Cassaschismic: Difference between revisions

'''Cassaschismic''' is a [[rank-3 temperament|rank-3]] [[regular temperament|temperament]] that expands [[gary]]'s [[chain of fifths]] into the full [[11-limit]] by adding an independent [[generator]] for [[5/1|5th]] [[harmonic]]. It is therefore a member of the [[garischismic family]] and [[olympic clan]].  

The generator for [[5/1|p5]] can be used for [[13/1|p13]] and [[19/1|p19]]. 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 p5, p13, p19.  

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

{{Databox|Generators needed to reach the aberschisma|Newt (41 & 270) finds it at -41 hemififths.<br>Cotoneum (41 & 217) finds it at -41 fifths, equating it with the 41-comma.<br>Gariwizmic (94 & 270) finds it at +53 fifths (mercator comma) - 1/2-pythcomma.<br>Vulture (53 & 217) finds it at -41 1/4-fifths.<br>Satin (94 & 217) finds it at -94 1/3-fourths.<br>Heptacot (12e & 311) finds it at 12 1/7-fifths. <br>Paramity (53 & 311) finds it at -53 1/5-elevenths.}}

Cassaschismic 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]]. [[12edo]] supports it trivially through the 12e [[val]], where both the comma step and the aberschisma step are tempered out.  

[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 isn't 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 p5, p13, p19.

{| class="wikitable" style="text-align:center; vertical-align:middle;"

!Ratio

!Example on C

|3/2

|C–G (perfect fifth)

|5/4

|C–^↓E (upsubmajor third)

|7/4

|C–↓Bb (subminor seventh)

|11/8

|C–↑↑F (hyperfourth)

|13/8

|C–v↑↑Ab (downhyperminor sixth)

|19/16

|C–^Eb (upminor third)