Cassaschismic: Difference between revisions

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'''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|major 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.

Other rank-2 temperaments of cassaschismic include [[cotoneum]], [[~~gariwizmic~~]], [[~~newt~~]], [[satin]], [[vulture]], [[paramity]] and [[heptacot]]; these temperaments, instead of tempering out the aberschisma, find it deep in the generator chain.

Other rank-2 temperaments of cassaschismic include [[cotoneum]], [[newt]], [[gariwizmic]], [[satin]], [[vulture]], [[paramity]] and [[heptacot]]; these temperaments, instead of tempering out the aberschisma, find it deep in the generator chain.

{{Databox|Generators needed to reach the aberschisma|

* Cotoneum (41 & 217): -41 fifths, equating it with the 41-comma;

* Newt (41 & 270): -41 hemififths;

* Cotoneum (41 & 217): -41 fifths, equating it with the 41-comma;

* Gariwizmic (94 & 270): +53 fifths (mercator comma) - 1/2 pythagorean comma;

* Satin (94 & 217): -94 1/3-fourths;

* Vulture (53 & 217): -41 1/4-fifths;

* Satin (94 & 217): -94 1/3-fourths;

* Paramity (53 & 311): -53 1/5-elevenths;

* Heptacot (12e & 311): 12 1/7-fifths.

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| C–^Eb (upminor third)

|}

== See also ==

* [[User:Eufalesio/Ultimate]] – An opinion-based derivation of cassaschismic

== External links ==

* [https://www.desmos.com/calculator/pbyqpjgrrn] A [https://www.desmos.com/calculator Desmos] graph showing cassaschismic edos up to 311, [[8afdo|harmonic mode 8]] (green), and [[5L 7s]] 6|5 (red). The purple line on 12 is the [[patent val]] prime 11, which is not used in cassaschismic. The blue dots indicate going up and down by Pythagorean commas in the 12L 29s scale, and the orange dots indicate the leftover edosteps. The jump from 94 to 270 is due to 135edo being next in line for cassandra; since halving it results in 270edo, it is used instead, and also to showcase the use of aberschismas to reach primes 5, 13, and 19.

* [https://www.desmos.com/calculator/pbyqpjgrrn Ultimate mapgraph] – A [https://www.desmos.com/calculator Desmos] graph by [[Eufalesio]] showing cassaschismic edos up to 311, [[8afdo|harmonic mode 8]] (green), and [[5L 7s]] 6|5 (red). The purple line on 12 is the [[patent val]] prime 11, which is not used in cassaschismic. The blue dots indicate going up and down by Pythagorean commas in the 12L 29s scale, and the orange dots indicate the leftover edosteps. The jump from 94 to 270 is due to 135edo being next in line for cassandra; since halving it results in 270edo, it is used instead, and also to showcase the use of aberschismas to reach primes 5, 13, and 19.

[[Category:Cassaschismic| ]]

@@ Line 13: / Line 13: @@
 '''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|major 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.
-Other rank-2 temperaments of cassaschismic include [[cotoneum]], [[gariwizmic]], [[newt]], [[satin]], [[vulture]], [[paramity]] and [[heptacot]]; these temperaments, instead of tempering out the aberschisma, find it deep in the generator chain.
+Other rank-2 temperaments of cassaschismic include [[cotoneum]], [[newt]], [[gariwizmic]], [[satin]], [[vulture]], [[paramity]] and [[heptacot]]; these temperaments, instead of tempering out the aberschisma, find it deep in the generator chain.
 {{Databox|Generators needed to reach the aberschisma|
+* Cotoneum (41 & 217): -41 fifths, equating it with the 41-comma;
 * Newt (41 & 270): -41 hemififths;
-* Cotoneum (41 & 217): -41 fifths, equating it with the 41-comma;
 * Gariwizmic (94 & 270): +53 fifths (mercator comma) - 1/2 pythagorean comma;
+* Satin (94 & 217): -94 1/3-fourths;
 * Vulture (53 & 217): -41 1/4-fifths;
-* Satin (94 & 217): -94 1/3-fourths;
 * Paramity (53 & 311): -53 1/5-elevenths;
 * Heptacot (12e & 311): 12 1/7-fifths.
@@ Line 224: / Line 224: @@
 | C–^Eb (upminor third)
 |}
+== See also ==
+* [[User:Eufalesio/Ultimate]] – An opinion-based derivation of cassaschismic
 == External links ==
-* [https://www.desmos.com/calculator/pbyqpjgrrn] A [https://www.desmos.com/calculator Desmos] graph showing cassaschismic edos up to 311, [[8afdo|harmonic mode 8]] (green), and [[5L 7s]] 6|5 (red). The purple line on 12 is the [[patent val]] prime 11, which is not used in cassaschismic. The blue dots indicate going up and down by Pythagorean commas in the 12L 29s scale, and the orange dots indicate the leftover edosteps. The jump from 94 to 270 is due to 135edo being next in line for cassandra; since halving it results in 270edo, it is used instead, and also to showcase the use of aberschismas to reach primes 5, 13, and 19.
+* [https://www.desmos.com/calculator/pbyqpjgrrn Ultimate mapgraph] – A [https://www.desmos.com/calculator Desmos] graph by [[Eufalesio]] showing cassaschismic edos up to 311, [[8afdo|harmonic mode 8]] (green), and [[5L 7s]] 6|5 (red). The purple line on 12 is the [[patent val]] prime 11, which is not used in cassaschismic. The blue dots indicate going up and down by Pythagorean commas in the 12L 29s scale, and the orange dots indicate the leftover edosteps. The jump from 94 to 270 is due to 135edo being next in line for cassandra; since halving it results in 270edo, it is used instead, and also to showcase the use of aberschismas to reach primes 5, 13, and 19.
 [[Category:Cassaschismic| ]] <!-- main article -->