9edo

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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

[Derivation by equal division of various key intervals or of interval qualities]

JI approximation

[Overview of viable vals, tuning tendencies, and accurate/structurally interesting subgroups. Always cover the patent val alongside any notable non-patent vals.]

Approximation of prime harmonics in 9edo

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	Intervals		Notation			Bame (optional)
		Group 1	Additional groups	Notation 1 (usually ups and downs)	Notation 2 (usually neutral diatonic notation, if different))	Additional notations	System 1 (usually ADIN)	System 2 (usually ups and downs, if meaningfully different)	System 3 (usually neutral diatonic, if different)	Additional systems
0	XXX	XX/XX	XX/XX	D	D	J	...	...	...	...
...	...	...	...	...	...	...

[Add solfege in a separate table, if applicable]

Tempering properties

Tempered commas

[List commas with S-expressions and examples of what they equate]

Notable structural chains

[Generator chains]

Compositional theory

Tertian structure

[Describe]

Thirds in Xedo

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]

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?]

Instruments

[Put instruments and isomorphic layouts here]