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
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -35.3 | +13.7 | -35.5 | +62.8 | -18.0 | -40.5 | -21.6 | +28.4 | -30.8 | +62.6 | +38.4 |
| Relative (%) | -26.5 | +10.3 | -26.6 | +47.1 | -13.5 | -30.4 | -16.2 | +21.3 | -23.1 | +46.9 | +28.8 | |
| Steps (reduced) |
14 (5) |
21 (3) |
25 (7) |
29 (2) |
31 (4) |
33 (6) |
35 (8) |
37 (1) |
38 (2) |
40 (4) |
41 (5) | |
The 9edo scale has the peculiar property of representing certain 7-limit intervals almost exactly. A 7-limit version of 9edo goes
1: [133.238|27/25 133.238] large limma, BP small semitone
2: 7/6 266.871 septimal minor third
3: [400.108|63/50 400.108] quasi-equal major third
4: [533.742|49/36 533.742] Arabic lute acute fourth
5: [666.258|72/49 666.258] Arabic lute grave fifth
6: [799.892|100/63 799.892] quasi-equal minor sixth
7: 12/7 933.129 septimal major sixth
8: [1066.762|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 - [- 12/7|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 |
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.
Scales
Pathological modes
2 1 1 1 1 1 1 1 1L 7s MOS
JI approximation
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[dead link] by Aaron Andrew Hunt
- Improvisation for Electric Guitar in 9EDO by Chris Vaisvil
- Comets Over Flatland 8[dead link] 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.