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| The '''cathartic''' temperament is a rank-3 [[regular temperament]] for the 2.3.5.7.17.19 [[subgroup]] (or any immediate restrictions thereof), defined by [[tempering out]] [[1216/1215]], [[1225/1224]], and [[1445/1444]]. It is generated by a perfect fifth (~[[3/2]]) and an ultramajor second (~[[81/70]]). It is first found and explored by [[Flora Canou]].
| | #redirect [[Cathartic family #Cathartic]] |
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| See [[Cathartic family #Cathartic]] for more technical data.
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| == Overview ==
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| The temperament was at first discovered as [[7-limit]] only, where it tempers out the [[cathartic comma]], 4802000/4782969 = {{monzo| 4 -14 3 4 }}. It passes as a [[microtemperament]] with the [[9-odd-limit]] [[minimax tuning|minimax]] error of 0.98[[¢]] (1/7 cathartic comma). Extending it to the 2.3.5.7.17.19 subgroup adds virtually no additional error as the number of minimax error is also characteristic of the [[tonality diamond]] of {1, 3, 5, 7, 9, 15, 17, 19, 21}.
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| {| class="wikitable"
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| ! Name
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| ! Ratio
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| ! Cents
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| ! Monzo
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| |-
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| | Password
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| | 1216/1215
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| | 1.42430
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| | {{monzo| 6 -5 -1 0 0 0 0 1 }}
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| |-
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| | Noellisma
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| | 1225/1224
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| | 1.41383
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| | {{monzo| -3 -2 2 2 0 0 -1 }}
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| |-
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| | Aureusma
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| | 1445/1444
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| | 1.19850
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| | {{monzo| -2 0 1 0 0 0 2 -2 }}
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| |}
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| The password equates 19/15 with 81/64, 20/19 with 256/243, and 81/80 with 76/75. It characterizes the temperament with a counterpyth fifth and enables 19-odd-limit essentially tempered chords. The aureusma splits the 5/4 major third into a stack of two 19/17's and enables 19-odd-limit essentially tempered chords. The noema equates 35/34 with 36/35 and enables 25-odd-limit essentially tempered chords. Finally, cathartic tempers out 165376/165375, the decimillisma, the difference between 1216/1215 and 1225/1224.
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| {| class="wikitable"
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| ! Name
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| ! Ratio
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| ! Cents
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| ! Monzo
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| |-
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| | Decimillisma
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| | 165376/165375
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| | 0.01047
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| | {{monzo| 9 -3 -3 -2 0 0 1 1 }}
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| |}
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| Recommendable edo tunings include 94, 99, 118, 193, 217, 311, 410 and 721, with 721 giving the OPV for the 2.3.5.7.17.19 subgroup.
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| Decent amount of harmonic resources are provided by a simple 9-note scale. Flora Canou commented:
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| :''— It sounds somewhat like a Phrygian scale but the abundance of small intervals of [[28/27]] makes it melodically active.''
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| 14- and 19-note scales are also highly concise for the temperament. See [[#scales]] for more information.
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| == Interval lattice ==
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| <gallery>
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| File:Lattice Orthocanousmic.png
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| File:Lattice Orthocanousmic Rearranged.png
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| </gallery>
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| == Scales ==
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| {{Main| Cathartic scales }}
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| Fokker blocks of 9, 14 and 19 tones are among the most useful cathartic scales.
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| [[Category:Cathartic| ]] <!-- main article --> | | [[Category:Cathartic| ]] <!-- main article --> |
| [[Category:Rank-3 temperaments]] | | [[Category:Rank-3 temperaments]] |
| | [[Category:Microtemperaments]] |
| [[Category:Cathartic family]] | | [[Category:Cathartic family]] |