13edo: Difference between revisions
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== Tuning theory == | == Tuning theory == | ||
{{Harmonics in ED|13|23|0}} | |||
=== Harmonic series approximations === | === Harmonic series approximations === | ||
13edo approximates the following harmonic series chord well: | 13edo approximates the following harmonic series chord well: | ||
* 34:36:38:40:42:52:55:58:68 | * 34:36:38:40:42:52:55:58:68 | ||
== One functional system for 13edo == | == One functional system for 13edo == | ||
The following system has been developed by Jaimbee and Inthar. | The following system has been developed by Jaimbee and Inthar. | ||
Revision as of 17:52, 4 January 2026
13edo, or 13 equal divisions of the octave, is the equal tuning featuring steps of (1200/13) ~= 92.308 cents, 13 of which stack to the octave 2/1. It does not approximate many small prime harmonics well at all, and the approximations it does have do not fit very well in a temperament (they fit better in a neji), so DR-based interpretations may be preferred among 13edo users.
13edo's greatest melodic strength is its proximity to 12edo, whose most important effect is providing an oneirotonic (5L3s, LLsLLsLs) MOS which is a compressed diatonic. A functional system for 13edo oneirotonic is provided below.
Tuning theory
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | 0.0 | +36.5 | -17.1 | -45.7 | +2.5 | -9.8 | -12.6 | -20.6 | +17.9 |
| Relative (%) | 0.0 | +39.5 | -18.5 | -49.6 | +2.7 | -10.6 | -13.7 | -22.3 | +19.4 | |
| Steps
(reduced) |
13
(0) |
21
(8) |
30
(4) |
36
(10) |
45
(6) |
48
(9) |
53
(1) |
55
(3) |
59
(7) | |
Harmonic series approximations
13edo approximates the following harmonic series chord well:
- 34:36:38:40:42:52:55:58:68
One functional system for 13edo
The following system has been developed by Jaimbee and Inthar.
13edo's melodically strongest scale is the oneirotonic MOS (preserving the diatonic property of having at least 2 semitones), so it behooves us to find harmonies that work for it.
The most basic chords in this functional harmony system are:
- Major +1+1 triad 0-4-7\13: A compressed major triad that sounds desaturated and somewhat bittersweet. Oneirotonic provides only two of these triads, so alterations are somewhat frequently used to get a major triad. The major triad has the following important tetrad supersets:
- 0-2-4-7\13: Reinforces the quasi-DR effect with an extra tone; approximately +1+1+2.
- 0-4-7-10\13: A compressed dominant tetrad; approximately +1+1+1.
- 0-4-7-12\13
- Minor +1+2 triad 0-3-8\13: A bright and brooding if somewhat hollow-sounding minor triad. The important supersets are:
- 0-3-8-10: Approximately +1+2+1
- 0-3-8-12
- 0-3-8-11
- 0-3-8-15
- 0-3-8-12-15: A concatenation of the minor +1+2 and major +1+1 triads.
