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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]], [[46edo]], [[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]]. |
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| The '''quartismic temperament''' or '''Saquinlu-azo temperament''' is the temperament that tempers out this comma. This page will also list various derived temperaments that temper out this comma and thus belong in the quartismic family.
| | == Quartismic == |
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| = 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
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| No-five POTE generators: ~3/2 = 701.9826, ~33/32 = 53.3748
| | [[Subgroup]]: 2.3.5.7.11 |
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| No-five mapping generator:
| | [[Comma list]]: 117440512/117406179 |
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| No-five Map: [<1 0 1 5|, <0 1 1 -1|, <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 }}] |
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| No-five EDOs: {{EDOs|21, 22, 24, 43, 46, 89, 135, 270, 359, 494, 629, 653, 742, 877, 1012, 1236, 1506, 2159, 2248, 2383, 2518, 7419}}
| | Mapping generators: ~2, ~3, ~5, ~33/32 |
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| Badness:
| | [[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.9742, ~5/4 = 386.3137, ~33/32 = 53.3683 |
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| The following scale tree has been found:
| | {{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 }} |
| * [http://www.microtonalsoftware.com/scale-tree.html?left=12&right=11&rr=1200&ioi=106.71461627796054 1200-106.71461627796054-12-11 Scale Tree]
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| The following rank-2 quartismic temperament MOS scales have been found:
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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=9&horiz=1&colors=&waveform=triangle&env=organ Rank 2 scale (53.37418112074753, 2/1), 13|9]
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| * [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]
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| == Full 11-limit extensions ==
| | [[Badness]]: 0.274 × 10<sup>-6</sup> |
| 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.
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| ===Shrutar extension=== | | == Tridecimal quartismic == |
| This is the 22&46 temperament. See [[Diaschismic_family#Shrutar|Shrutar]].
| | [[Subgroup]]: 2.3.5.7.11.13 |
| ===Escapade extension===
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| This is the 22&43 temperament. See [[Escapade_family|Escapade]].
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| ===Godzilla extension===
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| This is the 24&43 temperament. See [[Semaphore_and_Godzilla|Godzilla]].
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| = Altierran =
| | [[Comma list]]: 6656/6655, 123201/123200 |
| The Altierran clan is the temperament clan consisting of those temperaments in which both the schisma and the quartisma are tempered out.
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| Commas: 32805/32768, 117440512/117406179
| | [[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 }}] |
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| POTE generators: 701.7299, 53.3889
| | [[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.9695, ~5/4 = 386.3174, ~33/32 = 53.3698 |
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| Mapping generators: 2/1, 3/2, 33/32
| | {{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 }} |
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| Map: [<1 0 15 1 5|, <0 1 -8 1 -1|, <0 0 0 5 1|]
| | [[Badness]]: 1.739 × 10<sup>-6</sup> |
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| EDOs: {{EDOs|135, 159, 224, 472}}
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| Badness:
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| == 13-limit ==
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| Commas: 10985/10976, 32805/32768, 117440512/117406179
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| POTE generators: 166.0628, 53.4151
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| Mapping generators: 2/1, 11/10, 33/32
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| Map: [<1 2 -1 3 3 5|, <0 -3 24 -3 3 -11|, <0 0 0 5 1 5|]
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| EDOs: {{EDOs}}
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| Badness: | |
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| == 17-limit ==
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| Commas:
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| POTE generator:
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| Mapping generator:
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| Map:
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| EDOs:
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| Badness:
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| = Meanquarter =
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| The Meanquarter clan is the temperament clan consisting of those temperaments in which both the meantone comma and the quartisma are tempered out.
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| Commas: 81/80, 117440512/117406179
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| POTE generators: 697.3325, 54.1064
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| Mapping generators: 3/2, 33/32
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| Map: [<1 0 -4 1 5|, <0 1 4 1 -1|, <0 0 5 1|]
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| EDOs: {{EDOs|24, 43, 45}}
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| Badness:
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| = Coin =
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| The Coin clan is the temperament clan consisting of those temperaments in which both the magic comma and the quartisma are tempered out.
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| Commas: 3125/3072, 117440512/117406179
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| POTE generators: 380.3623, 433.3120
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| Mapping generators: 5/4, 9/7
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| Map: [<1 0 2 1 5|, <0 5 1 0 -6|, <0 0 0 5 1|]
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| EDOs: {{EDOs|22, 25}}, 139cdd
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| Badness:
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| = Kleirtismic =
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| The Kleirtismic clan is the temperament clan consisting of those temperaments in which both the kleisma and the quartisma are tempered out.
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| Commas: 15625/15552, 117440512/117406179
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| POTE generators: 317.0291, 370.2940
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| Mapping generators: 6/5, (16/13?)
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| Map: [<1 0 1 1 5|, <0 6 5 1 -7|, <0 0 0 5 1|]
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| EDOs: {{EDOs|159, 178, 246}}
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| Badness:
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| = Doublefour =
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| The Doublefour clan is the temperament clan consisting of those temperaments in which both the tetracot comma and the quartisma are tempered out.
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| Commas: 20000/19683, 117440512/117406179
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| POTE generators: 175.9566, 52.9708
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| Mapping generators: 10/9, 33/32
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| Map: [<1 1 1 2 4|, <0 4 9 4 -4|, <0 0 0 5 1|]
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| EDOs:
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| Badness:
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| | [[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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- 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