561/560: Difference between revisions
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'''561/560''', the '''monardisma''' is an [[unnoticeable comma|unnoticeable]] [[17-limit]] [[superparticular]] [[comma]] of about 3.1 [[cent]]s. It is the difference between [[33/32]] and [[35/34]]. | '''561/560''', the '''monardisma''' is an [[unnoticeable comma|unnoticeable]] [[17-limit]] [[superparticular]] [[comma]] of about 3.1 [[cent]]s. It is the difference between [[33/32]] and [[35/34]]. | ||
== Temperaments == | == Temperaments == | ||
[[Tempering out]] this comma in the 2.3.5.7.11.17 [[subgroup]] | [[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; | * [[14/11]] and [[20/17]] become [[fifth complement]]s; | ||
* [[11/10]] and [[17/14]] become [[fourth complement]]s. | * [[11/10]] and [[17/14]] become [[fourth complement]]s. | ||
=== Monardic === | |||
[[Subgroup]]: 2.3.5.7.11.17 | |||
[[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 === | === Monardismic === | ||
| Line 31: | Line 55: | ||
| ⟨ || 0 || 0 || 0 || 0 || 0 || 1 || 0 || ]] | | ⟨ || 0 || 0 || 0 || 0 || 0 || 1 || 0 || ]] | ||
|} | |} | ||
: mapping generators: ~2, ~3, ~5, ~7, ~11, ~13 | |||
: | |||
[[Optimal tuning]]s: | [[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, ~3/2 = 701.6204, ~5/4 = 386.3575, ~7/4 = 968.9591, ~11/8 = 550.2059, ~13/8 = 840.0917 | * [[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 }}* | {{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]] | <nowiki/>* [[optimal patent val]]: [[484edo|484]] | ||
[[Badness]] (Sintel): 0.398 | |||
== Etymology == | == Etymology == | ||