Cassaschismic: Difference between revisions

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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)

| Edo join 1 = 41 | Edo join 2 = 53 | Edo join 3 = 270

| Mapping = 1; 0 1 0 -14 23 12 5; 0 0 1 0 0 -1 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

| Optimization method = CWE

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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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Here is a quick compressed cheat sheet of octave-reduced intervals. This is a simplification with many (infinitely many) intervals left out for the sake of brevity. For every entry here, ratios here represent pitch-classes and their pitch class inverses; so for instance 8/5 pitch class is mapped to 8 fifths - 1 aberschisma step, being the octave inverse of 5/4 pitch class negates the mappings so it is found at -8 fifths + 1 aberschisma step. There are no octave reduced primes or prime inverses with positive fifth step and aberschisma step.

An image is provided instead showing copies of a [[Standard Lumatone mapping for Pythagorean]]-like arrangement of the chain of fifths, copied thrice to access most of the no-17 27-odd-limit and then some.

[[File:UltimateMapGraphOrtho.png|thumb|750x750px|A visualization of Ultimate (cassaschismic temperament), with 3 copies of a chain of fifths laid out in a Lumatone-like fashion.]]

{| class="wikitable center-1 right-2 right-4"

! rowspan="2" | #

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|}

<nowiki>*</nowiki> In 2.3.5.7.11.13.19-subgroup CWE tuning, octave reduced

[https://www.desmos.com/calculator/pbyqpjgrrn Here] is a Desmos graph showing how cassaschismic edos up to 311 [[8afdo|harmonic mode 8]] (green), and [[5L 7s]] 6|5 (red). The purple line on 12 is patent val p11, 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 the line of cassandra; since halving it results in 270edo, it is used instead, also to showcase the use of aberschismas to reach primes 5, 13, and 19.

== Notation ==

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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 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| ]]

[[Category:Rank-3 temperaments]]

@@ Line 4: / Line 4: @@
 | 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
-| Mapping = 1; 0 1 0 -14 23 12 5; 0 0 1 0 0 -1 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
 | Optimization method = CWE
@@ 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 34: / Line 34: @@
 Here is a quick compressed cheat sheet of octave-reduced intervals. This is a simplification with many (infinitely many) intervals left out for the sake of brevity. For every entry here, ratios here represent pitch-classes and their pitch class inverses; so for instance 8/5 pitch class is mapped to 8 fifths - 1 aberschisma step, being the octave inverse of 5/4 pitch class negates the mappings so it is found at -8 fifths + 1 aberschisma step. There are no octave reduced primes or prime inverses with positive fifth step and aberschisma step.
+An image is provided instead showing copies of a [[Standard Lumatone mapping for Pythagorean]]-like arrangement of the chain of fifths, copied thrice to access most of the no-17 27-odd-limit and then some.
+[[File:UltimateMapGraphOrtho.png|thumb|750x750px|A visualization of Ultimate (cassaschismic temperament), with 3 copies of a chain of fifths laid out in a Lumatone-like fashion.]]
 {| class="wikitable center-1 right-2 right-4"
 ! rowspan="2" | #
@@ Line 195: / Line 197: @@
 |}
 <nowiki>*</nowiki> In 2.3.5.7.11.13.19-subgroup CWE tuning, octave reduced
-[https://www.desmos.com/calculator/pbyqpjgrrn Here] is a Desmos graph showing how cassaschismic edos up to 311 [[8afdo|harmonic mode 8]] (green), and [[5L 7s]] 6|5 (red). The purple line on 12 is patent val p11, 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 the line of cassandra; since halving it results in 270edo, it is used instead, also to showcase the use of aberschismas to reach primes 5, 13, and 19.
 == Notation ==
@@ Line 226: / Line 226: @@
 | C–^Eb (upminor third)
 |}
+== See also ==
+* [[User:Eufalesio/Ultimate]] – An opinion-based derivation of cassaschismic
+== External links ==
+* [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 -->
 [[Category:Rank-3 temperaments]]