Some Background: About Ezra Sims’ Scale
In some cases the differences between the intervals of the harmonic series
(the “natural” or “just ” intervals) and the twelve traditional
equal-tempered intervals are almost imperceptible, and in other cases they
are quite noticeable. For example, a natural perfect fifth—frequency
ratio 3:2—is barely 2 cents (2/100 of a semitone) smaller than its equal-tempered
counterpart, while the natural minor seventh—frequency ratio 7:4—is
almost 31 cents (roughly 1/3 of a semitone, or 1/6 of a "tone")
smaller than the equal-tempered one.
By dividing the octave into 72 equally spaced intervals, one creates a minute
chromatic that includes the twelve traditional equal-tempered intervals and
adds 1/12 tones (and multiples of 1/12 tones, such as 1/6 tones, 1/4 tones
and 1/3 tones). Using this 72-note palette, one is able to approximate the
natural intervals very closely. For example, one can produce a minor seventh
that is exactly 1/6 of a tone smaller than the traditional equal-tempered
minor seventh, and will resemble very closely the natural minor seventh mentioned
above. (The difference between these two minor sevenths is just about 2 cents.)
In the early 1970’s Ezra Sims began using intervals derived from 72-note
equal temperament in order to create a 24-note, asymmetrical, transposable
scale. (Some transposition of this scale is in use at any moment of his music,
in the same way that some transposition of the diatonic scale is in use at
all times in tonal music.) The 24 pitches of his scale were selected to mimic
intervals from the harmonic series. Indicated in open noteheads are the “stable”
pitches [see Example C]. These are drawn from the lower harmonics, no.’s
1-16 [see Example B], and include the fundamental itself [the pitch C, in
this case]. The filled noteheads indicate the “less stable” pitches,
and they are drawn from much higher, more remote harmonics of the same fundamental.
To select which of the available less stable pitches he would use for
his scale, Sims relied upon a synthesis of his own musical instincts and certain
acoustical facts. For example, he wanted two pitches to break up the intervals
between the three lowest “stable” pitches of his scale [here, the
C, the D and the 1/12-low E], which form a major second and a 1/12-low major
third from the bottom. One simple way to find those pitches was to see how
another 1/12-low major third within the harmonic series gets broken up, and
to use that as a model. He found that the 1/12-low major third formed by the
12th-15th harmonics was broken up by 2/3 tones [see Example B, harmonics no.’s
12-13], so he used these intervals, and then split them into 1/3 tones.
This is how he arrived at the sequence of 1/3 tones which comprise the bottom
seven notes of his scale. (Later on he was hearing quarter-tones splitting
up the same three stable pitches, and found a model for that in the harmonic
series as well. These pitches create the alternate form of the bottom notes
of the scale shown in Example C.)
One more feature of Sims’ music which is important to mention in an introduction
such as this is his special treatment of “summation” and “difference”
tones. These are pitches perceived (though not actually sounding)
as the result of the mind’s adding or subtracting the frequencies of
two simultaneous notes. For example, an E (660 Hz) and an A (440 Hz) imply
a summation tone of 1100 Hz (the C-sharp above the E) and a difference tone
of 220 Hz (the A below the 440 A). (A practical example of difference tones
is the telephone receiver, which is too small to produce the fundamental tones
of the voice, and transmits only clusters of “overtones” from which
the mind intuits the resultant fundamentals.)
Example A
Sims' accidentals for 72-note equal temperament:
|
1/12 tone high
|
 |
1/12 tone low |  |
| 1/6 tone high |  |
1/6 tone low |  |
| 1/4 tone high |  |
1/4 tone low |  |
(A font with these symbols was created by cellist Ted Mook, and may be found at:http://www.mindspring.com/~tmook/micro.html.)
Example B
The first 16 pitches of the harmonic series, in Sims' notation
Example C
Ezra Sims’ scale
Interview by Julia Werntz
Originally published in The Micro-Tome
Vol.1, Issue 2
November 1996
Stability
W: Are the stable tones of your scale (marked in whole notes) classified according
to their closeness to the fundamental, so that the third harmonic (a P5),
for example, would be treated in your music as more stable than the fifth
harmonic (a M3)?
S: To try to put it into a nutshell: going by what my ear has demanded in
my own compositions, I have this suspicion that our concern now, for the next
little while (at least my concern), will be for the harmonics 6-12 as our
basic chord, so to speak, in the way that 3-6 used to be our basic chord.
Just as you might, in C major, end on a G and an E and it won’t be quite
as final as a C and a G, or a C, E and G, but it will establish itself
as the home harmony, I suspect the same thing happens with the other ones:
I could end (again in C) on a C, D, E and quarter-low F-sharp, (the eighth,
ninth, tenth and eleventh harmonics) and that will not be quite so solid
an ending as the lower harmonics would be, but it still will be a consonant
tonal ending.
What I suspect is the least stable among them, in any case, is no.
13 (in C, the raised A-flat). That’s because 72 notes won’t quite
describe it as accurately as it will the others.
Summation and Difference Tones
W: Are the “filled notehead” pitches the only dissonances you want
to use?
S: For a while that was the case. Now I’m using different kinds of dissonance,
such as summation and difference tones. But there’s one thing: those
notes are not inherently dissonant; they’re less stable,
they’re somewhat stable, somewhat unstable.
W: How are summation and difference tones felt and treated in the writing?
Are they "special" notes with a specific musical effect or structural
function?
S: Any two notes from the scale can imply a summation tone, say. Another voice
may have a note that is close to it but is not the summation tone, and it
will want to resolve to the summation tone. And that, of course, is the definition
of dissonance.
Once in a while I’ve been in the position where I had to go outside the
scale to get the consonance that’s needed because some very strong pair
of voices implied a summation tone that was not in the scale. (I’ve known
what it is, and I’ve written it, it sounds right, it obviously has a
theoretical implication that I’ve never quite explored.)
Compositional Practice
W: Is the fundamental, scale-generating tone of a work perceived and portrayed
as “home”? If so, how is this established on a “liminal”
level in the writing?
S: Take the clarinet quintet, first section. It opens with the violin playing
two notes: a D and a sixth-low C. This implies the key of D. Then the clarinet
comes in on a quarter-low C-sharp. This configuration will only happen in
my scale of G. The key of G becomes even more clear when the clarinet moves
to a sixth high D-flat. That makes it certain that we are in G.
Each measure has a idiosynchratic and obvious cluster that could only come
from some scale or other.
W: But you prefer not to actually give the fundamental, in most cases.
S: Well, because that would pin it to the ground and it would be rather dull.
It’s nice to have it… not ambiguous, but less weighted.
In diatonic music, particularly the late stuff, it’s not always confirmed
what key you’re in, but there are implications of melodic line and what
intervals are chosen that now tell us where we are. I’ve been working
with my method so long that the same thing happens in these more complicated
intervals. What gives me hope is that naïve listeners seems to hear it
very well. Trained ones sometimes have more trouble, because they’re
professionally deformed to expect only the classic twelve notes.
Performance Practice
W: Are performers informed, in their scores, of “parent keys” and
modulations, so that these keys will affect their intonation?
S: Well, I put them in partly for my own comfort in rehearsing and the like.
I started doing it and it pleased me, but I don’t do it always now. The
new piece will not have it. It’s so rich in the matter of keys and chromaticism
that I don’t know whether there will be much point in it.
I’ve always remembered a time when Janet Packer was playing one of my
pieces, I think it was an ensemble piece. She’d been practicing and a
note struck her as wrong. She’d noticed what key it was supposed to be
in and went to the chart at the back of the piece, and found that I’d
made a mistake.
W: Is distinction between the “true” just intonation and the 72-note,
equal-tempered version significant in the final product?
S: The thing is, I can play every piece that I have in the synthesizer either
in equal or just. I can’t tell the difference. I couldn’t
tell the difference if you sprung one or the other on me. When I’m working
on it I begin usually to feel that the just has a little stronger tonal quality,
character, or it might be just a little prettier.
My concern for the overtone relations between the notes I use—what
some people would call just intonation—is theoretical. It's not to coerce
the performers into the uncomfortable production of some Procrustean set of
frequencies. It's for me to use as a touchstone to help me understand and
to inform the harmonic shape of the piece. As such it acts an armature to
my composition.
But when we get to actual performance, lots of other things simply have to
happen. Nobody ever plays in tune, exactly. No wind instrument can play a
sraight, absolutely accurate chromatic scale. Each individual instrument has
its own little odd spots. The clarinet gets louder, its pitch drops, and a
flute gets louder and its pitch rises. All I ask is that people make it sound
right to me. I’m sure that if frequency counters were ever enough
to be able to isolate a single line, or really figure out what’s happening
in an ensemble, you’d find that they were not playing the note that’s
written down, exactly. That they are playing notes that, given the context,
sound like the notes I’ve written down. That’s all I want.
I think that any musician is going to try for pure intervals—just, harmonic
intervals—but any musician also is going to have, I hope, the intuition
to not let it lead him astray. Not let it get in the way of expressivity,
of making the piece sound right, of cooperating with his instrument so that
it works best, all that kind of thing.
For more about Ezra Sims, click here.