561/560: Difference between revisions

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 {{Infobox Interval
-| Name = monardisma, <br>tsaharuk comma
+| Name = monardisma
 | Color name = 17o1org1, <br>solorugu unison, <br>Solorugu comma
 | Comma = yes
 }}
+'''561/560''', the '''monardisma''' is an [[unnoticeable comma|unnoticeable]] [[17-limit]] [[superparticular]] [[comma]] of about 3.09 [[cent]]s. It is the difference between the pentacircle minor third, [[33/28]], and the diatismic minor third, [[20/17]], on account of it being the difference between the diatisma, [[136/135]], and the pentacircle comma, [[896/891]]. It also represents the difference between the supraminor thirds of [[17/14]] and [[40/33]], and between the quartertones of [[33/32]] and [[35/34]].
-'''561/560''', the '''monardisma''' or '''tsaharuk comma''' is a [[17-limit]] [[superparticular]] [[comma]] of about 3.1 [[cent]]s. It is the difference between [[33/32]] and [[35/34]]. Tempering it out defines the '''monardismic temperament''', which enables [[monardismic chords]].
+== Temperaments ==
+[[Tempering out]] this comma in the full 17-limit results in the rank-6 '''monardismic''' temperament, or in the 2.3.5.7.11.17 [[subgroup]], the rank-5 '''monardic''' temperament, enabling [[monardismic chords]]. This temperament has a number of notable properties:
+* [[14/11]] and [[20/17]] become [[fifth complement]]s;
+* [[11/10]] and [[17/14]] become [[fourth complement]]s.
+=== Monardic ===
+[[Subgroup]]: 2.3.5.7.11.17
+[[Subgroup val|Subgroup-val]] [[mapping]]: <br>
+{| class="right-all"
+|-
+| [⟨ || 1 || 0 || 0 || 0 || 0 || 4 || ],
+|-
+| ⟨ || 0 || 1 || 0 || 0 || 0 || -1 || ],
+|-
+| ⟨ || 0 || 0 || 1 || 0 || 0 || 1 || ],
+|-
+| ⟨ || 0 || 0 || 0 || 1 || 0 || 1 || ],
+|-
+| ⟨ || 0 || 0 || 0 || 0 || 1 || -1 || ]]
+|}
+: mapping generators: ~2, ~3, ~5, ~7, ~11
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.2042{{c}}, ~3/2 = 701.6222{{c}}, ~5/4 = 386.1804{{c}}, ~7/4 = 968.8197{{c}}, ~11/8 = 550.0934{{c}}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.6204{{c}}, ~5/4 = 386.3575{{c}}, ~7/4 = 968.9591{{c}}, ~11/8 = 550.2059{{c}}
+{{Optimal ET sequence|legend=1| 12, 14c, 15, 17cg, 19g, 22, 31, 41g, 43, 46, 58, 68, 72, 118, 183, 255, 301, 373g, 674eg, 768egg, 1290beeeggg }}
+[[Badness]] (Sintel): 0.204
+=== Monardismic ===
+[[Subgroup]]: 2.3.5.7.11.13.17
+[[Mapping]]: <br>
+{| class="right-all"
+|-
+| [⟨ || 1 || 0 || 0 || 0 || 0 || 0 || 4 || ],
+|-
+| ⟨ || 0 || 1 || 0 || 0 || 0 || 0 || -1 || ],
+|-
+| ⟨ || 0 || 0 || 1 || 0 || 0 || 0 || 1 || ],
+|-
+| ⟨ || 0 || 0 || 0 || 1 || 0 || 0 || 1 || ],
+|-
+| ⟨ || 0 || 0 || 0 || 0 || 1 || 0 || -1 || ],
+|-
+| ⟨ || 0 || 0 || 0 || 0 || 0 || 1 || 0 || ]]
+|}
+: mapping generators: ~2, ~3, ~5, ~7, ~11, ~13
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.2042{{c}}, ~3/2 = 701.6222{{c}}, ~5/4 = 386.1804{{c}}, ~7/4 = 968.8197{{c}}, ~11/8 = 550.0934{{c}}, ~13/8 = 839.9145{{c}}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.6204{{c}}, ~5/4 = 386.3575{{c}}, ~7/4 = 968.9591{{c}}, ~11/8 = 550.2059{{c}}, ~13/8 = 840.0917{{c}}
+{{Optimal ET sequence|legend=1| 12f, 14cf, 15, 17cg, 19g, 22, 26, 29g, 31, 38df, 41g, 43, 46, 58, 72, 103, 111, 130, 140, 183, 243e, 301, 373g, 886eefgg, 1362bdeeefgggg }} *
+<nowiki/>* [[optimal patent val]]: [[484edo|484]]
+[[Badness]] (Sintel): 0.398
+== Etymology ==
+This comma was named by [[Scott Dakota]] in 2023 after the [[Monarda]] scale, a 12-tone scale of [[tannic]], where this is one of the commas being tempered out.
 == See also ==
-* [[Unnoticeable comma]]
 * [[List of superparticular intervals]]
 [[Category:Monardismic]]
+[[Category:Commas with unknown etymology]]