wrapped up in lattice

[Erlich, TD 122-20:]
>
> Carl Lumma wrote,
>
> >1. Triangular is better than rectangular.
>
> For octave-equivalent lattices, I totally agree, and perhaps Joe Monzo
> is coming around on this too. But for octave-specific lattices, an
(n*d)
> measure of complexity would be handled by a rectangular lattice with
> prime axes (or, equivalently as far as the city-block metric is
> concerned, a Monzo-type lattice with an additional axis for octaves)
and
> lengths proportional to the log of the axes. This is equivalent to
> Tenney's harmonic distance if he included a 2-axis (anyone know if he
> did or didn't)? I now see where Graham Breed was coming from in
> supporting rectangular lattices for octave-specific lattices.
>
> The reason omitting the 2-axis forces one to make the lattice
triangular
> is that typically many more powers of two will be needed to bring a
> product of prime factors into close position than to bring a ratio of
> prime factors into close position. So the latter should be represented
> by a shorter distance than the former. Simply ignoring distances along
> the 2-axis and sticking with a rectangular (or Monzo) lattice is
> throwing away information.

I admit there are still modifications I'd like to make
to my lattices, but I'm not sure know how yet.

I really think the "crumpled napkin" idea is the best way
of making a harmonic lattice, with low--high-prime bridges
forming the wrinkles. But this is a geometry which
is far beyond my ability to draw

- but I'm working on it. ;-)

-monz
|\=/|.-"""-. Joseph L. Monzo...................monz@juno.com
/6 6\ \ http://www.ixpres.com/interval/monzo/homepage.html
=\_Y_/= (_ ;\ c/o Sonic Arts, PO Box 620027, San Diego, CA, USA
_U//_/-/__/// | "...I broke thru the lattice barrier..." |
/monz\ ((jgs; | - Erv Wilson |

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