Septischismic: Difference between revisions

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

}}

'''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]].

'''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]].

~~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 ~~3–5~~{{c}} 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~~.

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]], and [[~~vulture~~]]; these temperaments, instead of tempering out the aberschisma, find it deep in the generator chain.

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. Cotoneum (41 & 217) finds it at -41 fifths, equating it with the 41-comma. Gariwizmic (94 & 270) finds it at +53 fifths (mercator comma) ~~minus a half pythagorean comma~~. Vulture (53 & 217) finds it at -41 1/4-fifths. Satin (94 & 217) finds it at -94 1/3-fourths.}}

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

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.

See [[Garischismic family #Cassaschismic]] for technical data.

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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 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 ~~comes from the fact that~~ 135edo is next in the line of cassandra~~, but~~ halving it results in 270edo, ~~so the latter~~ is used.

[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.

== Notation ==

@@ Line 11: / Line 11: @@
 | Odd limit 1 = 11 | Mistuning 1 = 0.588 | Complexity 1 = ?
 }}
-'''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]].
+'''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]].
-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 3–5{{c}} 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.
+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]], and [[vulture]]; these temperaments, instead of tempering out the aberschisma, find it deep in the generator chain.
+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) minus a half pythagorean comma.<br>Vulture (53 & 217) finds it at -41 1/4-fifths.<br>Satin (94 & 217) finds it at -94 1/3-fourths.}}
+{{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]], 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.
+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.
 See [[Garischismic family #Cassaschismic]] for technical data.
@@ Line 188: / Line 188: @@
 <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 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 comes from the fact that 135edo is next in the line of cassandra, but halving it results in 270edo, so the latter is used.
+[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.
 == Notation ==