re: diamonds are forever retool

>>So it is tempting to say that the closest packing of N-dimensional spheres
>>is achieved by centering them on the vertices of the N-dimensional
>>triangular lattice,
>
>This doesn't work in 8 dimensions or higher. In 8 dimensions, there's so
>much space between the spheres in this packing that you can insert a whole
>other copy of the packing into the spaces, creating what I believe is
>called the Leech lattice. Paul Hahn probably knows more.

Interesting! Quoting the original version of the article...

>I believe this is also unproven (tho it works in 2 and 3 dimensions). In
>fact, I have read things that seem to suggest it is not true.

>>6. The one structure with the highest chord/note ratio at limit X?
>>
>>[#6.] I am not aware of a proof, but there can be no doubt.
>
>Sorry, it ain't so.
>
>5-limit diamond:
>6 consonant chords, 7 notes
>
>Weird 12-tone scale:
>13 consonant chords, 12 notes

Thanks for pointing that out! I was too hasty. What I should have said is
that they have the highest saturated-chord/note ratio of any scale with as
many or fewer notes. Hmm. Maybe the rule even holds for scales of
cardinality up to 2x, or something. . .

-Carl

>Maybe the rule even holds for scales of
>cardinality up to 2x, or something. . .

5-limit diamond: 7 notes, 6 consonant triads.
5-limit diamond+1 note: 8 notes, 7 consonant triads.