Cassaschismic: Difference between revisions

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| Comma basis = [[19712/19683]], [[41503/41472]] (11-limit); [[2080/2079]], [[4096/4095]], [[19712/19683]] (13-limit); [[1216/1215]], [[1540/1539]], [[1729/1728]], [[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

| Pergen = (P8, P5, ^1)

| Color name = Salozo & Sasaru + Yo Salozo & Sasaru (& Sathoyo (& Sanogu))

| Color name = Salozo & Sasaru + Ya Salozo & Sasaru (& Sathoyo (& Sanogu))

| 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]] 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]].

~~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 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.

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

{{Databox|~~title=~~Generators needed to reach the ~~aberschismas~~|~~contents=~~Newt ~~finds it at~~ -41 hemififths~~. ~~Cotoneum ~~finds~~ it ~~at +~~41 ~~fifths. ~~Gariwizmic ~~finds it at~~ +53 fifths ~~minus a half~~ pythagorean comma~~. ~~Vulture ~~finds it at~~ -41 1/4-fifths~~. ~~Satin ~~finds it at~~ -94 1/3-fourths.}}

{{Databox|Generators needed to reach the aberschisma|

* 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;

* 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.

}}

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. It can be used in any of those forms.

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

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|}

* In 2.3.5.7.11.13.19-subgroup CWE tuning, octave reduced

<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 ==

Cassaschismic is easily notated with [[chain-of-fifths notation]] with two extra pairs of accidentals: one for the comma step, and the other for the aberschisma step. It can therefore be seen as an addition to the cassandra chain of fifths, which itself can be seen as an addition to the 12edo chain of fifths, providing a layered-precision system of notation that ranges from rough (12), to moderately accurate (41, 53, 94), to highly accurate (217, 270, 311, …).

As an example, we can use up and down arrows with shafts (↑/↓) for the comma step, and arrows without shafts (^/v) for the aberschisma step~~. In this scheme, 4:5:6:7:9:11:13 on a C is notated as C–^↓E–G–↓B♭–D–↑↑F–v↑↑A♭~~.

As an example, we can use up and down arrows with shafts (↑/↓) for the comma step, and arrows without shafts (^/v) for the aberschisma step.

{| class="wikitable center-all"

|+Nomenclature of selected intervals

! 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)

|}

[[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
 | Pergen = (P8, P5, ^1)
-| Color name = Salozo & Sasaru + Yo<br>Salozo & Sasaru (& Sathoyo (& Sanogu))
+| Color name = Salozo & Sasaru + Ya<br>Salozo & Sasaru (& Sathoyo (& Sanogu))
 | 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]] 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]].
-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 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.
-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]], [[paramity]] and [[heptacot]]; these temperaments, instead of tempering out the aberschisma, find it deep in the generator chain.
-{{Databox|title=Generators needed to reach the aberschismas|contents=Newt finds it at -41 hemififths.<br>Cotoneum finds it at +41 fifths.<br>Gariwizmic finds it at +53 fifths minus a half pythagorean comma.<br>Vulture finds it at -41 1/4-fifths.<br>Satin finds it at -94 1/3-fourths.}}
+{{Databox|Generators needed to reach the aberschisma|
+* 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;
+* 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.
+}}
-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. It can be used in any of those forms.
 See [[Garischismic family #Cassaschismic]] for technical data.
@@ Line 186: / Line 194: @@
 | 33/32
 |}
-* In 2.3.5.7.11.13.19-subgroup CWE tuning, octave reduced
+<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 ==
 Cassaschismic is easily notated with [[chain-of-fifths notation]] with two extra pairs of accidentals: one for the comma step, and the other for the aberschisma step. It can therefore be seen as an addition to the cassandra chain of fifths, which itself can be seen as an addition to the 12edo chain of fifths, providing a layered-precision system of notation that ranges from rough (12), to moderately accurate (41, 53, 94), to highly accurate (217, 270, 311, …).
-As an example, we can use up and down arrows with shafts (↑/↓) for the comma step, and arrows without shafts (^/v) for the aberschisma step. In this scheme, 4:5:6:7:9:11:13 on a C is notated as C–^↓E–G–↓B♭–D–↑↑F–v↑↑A♭.
+As an example, we can use up and down arrows with shafts (↑/↓) for the comma step, and arrows without shafts (^/v) for the aberschisma step.
+{| class="wikitable center-all"
+|+Nomenclature of selected intervals
+! 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)
+|}
 [[Category:Cassaschismic| ]] <!-- main article -->
 [[Category:Rank-3 temperaments]]