Quartismic family: Difference between revisions

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The '''quartisma''' or '''Saquinlu-azo comma''' is a comma with a ratio of '''117440512/117406179''' and a [[monzo]] of {{monzo|24 -6 0 1 -5}}.  It is an [[unnoticeable comma]] of the [[11-limit]]- specifically one of the the 2.9.7.11 subgroup- with a value of approximately 0.50619 cents. The quartisma is significant on account of it being the difference between a stack of five [[33/32]] quartertones and one [[7/6]] subminor third in Just Intonation.  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.  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 for 33/32 and 7/6, yet, upon closer examination, actually fail to temper out the comma, 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]], [[68edo]], [[90edo]], [[91edo]], [[92edo]], [[159edo]], 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]].


The rank-3 '''quartismic temperament''' or '''Saquinlu-azo temperament''' is the rank-3 2.3.7.11 temperament that tempers out this comma.  This page will also list various rank-2 temperaments that temper out this comma and thus belong in the quartismic family.
== 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 {{monzo| 24 -6 0 1 -5 }}. As the quartisma is an unnoticeable comma, this rank-4 temperament is a [[microtemperament]].


Comma: 117440512/117406179
[[Subgroup]]: 2.3.5.7.11


No-five Map: [<1 0 1 5], <0 1 1 -1], <0 0 5 1]]
[[Comma list]]: 117440512/117406179


No-five POTE generators: ~3/2 = 701.9826, ~33/32 = 53.3748
[[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 }}]


No-five EDOs: {{EDOs|21, 22, 24, 43, 46, 89, 135, 359, 494, 629, 742, 877, 1012, 1506, 2248, 2383, 2518, 7419}}
Mapping generators: ~2, ~3, ~5, ~33/32


The following scale tree has been found:
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.9742, ~5/4 = 386.3137, ~33/32 = 53.3683
* [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 quartismic 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=5&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]


== Full 11-limit extensions ==
{{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 }}


===Shrutar===
[[Badness]]: 0.274 × 10<sup>-6</sup>
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]].


== 13-limit extensions ==
== Tridecimal quartismic ==
For 13-limit extensions, one could temper out 10985/10976. However, there are other possibilities as well.
[[Subgroup]]: 2.3.5.7.11.13


[[Category:Quartismic]]
[[Comma list]]: 6656/6655, 123201/123200
[[Category:Rank 2]]
[[Category:Temperament]]


= 5-limit =
[[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 }}]
Arguably one of the better choices for 5-limit representation among quartismic temperaments involves additionally tempering out 32805/32768- the [[schisma]].


Commas: 117440512/117406179, 32805/32768
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.9695, ~5/4 = 386.3174, ~33/32 = 53.3698


Map:
{{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 }}


POTE generators:
[[Badness]]: 1.739 × 10<sup>-6</sup>


EDOs:
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Microtemperaments]]
[[Category:Quartismic]]
[[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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