9edo

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9 EDO divides the octave into nine equal parts each of 133+1/3 cents precisely.

Theory

The 9-EDO scale 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 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
Melodic notation
Major wider than minor
Harmonic notation
Major narrower than minor
0 0.00 1/1 perfect unison D perfect unison D
1 133.33 14/13, 13/12, 12/11 minor 2nd E major 2nd E
2 266.67 7/6 major 2nd, minor 3rd E#, Fb minor 2nd, major 3rd Eb, F#
3 400.00 5/4, 14/11, 9/7 major 3rd F minor 3rd F
4 533.33 4/3, 11/8 perfect 4th G perfect 4th G
5 666.67 16/11, 3/2 perfect 5th A perfect 5th A
6 800.00 14/9, 11/7, 8/5 minor 6th B major 6th B
7 933.33 12/7 major 6th, minor 7th B#, Cb minor 6th, major 7th Bb, C#
8 1066.67 11/6, 13/7 major 7th C minor 7th C
9 1200.00 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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9ed2-001.svg

Commas

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

Ratio Monzo Cents Color Name Name 1 Name 2 Name 3
27/25 [0 3 -2 133.24 Gugu Large Limma Large Semitone Bug Comma
135/128 [-7 3 1 92.18 Layobi Major Chroma Major Limma Pelogic Comma
16875/16384 [-14 3 4 51.12 Laquadyo Negri Comma Double Augmentation Diesis
128/125 [7 0 -3 41.06 Trigu Diesis Augmented Comma
2109375/2097152 [-21 3 7 10.06 Lasepyo Semicomma Fokker Comma
36/35 [2 2 -1 -1 48.77 Rugu Septimal Quarter Tone
525/512 [-9 1 2 1 43.41 Lazoyoyo Avicenna Avicenna's Enharmonic Diesis
49/48 [-4 -1 0 2 35.70 Zozo Slendro Diesis
686/675 [1 -3 -2 3 27.99 Trizo-agugu Senga
2430/2401 [1 5 1 -4 20.79 Quadru-ayo Nuwell
1728/1715 [6 3 -1 -3 13.07 Triru-agu Orwellisma Orwell Comma
225/224 [-5 2 2 -1 7.71 Ruyoyo Septimal Kleisma Marvel Comma
6144/6125 [11 1 -3 -2 5.36 Sarurutrigu Porwell
65625/65536 [-16 1 5 1 2.35 Lazoquinyo Horwell
99/98 [-1 2 0 -2 1 17.58 Loruru Mothwellsma
121/120 [-3 -1 -1 0 2 14.37 Lologu Biyatisma
176/175 [4 0 -2 -1 1 9.86 Lorugugu Valinorsma
385/384 [-7 -1 1 1 1 4.50 Lozoyo Keenanisma
540/539 [2 3 1 -2 -1 3.21 Lururuyo Swetisma
91/90 [-1 -2 -1 1 0 1 19.13 Thozogu Superleap
676/675 [2 -3 -2 0 0 2 2.56 Bithogu Parizeksma

Images

9edo wheel.png

Instruments

IMG_2223-800x600.jpg

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

Music

Ear Training

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