Cassaschismic: Difference between revisions

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-#redirect [[Garischismic family #Cassaschismic]]
 [[Category:Rank-3 temperaments]]
 [[Category:Microtemperaments]]
 [[Category:Garischismic family]]
 [[Category:Olympic clan]]
+{{Infobox regtemp|Title=Cassaschismic|Subgroups=2.3.5.7.11.13.19|Comma basis=[[19712/19683]], [[41503/41472]]|Edo join 1=217|Edo join 2=270|Mapping=1; 0 1 0 -14 23 12 5; 0 0 1 0 0 -1 1|Generators=3/2|Generators tuning=701.731|Optimization method=TBA|MOS scales=[[2L&nbsp;3s]], [[5L&nbsp;2s]], [[5L&nbsp;7s]], [[12L&nbsp;5s]]|Pergen=(P8, P5)|Color name=Salozo & Saruru|Odd limit 1=15|Mistuning 1=TBA|Complexity 1=12|Odd limit 2=13-limit 225|Mistuning 2=TBA|Complexity 2=29|Edo join 3=311}}{{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.}}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.
+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>
+<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 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.
+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 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=""
+!
+! colspan="2" |MC-span
+|-
+!'''Fifth-span'''
+!-1
+!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
+|}
+== Scales ==
+* [[5L 7s]] (p-chromatic)
+* [[12L 5s]] (p-enharmonic)
+* [[12L 17s]] (pythagotonic)
+* [[12L 29s]] (pythamystonic)
+* [[41L 12s]] (pythomerc)
+* [[41L 53s]] (''garytonic'')
+== 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>
+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...).
+== Tunings ==
+TBA