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

Prime harmonic approximations

Approximation of prime harmonics in 13edo

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

Template:Idiosyncratic 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.