9edo
UNDER CONSTRUCTION
9edo, or 9 equal divisions of the octave (sometimes called 9-TET or 9-tone equal temperament), is the equal tuning featuring steps of (1200/9) = 133.333… cents exactly, 9 of which stack to the perfect octave 2/1.
9edo is probably best known for its very flat, muddy-sounding fifth interval and antidiatonic (2L 5s) scale, a version of the diatonic scale with inverted harmonic properties (such as major and minor intervals being flipped) when you use the circle of fifths. It does not represent small harmonics that well, but it has extremely accurate renditions of the just intonation intervals 27/25 and 7/6, which forms the basis of an ultra-precise regular temperament called ennealimmal.
General theory
Derivation
9edo is the equal division corresponding to the 9-form, which may be understood from a polychordal point of view as dividing each perfect fourth into four, creating pentachords, while leaving the whole tone between them undivided (as opposed to the 10-form, which divides the whole tone in two).
JI approximation
[Overview of viable vals, tuning tendencies, and accurate/structurally interesting subgroups. Always cover the patent val alongside any notable non-patent vals.]
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | 0.0 | -35.3 | +13.7 | -35.5 | -18.0 | -40.5 | +28.4 | -30.8 | +38.4 | +37.1 | +55.0 |
| Relative (%) | 0.0 | -26.5 | +10.3 | -26.6 | -13.5 | -30.4 | +21.3 | -23.1 | +28.8 | +27.8 | +41.2 | |
| Steps (reduced) |
9 (0) |
14 (5) |
21 (3) |
25 (7) |
31 (4) |
33 (6) |
37 (1) |
38 (2) |
41 (5) |
44 (8) |
45 (9) | |
Edostep interpretations
[Cover these in a list]
Intervals and notation
| Edostep | Cents | JI approximation | Notation | |||
|---|---|---|---|---|---|---|
| 7-limit Ennealimmal-based (accurate) | Coarse | Melodic antidiatonic | Harmonic antidiatonic | 36edo notation (ups and downs) | ||
| 0 | 0 | 1/1 | D | D | D | |
| 1 | 133.33 | 27/25 | 16/15 | E | E | ^Eb |
| 2 | 266.67 | 7/6 | 8/7 | E#, Fb | Eb, F# | vF |
| 3 | 400 | 63/50 | 5/4 | F | F | F# |
| 5 | 533.33 | 49/36 | 4/3, 11/8 | G | G | ^G |
| 6 | 666.67 | 72/49 | 3/2, 16/11 | A | A | vA |
| 6 | 800 | 100/63 | 8/5 | B | B | Bb |
| 7 | 933.33 | 12/7 | 7/4 | B#, Cb | Bb, C# | ^B |
| 6 | 1066.67 | 50/27 | 15/8 | C | C | vC# |
| 9 | 1200 | 2/1 | D | D | D | |
Tempering properties
Tempered commas
[List commas with S-expressions and examples of what they equate]
Arithmetic progressions
Notable structural chains
[Generator chains]
Compositional theory
Tertian structure
[Describe]
| Quality (ADIN) | Mosdiatonic Quality | Quality |
|---|---|---|
| Cents | XXX | XXX |
| Just interpretation | X/X | X/X |
| Steps | X | X |
Diatonic thirds are bolded.
Scales
[if edo is composite, link to subset edos and discuss scales of subset edos there]
[List scales and scale descriptions including structure (generators if applicable), notable intervals available, associated temperaments, relationships to other scales, and logic for derivation.]
[Explain more complex scale theory topics here]
Tables of scales
Harmony
[explain notable JI and DR chords. TRY TO PROVIDE COHERENT CHORD SYSTEMS RATHER THAN JUST LISTING RANDOM CHORDS WITH NO RELATION. Explain tunings of familiar chords]
[Explain systems of harmony here, this is how you put the chords together to make music]
[Sections such as "functional harmony", "modal harmony", etc - varies based on the edo and the personal composition style]
[Use individual voices maybe?]
Tables of chords
Instruments
[Put instruments and isomorphic layouts here]
Supersets and subsets
Comparisons to other tuning systems
Music in 9edo
See also
