sphere packing and music

Ray Tomes wrote,

>This shape is interesting because it is the same as Kepler's sphere
>stacking problem. . . .For
>those that don't know this is a famous unproven conjecture about the
>most efficient way of packing spheres.

It was proved in 1998! See http://www.math.lsa.umich.edu/~hales/countdown/

But Gauss had proved in the 19th century that this way of packing spheres is
most efficient among lattice packings, and it is doubtful that a non-lattice
packing would find application in music theory . . .

Known as the "face-centered cubic" configuration of spheres, it has the same
efficiency as the "hexagonal close" configuration, which has only two
staggered positions of the triangular grid rather than three. An infinite
number of ways of combining the three staggered positions of the triangular
grid exist, and they all have the same efficiency. Efficiency, by the way,
is defined as the proportion of the volume that is inside the spheres.

Getting back to the original configuration, any sphere and the 12 spheres
touching it form a cuboctahedron corresponding to the 13 notes of the
Partchian 7-limit tonality diamond (the central sphere is the 1/1). Erv
Wilson used this correspondence long ago, and the way he, I, and many people
on this list draw the 7-limit tonality diamond is identical to the way
Buckminster Fuller drew his "vector equilibrium" (see
http://euch3i.chem.emory.edu/proposal/www.servtech.com/public/rwgray/synerge
tics/s04/figs/f1301.html for this and how it arises from sphere packing). If
you "mash" the spheres in this packing together, each sphere gets deformed
into a rhombic dodecahedron.

See http://www.treasure-troves.com/math/SpherePacking.html for more --
notice the "Stella Octangula" which corresponds in to Wilson's Stellated
Hexany.