RE: To Paul Erlich

You wrote:

>Fourier proved that any periodic waveform can be expressed as the sum
>of sine waves whose frequencies form an EXACT harmonic series.

I quote from the Encyclopedic Dictionary of Mathematics (MIT 2nd Ed 1993)
''Fourier also stated (without rigorous proof) that an arbitrary function
could be represented by trigonometric series, a statement that gave rise
to subsequent developments in analysis. (Vol 1, p 620)''


At Santa Cruz (way back when) we did a series of sprectral analyses of
real instruments and deviations from harmonic seria (mostly stretching
but also some more eccentric behavior) were the norm. (This is certainly
familar to anyone who has worked with real string instrument harmonics).
At Darmstadt in '90 I saw some similar work from IRCAM done with extreme
amplification so that the deviations through - say - the 64th partial were
evident. Again: whether voices, bowed strings, brass, or woodwind, all had
harmonics deviating from the expected integer ratios.

Deviations and stretching in the strings and vocal chords seemed to
account for the situation there, but what is with the wind instruments?

A possible explanation:

In a trombone, for example, you have several systems working at once whose
net effect is quite complex, indeed aperiodic. The basic tube, and the bell have
separate resonances (which vary with the slide and with muting), the mouth
cavity of the player has its own variable resonances, and the oscillating
lips are a very complicated sound with a large noise component and are
variable in the extreme. The upshot of all this is a constant negotiation
between the elements to get something acceptable. One short example: if I
am to play d1 as the tonic of a major triad on a tenor trombone, I have to
choose between buzzing d1 and trying to resonate the 5th partial of Bb in
the first position or the third partial of G in the fourth position. Each
of which produces a distict timbre and both of which may be problematic in
the given triad, so I can change the shape of my mouth, or mute, or - yes - add
either slide or lip vibrato (and I have heard players who could lip vibrato
over an octave), or add or remove noise from the lip (''growl''). (I know
this will damage my reputation, but I recommend listening to any recordings
of Jackie Gleason's ''Velvet Brass'', of Spike Jones recordings with Frank
Rehak, or of Stuart Dempster's performance of Robert Erickson's _General
Speech_ to hear what trombonists can do with pitch and timbre.)

About vocal ''subharmonics'', I find the easiest way to learn to produce them
is to start by singing while inhaling (the ''fry''), eventually adding ex-
haling, so that one can eventually do both and sing continuously (I remem-
ber Robert Ashley doing something similar in the early 70s so that he could
speak without pausing. The Tuvan singers here in Frankfurt on the main
shopping street do the same.

You wrote:

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

Yes, I agree with La Monte (while recognizing that some people have excellent
pitch height memory) and thus combination tones are indispensible - if you
are making reference to my spacing model, the harmonic series is the model and
structures are described in terms of just intonation but only to simplify
matters as I assume some degree of tolerance, likewise I assumed sine waves
(need I repeat it again) to avoid masking.

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