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[edit]
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[edit]
Commas[edit]
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[edit]
- 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[edit]
9 EDO ear-training exercises by Alex Ness available here.
Instruments[edit]
Ukulele (MicroUke 1.2) set to 9 EDO with 40 lb. test fishing line (by cenobyte)