|
|
| (58 intermediate revisions by 9 users not shown) |
| Line 1: |
Line 1: |
| 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- 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, 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, and thus, all such EDOs not suitable for temperaments in this family. Examples of edos that actually ''do'' temper out the quartisma are [[22edo]], [[24edo]], [[46edo]], [[68edo]], [[159edo]], [[224edo]] and [[3125edo]].
| | #redirect [[Catalog of rank-4 temperaments #Quartismic (117440512/117406179)]] |
| | |
| = 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
| |
| | |
| POTE generators:
| |
| | |
| Mapping generator:
| |
| | |
| Map:
| |
| | |
| 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}}
| |
| | |
| Badness:
| |
| | |
| == No-five Children ==
| |
| 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
| |
| | |
| POTE generators: ~3/2 = 701.9826, ~33/32 = 53.3748
| |
| | |
| Mapping generator:
| |
| | |
| Map: [<1 0 1 5|, <0 1 1 -1|, <0 0 5 1|]
| |
| | |
| EDOs: {{EDOs|21, 22, 24, 43, 46, 89, 135, 270, 359, 494, 629, 653, 742, 877, 1012, 1236, 1506, 2159, 2248, 2383, 2518, 7419}}
| |
| | |
| Badness:
| |
| | |
| The following scale tree has been found:
| |
| * [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&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]
| |
| | |
| == Full 11-limit extensions ==
| |
| 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:Quartismic]]
| |
| [[Category:Microtemperament]]
| |
| [[Category:Rank 2]]
| |
| [[Category:Temperament]]
| |