Cassaschismic: Difference between revisions

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| Optimization method = CWE

| Pergen = (P8, P5, ^1)

| Color name = Salozo & Sasaru

| Color name = Salozo & Sasaru (& Sathurugu)

| Odd limit 1 = 11 | Mistuning 1 = 0.588 | Complexity 1 = ?

| Odd limit 1 = 11 | Mistuning 1 = 0.588 | Complexity 1 = 74

}}

'''Cassaschismic''' is a [[rank-3 temperament|rank-3]] [[regular temperament|temperament]] that expands the [[chain of fifths]] of [[gary]] 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]].

By moving the generators around, the generator for 5 can be used for [[13/1|13]] and [[19/1|19]]. It can also be taken to be a generic aberschisma, which represents the [[schisma]], the [[aberschisma]], the [[undevicesimal schisma]], and many other important commas around that size. [[Tempering out]] this tiny interval results in [[cassandra]], so cassaschismic may be viewed as a rank-3 [[detemperament]] thereof, modifying its mapping by ±1 aberschisma step to reach the rest of primes.

By moving the generators around, the generator for 5 can be used for [[13/1|13]] and [[19/1|19]]. It can also be taken to be a generic 3~5c aberschisma, which represents the [[schisma]], the [[aberschisma]], the [[undevicesimal schisma]], and many other important commas around that size. [[Tempering out]] this tiny interval results in [[cassandra]], so cassaschismic may be viewed as a rank-3 [[detemperament]] thereof, modifying its mapping by ±1 aberschisma step to reach the rest of primes.

Other subsets include [[gariwizmic]], [[cotoneum]], [[newt]], [[vulture]], [[heptacot]], [[satin]], [[witcher]]; which instead of tempering out the aberschisma, find it deep in the generator chain.

It is [[support]]ed by notable [[equal temperament]]s such as {{EDOs| 217, 270, 311, and 364 }}, where the aberschisma step is well represented by one edostep. It is also trivially supported by edos of cassandra, these being [[41edo|41]], [[53edo|53]], [[94edo|94]], and of course, [[12edo]] through the 12e [[val]], where both the comma step and the aberschisma step are tempered out, so it can be used in any of those forms as well.

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== 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 highly accurate (217, 270, 311, …).

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 for the aberschisma step. In this scheme, 4:5:6:7:9:11:13 on C is notated as C–^↓E–G–↓B♭–D–↑↑F–v↑↑A♭.

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. In this scheme, 4:5:6:7:9:11:13 on a C is notated as C–^↓E–G–↓B♭–D–↑↑F–v↑↑A♭.

[[Category:Cassaschismic| ]]

@@ Line 8: / Line 8: @@
 | Optimization method = CWE
 | Pergen = (P8, P5, ^1)
-| Color name = Salozo & Sasaru
+| Color name = Salozo & Sasaru (& Sathurugu)
-| Odd limit 1 = 11 | Mistuning 1 = 0.588 | Complexity 1 = ?
+| Odd limit 1 = 11 | Mistuning 1 = 0.588 | Complexity 1 = 74
 }}
 '''Cassaschismic''' is a [[rank-3 temperament|rank-3]] [[regular temperament|temperament]] that expands the [[chain of fifths]] of [[gary]] 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]].
-By moving the generators around, the generator for 5 can be used for [[13/1|13]] and [[19/1|19]]. It can also be taken to be a generic aberschisma, which represents the [[schisma]], the [[aberschisma]], the [[undevicesimal schisma]], and many other important commas around that size. [[Tempering out]] this tiny interval results in [[cassandra]], so cassaschismic may be viewed as a rank-3 [[detemperament]] thereof, modifying its mapping by ±1 aberschisma step to reach the rest of primes.
+By moving the generators around, the generator for 5 can be used for [[13/1|13]] and [[19/1|19]]. It can also be taken to be a generic 3~5c aberschisma, which represents the [[schisma]], the [[aberschisma]], the [[undevicesimal schisma]], and many other important commas around that size. [[Tempering out]] this tiny interval results in [[cassandra]], so cassaschismic may be viewed as a rank-3 [[detemperament]] thereof, modifying its mapping by ±1 aberschisma step to reach the rest of primes.
+Other subsets include [[gariwizmic]], [[cotoneum]], [[newt]], [[vulture]], [[heptacot]], [[satin]], [[witcher]]; which instead of tempering out the aberschisma, find it deep in the generator chain.
 It is [[support]]ed by notable [[equal temperament]]s such as {{EDOs| 217, 270, 311, and 364 }}, where the aberschisma step is well represented by one edostep. It is also trivially supported by edos of cassandra, these being [[41edo|41]], [[53edo|53]], [[94edo|94]], and of course, [[12edo]] through the 12e [[val]], where both the comma step and the aberschisma step are tempered out, so it can be used in any of those forms as well.
@@ Line 185: / Line 187: @@
 == 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 highly accurate (217, 270, 311, …).
+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 for the aberschisma step. In this scheme, 4:5:6:7:9:11:13 on C is notated as C–^↓E–G–↓B♭–D–↑↑F–v↑↑A♭.
+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. In this scheme, 4:5:6:7:9:11:13 on a C is notated as C–^↓E–G–↓B♭–D–↑↑F–v↑↑A♭.
 [[Category:Cassaschismic| ]] <!-- main article -->