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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 '''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.