Cassaschismic: Difference between revisions

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| Odd limit 1 = 11 | Mistuning 1 = 0.588 | Complexity 1 = ?
| Odd limit 1 = 11 | Mistuning 1 = 0.588 | Complexity 1 = ?
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Cassaschismic is a rank-3 temperament for the 11-, 13-limit and beyond. It is a member of [[garischismic clan]] and [[olympic clan]]. It extends the chain-of-fifths sequence of [[gary]] into the full 11-limit by adding an independent generator for 5/4, which naturally can be then used for 13/8 and 19/16.  
'''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]].  


By moving the generators around, this generator can be instead taken to be a tiny <span data-darkreader-inline-color="">3~5c ''minicomma'' (from this point forward refered to as "MC") that represents 385/384, 352/351, 5120/5103, 513/512, the schisma... etc; acting as a rank-3 detemper of [[cassandra]], where cassandra mappings are modified by ±1 MC to reach the rest of primes.</span>
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 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.  


<span data-darkreader-inline-color="">The</span> [[pergen]] <span data-darkreader-inline-color="">is (P8, P5, ^1), where ^1 is the MC. 4:5:6:7:9:11:13 is notated as P1 ^</span>'''↓'''<span data-darkreader-inline-color="">M3 P5</span> '''↓'''<span data-darkreader-inline-color="">m7 M9 ↑↑11 v↑↑m13, where ↑/'''↓''' represents alteration by a [[Pythagorean comma|pyth]]-[[64/63|septimal]] comma (from this point forward refered to as "PC")</span>
It 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.


It is supported by notable edos such as {{EDOs|41, 53, 94, 217, 270, 311}}. 41 and 53 are known for their incredibly accurate fifths; 270edo provides an astonishingly accurate equal tuning for cassaschismic in the no-17 19-odd-limit; 311edo notably provides an optimized extension to the full 41-odd-limit with a slightly worse 13-limit. It is also trivially supported by 12edo through the 12e val, where the pythagorean comma and MC are tempered out. See [[Garischismic family|Garischismic family#Cassaschismic]] for technical data.
See [[Garischismic family #Cassaschismic]] for technical data.


The temperament is also known by [[Kite Giedraitis]] and [[Eufalesio]] as ''[[User:Eufalesio/Ultimate|Ultimate]]'', being nicknamed as such in 2026.
== Interval chain ==
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.


== Interval chain ==
{| class="wikitable center-1"
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 MC, being the octave inverse of 5/4 pitch class negates the mappings so it is found at -8 fifths + 1 MC. There are no octave reduced primes or prime inverses with positive fifth-span and MC-span.
! #
{| class="wikitable" data-darkreader-inline-color=""
! Aberschisma offset -1  
!
! Aberschisma offset 0
! colspan="2" |MC-span
|-
!'''Fifth-span'''
!-1
!0
|-
|-
!0
| 0
|720/361
| 720/361
|1/1
| '''1/1'''
|-
|-
!1
| 1
|256/171
| 256/171
|'''3/2'''
| '''3/2'''
|-
|-
!2
| 2
|64/57
| 64/57
|'''9/8'''
| '''9/8'''
|-
|-
!3
| 3
|'''32/19'''
| '''32/19'''
|27/16
| 27/16
|-
|-
!4
| 4
|24/19
| 24/19
|81/64
| 81/64
|-
|-
!5
| 5
|36/19
| 36/19
|243/128
| 243/128
|-
|-
!6
| 6
|64/45
| 64/45
|729/512
| 729/512
|-
|-
!7
| 7
|'''16/15'''
| '''16/15'''
|77/72
| 77/72
|-
|-
!8
| 8
|'''8/5'''
| '''8/5'''
|77/48
| 77/48
|-
|-
!9
| 9
|6/5
| 6/5
|77/64
| 77/64
|-
|-
!10
| 10
|9/5
| 9/5
|65/36
| 65/36
|-
|-
!11
| 11
|27/20
| 27/20
|65/48
| 65/48
|-
|-
!12
| 12
|81/80
| 81/80
|64/63
| 64/63
|-
|-
!13
| 13
|243/160
| 243/160
|32/21
| 32/21
|-
|-
!14
| 14
|729/640
| 729/640
|'''8/7'''
| '''8/7'''
|-
|-
!15
| 15
|416/243
| 416/243
|12/7
| 12/7
|-
|-
!16
| 16
|104/81
| 104/81
|9/7
| 9/7
|-
|-
!17
| 17
|52/27
| 52/27
|27/14
| 27/14
|-
|-
!18
| 18
|13/9
| 13/9
|81/56
| 81/56
|-
|-
!19
| 19
|13/12
| 13/12
|88/81
| 88/81
|-
|-
!20
| 20
|'''13/8'''
| '''13/8'''
|44/27
| 44/27
|-
|-
!21
| 21
|39/32
| 39/32
|11/9
| 11/9
|-
|-
!22
| 22
|64/35
| 64/35
|11/6
| 11/6
|-
|-
!23
| 23
|48/35
| 48/35
|'''11/8'''
| '''11/8'''
|-
|-
!24
| 24
|36/35
| 36/35
|33/32
| 33/32
|}
|}


== Notation ==
== Notation ==
Much like [[schismic]], using cassaschismic <span data-darkreader-inline-color="">can be a challenge because it defies the tradition of diatonic</span> {{w|tertian harmony}} <span data-darkreader-inline-color="">in</span> [[chain-of-fifths notation]]<span data-darkreader-inline-color="">; The just major triad on C is not C–E–G like in</span> [[meantone]]<span data-darkreader-inline-color="">, and it isn't C–F♭–G or C-'''↓'''E-G like in schismic either. Because it is a rank-3 temperament, it needs two extra pairs of accidentals, one for the PC (like ↑/'''↓)''', and one for the MC (like ^/v).</span>
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, …).  


It can be instead see as an "addition" to the schismic/garibaldi/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 accurate (41,53,94), to incredibly accurate (217,270,311...).
As an example, we can use up and down arrows with shafts (↑/↓) for the comma step, and arrows without shafts for the aberschisma step. In this scheme, 4:5:6:7:9:11:13 on C is notated as C–^↓E–G–↓B♭–D–↑↑F–v↑↑A♭.  


[[Category:Rank-3 temperaments]]
[[Category:Rank-3 temperaments]]

Revision as of 08:08, 18 May 2026

This article or section contains multiple idiosyncratic terms. Such terms are used by only a few people and are not regularly used within the community.

Terms: Many of the information and terminology here has been ported and adapted from Eufalesio's user page "Ultimate".

Todo: review

Replace idiosyncratic terms with conventional ones.

Cassaschismic
Subgroups 2.3.5.7.11, 2.3.5.7.11.13, 2.3.5.7.11.13.19
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)
Reduced mapping ⟨1; 0 1 0 -14 23 12 5; 0 0 1 0 0 -1 1]
ET join 41 & 53 & 270
Generators (CWE) ~3/2 = 702.2307 ¢, ~5/4 = 386.3245 ¢
MOS scales n/a
Ploidacot n/a
Pergen (P8, P5, ^1)
Color name Salozo & Sasaru
Minimax error 11-odd-limit: 0.588 ¢
Target scale size 11-odd-limit: ? notes

Cassaschismic is a rank-3 temperament that expands the chain of fifths of gary into the full 11-limit by adding an independent generator for the 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 and 19. It can also be taken to be a 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.

It is supported by notable equal temperaments such as 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 41, 53, 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.

See Garischismic family #Cassaschismic for technical data.

Interval chain

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.

# Aberschisma offset -1 Aberschisma offset 0
0 720/361 1/1
1 256/171 3/2
2 64/57 9/8
3 32/19 27/16
4 24/19 81/64
5 36/19 243/128
6 64/45 729/512
7 16/15 77/72
8 8/5 77/48
9 6/5 77/64
10 9/5 65/36
11 27/20 65/48
12 81/80 64/63
13 243/160 32/21
14 729/640 8/7
15 416/243 12/7
16 104/81 9/7
17 52/27 27/14
18 13/9 81/56
19 13/12 88/81
20 13/8 44/27
21 39/32 11/9
22 64/35 11/6
23 48/35 11/8
24 36/35 33/32

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 for the aberschisma step. In this scheme, 4:5:6:7:9:11:13 on C is notated as C–^↓E–G–↓B♭–D–↑↑F–v↑↑A♭.

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