source file: m1395.txt
Date: Fri, 24 Apr 1998 21:18:49 -0500 (CDT)

Subject: Orchestral-Instrument Aperiodicity

From: mr88cet@texas.net (Gary Morrison)

Any of you folks who are interested in the discussion on slight
aperiodicities (i.e., not quite exact harmonic overtone structures) in the
tones of orchestral instruments might want to check out:

        http://lonestar.texas.net/~mr88cet/tuningStuff/BClarWave.GIF

(I hope that URL works;  I confess that I'm still a little new to web stuff.)

   Some things are notable considerations in or about this picture:
1.  This is a bass-clarinet (concert) C# at about 139Hz.
2.  It is clearly not COMPLETELY periodic, despite the fact that the tone
    itself sounds almost completely unvarying.
3.  If you look at the heights of various peaks and troughs in the waves, you
    will see that they vary by now means randomly.  As the red, hand-drawn
    curve shows, there's a definite periodicity to the depth of the lowest
    trough.  That frequency does not seem to match the fundamental or any other
    harmonic present in the wave.  (This fact is easier to see when you zoom
    out on the wave a little more.)
4.  Since the entire wave does not move up and down with this curve, this can't
    be due to a subharmonic.  (Also, it was run through a subsonic filter before
    being drawn here, and this particular pitch is in the lowest register of
    the bass clarinet, so it subharmonics are inherently unlikely.)
5.  A slightly-detuned upper partial seems to be the only possible explanation
    for this sort of motion of that lower trough, as well as the other peaks
    and troughs.
6.  It is sampled at 44.1KHz, which works out to about 317 samples per cycle.
7.  It is displayed at about 3 1/2 samples per pixel.
8.  A Fourier analysis shows that the highest harmonic that is loud enough to
    have any chance of being visible on this display is the 38th, which has a
    frequency of about 5.3KHz.
9.  That 38th harmonic is represented by about 8 samples, so it should have
    no problem whatsoever being accurately represented here.
10. If you zoom in on the wave closer, you find that all of the various wiggles
    are accurately represented here.  None have been spacially aliased by
    insufficient display resolution, for example.
11. The troughs circled in green MAY reflect the peaks and troughs of a high
    harmonic walking through that particular slightly-lower-harmonic-dictated
    trough.  I'm not sure of that by any means though.