Re: Digest Number 229

I wrote,

>>There seem to be some "ghost" triangles there that shouldn't be there.
This
>>is the way I do ASCII lattices:

>>$ A---------E---------B
>> /|\ /|\ / \
>> / | \ / | \ / \
>> / Eb \ / Bb \ / * \
>> /,' `.\ /,' `.\ / \
>> F---------C---------G---------D

Ray Tomes wrote,

>But you have missed out the line from Eb to Bb which should be there.
>Otherwise more nicely drawn than mine.

I am very embarassed. That's not a mistake I usually make, and I've drawn
tons of these diagrams! Here's the corrected version:

$ A---------E---------B
/|\ /|\ / \
/ | \ / | \ / \
/ Eb--------Bb \ / * \
/,' `.\ /,' `.\ / \
F---------C---------G---------D

>This shape is interesting because it is the same as Kepler's sphere
>stacking problem. It can be represented as a triangular lattice with
>another one on top or a square lattice with another one on top.

That is, one can represent it by stacking triagular lattices in _three_
staggered positions, or square lattices in two staggered positions. For
viewing more than three layers of this lattice at once, an oblique view is
necessary.

>For
>those that don't know this is a famous unproven conjecture about the
>most efficient way of packing spheres.

If we regard these as equilateral triangles, then each note is at the center
of an imaginary sphere, and the spheres are (probably) packed as efficiently
as possible. However, Ray, I think you and I would both prefer not to regard
the triangles as equilateral, but as isosceles with, e.g., the 3:1 side
shorter than the 5:1 side or the 5:3 side.

Paul/ Ray/ Dan!
Since this set was the basis of my scale Centaur , i thought to throw it out.
Erv had done a whole series of scales based on harmonic (or subharmonic) series a
3/2 apart called helixsongs. I believe Lydia Ayers was one of the first to
compose on this scale. Anyway having my first pump organ and a beginner in the
70's, I needed a scale for ear training. This scale had more properties than
most. Remember that lattices are bias to the harmonic properties at the expense
of the melodic.Scale construction seems best when both are taken into
concideration! More on this scale to come!

"Paul H. Erlich" wrote:

> From: "Paul H. Erlich" <PErlich@Acadian-Asset.com>
>
> I wrote,
>
> >>There seem to be some "ghost" triangles there that shouldn't be there.
> This
> >>is the way I do ASCII lattices:
>
> >>$ A---------E---------B
> >> /|\ /|\ / \
> >> / | \ / | \ / \
> >> / Eb \ / Bb \ / * \
> >> /,' `.\ /,' `.\ / \
> >> F---------C---------G---------D
>
> Ray Tomes wrote,
>
> >But you have missed out the line from Eb to Bb which should be there.
> >Otherwise more nicely drawn than mine.
>
> I am very embarassed. That's not a mistake I usually make, and I've drawn
> tons of these diagrams! Here's the corrected version:
>
> $ A---------E---------B
> /|\ /|\ / \
> / | \ / | \ / \
> / Eb--------Bb \ / * \
> /,' `.\ /,' `.\ / \
> F---------C---------G---------D
>
> >This shape is interesting because it is the same as Kepler's sphere
> >stacking problem. It can be represented as a triangular lattice with
> >another one on top or a square lattice with another one on top.
>
> That is, one can represent it by stacking triagular lattices in _three_
> staggered positions, or square lattices in two staggered positions. For
> viewing more than three layers of this lattice at once, an oblique view is
> necessary.
>
> >For
> >those that don't know this is a famous unproven conjecture about the
> >most efficient way of packing spheres.
>
> If we regard these as equilateral triangles, then each note is at the center
> of an imaginary sphere, and the spheres are (probably) packed as efficiently
> as possible. However, Ray, I think you and I would both prefer not to regard
> the triangles as equilateral, but as isosceles with, e.g., the 3:1 side
> shorter than the 5:1 side or the 5:3 side.
>
>

-- Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com