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Cassaschismic: Difference between revisions

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{{Idiosyncratic terms|Many of the information and terminology here has been ported and adapted from Eufalesio's user page "Ultimate" and his own TAMNAMS extensions.}}
{{Todo|inline=1|review|comment= Replace idiosyncratic terms with conventional ones. }}
{{Infobox regtemp
{{Infobox regtemp
| Title = Cassaschismic
| Title = Cassaschismic
Line 6: 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)
| 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
| 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
| Generators = 3/2; 5/4 | Generators tuning = 702.2307; 386.3245
| Optimization method = CWE
| Optimization method = CWE
| Pergen = (P8, P5, ^1)
| Pergen = (P8, P5, ^1)
| Color name = Salozo & Sasaru
| Color name = Salozo & Sasaru + Ya<br>Salozo & Sasaru (& Sathoyo (& Sanogu))
| Odd limit 1 = 11 | Mistuning 1 = 0.588 | Complexity 1 = ?
| Odd limit 1 = 11 | Mistuning 1 = 0.588 | Complexity 1 = ?
}}
}}
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]] 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, 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>
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.  


<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>
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.


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.
{{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]]. [[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.


The temperament is also known by [[Kite Giedraitis]] and [[Eufalesio]] as ''[[User:Eufalesio/Ultimate|Ultimate]]'', being nicknamed as such in 2026.
== Interval lattice ==
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 right-2 right-4"
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.
! rowspan="2" | #
{| class="wikitable" data-darkreader-inline-color=""
! colspan="2" | Aberschisma offset -1
!
! colspan="2" | Aberschisma offset 0
! colspan="2" |MC-span
|-
|-
!'''Fifth-span'''
! Cents*
!-1
! Approx. ratios
!0
! Cents*
! Approx. ratios
|-
|-
!0
| 0
|720/361
| 1195.83
|1/1
| 351/176
| 0.00
| '''1/1'''
|-
|-
!1
| 1
|256/171
| 698.06
|'''3/2'''
| 256/171
| 702.23
| '''3/2'''
|-
|-
!2
| 2
|64/57
| 200.29
|'''9/8'''
| 64/57
| 204.46
| '''9/8'''
|-
|-
!3
| 3
|'''32/19'''
| 902.52
|27/16
| '''32/19'''
| 906.69
| 27/16
|-
|-
!4
| 4
|24/19
| 404.75
|81/64
| 24/19
| 408.92
| 19/15
|-
|-
!5
| 5
|36/19
| 1106.98
|243/128
| 36/19
| 1111.15
| 19/10
|-
|-
!6
| 6
|64/45
| 609.21
|729/512
| 27/19
| 613.38
| 57/40
|-
|-
!7
| 7
|'''16/15'''
| 111.44
|77/72
| '''16/15'''
| 115.62
| 77/72
|-
|-
!8
| 8
|'''8/5'''
| 813.68
|77/48
| '''8/5'''
| 817.85
| 77/48
|-
|-
!9
| 9
|6/5
| 315.91
|77/64
| 6/5
| 320.08
| 77/64
|-
|-
!10
| 10
|9/5
| 1018.14
|65/36
| 9/5
| 1022.31
| 65/36
|-
|-
!11
| 11
|27/20
| 520.37
|65/48
| 27/20
| 524.54
| 65/48
|-
|-
!12
| 12
|81/80
| 22.60
|64/63
| 81/80
| 26.77
| 64/63
|-
|-
!13
| 13
|243/160
| 724.83
|32/21
| 38/25
| 729.00
| '''32/21'''
|-
|-
!14
| 14
|729/640
| 227.06
|'''8/7'''
| 57/50
| 231.23
| '''8/7'''
|-
|-
!15
| 15
|416/243
| 929.29
|12/7
| 77/45
| 933.46
| 12/7
|-
|-
!16
| 16
|104/81
| 431.52
|9/7
| 77/60
| 435.69
| 9/7
|-
|-
!17
| 17
|52/27
| 1133.75
|27/14
| 52/27
| 1137.92
| 27/14
|-
|-
!18
| 18
|13/9
| 635.98
|81/56
| 13/9
| 640.15
| 81/56
|-
|-
!19
| 19
|13/12
| 138.21
|88/81
| 13/12
| 142.38
| 88/81
|-
|-
!20
| 20
|'''13/8'''
| 840.44
|44/27
| '''13/8'''
| 844.61
| 44/27
|-
|-
!21
| 21
|39/32
| 342.67
|11/9
| 39/32
| 346.85
| 11/9
|-
|-
!22
| 22
|64/35
| 1044.91
|11/6
| 64/35
| 1049.08
| 11/6
|-
|-
!23
| 23
|48/35
| 547.14
|'''11/8'''
| 48/35
| 551.31
| '''11/8'''
|-
|-
!24
| 24
|36/35
| 49.37
|33/32
| 36/35
| 53.54
| 33/32
|}
|}
<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 ==
== 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 (^/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]]
[[Category:Rank-3 temperaments]]
[[Category:Microtemperaments]]
[[Category:Microtemperaments]]
[[Category:Garischismic family]]
[[Category:Garischismic family]]
[[Category:Olympic clan]]
[[Category:Olympic clan]]
Retrieved from "https://en.xen.wiki/w/Cassaschismic"