way to express mixed o and u tonalities

> [Dan Stearns, TD 702.17]
> BTW, was any clean and clear way to express mixed o and
> u tonalities ever put forward?

Dan,

In my case, the use of a lattice is actually an outgrowth
of the attempt to do exactly this.

I devised a method of notation in the mid-90s, which was
published in the first version of my book in 1995, and which
was expressly designed to portray o- and u-tonalities
simultaneously. You can read about it in my introductory
article to my theory:
http://www.ixpres.com/interval/monzo/article/article.htm

In brief:

I use n^0 as my basic reference and calculate all ratios from
that, using prime-factor notation. If letter-names are
required, I use either 'C n^0' as an absolute reference,
or whatever letter-name-plus-accidental I need for the 'tonic'.

Each harmony simply has a fractional expression written under it,
with a (optional letter-name and) prime-factor notation designating
the 'root' - which is the numerary nexus - on one side of the
dividing-line (below it for otonalities, above it for utonalities),
and the integers designating identities on the other side.

The identities can also be either relative (i.e., 'octave'-
equivalent) or absolute.

For example:

'C major' otonal triad
3
5
1
-----
C n^0

'C minor' utonal triad

G 3^1
-----
1
5
3

The progression from one to the other would be:

3
5
1 G 3^1
----- -----
C n^0 1
5
3

I think that's the clearest way to represent a rational
(or implied rational) harmonic analysis, short of using
the lattice diagrams.

-monz

Joseph L. Monzo San Diego monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
| 'I had broken thru the lattice barrier...' |
| -Erv Wilson |
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