9edo
| ← 8edo | 9edo | 10edo → |
9 equal divisions of the octave (9EDO) is the tuning system derived by dividing the octave into 9 equal steps of 133+1/3 cents each precisely. It is also the first odd composite EDO.
Theory
| prime 2 | prime 3 | prime 5 | prime 7 | prime 11 | prime 13 | prime 17 | prime 19 | ||
|---|---|---|---|---|---|---|---|---|---|
| error | absolute (¢) | 0.0 | -35.3 | +13.7 | -35.5 | -18.0 | -40.5 | +28.4 | -30.8 |
| relative (%) | 0 | -26 | +10 | -27 | -13 | -30 | +21 | -23 | |
| nearest EDO-mapping | 9 | 5 | 3 | 7 | 4 | 6 | 1 | 2 | |
| fifthspan | 0 | +1 | -3 | -4 | -1 | +3 | +2 | +4 | |
The 9EDO 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.
Differences between distributionally-even scales and smaller EDOs
| N | L-Nedo | s-Nedo |
|---|---|---|
| 2 | 66.667¢ | -66.667¢ |
| 4 | 100¢ | -33.333¢ |
| 5 | 26.667¢ | -106.667¢ |
| 6 | 66.667¢ | -66.667¢ |
| 7 | 95.238¢ | -38.095¢ |
| 8 | 116.667¢ | -16.667¢ |
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 |
Commas
9EDO tempers out the following commas. (Note: This assumes val ⟨9 14 21 25 31 33].)
| Prime Limit |
Ratio[1] | Monzo | Cents | Color name | Name(s) |
|---|---|---|---|---|---|
| 5 | 27/25 | [0 3 -2⟩ | 133.24 | Gugu | Large Limma, Large Semitone, Bug Comma |
| 5 | 135/128 | [-7 3 1⟩ | 92.18 | Layobi | Major Chroma, Major Limma, Pelogic Comma |
| 5 | 16875/16384 | [-14 3 4⟩ | 51.12 | Laquadyo | Negri Comma, Double Augmentation Diesis |
| 5 | 128/125 | [7 0 -3⟩ | 41.06 | Trigu | Diesis, Augmented Comma |
| 5 | (14 digits) | [-21 3 7⟩ | 10.06 | Lasepyo | Semicomma, Fokker Comma |
| 7 | 36/35 | [2 2 -1 -1⟩ | 48.77 | Rugu | Septimal Quarter Tone |
| 7 | 525/512 | [-9 1 2 1⟩ | 43.41 | Lazoyoyo | Avicenna, Avicenna's Enharmonic Diesis |
| 7 | 49/48 | [-4 -1 0 2⟩ | 35.70 | Zozo | Slendro Diesis |
| 7 | 686/675 | [1 -3 -2 3⟩ | 27.99 | Trizo-agugu | Senga |
| 7 | 2430/2401 | [1 5 1 -4⟩ | 20.79 | Quadru-ayo | Nuwell |
| 7 | 1728/1715 | [6 3 -1 -3⟩ | 13.07 | Triru-agu | Orwellisma, Orwell Comma |
| 7 | 225/224 | [-5 2 2 -1⟩ | 7.71 | Ruyoyo | Septimal Kleisma, Marvel Comma |
| 7 | 6144/6125 | [11 1 -3 -2⟩ | 5.36 | Sarurutrigu | Porwell |
| 7 | 65625/65536 | [-16 1 5 1⟩ | 2.35 | Lazoquinyo | Horwell |
| 7 | (16 digits) | [-11 -9 0 9⟩ | 1.84 | Tritrizo | No-fives ennealimma |
| 11 | 99/98 | [-1 2 0 -2 1⟩ | 17.58 | Loruru | Mothwellsma |
| 11 | 121/120 | [-3 -1 -1 0 2⟩ | 14.37 | Lologu | Biyatisma |
| 11 | 176/175 | [4 0 -2 -1 1⟩ | 9.86 | Lorugugu | Valinorsma |
| 11 | 385/384 | [-7 -1 1 1 1⟩ | 4.50 | Lozoyo | Keenanisma |
| 11 | 540/539 | [2 3 1 -2 -1⟩ | 3.21 | Lururuyo | Swetisma |
| 13 | 91/90 | [-1 -2 -1 1 0 1⟩ | 19.13 | Thozogu | Superleap |
| 13 | 676/675 | [2 -3 -2 0 0 2⟩ | 2.56 | Bithogu | Parizeksma |
- ↑ Ratios longer than 10 digits are presented by placeholders with informative hints
Linear temperaments
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.
Pathological Modes
2 1 1 1 1 1 1 1 1L 7s MOS
Selected Just Intervals
Images
Instruments
Ukulele (MicroUke 1.2) set to 9EDO with 40 lb. test fishing line (by cenobyte)
Music
- Tenacious Chorale (only movement I is in 9EDO) by Stephen Weigel
- Zones of Lasting Novelty (Un12 2019) by Stephen Weigel; perf. Hans Gunter-Lock, Jacob Barton, and Stephen Weigel
- Gamelan, Origin, Creation by Stephen Weigel (Beo String Quartet, dedicated to Lou Harrison)
- 69 by Tabytha
- 69 Pentangled by Tabytha
- Nocturne in 9EDO by Daniel Wolf
- Prelude in 9ET by Aaron Andrew Hunt
- Improvisation for Electric Guitar in 9EDO 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 EDOsimultaneously)
- New World by Carlo Serafini (blog entry)
- Interdimensional Train Ride by Santiago Cosentino
Ear Training
9EDO ear-training exercises by Alex Ness available here.