I believe that space is closer to the surface of a klein bottle, than that of a mobius strip. (at roughly the 1 hour mark) Because two mirrored mobius strips glued together on their edge form a klein bottle.
This is very helpful being able to see his presentation (as fuzzy as this is). It's probably worth getting Tymoczko's book especially since hypercube algorithms are already well-known programming chores, and a minor remapping would make for a workable tonal tesseract or 4-D tonal mobius.
Great lecture. Btw, major fifth *does* divide the octave in half, but because of the log scale and equal temperament division, it appears above half... So although many ideas sound beautiful in this lecture, not all reasoning is 100% mathematical... Especially the argument about the good chords dividing the octave “nearly but not precisely evenly” - which is an artefact of the equal temperament scale. I’m gonna try to rework these circles in log frequency space. P.S. Oh, no, he explained that problem with piano keyboard in the first answer to the audience question ;)
