125edo]].
{{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 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:
[[Subgroup]]: 2.3.5.7.11


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


Map:  
[[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 }}]


EDOs: {{EDOs|21, 22, 24, 25, 43, 45, 46, 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, 7419}}
Mapping generators: ~2, ~3, ~5, ~33/32


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


== No-five Children ==
{{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 }}
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
[[Badness]]: 0.274 × 10<sup>-6</sup>


POTE generators: ~3/2 = 701.9826, ~33/32 = 53.3748
== Tridecimal quartismic ==
[[Subgroup]]: 2.3.5.7.11.13


Mapping generator:
[[Comma list]]: 6656/6655, 123201/123200


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


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


Badness:
{{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 }}


The following scale tree has been found:
[[Badness]]: 1.739 × 10<sup>-6</sup>
* [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]].
===Godzilla extension===
This is the 24&43 temperament.  See [[Semaphore_and_Godzilla|Godzilla]].
 
= 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: 701.7299, 53.3889
 
Mapping generators: 2/1, 3/2, 33/32
 
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: 166.0628, 53.4151
 
Mapping generators: 2/1, 11/10, 33/32
 
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.
 
Commas: 81/80, 117440512/117406179
 
POTE generators: 697.3325, 54.1064
 
Mapping generators: 3/2, 33/32
 
Map: [<1 0 -4 1 5|, <0 1 4 1 -1|, <0 0 5 1|]
 
EDOs: {{EDOs|24, 43, 45}}
 
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: 380.3623, 433.3120
 
Mapping generators: 5/4, 9/7
 
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: 317.0291, 370.2940
 
Mapping generators: 6/5, (16/13?)
 
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: 175.9566, 52.9708
 
Mapping generators: 10/9, 33/32
 
Map: [<1 1 1 2 4|, <0 4 9 4 -4|, <0 0 0 5 1|]
 
EDOs: 48d, {{EDOs|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]]
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