9th-octave temperaments: Difference between revisions

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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 ==

The main 9th-octave temperament of interest is [[ennealimmal]] (temperament data given ~~there~~), 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).

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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Therefore, one can consider it as interpreting [[9edo]] as a circle of [[7/6]]'s (corresponding to tempering the [[septimal ennealimma]]) and as a circle of [[27/25]]'s (corresponding to tempering the [[ennealimma]]), which is an equivalent description which implies tempering the [[landscape comma]] which makes [[63/50]] equal to exactly [[3edo|a third of an octave]].

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 ~~for~~ this ~~strategy~~ is [[63edo]], a tuning doing very well in the no-17's no-19'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)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]]).~~

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-{{Stub}}
+{{Infobox fractional-octave|9}}
-{{Fractional-octave navigation|9}}
+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 there), 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).
 It can be thought of as leveraging the most accurate [[JI]] interpretations of [[9edo]], which surprisingly are all 7-limit:
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 Therefore, one can consider it as interpreting [[9edo]] as a circle of [[7/6]]'s (corresponding to tempering the [[septimal ennealimma]]) and as a circle of [[27/25]]'s (corresponding to tempering the [[ennealimma]]), which is an equivalent description which implies tempering the [[landscape comma]] which makes [[63/50]] equal to exactly [[3edo|a third of an octave]].
-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 for this strategy is [[63edo]], a tuning doing very well in the no-17's no-19'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]]).
+{{Navbox fractional-octave}}
-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]]).
+{{todo|inline=1|expand|add examples}}