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日本語

The 9EDO scale, which divides the octave into nine equal parts each of 133+1/3 cents precisely, has the peculiar property of representing certain 7-limit intervals almost exactly. A 7-limit version of 9EDO goes

1: 27/25 133.238 large limma, BP small semitone
2: 7/6 266.871 septimal minor third
3: 63/50 400.108 quasi-equal major third
4: 49/36 533.742 Arabic lute acute fourth
5: 72/49 666.258 Arabic lute grave fifth
6: 100/63 799.892 quasi-equal minor sixth
7: 12/7 933.129 septimal major sixth
8: 50/27 1066.762 grave major seventh
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 1/1 - 7/6 - 49/36 - 12/7/1 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.

Notation


9edo can be notated with conventional notation, including the staff, note names, relative notation, etc. in two ways. The first preserves the melodic meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the harmonic meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 9edo "on the fly".

Degree
Cents
Approximate
Ratios
Major wider
than minor
Major narrower
than minor
0
0
1/1
perfect unison
D
perfect unison
D
1
133
13/12, 12/11
minor 2nd
E
major 2nd
E
2
267
7/6
major 2nd, minor 3rd
E#, Fb
minor 2nd, major 3rd
Eb, F#
3
400
5/4, 9/7
major 3rd
F
minor 3rd
F
4
533
4/3, 11/8
perfect 4th
G
perfect 4th
G
5
667
16/11, 3/2
perfect 5th
A
perfect 5th
A
6
800
14/9, 8/5
minor 6th
B
major 6th
B
7
933
12/7
major 6th, minor 7th
B#, Cb
minor 6th, major 7th
Bb, C#
8
1067
11/6, 24/13
major 7th
C
minor 7th
C
9
1200
2/1
octave
D
octave
D


9EDO contains a pentatonic MOS scale -- 2L 3s (1 3 1 3 1) -- with a heptatonic extension -- 2L 5s (1 1 2 1 1 2 1, sometimes called "mavila" or "antidiatonic"). Indonesian pelog scales sometimes use five-tone subsets of a seven-tone superset in a similar way, and it has been suggested that Indonesian gamelan music stems from a 9EDO tradition.

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Images


9edo wheel.png

Commas

9 EDO tempers out the following commas. (Note: This assumes val < 9 14 21 25 31 33 |.)

Comma
Monzo
Value (Cents)
Name 1
Name 2
Name 3
135/128
| -7 3 1 >
92.18
Major Chroma
Major Limma
Pelogic Comma
16875/16384
| -14 3 4 >
51.12
Negri Comma
Double Augmentation Diesis

128/125
| 7 0 -3 >
41.06
Diesis
Augmented Comma

2109375/2097152
| -21 3 7 >
10.06
Semicomma
Fokker Comma

36/35
| 2 2 -1 -1 >
48.77
Septimal Quarter Tone


525/512
| -9 1 2 1 >
43.41
Avicenna
Avicenna's Enharmonic Diesis

49/48
| -4 -1 0 2 >
35.70
Slendro Diesis


686/675
| 1 -3 -2 3 >
27.99
Senga


2430/2401
| 1 5 1 -4 >
20.79
Nuwell


1728/1715
| 6 3 -1 -3 >
13.07
Orwellisma
Orwell Comma

225/224
| -5 2 2 -1 >
7.71
Septimal Kleisma
Marvel Comma

6144/6125
| 11 1 -3 -2 >
5.36
Porwell


65625/65536
| -16 1 5 1 >
2.35
Horwell


99/98
| -1 2 0 -2 1 >
17.58
Mothwellsma


121/120
| -3 -1 -1 0 2 >
14.37
Biyatisma


176/175
| 4 0 -2 -1 1 >
9.86
Valinorsma


385/384
| -7 -1 1 1 1 >
4.50
Keenanisma


540/539
| 2 3 1 -2 -1 >
3.21
Swetisma


91/90
| -1 -2 -1 1 0 1 >
19.13
Superleap


676/675
| 2 -3 -2 0 0 2 >
2.56
Parizeksma



Compositions



Ear Training

9 EDO ear-training exercises by Alex Ness available here.

Instruments


IMG_2223-800x600.jpg

Ukulele (MicroUke 1.2) set to 9 EDO with 40 lb. test fishing line (by cenobyte)