9th-octave temperaments: Difference between revisions
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{{ | {{Infobox fractional-octave|9}} | ||
Although 9edo itself is not particularly accurate for low-complexity harmonics, some temperaments which are multiples of 9 are. | Although 9edo itself is not particularly accurate for low-complexity harmonics, some temperaments which are multiples of 9 are. | ||
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== Ennealimmal == | == Ennealimmal == | ||
{{See also | Ennealimmal}} | {{See also | Ennealimmal}} | ||
{{See also | Tritrizo clan #Ennealimmal}} | |||
The main 9th-octave temperament of interest is [[ennealimmal]] (temperament data given at [[Tritrizo clan#Ennealimmal|Tritrizo clan]]), notable for being the [[7-limit]] [[microtemperament]] tempering the two smallest [[superparticular interval]]s of the 7-limit, [[2401/2400]] = S49 = ([[49/40]])/([[60/49]]) and [[4375/4374]] = S25/S27 = ([[7/6|28/24]])/([[27/25]])<sup>2</sup>, with the smallest [[patent val]] [[edo]] tunings being [[27edo]] (a sharp [[superpyth]] tuning supporting [[modus]] and [[augene]]) and [[45edo]] (the [[optimal patent val]] of [[flattone]]), which sum to [[72edo]] (the smallest edo tuning that starts to show the accuracy of ennealimmal, with a mild flat tendency) and relatedly [[99edo]] (the second such tuning, with a mild sharp tendency instead). | The main 9th-octave temperament of interest is [[ennealimmal]] (temperament data given at [[Tritrizo clan#Ennealimmal|Tritrizo clan]]), notable for being the [[7-limit]] [[microtemperament]] tempering the two smallest [[superparticular interval]]s of the 7-limit, [[2401/2400]] = S49 = ([[49/40]])/([[60/49]]) and [[4375/4374]] = S25/S27 = ([[7/6|28/24]])/([[27/25]])<sup>2</sup>, with the smallest [[patent val]] [[edo]] tunings being [[27edo]] (a sharp [[superpyth]] tuning supporting [[modus]] and [[augene]]) and [[45edo]] (the [[optimal patent val]] of [[flattone]]), which sum to [[72edo]] (the smallest edo tuning that starts to show the accuracy of ennealimmal, with a mild flat tendency) and relatedly [[99edo]] (the second such tuning, with a mild sharp tendency instead). | ||
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An alternative 7-limit 9th-octave temperament supported by more edos is to preserve the mapping of 7/6 but not that of 27/25, resulting in [[septiennealimmal]], with many extensions possible. An important edo of interest that takes this route is [[63edo]], a tuning doing very well in the no-17's no-19's (no-37's) no-41's [[47-limit]] if you forgive inconsistencies arising from its [[magic]]-tempered ~[[5/4]]. | An alternative 7-limit 9th-octave temperament supported by more edos is to preserve the mapping of 7/6 but not that of 27/25, resulting in [[septiennealimmal]], with many extensions possible. An important edo of interest that takes this route is [[63edo]], a tuning doing very well in the no-17's no-19's (no-37's) no-41's [[47-limit]] if you forgive inconsistencies arising from its [[magic]]-tempered ~[[5/4]]. | ||
Some higher-limit interpretations of interest for both routes are [[14/13]]~[[13/12]] (tempering [[169/168|S13]]) for lower-complexity interpretations of 1\9 [[34/27]] for 1\3 (tempering [[19683/19652]] to give an interpretation to [[3edo]]) and the "rooted/harmonic wolf fifth" [[47/32]] for 5\9, by tempering ([[64/47]])/[[49/36|(7/6)<sup>2</sup>]] = [[2304/2303|S48]] = ([[48/47]])/([[49/48]]). | Some higher-limit interpretations of interest for both routes are [[14/13]]~[[13/12]] (tempering [[169/168|S13]]) for lower-complexity interpretations of 1\9, [[34/27]] for 1\3 (tempering [[19683/19652]] to give an interpretation to [[3edo]]), and the "rooted/harmonic wolf fifth" [[47/32]] for 5\9, by tempering ([[64/47]])/[[49/36|(7/6)<sup>2</sup>]] = [[2304/2303|S48]] = ([[48/47]])/([[49/48]]). | ||
{{Navbox fractional-octave}} | {{Navbox fractional-octave}} | ||