Minor and sharp major eleventh chords in 5 limit just intonation modulate in a variety of ways. It doesn't stick to a fixed set of pitches, so it can modulate arbitrarily without intonation problems.

As usual it's all PD with no samples or VSTs or anything.

Basically, I'm constructing minor and sharp major eleventh chords from two chains of fifths (1/1, 3/2, 9/4) separated by a third (major or minor). The size of the major third can be set arbitrarily, and the minor third is derived from that. I can modulate to other chords in a number of ways:

-Transpose the base pitch by 3/2 or 4/3 (dominant/subdominant)
-Transpose the base pitch up or down by the third and toggle between major/minor thirds (diatonic mediant/submediant)
-Transpose the base pitch up or down by the third (chromatic mediant/submediant)
-Toggle between major/minor (parallel transformation)
-Transpose by the interval between the major/minor thirds and toggle between major/minor (slide transformation).

Sometimes several of these are combined (up to 4). This has to be done carefully to keep it from sounding too weird. I tried a couple other things besides that (hexatonic poles, etc.), but I thought they didn't work very well.

Originally the idea was to use some exotic interval for the major third, but I found that most of them sounded too dissonant. For major thirds larger than about 32/25 (427 cents), some of the resulting intervals get too small. And for major thirds smaller than 5/4 (386 cents), the major and minor intervals are too close together. So the useful range is pretty small, and a lot of the ones I wanted to try (9/7, 11/9) don't really work. I thought the only major third sizes that worked well were 14/11, 81/64 (Pythagorean) and 5/4. I went with 5/4. But I want to revisit this idea with more exotic intervals. I'll just have to construct the chord differently so that it works better.

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