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. | 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.}} ( | 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 == | == Ennealimmal == | ||
{{See also | Ennealimmal}} | {{See also | Ennealimmal}} | ||
{{See also | Tritrizo clan #Ennealimmal}} | |||
The main 9th-octave temperament of interest is [[ennealimmal]] (temperament data given | 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: | 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]]. | 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]]). | ||
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Latest revision as of 12:01, 27 December 2025
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: 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 at Tritrizo clan), notable for being the 7-limit microtemperament tempering the two smallest superparticular intervals of the 7-limit, 2401/2400 = S49 = (49/40)/(60/49) and 4375/4374 = S25/S27 = (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:
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 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 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 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)/(7/6)2 = S48 = (48/47)/(49/48).
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Todo: expand, add examples |