Re: Newbie questions - "modulation"

Jim Savage, TD 384.10, touches upon a very basic and potentially
difficult question in adaptive tuning: when and by how much should
the tunings of individual notes change as the harmony modulates?

I have beat my head against the wall for considerable amounts of time on
this question, but have yet to find a better answer than the method I
use in my dynamic tunings (http://www.idcomm.com/personal/jadl/).

At any given moment in time, with a certain set of pitches sounding,
I first determine the tuning to use: which notes to tune sharp, which
flat. But for the absolute placement, I simply try for least average
deviation from 12-tET. I do take a weighted average, considering the
relative volumes of the notes sounding, so that louder notes drift less.

Let me illustrate with a concrete example, the classic "comma pump"
sequence, I-vi-ii-V-I. Assume all notes have equal volume. The
following charts how to tune JI relative to 12-tET:

Absolute Centered
shift shift
(cents) * (cents)
-------- --------
C major (I)
C 0.00 +3.91
E -13.69 -9.78
G +1.96 +5.87
A minor (vi)
A 0.00 -5.87
C +15.65 +9.78
E +1.96 -3.91
D minor (ii)
D 0.00 -5.87
F +15.65 +9.78
A +1.96 -3.91
G major (V)
G 0.00 +3.91
B -13.69 -9.78
D +1.96 +5.87
C major (I)
C 0.00 +3.91
E -13.69 -9.78
G +1.96 +5.87

*There could easily be disagreement about which notes should be retuned
for "absolute" tuning; this is one example. No such ambiguity exists
for centered shift.

This simple technique, while ugly from a theoretical point of view,
nicely finesses the Pythagorean comma, while doing its best to center
the tuning around syntonic comma shifts.

Whether the "ideal" drift is calculated by this method or another, in
actual sequences further consideration should be given to reduce
the jerking around of the tunings of continuously sounding notes (as
other notes, and the underlying harmony, modulate). This kind of
optimization is very difficult in anything but a program (or human)
which has access to an entire sequence; in other words, it is difficult
to do in real-time playing.

JdL

> [Paul Erlich, TD 385.6]
>
> My view is that works of classical music operate on a sort
> of medium-term pitch memory (of which perfect pitch would
> be the long-term analogy) and so when the music returns to
> the home key, the musical effect is lost unless the final
> pitch is quite close to the original pitch. There are works
> of Mozart which would drift downward by almost half an
> octave if strict JI were adhered to in all comma-dependent
> progressions (including, but not limited to, modulations)
> and that drift would certainly destroy the feeling of "return,"
> as well as a host of other pitch-memory dependent features
> on shorter scales.

I just found an article yesterday in _Music Theory Spectrum_
that might shed some interesting light on this question.
Unfortunately, I didn't make a copy of it, so I don't yet
have the citation. It was a relatively recent issue,
from within the last few years.

It describes some formal aspects of a Schubert quartet
(I believe it was _Death and the Maiden_), and places
great emphasis on how scrupulous adherence to a 5-limit
JI intonation in a performance of the piece would bring
out tonal/formal strategies implemented by Schubert that
otherwise would not be as clearly disclosed.

Perhaps someone else has access to the article and can respond
in more detail. I'm very interested in how a JI performance
of this piece would deal with the intonational drift
mentioned by Paul.

BTW, do any of you keep up with the _Journal of Music Theory_?
The latest issue (or at least the latest one *I've* seen:
vol 42 # 2, Fall 1998) is entirely devoted to papers
elaborating neo-Riemannian theory. It seems that the
12-eq guys are appropriating 'our' lattice diagrams to
explain their own theories of 12-tET pitch-class sets.
Some of their diagrams look very similar indeed to the
triangular 3-D 7-limit lattices made by Pauls E. & H. et al.

One of the articles (altho this particular one has no
diagrams, but lots of mathematical formulae) is by Steven
Soderberg, who was an occasional contributor to this List
about a year ago.

-monz

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

___________________________________________________________________
Get the Internet just the way you want it.
Free software, free e-mail, and free Internet access for a month!
Try Juno Web: http://dl.www.juno.com/dynoget/tagj.