Septischismic: Difference between revisions
Acknowledge there are add-17,29,31 extensions, TBA |
Shift the 13-limit comma basis to the 11-limit in the infobox as the latter is nullity-2. Reduce decimal places to 2 since the infobox is meant to only give a general impression |
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{{Infobox regtemp | {{Infobox regtemp | ||
| Title = Septischismic | | Title = Septischismic | ||
| Subgroups = 2.3.5.7 | | Subgroups = 2.3.5.7, 2.3.5.7.11, 2.3.5.7.11.13.19 | ||
| Comma basis = [[ | | Comma basis = [[33554432/33480783]] (7-limit); <br>[[19712/19683]], [[41503/41472]] (11-limit); <br>[[1216/1215]], [[1540/1539]], [[1729/1728]], <br>[[2080/2079]] (2.3.5.7.11.13.19) | ||
| Mapping = 1; 1 0 -14 23 12 5; 0 1 0 0 -1 1 | |||
| Edo join 1 = 41 | Edo join 2 = 53 | Edo join 3 = 270 | | Edo join 1 = 41 | Edo join 2 = 53 | Edo join 3 = 270 | ||
| Generators = 3/2; 5/4 | Generators tuning = 702.23; 386.32 | |||
| Generators = 3/2; 5/4 | Generators tuning = 702. | |||
| Optimization method = CWE | | Optimization method = CWE | ||
| Pergen = (P8, P5, ^1) | | Pergen = (P8, P5, ^1) | ||
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| Odd limit 1 = 11 | Mistuning 1 = 0.588 | Complexity 1 = ? | | Odd limit 1 = 11 | Mistuning 1 = 0.588 | Complexity 1 = ? | ||
}} | }} | ||
'''Septischismic''' is a [[rank-3 temperament]] that expands [[gary]]'s [[chain of fifths]] into the full [[ | '''Septischismic''' is a [[rank-3 temperament]] that expands [[gary]]'s [[chain of fifths]] into the full [[7-limit]] by adding an independent [[generator]] for the [[5/1|5th]] [[harmonic]]. It is therefore a member of the [[septischismic family]]. The [[11-limit]] version, sometimes called '''cassaschismic''', inherits gary's mapping for [[11/1|11]], making it a member of the [[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|major minthma]] and many other important commas around that size. [[Tempering out]] this aberschisma results in [[cassandra]], so septischismic 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|major minthma]] and many other important commas around that size. [[Tempering out]] this aberschisma results in [[cassandra]], so septischismic is a rank-3 [[detemperament]] of it, modifying its mapping by ±1 aberschisma to reach primes 5, 13, and 19. | ||
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Septischismic is [[support]]ed by notable [[equal temperament]]s such as {{EDOs| 217, 270, 311, 364, 581, 634}}, where the aberschisma step is well represented by one or two edosteps. It is also trivially supported by cassandra edos, these being [[41edo|41]], [[53edo|53]], [[94edo|94]]. [[12edo]] supports it trivially through the 12e [[val]], where both the comma step and the aberschisma step are tempered out. It can be used in any of those forms. | Septischismic is [[support]]ed by notable [[equal temperament]]s such as {{EDOs| 217, 270, 311, 364, 581, 634}}, where the aberschisma step is well represented by one or two edosteps. It is also trivially supported by cassandra edos, these being [[41edo|41]], [[53edo|53]], [[94edo|94]]. [[12edo]] supports it trivially through the 12e [[val]], where both the comma step and the aberschisma step are tempered out. It can be used in any of those forms. | ||
There are natural strong extensions available to add primes 17 and 31, but they are | There are two natural strong extensions available to add primes 17, 29 and 31, but they are more complex and in the case of 29 and 31, accrue considerably more error into the temperament than the rest of primes. | ||
See [[Septischismic family #Septischismic]] for technical data. | See [[Septischismic family #Septischismic]] for technical data and extensions. | ||
== Interval lattice == | == Interval lattice == | ||