9edo: Difference between revisions

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[[File:9edo scale.mp3|thumb|A chromatic 9edo scale on C.]]

~~The~~ 9edo scale has the peculiar property of representing certain [[7-limit]] intervals almost exactly. A 7-limit version of 9edo goes

9edo is the most basic tuning which supports an [[antidiatonic]] scale. Its fifth is considerably flatter than just, but still falls into the category of "fifth" despite this. 9edo is also the first edo to have distinct major and minor chords (if 5edo's tendo and arto chords are ignored).

9edo splits the octave into three parts, each representing the major third 5/4, similarly to 12edo, which is of moderate accuracy. A similarly crude approximation of 11/8 (a sharp fourth) is available at the perfect fourth of 4 steps, which means 9edo can be seen as a simple 2.5.11 system. Looking at the intervals in this subgroup, the submajor second 11/10 is tuned to 133 cents (extremely flat) and 25/22 is even worse (but still consistent); the supermajor sixth 55/32 is tuned very accurately at 933 cents (only slightly flat). Overall, 9edo is not a great system for approximating low-complexity JI intervals consistently. However, if we turn to inconsistent representations, we see quite a few options before us. In particular, the 9edo scale has the peculiar property of representing certain [[7-limit]] intervals almost exactly, but not the harmonic 7/4 (a subminor seventh) itself (unless [[semaphore]], which equates it with the supermajor sixth 12/7, is taken as an acceptable temperament in this tuning). A 7-limit version of 9edo goes

1: [[27/25]] 133.238 large limma, BP small semitone

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9: [[2/1]] 1200.000 octave

~~Here the characterizations are taken from [[Scala]], which also describes the scale itself as "Pelog Nawanada: Sunda".~~ Chords such as {{dash|1/1, 7/6, 49/36, 12/7|med}} are therefore natural ones for 9edo. The above scale generates the [[just intonation subgroup]] 2.27/25.7/3, which is closely related to 9edo.

Chords such as {{dash|1/1, 7/6, 49/36, 12/7|med}} are therefore natural ones for 9edo. The above scale generates the [[just intonation subgroup]] 2.27/25.7/3, which is closely related to 9edo.

9edo has very little to offer in terms of accuracy for harmonics. Most other systems that lack good perfect fifths such as [[6edo]], [[11edo]], [[13edo]] and [[18edo]] at least contain a reasonable approximation of 9/8 (or (3/2)<sup>2</sup>), so they can be viewed as temperaments in subgroups like 2.9.5 and 2.9.7. 9edo, on the other hand, completely misses both 3/2 and 9/8. You could then try to treat 9edo as an entirely no-threes system without any 3/2 or 9/8, in a subgroup like 2.5.7.11, but even here 9edo performs somewhat poorly, because its best [[7/4]] is much closer to 12/7 and is off by 36 cents, while its best [[11/8]] is off by 18 cents. The 13th harmonic is also entirely missed by 9edo.

This being said, 9edo's fifth does approximate [[47/32]] to within about 1.2 cents, and remains near enough the boundary of [[perfect fifth]] and [[subfifth]], so it sounds quite dirty but still recognizable. 9 is the first edo to include the [[2L 5s|antidiatonic (2L 5s)]] scale, which this fifth generates as well.

=== Odd harmonics ===

@@ Line 11: / Line 11: @@
 [[File:9edo scale.mp3|thumb|A chromatic 9edo scale on C.]]
-The 9edo scale has the peculiar property of representing certain [[7-limit]] intervals almost exactly. A 7-limit version of 9edo goes
+edo is the most basic tuning which supports an [[antidiatonic]] scale. Its fifth is considerably flatter than just, but still falls into the category of "fifth" despite this. 9edo is also the first edo to have distinct major and minor chords (if 5edo's tendo and arto chords are ignored).
+edo splits the octave into three parts, each representing the major third 5/4, similarly to 12edo, which is of moderate accuracy. A similarly crude approximation of 11/8 (a sharp fourth) is available at the perfect fourth of 4 steps, which means 9edo can be seen as a simple 2.5.11 system. Looking at the intervals in this subgroup, the submajor second 11/10 is tuned to 133 cents (extremely flat) and 25/22 is even worse (but still consistent); the supermajor sixth 55/32 is tuned very accurately at 933 cents (only slightly flat). Overall, 9edo is not a great system for approximating low-complexity JI intervals consistently. However, if we turn to inconsistent representations, we see quite a few options before us. In particular, the 9edo scale has the peculiar property of representing certain [[7-limit]] intervals almost exactly, but not the harmonic 7/4 (a subminor seventh) itself (unless [[semaphore]], which equates it with the supermajor sixth 12/7, is taken as an acceptable temperament in this tuning). A 7-limit version of 9edo goes
 : [[27/25]] 133.238 large limma, BP small semitone
@@ Line 31: / Line 33: @@
 : [[2/1]] 1200.000 octave
-Here the characterizations are taken from [[Scala]], which also describes the scale itself as "Pelog Nawanada: Sunda". Chords such as {{dash|1/1, 7/6, 49/36, 12/7|med}} are therefore natural ones for 9edo. The above scale generates the [[just intonation subgroup]] 2.27/25.7/3, which is closely related to 9edo.
+Chords such as {{dash|1/1, 7/6, 49/36, 12/7|med}} are therefore natural ones for 9edo. The above scale generates the [[just intonation subgroup]] 2.27/25.7/3, which is closely related to 9edo.
-edo has very little to offer in terms of accuracy for harmonics. Most other systems that lack good perfect fifths such as [[6edo]], [[11edo]], [[13edo]] and [[18edo]] at least contain a reasonable approximation of 9/8 (or (3/2)<sup>2</sup>), so they can be viewed as temperaments in subgroups like 2.9.5 and 2.9.7. 9edo, on the other hand, completely misses both 3/2 and 9/8. You could then try to treat 9edo as an entirely no-threes system without any 3/2 or 9/8, in a subgroup like 2.5.7.11, but even here 9edo performs somewhat poorly, because its best [[7/4]] is much closer to 12/7 and is off by 36 cents, while its best [[11/8]] is off by 18 cents. The 13th harmonic is also entirely missed by 9edo.
-This being said, 9edo's fifth does approximate [[47/32]] to within about 1.2 cents, and remains near enough the boundary of [[perfect fifth]] and [[subfifth]], so it sounds quite dirty but still recognizable. 9 is the first edo to include the [[2L 5s|antidiatonic (2L 5s)]] scale, which this fifth generates as well.
 === Odd harmonics ===