The Bohlen-Pierce musical scale is based on odd harmonics only, and uses 13 equal tempered steps to the 3:1 interval (the 12th of the conventional scale). Apparently you can develop the harmony along the same lines as Western harmony, using the mathematically natural generalizations. Here's a BP version of the "Pachabel canon", a work which the uploader says "constitutes an elegant mathematical proof that the diatonic scale is emergent, not a cultural or academic contrivance". It actually starts to sound okay, after a minute or two. A little longer, and I realize that there are fewer BP notes: a tritave, the 3:1 BP replacement for the 2:1 octave, is 50% wider than an octave, and you can't fit as many in the useful range of human hearing. Hence the contra-bass and squeaky treble.
Even nicer is this brief improvisation on a BP clarinet. The clarinet is almost completely missing the second harmonic naturally, so the BP scale doesn't have overtone clashes.