88CET #24: 88CET-Mapped Timbres

At various times in these posts, I've attached a caveat to my statements
about 88CET, that they apply to essentially harmonic timbres. Many of those
statements relate to the "taboo", off perfect consonances, like the off octaves,
and how important it is to avoid them. I also briefly aluded to Bill Sethares'
work with 88CET using timbres optimized for 88CET.

That's what I'm going to discuss briefly here.

As most tuning-listers are well aware, almost all musical tones consist of
composite of frequency components (partials). Although we often speak in terms
of relationships between the fundamental pitches of the notes in a chord, all of
the frequency components in all notes within a chord ultimately interact. The
pitches of all of the partials ultimately determine how a chord will sound.
So my repeated statements then that the "taboo" off perfect consonances sound
awful, are all relative to the timbre in which you play them. If the partials
are essentially harmonic, then those "taboo" intervals are realistically
unusable.

But Sethares took this tuning-timbre relationship in reverse. Rather than
taking the overtone structure for granted, and accepting certain intervals as
unfortunate casualties, Sethares instead took the 14-step stretched octave as a
basic building block of his 88CET music, and remapped the partials so that
sounds good. In particular, he came up with a mapping so that the 1232c
off-octave creates an identity sensation much like that of an octave for the
exact harmonic timbres. Not surprisingly, Sethares has this process down to a
science, or at least to an art, and 88CET is just one of many tunings to which
he's applied the technique.

Two general comments are in order. First, this pseudo-octave aside, this
mapping process produces some really fascinating-sounding timbres. They sound
somewhat like hybrid timbres - hybrids of whatever they were before mapping,
with a bell-like sound. The commonly-held belief that only harmonic overtone
structures "fuse" to what our ears perceive as single tones, turns out to be
mostly myth. Secondly, it does work; on their own terms, these timbres really
do produce an octave-like identity sensation.

This paper is not the correct forum for explaining the theory behind
Sethares' mapping process, but this much is worth telling: The mapping is built
upon empirical curves elating the dissonance of two sinewaves to frequency
differences, measured by Plomp and Levelt in 1965. Their experimental data was
based on statistical surveys of many people. Sethares curve-fitted this
experimental data to a mathematical relationship, and derived a procedure for
applying the results to all partials of a complex tone, thereby measuring the
overall dissonance of an given interval as rendered in a given tonal spectrum.
He then turned the process around so that he could derive tonal spectra for
minimal dissonance of a given set of intervals.

In the dissonance curve for 88CET, the troughs (local minima) in the curve
are places of minimum dissonance, and thus maximal consonance. The overtone
retuning for making 88CET's 1232c off-octave into a pseudo-octave, starting from
a hypothetical purely-harmonic tone, uses the following frequency multipliers
for the first ten partials: 1, 2.03, 3.06, 3.94, 5.087, 5.92, 6.9, 8.04, 9.36,
10.36.

Perhaps not surprisingly, this isn't without its trade-offs, both good and
bad. To my ears, these nonharmonic overtone structures make the exact pitch of
the notes a little hard to pinpoint. As such, they somewhat blur 88CET's
interval approximations that are nearly just. The fifth loses some of its
steely-cold simplicity, the subminor third and seventh some of their septimal
"zap", and the major tenth some of its sweet richness. But at least one
intervals benefits: the 9:7 supramajor third, often cited as harsh, loses some
of its edge. The dissonance curve has a trough on 88CET's 9:7 approximation.


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Since the clavichord was introduced, this is no basis for equal
temperament. As many know, the clavichord is structurally designed so
that the pitch rises when a key is pressed harder. Essentially, pitch is
substituted for dynamics when the player increases the pressure on the keys.

If the Chin - a Chinese Zither hanging on the wall of Chinese philosphers
- with its ancient notation that often leaves the player playing even
after the sound has ceased to become audible, was matched with the barely
audible clavichord, it is probably on this esoteric level that the
clavichord was introduced.

Johnny Reinhard
Director
American Festival of Microtonal Music
318 East 70th Street, Suite 5FW
New York, New York 10021 USA
(212)517-3550/fax (212) 517-5495
reinhard@ios.com


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