13edo
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 JI 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
Intervals
Note: The logic of ground's notation is to preserve the diatonic order of nominals for the stacked oneirotonic subfourth generators, with one additional note: BEADGCFX
| Edostep | Cents | Interval region name | Oneirotonic TAMNAMS name | Fox-Raven notation (on J = 180 Hz) | Ground's notation (on A = 440 Hz) | 26edo subset notation (on A = 440 Hz) |
|---|---|---|---|---|---|---|
| 0 | 0 | Unison | Perfect 0-(oneiro)step (P0oneis) | J | A | A |
| 1 | 92.3 | Minor 2nd | Minor 1-(oneiro)step (m1oneis) | J# / Kb | A# / Cb | Ax / Bbb |
| 2 | 184.6 | Major 2nd | Major 1-(oneiro)step (M1oneis) | K | C | B |
| 3 | 276.9 | (Sub)minor 3rd | Minor 2-(oneiro)step (m2oneis) | L | B | Bx / Cb |
| 4 | 369.2 | (Sub)major 3rd | Major 2-(oneiro)step (M2oneis) Diminished 3-(oneiro)step (d3oneis) |
L# / Mb | B# / Db | C# |
| 5 | 461.5 | Subfourth | Perfect 3-(oneiro)step (P3oneis) | M | D | Db |
| 6 | 553.8 | Ultrafourth / Infratritone | Minor 4-(oneiro)step (m4oneis) | Nb | Fb | D# |
| 7 | 647.2 | Ultratritone / Infrafifth | Major 4-(oneiro)step (M4oneis) | N | F | Eb |
| 8 | 738.5 | Superfifth | Perfect 5-(oneiro)step (P5oneis) | O | E | E# / Fbb |
| 9 | 830.8 | (Super)minor 6th | Augmented 5-(oneiro)step (A5oneis) Minor 6-(oneiro)step (m6oneis) |
O# / Pb | E# / Gb | F |
| 10 | 923.1 | (Super)major 6th | Major 6-(oneiro)step (M6oneis) | P | G | Fx / Gbb |
| 11 | 1015.4 | Minor 7th | Minor 7-(oneiro)step (m7oneis) | Qb | Xb | G |
| 12 | 1107.7 | Major 7th | Major 7-(oneiro)step (M7oneis) | Q | X | Gx / Abb |
| 13 | 1200 | Octave | Perfect 8-(oneiro)step (P8oneis) | J | A | A |
Prime harmonic approximations
| 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) | |
Edostep interpretations
13edo's edostep functions in the 2.5.11.13 subgroup as:
- 26/25 (the interval between 5/4 and 13/10)
- 55/52 (the interval between 11/8 and 13/10, and between 5/4 and 13/11)
- 128/121 (the interval between 11/8 and 16/11)
Harmonic series approximations
13edo approximates the following harmonic series chord well (x indicates notes that are harder to approximate):
34:36:38:40:42:x:47:x:52:55:58:61:x:68
Making an over-17 13edo neji thus requires you to choose those three notes:
- The notes resulting in lowest pairwise error in mode 34 are 44, 49, and 64.
- The closest notes in mode 68 are 89, 99, and 129 (which are significantly more complex).
- A less accurate but lower-complexity neji (limited to oneirotonic) is 22:25:26:29:32:34:38:42:44, so one could specifically choose 44, 50, and 64.
Jaimbee and Inthar's 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.
For a DR-forward framework like this, prefer mellow timbres to bright ones to bring out the DR effect.
Basic chords
The most basic chords in this functional harmony system are:
- Major triad 0-4-7\13: A compressed major triad that sounds desaturated and somewhat bittersweet. Somewhat dubiously +1+1. 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.
- 0-4-7-12\13
- Minor +1+2 triad 0-3-8\13: A bright and brooding if somewhat hollow-sounding minor triad. Approximately 17:20:26. The important supersets are:
- 0-3-8-10: Approximately +1+2+1.
- 0-3-8-12: Approximately +1+2+2.
- 0-3-8-11: Something like a minor 7th tetrad.
- 0-3-8-15
- 0-3-8-12-15: A concatenation of the minor +1+2 and major +1+1 triads.
- 0-5-9\13: A +1+1 triad and a compressed 2nd inversion major triad. Approximately 13:17:21.
- 0-5-7-9: Approximately +2+1+1.
- 0-5-9-12: A compressed major triad on top of a subfourth.
- 0-5-9-12-15
- 0-5-7-9-12-15-17
- 0-5-7\13: Compressed sus4. Approximately +2+1.
- 0-4-8\13: "Submajor augmented" triad.
- 0-3-6\13: The most diminished-like triad.
