re undertones/subharmonics by Jim Cole

Hello Jim - I found your descriptive post most interesting. I'll briefly
mention here some things I found when doing additive analysis of
recorded musical sounds in the early 1990s which may be of interest to you.

In 1990 I developed an additive analysis system which could be applied
to digitized recorded sounds. I first did a frequency analysis on the
sound yielding a frequency vs time trajectory. I then did a modified
Fourier analysis on the sound, taking for current window analysis frequency
that read from the previously determined frequency vs time trajectory.
I did a frequency analysis on a trombone Ab3 note and found the frequency
trajectory to have a lot of "wobble". If I did the analysis assuming
a frequency approximately at Ab1, two octaves below the heard pitch of
that note, I obtained a smoother frequency trajectory. I did additive
analyses for the two cases: (1) assuming the frequency trajectory
obtained for Ab3 and (2) assuming that obtained for Ab1. Both analyses
yielded the amplitude and frequency trajectories for a series of partials
having 1, 2, 3, 4, etc. times the frequency of Ab3. However, in addition,
there were additional weak partials having 1, 2, 3, 5, 6, 7, 9, 10, 11, etc.
times the frequency of Ab1. I resynthesized sounds based on both the
analysis
on Ab3 and that on Ab1. Both closely resembled the original recorded
trombone note, but the resynthesis on Ab1 had a kind of breathy "presence"
which made it seem almost indistinguishable from the original recorded note.
Thus I concluded that some sounds produced by musical instruments, certainly
horns, may contain a "subfundamental" at 1/2 or 1/3 or 1/4, etc. the
frequency of
the nominal fundamental and faint overtones of this "subfundamental".

I analyzed voice sung notes and often found a weak subfundamental an octave
below
the apparent fundamental. Basing analysis on the subfundamental usually
yielded
a more realistic additive resynthesis of the sound. I analyzed the notes C4,
F4,
Bb4, Eb5, and Ab5 sung by a soprano voice student singing the vowel sound
"ah".
There was a prominent subfundamental at C3 for her lowest sung note, which
had
a rather breathy character. I believe I recall there being a weaker
subfundamental
for her F4, while for the higher notes the subfundamental was scarcely
detectable.

I did not pursue the question of determining the physical explanation for
there
being subfundamentals, but given the physics of standing waves in cavities
such as
horns, it seems plausible that there should be faint vibrations at one half
or
possibly another submultiple of the fundamental frequency.

Dave Hill La Mesa, CA