Lattices

Here's a messge I sent privately to Graham Breed
(newly edited somewhat). Since we're ironing out
the history of using lattice diagrams, I thought I'd
make it public.
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When I first designed a lattice graph to represent pitches
back around 1988, I used exactly the same triangular
format used by you and Erlich, with 5/4 above and to
the right of 1/1, and 6/5 below and to the right. It was
something that I came up with by playing around with
Partch's Tonality Diamond scheme.

I was shocked to find out that diagrams exactly like this
appeared in a book called "A Graphic Introduction to the
Harmon" by Esther Watson Tipple & Royal Merrill Frye,
and published back in 1942!

I just read Mandelbaum's (unpublished) book for the first
time, and have found out that Fokker used diagrams very
much like mine. In fact, Riemann's harmonic ideas [c. 1890]
are based on a 3^x * 5^y lattice structure.

Over the past four years, doing research for my book, I've
seen almost all of the various visual pitch representations
that I thought I invented, in books that were written as far
back as 1600!

So Gregg Gibson [TD 1278] certainly had a point about how
we moderns think we're doing something new that's already
been done countless times before. But of course, technology
has given us many new capabilities, and we can adapt these
old ideas to new technologies.

I think the 3-dimensional ladder-type lattices I use (see
some of my recent postings) are the best way to represent
JI systems of more than 3 prime factors.

Joseph L. Monzo
monz@juno.com

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