3edo

While 3EDO is more chord than scale (being identical for what 12EDO uses as an augmented triad, it is familiar in the former capacity) it is nonetheless possible to make music in it. But its real significance is theoretical. Its associated 5-limit tuning map, or val, is <3 5 7|, meaning octaves are mapped to 3 steps (hence 3EDO), 3s to 5 steps, and 5s to 7 steps. It represents the 5-limit consistently and modulo three, the mappings of 2, 3, and 5 are distinct, and both of 1-5/4-3/2 and 1-6/5-3/2 are mapped to 0-1-2. 3EDO therefore erases the distinction between major and minor, and between the root, third and fifth of the chord, while keeping the basic outline of the triad.

It also erases leading tones in the sense that 10/9, 16/15 and 25/24 are all mapped to the unison, or in other words to 0 steps. It can therefore be seen as related to 19th century musical theories such as those of Carl Friedrich Weitzmann, who classified triads in terms of the associated augmented triad of 12EDO as semitonal displacements. Moreover if we encode the kind of triad (major or minor, or even including augmented and diminished) along with the 3EDO note, we can reconstruct a 5-limit or even 7-limit version of the music. Changing the code will change the music, but preserve its 3EDO skeleton. And because of the way 3EDO relates to 10/9, 16/15 and 25/24, there is a tendency for voice-leading to work itself out correctly when we do.