Some scale conjectures

I just thought I'd try to share a connection between what I've been doing and what you guys have been doing.

In Example 11 of my paper "Scale Networks and Debussy" I describe a cubic lattice that shows efficient voice leading among structurally similar scales -- diatonic, acoustic, harmonic major, and harmonic minor.

In Appendix II(a), I ask -- "what scales are such that you can build a lattice of this sort?" There I answer "any maximally even scale that is not transpositionally symmetrical." (NB: I've been saying "Appendix I" on this list, but really I mean "Appendix II(a)."

But, after our discussion here, I realize that a slightly more general answer can be given: such lattices exist for any MOS scale whose period is an octave. (You can even build a closely analogous lattice for MOS scales with suboctave periodicity, in which several voices move at a time.)

Furthermore, I'm pretty sure that there will be an analogue to the "Diatonic Thirds" property as long as the MOS scale is sufficiently large and not "obviously generated" in the sense I described a few emails ago. That is, for all the scales on the lattice, there will be some scalar interval that comes in precisely two sizes.

In the case of the familiar diatonic/acoustic/harmonic major/harmonic minor lattice, this interval is the third. But in the general case, I think, there will always be an analogous interval.

What this means is that the two descriptions of the diatonic/acoustic/harmonic major/minor scales as "meantone compact" and as "nearly diatonic but sharing thirds of the same size" are really equivalent. Each description implies the other.

That's pretty cool! I haven't triple-checked the math, but I'm pretty sure this is right.

DT
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Dmitri Tymoczko
Assistant Professor of Music, Princeton University
Radcliffe Institute for Advanced Study
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