Cassaschismic: Difference between revisions
What on earth are "p5" "p13" "p19"? Plz don't invent language here |
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| Comma basis = [[19712/19683]], [[41503/41472]] (11-limit); <br>[[2080/2079]], [[4096/4095]], [[19712/19683]] (13-limit); <br>[[1216/1215]], [[1540/1539]], [[1729/1728]], <br>[[2080/2079]] (2.3.5.7.11.13.19) | | Comma basis = [[19712/19683]], [[41503/41472]] (11-limit); <br>[[2080/2079]], [[4096/4095]], [[19712/19683]] (13-limit); <br>[[1216/1215]], [[1540/1539]], [[1729/1728]], <br>[[2080/2079]] (2.3.5.7.11.13.19) | ||
| Edo join 1 = 41 | Edo join 2 = 53 | Edo join 3 = 270 | | Edo join 1 = 41 | Edo join 2 = 53 | Edo join 3 = 270 | ||
| Mapping = 1; | | Mapping = 1; 1 0 -14 23 12 5; 0 1 0 0 -1 1 | ||
| Generators = 3/2; 5/4 | Generators tuning = 702.2307; 386.3245 | | Generators = 3/2; 5/4 | Generators tuning = 702.2307; 386.3245 | ||
| Optimization method = CWE | | Optimization method = CWE | ||
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| Odd limit 1 = 11 | Mistuning 1 = 0.588 | Complexity 1 = ? | | Odd limit 1 = 11 | Mistuning 1 = 0.588 | Complexity 1 = ? | ||
}} | }} | ||
'''Cassaschismic''' is a [[rank-3 | '''Cassaschismic''' is a [[rank-3 temperament]] that expands [[gary]]'s [[chain of fifths]] into the full [[11-limit]] by adding an independent [[generator]] for the [[5/1|5th]] [[harmonic]]. It is therefore a member of the [[garischismic family]] and [[olympic clan]]. | ||
The generator for 5 can be used for [[13/1|13]] and [[19/1|19]]. By moving the generators around, it can also be taken to be a ~4.5{{c}} generic aberschisma, which represents the [[schisma]], the [[aberschisma]], the [[undevicesimal schisma]], the [[352/351|minor minthma]] and many other important commas around that size. [[Tempering out]] this aberschisma results in [[cassandra]], so cassaschismic is a rank-3 [[detemperament]] of it, modifying its mapping by ±1 aberschisma to reach primes 5, 13, and 19. | The generator for 5 can be used for [[13/1|13]] and [[19/1|19]]. By moving the generators around, it can also be taken to be a ~4.5{{c}} generic aberschisma, which represents the [[schisma]], the [[aberschisma]], the [[undevicesimal schisma]], the [[352/351|minor minthma]] and many other important commas around that size. [[Tempering out]] this aberschisma results in [[cassandra]], so cassaschismic is a rank-3 [[detemperament]] of it, modifying its mapping by ±1 aberschisma to reach primes 5, 13, and 19. | ||
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== Notation == | == Notation == | ||
Cassaschismic is easily notated with [[chain-of-fifths notation]] with two extra pairs of accidentals: one for the comma step, and the other for the aberschisma step. It can therefore be seen as an addition to the cassandra chain of fifths, which itself can be seen as an addition to the 12edo chain of fifths, providing a layered-precision system of notation that ranges from rough (12), to moderately accurate (41, 53, 94), to | Cassaschismic is easily notated with [[chain-of-fifths notation]] with two extra pairs of accidentals: one for the comma step, and the other for the aberschisma step. It can therefore be seen as an addition to the cassandra chain of fifths, which itself can be seen as an addition to the 12edo chain of fifths, providing a layered-precision system of notation that ranges from rough (12), to moderately accurate (41, 53, 94), to highly accurate (217, 270, 311, …). | ||
As an example, we can use up and down arrows with shafts (↑/↓) for the comma step, and arrows without shafts (^/v) for the aberschisma step. | As an example, we can use up and down arrows with shafts (↑/↓) for the comma step, and arrows without shafts (^/v) for the aberschisma step. | ||