11edo: Difference between revisions
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11edo, or 11 equal divisions of the octave, is the equal tuning featuring steps of (1200/11) ~= 109.09 cents, 11 of which stack to the perfect octave 2/1 | '''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]]) [[perfect fifth|fifth]], or even an [[7L 2s|armotonic]] or [[5L 3s|oneirotonic]] fifth, with the 3rd harmonic nearly halfway between its steps. The [[5/4|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]]. | |||
==== Edostep interpretations | 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 comprises 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-limit|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{{c}} 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: | 11edo's edostep has the following interpretations in the 2.7.11.17 subgroup: | ||
| Line 11: | Line 16: | ||
* 128/119 (the interval between 17/16 and 8/7) | * 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. | 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. | ||
==== Chords | === Intervals and notation === | ||
[WIP] | |||
{| class="wikitable" | |||
|+ | |||
! rowspan="2" | Edostep !! rowspan="2" | Cents !! rowspan="2" | 2.9.15.7.11.17 <br> JI approximation !! colspan="2" | Notation !! rowspan="2" | Smitonic interval quality !! rowspan="2" | Interval category <br> (ADIN) | |||
|- | |||
! rowspan="1" | 22edo subset notation <br> (ups & downs) !! rowspan="1" | 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 | |||
| | |||
|Nearminor second | |||
|- | |||
|2 | |||
|218 | |||
|8/7, '''9/8''', 17/15 | |||
|D | |||
|K | |||
| | |||
|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 | |||
| | |||
|Supermajor third | |||
|- | |||
|5 | |||
|545 | |||
|'''11/8''', 15/11 | |||
|^F, Gb | |||
|M | |||
| | |||
|Near fourth | |||
|- | |||
|6 | |||
|655 | |||
|16/11, 22/15 | |||
|F#, vG | |||
|M#, Nb | |||
| | |||
|Near fifth | |||
|- | |||
|7 | |||
|764 | |||
|[14/9], 11/7, '''25/16''' | |||
|^G, Ab | |||
|N | |||
| | |||
|Subminor sixth | |||
|- | |||
|8 | |||
|873 | |||
|5/3, 18/11, 28/17 | |||
|G#, vA | |||
|P | |||
| | |||
|Nearmajor sixth | |||
|- | |||
|9 | |||
|982 | |||
|'''7/4''', 16/9, 30/17 | |||
|^A, Bb | |||
|P#, Qb | |||
| | |||
|Subminor seventh | |||
|- | |||
|10 | |||
|1091 | |||
|['''15/8'''], 28/15, 17/9, [32/17] | |||
|A#, vB | |||
|Q | |||
| | |||
|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. | 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. | 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. | ||
Revision as of 16:00, 16 February 2026
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 comprises 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.
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 | Nearminor second | |
| 2 | 218 | 8/7, 9/8, 17/15 | D | K | 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 | Supermajor third | |
| 5 | 545 | 11/8, 15/11 | ^F, Gb | M | Near fourth | |
| 6 | 655 | 16/11, 22/15 | F#, vG | M#, Nb | Near fifth | |
| 7 | 764 | [14/9], 11/7, 25/16 | ^G, Ab | N | Subminor sixth | |
| 8 | 873 | 5/3, 18/11, 28/17 | G#, vA | P | Nearmajor sixth | |
| 9 | 982 | 7/4, 16/9, 30/17 | ^A, Bb | P#, Qb | Subminor seventh | |
| 10 | 1091 | [15/8], 28/15, 17/9, [32/17] | A#, vB | Q | 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.
