88CET #10: Note Doubling

Probably the easiest way in a traditional tuning to extend a chord is by
doubling one or more notes in octaves from its basic position. One of the main
goals of Sethares' 88CET-optimized partial mappings is to create a pseudo-octave
with a pitch-duplicating sensation kind of like the real octave's. That 14-step
pseudo-octave works great for doubling on these mapped timbres. On
approximately harmonic timbres however, this and the other false perfect
consonances don't work at all for doubling. The rest of this posting addresses
doubling in reference to the usual unmapped timbres.

Can the octave-doubling principle be extended to tunings that don't even have
an octave, by doubling by an interval other than an octave? Brian McLaren, who
has worked with a variety of nonoctave tunings, believes that whatever your
cycle-interval may be can produce a doubling effect. He reports, for example,
that doubling notes in 3:1 perfect twelfths in Pierce-Bohlen tuning works just
great.

What exactly the cycle interval is in 88CET, or in Carlos' alpha and beta
tunings for that matter, is not immediately apparent. The step size for these
tunings, unlike Pierce-Bohlen, are not based on making any particular interval
exactly just. They are instead based upon making several target intervals as
accurate as possible.

To my way of thinking, for a tone added to a chord to be thought of as a
double, that tone should not have much effect on the underlying character of the
entire chord. If it changes the character of the chord, then it clearly has a
significant function in the new chord, so it doesn't make sense to think of it
as a functionally equivalent double of a another chord tone.

The idea of nonoctave doubling then has meaning provided that:
1. The interval by which you double a note is more "bland" than the intervals
in the chord itself. Consider for example the traditional 4:5:6 close-
voiced, root-position, major triad. Doubling the root an octave higher
does not significantly change the underlying character of the resultant
chord from its undoubled form. The octave added into that chord is
comparatively bland; doesn't attract our ears' attention like the thirds.
On the other hand, if we were to instead to try to double its root a 7:4
ratio higher, the underlying character of the chord does significantly
change. It becomes a dominant seventh chord. The subminor seventh
attracts attention as much as the thirds in the chord, if not more.
2. No other resultant interval is more interesting than the intervals in the
undoubled chord. Using the above example of doubling the root of 4:5:6
major triad up an octave, the resultant chord has the ratio 4:5:6:8. That
clearly creates the interval of an octave between the root and its double
(4:8), but it also creates a perfect fourth (6:8), and a minor sixth (5:8).
The underlying character of the chord does not change also because those
intervals are no more attention-getting than the major and minor thirds
already in the undoubled chord. Obviously you would get a very different
sensation if you were to start with a 3:4:5 second-inversion triad, and
attempt to double the bass up a perfect fifth. Although the interval of
doubling, the fifth, is bland in comparison with the other intervals in the
original chord, this also produces a 9:8 major second, which is clearly
anything but bland, so it definitely attracts attention away from the
constituents of the original chord. It changes the chord's underlying
character.


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Thanks for correcting me on that.

Oh, by the way, here's a side-note for you clarinet fans: I picked up a
recording some of you might be interested in. A little known fact is that
Mozart's Clarinet Concerto and the Quintets were written for an instrument
somewhat different from the modern Bb/A soprano. That instrument could perhaps
be described as a hybrid of the usual soprano with a basset horn. (The basset
horn is kind of like the modern alto clarinet except that it's in F instead of
Eb, and - much more importantly - its bore diameter is essentially the same as
that of the soprano.)

The instrument that Mozart wrote these works for was essentially a soprano
clarinet with its low end range extended down something like a major third or
perhaps a P4 (it's hard to tell from the pictures). I already had a CD of the
quintets done on an original instruments' implementation of this instrument
(AMON RA CD-SAR-17), but I hadn't heard the Concerto performed this way. I
don't have the CD with me, so I can't locate the numbers and such, but it's
performed by David Shifron as I recall.

The performance of the Concerto was pretty good, and was interesting because
of the spots where the usual score goes up an octave to accomodate the smaller
range of the modern instrument - he doesn't do that here. But the performance
of the quintets isn't as impressive on my opinion. There are places where the
expressivity is kind of lacking, and I got the frequent sense that the quartet
was trying to slow Shifron down and the reverse.


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I have just signed on to this group, after hearing that my dissertation on
clarinet microtones was mentioned. This information (478 microtones) has been
included in a book ("The Clarinet of the Twenty-First Century") that also
describes 213 altissimo register fingerings and 463 multiple sounds - I also
have a somewhat shorter but similar book on bass clarinet. Please send me a
private email if you would like further information.

The German fingering system still exists (in fact many German orchestras will
not even admit a Boehm-system player to an audition), although almost all
American clarinetists use Boehm system instruments (one notable exception is
Michelle Zukovsky of the LA Philharmonic, who uses a Wurlitzer clarinet) -
multiphonics are possible, but, naturally involve different fingerings (Joji
Yuasa details multiphonic fingerings for both French/American/Boehm and German
systems in his "Clarinet Solitude") - it is not always possible to generate the
same pitches from both systems.

E. MIchael Richards
Associate Professor of Music
Hamilton College
Clinton, NY
76400.1435@CompuServe.com


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