User:Godtone/Ennealimmal: Difference between revisions

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~~{{Infobox fractional-octave|9}}~~

The main 9th-octave temperament of interest is [[ennealimmal]] (temperament data given at [[Septiennealimmal clan #Ennealimmal|Septiennealimmal 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]])2, 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).

~~Although 9edo itself is not particularly accurate for low-complexity harmonics, some temperaments which are multiples of 9 are.~~

~~For example, these multiple-of-9 EDOs appear in some [[zeta]] edo lists: {{EDOs|27, 72, 99, 171, 270, 342 and 441.}} (List is not exhaustive.)~~

~~== 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]])2, 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).

It can be thought of as leveraging the most accurate [[JI]] interpretations of [[9edo]], which surprisingly are all 7-limit:

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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]].

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)2]] = [[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)2]] = [[2304/2303|S48]] = ([[48/47]])/([[49/48]]).

~~{{Navbox fractional-octave}}~~

~~{{todo|inline=1|expand|add examples}}~~

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-{{Infobox fractional-octave|9}}
+The main 9th-octave temperament of interest is [[ennealimmal]] (temperament data given at [[Septiennealimmal clan #Ennealimmal|Septiennealimmal 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).
-Although 9edo itself is not particularly accurate for low-complexity harmonics, some temperaments which are multiples of 9 are.
-For example, these multiple-of-9 EDOs appear in some [[zeta]] edo lists: {{EDOs|27, 72, 99, 171, 270, 342 and 441.}} (List is not exhaustive.)
-== 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).
 It can be thought of as leveraging the most accurate [[JI]] interpretations of [[9edo]], which surprisingly are all 7-limit:
@@ Line 19: / Line 10: @@
 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}}
-{{todo|inline=1|expand|add examples}}