11edo

11edo, or 11 equal divisions of the octave (sometimes called 11-TET or 11-tone equal temperament), is the equal tuning featuring steps of (1200/11) ~= 109.09 cents, 11 of which stack to the perfect octave 2/1.

11edo lacks a diatonic (5L 2s) fifth, or even an armotonic or oneirotonic fifth, with the 3rd harmonic nearly halfway between its steps. The 5th harmonic, as well, lies right in between its steps. However, the errors cancel to allow 11edo representations of intervals such as 9/8, 5/3, and 15/8.

11edo can also be considered to approximate the 7th, 11th, and 17th harmonics, with 9\11 approximating 7/4, 5\11 approximating 11/8, and 1\11 approximating 17/16. In fact, 22edo's approximations to intervals of 7, 9, 11, 15, and 17 all come from 11edo. Given the logic of 22edo as an archy system, the same interval represents both 9/8 and 8/7.

The result of all this is that while 11edo forces harmony to arrange in ways extremely alien to a perspective based on the 3- or 5-limit, it still retains a breadth of approximation that allows for a complete tonal system to be built around 11edo's properties. Smitonic (4L 3s), generated by 3\11 (representing 6/5 and 11/9), can serve as a useful basis scale for navigating 11edo. The simple JI interval approximated best by 11edo is 9/7, at 1.3¢ sharp, while 5/3 and 7/4 can be used as bounding consonances for chords.

General theory

Edostep interpretations

11edo's edostep has the following interpretations in the 2.7.11.17 subgroup:

• 128/121 (the interval between 11/8 and 16/11)
• 121/112 (the interval between 11/8 and 14/11)
• 17/16 (the octave-reduced 17th harmonic)
• 128/119 (the interval between 17/16 and 8/7)

JI Approximation

11edo does not approximate harmonics 3 or 5 at all. Unless one considers ~50c of error acceptable. However, 11edo approximates 7/4 acceptably well and approximates 11/8 very well. As well as the supermajor third 9/7 exceptionally well.

Approximation of prime harmonics in 11edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	0.0	-47.4	+50.0	+13.0	-5.9	+32.2	+4.1	+29.8	+26.3	-47.8	-54.1
	Relative (%)	0.0	-43.5	+45.9	+11.9	-5.4	+29.5	+3.8	+27.3	+24.1	-43.8	-49.6
Steps (reduced)		11 (0)	17 (6)	26 (4)	31 (9)	38 (5)	41 (8)	45 (1)	47 (3)	50 (6)	53 (9)	54 (10)

Intervals and notation

[WIP]

Edostep	Cents	2.9.15.7.11.17 JI approximation	Notation		Smitonic interval quality	Interval category (ADIN)
			22edo subset notation (ups & downs)	Smitonic
0	0	1/1	C	J	Perfect unison	Perfect unison
1	109	[16/15], 15/14, 18/17, [17/16]	vC#, ^Db	J#, Kb	Minor second	Nearminor second
2	218	8/7, 9/8, 17/15	D	K	Major second	Supermajor second
3	327	6/5, 11/9, 17/14	vD#, ^Eb	L	Perfect third	Nearminor third
4	436	[9/7], 14/11, 32/25	E	L#, Mb	Minor fourth, augmented third	Supermajor third
5	545	11/8, 15/11	^F, Gb	M	Major fourth	Near fourth
6	655	16/11, 22/15	F#, vG	M#, Nb	Minor fifth	Near fifth
7	764	[14/9], 11/7, 25/16	^G, Ab	N	Major fifth, diminished sixth	Subminor sixth
8	873	5/3, 18/11, 28/17	G#, vA	P	Perfect sixth	Nearmajor sixth
9	982	7/4, 16/9, 30/17	^A, Bb	P#, Qb	Minor seventh	Subminor seventh
10	1091	[15/8], 28/15, 17/9, [32/17]	A#, vB	Q	Major seventh	Nearmajor seventh
11	1200	2/1	C	J	Perfect octave	Octave

Compositional theory

Chords

11edo has numerous xenharmonic sounding chords. An approach to chords in 11edo is stacking the interval 9/7 for very accurate 14:18:23 chords. Another is to take advantage of the duel fifths.

Scales

Arguably the main MOS scale in 11edo is the smitonic Scale 2-1-2-2-1-2-1. The smitonic scale is melodically rather similar in some regards to the 12edo diatonic scale. However, it is harmonically totally different due to the absence of harmonics 3 and 5 in 11edo.

There are others including the Checkertonic scale 2-1-1-2-1-1-2-1.