9edo
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.
Images
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
- Tenacious Chorale (only movement I is in 9-EDO) by Stephen Weigel
- In Our Own Lonely Worlds by Stephen Weigel
- Nocturne in 9tet by Daniel Wolf
- Prelude in 9ET by Aaron Andrew Hunt
- Improvisation for Electric Guitar in 9 edo by Chris Vaisvil
- Comets Over Flatland 8 by Randy Winchester
- Nine tones per Octave (9-EDO / 9-TET) by Ivor Darreg
- Gerbils at the Wheel of Government by Chris Vaisvil (in 9 and 18 edo simultaneously)
- New World by Carlo Serafini (blog entry)
- Interdimensional Train Ride by Santiago Cosentino
Ear Training
9 EDO ear-training exercises by Alex Ness available here.
Instruments
Ukulele (MicroUke 1.2) set to 9 EDO with 40 lb. test fishing line (by cenobyte)