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| The '''quartismic family''' is built up from temperaments that tempers 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]]. |
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| = 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]].
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| 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]]. |
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| POTE generators:
| | [[Subgroup]]: 2.3.5.7.11 |
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| Mapping generator:
| | [[Comma list]]: 117440512/117406179 |
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| 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 }}] |
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| 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 |
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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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| == 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.
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| Comma: 117440512/117406179
| | [[Badness]]: 0.274 × 10<sup>-6</sup> |
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| POTE generators: ~3/2 = 701.9826, ~33/32 = 53.3748
| | == Tridecimal quartismic == |
| | [[Subgroup]]: 2.3.5.7.11.13 |
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| Mapping generator:
| | [[Comma list]]: 6656/6655, 123201/123200 |
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| 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 }}] |
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| 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 |
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| 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 }} |
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| 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]
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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 ==
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| 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===
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| This is the 22&46 temperament. See [[Diaschismic_family#Shrutar|Shrutar]].
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| ===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 =
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| 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
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| POTE generators: 701.7299, 53.3889
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| Mapping generators: 2/1, 3/2, 33/32
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| Map: [<1 0 15 1 5|, <0 1 -8 1 -1|, <0 0 0 5 1|]
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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: 48d, {{EDOs|68}}, 89c
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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