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]]. | 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]]. | ||
The rank-3 '''quartismic''' or '''Saquinlu-azo temperament''' is the rank-3 2.3.7.11 temperament that tempers out this comma | 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 = | |||
Comma: 117440512/117406179 | |||
No-five map: [<1 0 1 5], <0 1 1 -1], <0 0 5 1]] | No-five map: [<1 0 1 5], <0 1 1 -1], <0 0 5 1]] | ||
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* [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&env=organ Rank 2 scale (53.37418112074753, 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=5&horiz=1&colors=&waveform=triangle&env=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&env=organ Rank 2 scale (106.71461627796054, 1200.0), 5|5] | * [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&env=organ Rank 2 scale (106.71461627796054, 1200.0), 5|5] | ||
== 13-limit children == | |||
For 13-limit extensions, one could easily temper out 10985/10976. However, there are other possibilities as well | |||
[[Category:Quartismic]] | [[Category:Quartismic]] | ||
[[Category:Rank 2]] | [[Category:Rank 2]] | ||
[[Category:Temperament]] | [[Category:Temperament]] | ||
Revision as of 15:59, 10 September 2020
The quartisma or Saquinlu-azo comma is a comma with a ratio of 117440512/117406179 and a monzo of [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 these calculations prove. Examples of edos that actually do temper out the quartisma are 22edo, 24edo, 68edo, 90edo, 91edo, 92edo, 159edo, and 3125edo.
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
Comma: 117440512/117406179
No-five map: [<1 0 1 5], <0 1 1 -1], <0 0 5 1]]
No-five POTE generators: ~3/2 = 701.9826, ~33/32 = 53.3748
No-five edos: 21, 22, 24, 43, 46, 89, 135, 359, 494, 629, 742, 877, 1012, 1506, 2248, 2383, 2518, 7419
The following scale tree has been found:
The following quartismic MOS scales have been found:
13-limit children
For 13-limit extensions, one could easily temper out 10985/10976. However, there are other possibilities as well