Ball scales (was RE: Digest Number 3438-the diamond)

Gene,

What immediately occured to me was that you can transform any
ellipsoid in n dimensions into an n-dimensional sphere, by suitably
scaling its axes.

Apart from the fact that a "round ball" is preferred by many
(millions) more people than an "oval ball" (especially in football!) -

Perhaps performing such a transformation on your ellipsoid
would shed some light on an alternative (and possibly more
"natural") metric for harmonic distance?

Regards,
Yahya

-----Original Message-----
________________________________________________________________________
Date: Fri, 11 Mar 2005 02:40:02 -0000
From: "Gene Ward Smith" <genewardsmith@coolgoose.com>
Subject: Re: Digest Number 3438-the diamond

--- In tuning@yahoogroups.com, John Chalmers <JHCHALMERS@U...> wrote:

> I was collaborating with Erv at the time he came up with the CPS and my
> recollection is that
> he was thinking of ways to relate chords as well as trying to find the
> most useful set of
> pitch bases (or 'keys') in which to print out the set of just intonation
> tables I was computing for him at the USCD computer center.

By the way, I just showed over on tuning-math that the eikosany is a
scale of ball type, another example of which is the diamond. One way
to describe what that means is that you can take the centroid, or
center point, of the eikosany, which is |0 3/2 1/2 1/2 1/2>, and draw
an ellipsoidal region around it which contains all of the notes of the
eikosany and no other notes. This was my project for the day and it
would be interesting to find a less laborious way of determining what
is, and what is not, a scale of ball type.

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