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 well at all, both ratios falling almost directly between the step sizes; however, this makes their difference tone 5/3 somewhat accurate. In addition, 11edo approximates 7/4, 11/8, and 17/16 quite well, and using dual fifths allows one to approximate ratios of 9 such as 9/7.
| 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) | |
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, 25/22 | 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, 34/25 | ^F, Gb | M | Major fourth | Near fourth |
| 6 | 655 | 16/11, 22/15, 25/17 | 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, 44/25 | ^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 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. The chords that this begets are [0 3 11] and [0 7 11].
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.
