Universal notation

Johnny Reinhard wrote,

>Re: systems, I don't believe there is any virtue in sticking to a
>particular system of tuning, per se. Anyone care to comment?

If this means sticking to one system of tuning for one's entire musical
career, well, the rest of the musical world outside of this list provides
many fine examples of composers and musicians who have achieved excellence
through focused hard work, or pure inspiration, with one tuning system. If
someone wishes to devote his or her life to an alternate tuning system, and
that person is extremely talented, we may have a small step towards breaking
out of the 12-tet monopoly. The steps taken by those who have devoted
themselves to various incarnations of just intonation, or have worked
extensively with 19- or 31-tone equal temperament, have been, no disrespect
intended, tiny. Spreading oneself too thin in the world of tunings, one ends

up mostly approximating a pre-existing style or set of tendencies, rather
than developing a new set of musical thought processes that exploit the
resources of a particular tuning.

Why are systems important? There are various systems of tuning in use,
mainly because melodies tend to gravitate around a set of fixed pitches, and
also to provide either a specification that all harmonies be just, or a
framework in which many usable harmonies are constructed from a small set of
melodically important tones. An additional feature of many systems is to
allow transposition. The problem, of course, is that these systems are
varied and theoretically incommensurable.

I must say that the tape of "Cosmic Rays" that Johnny sent me demonstrates
that his performers have an ability to reproduce the same dissonant
intervals at different pitch levels with amazing accuracy. Considering that
world-class string players such as Isaac Stern are often 50 cents out of
tune, I'm duly impressed. However, I am very skeptical that these
"quadratic-prime just" intervals are often played within 5-10 cents of their
written value, unless some trick of relating the beat frequency to the tempo
is used. My apologies if I am underestimating an obviously super-talented
ensemble, but for the rest of us, I doubt if jumping right into cents is the
best idea. 72-tet might be better as a first step, as it approximates
11-limit JI consistently in addition to containing standard 12-tet. On the
other hand, if one adopted 72-tet, distinctions of 1/4-septimal comma (such
as in my music) would require a bisection of the 72-tet steps. The result is
144-tet, which, as it happens, is still a consistent representation of
11-limit JI (the approximations are the same as in 72-tet). Alterations of
1/4-syntonic comma would require an additional bisection (the minimum to
perform Renaissance meantone, a wonderful tuning for all triadic diatonic
music), although there are those who like Renaissance music just fine in JI
or even 12-tet. But for the purists (like me), the final result is 288-tet,
or steps of just over 4 cents, which is the usual jnd or "just-noticeable
difference." It's too bad we're stuck with the lousy decimal system, as it's
mentally and physically easier to approximate bisections and a trisection
than the quintisections required by decimals.

If I had to train string players to play microtones, I'd have them learn
thirds of semitones (sixth-tones) first, using 7/4 and 7/3 as "landmarks,"
and then I'd have them learn quartertones, using 11/8 and 11/6 and Arabic
scales as guides, and then show then how to fill in the missing
twelfth-tones by playing 5/4, 5/3, and 7/5 "just". Once they had
internalized the 72-tet system, I'd have them play along with electronically
recorded examples to fine-tune their abilities, and then to motivate further
divisions into halves and quarters with septimal and meantone musical
examples. The accuracy of the latter divisions would be questionable and
lots of electronic recordings would help, but in theory, at least, the
players would be familiar with 288-tet. To ask for finer gradations than
this seems utterly infeasible, as the distinctions can no longer be heard by
ear. Having been arrived at by an ear-training of successive bisections and
trisections, with reference to musical examples at each step, one would
expect far more accuracy from this system than by simply asking for cents
deviations (play a 37-cents-sharp b-flat!)

On fixed-pitch instruments, it may be that none of this is important, as
some sort of tablature can always be used. On the other hand, using one's
ear is important even on fixed-pitch instruments, so there may be some value
in trying to standardize a 72-tet notation with additional alterations for
half-way and quarter-way notes.

Besides Ezra Sims, Franz Richter Herf has published a 72-tet notation, but I
doubt the two are materially different.

Anyhow, seeing that Paul Rapoport is back with us, I'd like to continue my
discussion with him. Although for practical performing purposes there is no
question that Johnny is right that a single, all-encompassing, and
backwards- (or should I say sideways-) compatible system is desirable for
performance purposes, for theoretical and historical purposes other systems
may be more convenient. Although there is no doubt that to get a modern
player to play meantone, you need a different alteration for every scale
degree, meantone was once the default. In Paul Rapoport's scheme, as long as
a meantone tuning (such as 19-tet or 31-tet) is specified, no alterations
are needed to render a diatonic passage. But as the number of
pre-specifiable tunings approaches 171, the practical usefulness of his
notation for, say, string players, approaches mud.

Which brings me to the problems with Rapoport's notation viewed purely as a
theoretical tool. Here's something I wrote up on August 16:

Paul R.:
You are correct that your view of "best" commas in equal temperaments is
based on using 1:3 and 1:5 as the basic building blocks of a system (we are
ignoring octaves here). You are in good company here, but this is why I
object to this view (I'm playing devil's advocate against myself at every
question):

Q. Why do we treat 1:5 and 1:3 as basic and fundamental?

A. Because they are consonant.

Q. Are any other intervals consonant?

A. Yes, 3:5 is consonant.

Q. Is 3:5 just a by-product of building 1:5 and 1:3 from the same note?

A. No. It is inherently consonant, and needs to be tuned accurately.

Q. How do you know?

A. Listen.

Q. I'm not convinced. In a triad, don't the 1:5 and the 1:3 determine
consonance?

A. No. Compare triads in 15-tet and 20-tet, built from the best 1:3 and the
best 1:5. They have the same 1:3, and the 1:5 is better in 20-tet. However,
the 15-tet triad is clearly more consonant.

Q. I hear it, but how can that be?

A. The 3:5 is tuned much worse in 20-tet. In fact, there is a closer
approximation to 3:5 in 20-tet than the difference between the best 1:5 and
the best 1:3.

Q. Well, why not use that 3:5 in the triad?

A. Because then one of the other two intervals, which you defined as basic,
would have to change.

Q. But isn't it true that in any tuning, the 1:3 and the 1:5 determine the
best triad?

A. No. Besides 20-tet, try 11-tet and 14-tet. In all three tunings, you'll
find better triads.

Q. But those aren't even recognizable!

A. Ok, try 64-tet. There the best perfect fifth is 37 steps, and the best
major third is 21 steps. Changing either of these by 1 to give a better
minor third of 17 steps improves the entire triad.

Q. Well, who said there could be only one good triad in a system? I mean,
with a large number of notes, of course you're going to have a lot of good
triads! Can't they all be musically useful?

A. Yes, but when the "best" is really only third best, your system does not
come close to exhibiting the potential of each tuning.

Q. Isn't potential of the tuning something for the composer to discover?

A. The tonal resources of 64-tet are difficult enough to fathom. A notation
system which obscures the basic consonance-dissonance relations of the
tuning is not serving the interests of tonality.

Anyway, now it's October, and all I can add is that Rapoport's scheme might
be saved by having a more complex derivation of the "best" intervals, one
which would be based on finding the best triad rather than the best fifth
and major third. Here a criterion like mean-squared error might be useful,
but the fifth can tolerate a bit more mistuning than the thirds . . .

-Paul E.


Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl
with SMTP-OpenVMS via TCP/IP; Tue, 29 Oct 1996 00:17 +0200
Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA04613; Sat, 26 Oct 1996 15:14:50 +0200
Received: from eartha.mills.edu by ns (smtpxd); id XA05672
Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI)
for id GAA00010; Sat, 26 Oct 1996 06:14:47 -0700
Date: Sat, 26 Oct 1996 06:14:47 -0700
Message-Id: <199610261510.PAA26923@teaser.teaser.fr>
Errors-To: madole@ella.mills.edu
Reply-To: tuning@eartha.mills.edu
Originator: tuning@eartha.mills.edu
Sender: tuning@eartha.mills.edu