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 perfect octave 2/1. It does not approximate many small prime harmonics well at all but approximates 10/9, 11/8, 17/13, 20/17, and 21/13 well for its size. But the approximations do not fit very well in a rank-2 temperament (they fit better in an over-34 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) MOS which is a compressed diatonic. A functional system for 13edo oneirotonic is provided below.

One functional system for 13edo

13edo's melodically strongest scale is the oneirotonic MOS, 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 if somewhat hollow-sounding minor triad. The important supersets are:
  • 0-3-8-10: Approximately +1+2+1
  • 0-3-8-12
  • 0-3-8-15
  • 0-3-8-10-12-15: A concatenation of the minor +1+2 and major +1+1 triads.