Re Tymoczko: As far as I can tell, 2-note chords form a 2-D surface (in this case, a Mobius strip), where "closeness" is defined by mod 12 arithmetic around the band. The problem comes with the edges of the band, where Tymoczko claims that the chords "bounce off" the "singularity" on the edge. 3-note chords form a 3-D volume with similarly weird topology, and 4-note chords form a 4-D "volume" with weird topology. Tymoczko "closenss" is required in order to try to figure out which individual note in each chord goes with which "voice"; the technical musical term apparently being "voice leading" (i.e., really "voice following" by a listener). At 09:21 AM 9/6/2015, Fred Lunnon wrote:
The Hart video is well worth a look!
It also raises an elementary question which has me scratching my head --- what is the connection between the Moebius band and the orbifold described by Tymoczko?
WFL
On 9/6/15, Henry Baker <hbaker1@pipeline.com> wrote:
As a lifelong lover of music, I'm amazed that I never knew that 2-note chords formed a Mobius strip. The George Hart video shows this the best.
Octaves in Modular Mobius Music - YouTube
https://www.youtube.com/watch?v=Q8moAKBegFg
Perhaps everyone else on this list already knew this, but it was new (& cool) to me.
It might be fun to animate Bach's 2-part counterpoint on such a Mobius strip from a midi score, although I suspect that others have already done this -- perhaps Hart himself?
At 06:05 AM 9/6/2015, Warren D Smith wrote:
I found Dmitri Tymoczko's paper THE GEOMETRY OF MUSICAL CHORDS here http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.215.7449&rep=rep1&t...