Macrotonal edonois

A macrotonal edonoi would be, by definition, a scale which meets two constraints:

• macrotonal - all steps are larger than a semitone
• edonoi - short for "equal divisions of a non-octave interval" - the scale consists of a single step stacked over and over which does not repeat at an octave

Examples include equal-tempered Bohlen Pierce (a.k.a. the 13th root of 3), the square root of 13:10, the 12th root of 3, the 4th root of 3:2, and the 6th root of 3:2.

Macrotonal edos

Macrotonal edonoi are related, in step-size and equality of steps, to macrotonal edos, but while macrotonal edos are a finite set, macrotonal edonoi are theoretically infinite. Macrotonal edos are extremely redundant systems. Not only is there a very limited set of intervals in one octave of a macrotonal edo, but thanks to octave equivalency, that small set repeats at every octave.

Macrotonal edonoi, by not containing octaves at all, take away the redundancy of octave equivalence, and are thus much more complex systems to compose in. Each new step further out produces a brand new interval, with no octave-equivalent complement that came before it.

If we consider macrotonal edos as distinct stopping-places in a continuum of scales with decreasing step size (from the 1200-cent step of 1edo, down to the 100-cent step of 12edo which defines the edge of "macrotonal"), then macrotonal edonoi represent unique universes that are "in the cracks".

Equal Divisions of Compound Octaves

What about dividing a compound octave, say, 4:1 or 8:1? Examples of this kind of scale would include the 15th root of 4 and the 22nd root of 8. I don't know whether or not we should use the term edonoi for these.

These kinds of scales, equal divisions of compound octaves, represent a middle-ground in terms of redundancy and complexity of an equal-step system. For instance, the 15th root of 4 can be arrived at by taking every other tone in 15edo. It doesn't repeat at one octave, but it repeats at two octaves, after having generated 15 tones. From there, the system is redundant with itself, as it now produces the same intervals two octaves higher than where they first appeared.