RE: First Published Treatise in English

>One phenomenon that I recall from trombone playing was the constant
>correction of pitch (by lip and slide) so that the deficient harmonic
>series of the tube would sound better. This may be taking place in vocal
>performance as well.

Daniel,
Fourier proved that any periodic waveform can be expressed as the sum
of sine waves whose frequencies form an EXACT harmonic series. A brass
instrument is crafted so that its RESONANCES approximate a harmonic series,
but this relationship is not exact (a lot of further misconceptions surround
brass instrument physics). However, when playing a single note, the
resonances are DRIVEN by the lips of the player, which normally produce a
periodic waveform during the "sustained" part of the note. Therefore, an
EXACT harmonic series is available to drive the resonances of the
instrument. The closer these resonances are to harmonic, the more the

overtones will be amplified -- this is the principle behind successful brass
instrument design.
Likewise, the human vocal cords, when employed in a steady, sustained
manner, produce a periodic waveform (though recently I have been able to
DOUBLE or occasionally TRIPLE the period of this waveform and sing an octave
or twelfth lower than would be normal -- undertone singing! I can then
emphasize different overtones of these really low notes for an amateur
version of throat singing). Overtone singing is just the process of shifting
the resonance to one or another of the harmonic overtones -- again, an exact
harmonic series results (of course, no one can maintain the pitch of the
fundamental perfectly, but that is a separate issue). Inharmonic series are
possible in large, essentially infinite-element vibrating systems like a
piano string, but brass and wind instruments and human voices are
essentially elaborate amplification devices for a single-element periodic
oscillator.

>When ''overtone singing'' had its heydey, I recall
>being disturbed by how unstable the fundamental was, and the constant
>intrusion of beats, perhaps resulting from attempts to tune the harmonics
>at the expense of the fundamentals.

In light of the above, this doesn't really make sense.

>La Monte
>contends that one can sing Just intervals only through elimination of
>beating

Daniel, do you agree with LaMonte and if so, how do you justify doing JI
with beatless pure tones without invoking combination tones?


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Date: Thu, 2 Jan 1997 10:29:21 -0800
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