Re: [math-fun] musical chords geometry
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...
Really nice Wikipedia article about orbifolds. The Moebius band is the underlying 2D-manifold for one of the 17 "parabolic 2D orbifolds". The Moebius band is, of course, one of the 17 wallpaper groups. Hilarie
On Sep 6, 2015, at 11:00 AM, Hilarie Orman <ho@alum.mit.edu> wrote:
Really nice Wikipedia article about orbifolds.
The Moebius band is the underlying 2D-manifold for one of the 17 "parabolic 2D orbifolds". The Moebius band is, of course, one of the 17 wallpaper groups.
I didn't know that about the Moebius band (which is my favorite band), and it was fun figuring out which wallpaper group that is. Some more cheerful facts about the Moebius band are * It is the configuration space of all lines in the plane (not necessarily through the origin). * The open Moebius band can be given a complete metric* of constant curvature 0 or one of constant curvature -1, but not one of constant curvature +1. * There is a cubic polynomial P(x,y,z) such that (a portion of) the locus P(x,y,z) = 0 defines a Moebius band in R^3. (I mentioned this here a while ago.) * The Moebius band is (part of) a minimal surface in the 3-sphere S^3 with its standard "round" metric. —Dan _________________________________________________________________________ * A metric is complete when every geodesic can be extended indefinitely. For example, every compact (smooth) surface with a smooth metric on it.
participants (3)
-
Dan Asimov -
Henry Baker -
Hilarie Orman