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

Gene,

Thanks for your message.

I must have missed your earlier assertion that you wanted -
"a metric on pitch classes with 5,7,9 and 11 the same length
from 1, and forming a regular simplex". Your results certainly
make sense in that context.

However, I feel that it may be more "natural" for
3, 5, 7, 9, and 11 each to be further from 1 than its
predecessor odd number. If that were the case, and we
still set d(1,3) = 1, we would have -

d(1,a) = 0 < d(1,3) = 1 < d(1,5) < d(1,7) < d(1,9) < d(1,11)

Now if you want d(1,9) = d(1,3^2) = 2 d(1,3) = 2,
that puts 5 a nd 7 between 1 and 2, and 11 more than 2,
units from 1.

-----

-----Original Message-----
From: Gene Ward Smith
Sent: Sunday 13 March 2005 7:45 am
To: Yahya Abdal-Aziz
Subject: Re: Ball scales (was RE: Digest Number 3438-the diamond)

--- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:

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

I actually started from the natural metric. As I said, we want a
metric on pitch classes with 5,7,9 and 11 the same length from 1, and
forming a regular simplex, and we do this by setting

|| |0 b c d e> || = sqrt(b^2 + 2b(c+d+e) + 4(c^2+d^2+e^2+cd+de+ec))

This gives 3 a length of 1, and 5,7,9,11 a length of 2. If we write
11-limit intervals as 9^w 5^x 7^y 11^z, then the quadratic form
becomes symetrical, w^2+x^2+y^2+z^2+wx+wy+wz+xy+xz+yz. This metric
gives the A4 lattice, so we have lattice points consisting of this
lattice, plus this lattice translated by 3.

Another way to do it is with reduced representives of the pitch
classes--choosing particular pitch class representatives to measure
distance between. If |a b c d e> is an 11-limit monzo, then

red(|a b c d e>) = |-b-2c-2d-2e b c d e>

gives a reduction to a particular pitch class representative, which
is a subgroup of the entire 11-limit. Then a^2+b^2/4+c^2+d^2+e^2,
which gives the same length to 2,9,5,7 and 11, and gives a mesh
lattice squashed along the 3 axis, will give the correct geometry.

This seems like a topic for tuning-math if you want to move more
questions there.

-----

--- In tuning-math@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:

> However, I feel that it may be more "natural" for
> 3, 5, 7, 9, and 11 each to be further from 1 than its
> predecessor odd number.

In some connections that might be best; for instance, we could make
3 of length log 3, 5 of length log 5 and so forth. One context in
which that is useful is discussed here:

http://66.98.148.43/~xenharmo/top.htm

However, at other times we want to maximize symmetry, and in this
case, we want to see if the eikosany, a symmetrical 11-limit scale,
can be seen as a scale of ball type, and the above metric would not
help us there.