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

JI approximation

11edo does not approximate harmonics 3 or 5 well at all, both ratios falling almost directly between the step sizes; however, this makes their difference tone 5/3 somewhat accurate. Using dual fifths, 11edo can approximate ratios of 9 such as 9/7, as well as ratios of 15 and 25. In addition, 11edo approximates 7/4, 11/8, and 17/16 quite well, making it a 2.9.15.7.11.17 subgroup system.

One may also note that the chord 13:19:23:26 is well-tuned in 11edo, even if the individual primes 13, 19, and 23 are not.

Approximation of odd harmonics in 11edo

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25
Error	Absolute (¢)	-47.4	+50.0	+13.0	+14.3	-5.9	+32.2	+2.6	+4.1	+29.8	-34.4	+26.3	-9.0
	Relative (%)	-43.5	+45.9	+11.9	+13.1	-5.4	+29.5	+2.4	+3.8	+27.3	-31.5	+24.1	-8.2
Steps (reduced)		17 (6)	26 (4)	31 (9)	35 (2)	38 (5)	41 (8)	43 (10)	45 (1)	47 (3)	48 (4)	50 (6)	51 (7)

Edostep interpretations

One step of 11edo can be interpreted in the 2.9.15.7.17 subgroup as:

• 18/17 (the interval between 17/16 and 9/8, or 17/14 and 9/7)
• 17/16 (the octave-reduced 17th harmonic)
• 16/15 (the interval between 9/8 and 6/5)
• 15/14 (the interval between 6/5 and 9/7).

11edo's edostep carries the following additional interpretations in the 2.5/3.7.11.17 subgroup:

• 35/33 (the interval between 6/5 and 14/11)
• 128/121 (the interval between 11/8 and 16/11)
• 128/119 (the interval between 17/16 and 8/7)
• 121/112 (the interval between 14/11 and 11/8).

Intervals and notation

As 11edo does not have a chain of fifths (unless one counts the hard 2L 5s scale generated by 6\11) suitable for notation, the task of notating 11edo proves challenging.

One approach is to treat 11edo as a subset of 22edo. The native-fifths notation for 22edo is derived through stacking 22edo's tempered version of 3/2 (13\22) and assigning names accordingly. Only every other position in the chain of fifths is a note of 11edo, so therefore starting from C, the notes C, D, E, Gb, F#, Ab, G#, Bb, and A# are part of 11edo.

In 22edo, as the diatonic MOS has hardness 4:1, a sharp corresponds to +3 steps of 22edo while a flat corresponds to -3 (representing the diatonic chroma in each case). In addition, the accidentals ^ and v raise and lower by one step of 22edo, respectively. The motion of one step in 11edo is therefore represented by the combination of the down and sharp accidentals: vC# is the step above C.

This notation can be seen as rather awkward, however, as it forces inconvenient representations, leaves glaring gaps in its system of nominals, and in general fails to reflect the radically non-diatonic structure of 11edo. Therefore, the other approach is to use a notational scale that is native to 11edo. The most obvious option here is smitonic (4L 3s), which not only is heptatonic, but is generated by 5/3, arguably 11edo's most important consonance.

For the nominals of smitonic, we opt to use the numbers 1 through 7, to avoid confusion with diatonic notes; 1 is identified here with C, and 1234567 follow the LsLLsLs mode of smitonic. As the MOS chroma is 1 step of 11edo, sharps and flats simply alter by 1 step in the smitonic-based system.

"Interval categories" are based on 22edo usage of the ADIN system; "nearminor/major" and "subminor/supermajor" correspond here to simple 5-limit and 7-limit qualities.

JI approximations (within the 25-odd-limit) of steps in 11edo, as well as the aforementioned ways of notating 11edo, are detailed in the table below. Intervals within 5 cents are in [brackets], and odd harmonics are bolded.

Edostep	Cents	2.9.15.7.11.17 JI approximation	Notation		Smitonic interval quality	Interval category (ADIN)
			22edo subset	Smitonic
0	0	1/1	C	1	Perfect unison	Perfect unison
1	109.1	[16/15], 15/14, 18/17, [17/16]	vC#, ^Db	1#, 2b	Minor second	Nearminor second
2	218.2	8/7, 9/8, [17/15], [25/22]	D	2	Major second	Supermajor second
3	327.3	6/5, 11/9, 17/14	vD#, ^Eb	3	Perfect third	Nearminor third
4	436.4	[9/7], 14/11, 32/25	E	3#, 4b	Minor fourth, augmented third	Supermajor third
5	545.5	11/8, 15/11	^F, Gb	4	Major fourth	Near fourth
6	654.5	16/11, 22/15	F#, vG	4#, 5b	Minor fifth	Near fifth
7	763.6	[14/9], 11/7, 25/16	^G, Ab	5	Major fifth, diminished sixth	Subminor sixth
8	872.7	5/3, 18/11, 28/17	G#, vA	6	Perfect sixth	Nearmajor sixth
9	981.8	7/4, 16/9, [30/17], [44/25]	^A, Bb	6#, 7b	Minor seventh	Subminor seventh
10	1090.9	[15/8], 28/15, 17/9, [32/17]	A#, vB	7	Major seventh	Nearmajor seventh
11	1200	2/1	C	1	Perfect octave	Octave

Compositional theory

Chords

11edo has numerous xenharmonic sounding chords.

One approach to harmony in 11edo is to take advantage of the dual fifths. 11edo has two fifth-like intervals roughly equidistant from 3/2, allowing for triadic-like harmony to function by changing the quality of the fifth while the third remains constant, the exact opposite of diatonic systems where the third changes quality and the fifth remains constant. If 3\11 (the nearminor third) is chosen as the third in question, these chords are present in the smitonic scale, for which 3\11 serves as the generator, and the chords that this begets are [0 3 6] and [0 3 7]\11. Notably, these can be inverted to [0 3 8], [0 5 8], and [0 4 8]\11, all of which are triads bounded by 8\11, which represents the consonance 5/3. The last of these also constitutes a stack of the very well-tuned interval 9/7.

Another important bounding interval in 11edo is 7/4, represented by 9\11, and important chords bounded by it include [0 5 9] and [0 4 9]\11, which represent 8:11:14 and its complement.

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, which can be viewed as a stretched augmented/tcherepnin scale.