[sdiy] Harmonic bandwidth

[Peter Keller]

On Sun, Jan 06, 2008 at 04:17:40PM -0800, Ben Lincoln wrote:
> Besides the obvious difference in sound, If you look at it in a spectral 
> view, you can see that the two sections are very similar in terms of 
> their basic content, but the second section appears "smeared". So if I 
> am using the incorrect terminology, I think that the author of the 
> PADsynth software is using it in the same incorrect way, because that is 
> what I took away from his discussion of "harmonic bandwidth".

The only thing I know of concerning "smearing" in a DFT is something
called "spectral leakage". It is what happens when you have a strong
frequency component that doesn't fall nicely into one of the specific
frequency channels the DFT is producing. The power of the original has
to go somewhere so it is spread out around the logical place where the
frequency should be in the DFT. As for how wide the spread is, I couldn't
tell you since I'm not that intimate with the mathematics.

However, he (the author of zynaddsubfx) does say this:
"I consider one harmonic(overtone) as being composed of many
frequencies. These sine components of one harmonic are spread over a
certain band of frequencies."

That says to me he could very well be misinterpreting spectral
leakage--but that seems so strange to me given the fantastically awesome
mathematical simulations he has written in his flagship product. I mean,
him, if anyone, should know about spectral leakage....

So, I would interpret his meaning as saying that he is sort of redefining
a harmonic from being instead a single frequency to being a band of
frequencies centered around the traditional definition of the single
frequency harmonic. Mathematically, what he describes doesn't have to
be the result of a DFT exhibiting spectral leakage and can exist in
continuous mathematics as real functions.

I think the above statement because of his algorithm:

------
1.  Make a very large array that represents the amplitude spectrum of
the sound (default all values are zero)

2. Generates the distribution of each harmonic in frequency and add it
to the array

3. Put random phases to each frequency of the spectrum

4. Do a single Inverse Fourier Transform of the whole spectrum. Here
is no need of any overlapping windows, because there is only one single
IFFT for the whole sample.
------

The fact that he generated a distribution into the DFT space, and then
did the inverse of it to get a perfectly periodic sound which contained
the "fat" harmonics is evidence (to me at least) that he is working with
his different definition of the word harmonic.

As for why this sounds better, he makes the claim that traditionally
choirs and orchestras were detuned either through intent or accident
and that resulted in a richer sound since people were always "nearby"
(in both a positive and negative cent direction) the correct harmonics.
I can hypothesize about why the spread factor of the distribution
increases as you go higher and higher in frequencies--instruments have
more and more complex disturbances in their higher vibrational modes
due to materials used and their construction methods.

However, I don't know enough about psychoacoustics to know why a "fat"
sound (a spread harmonic) is considered richer than a "thin" sound
(a single frequency harmonic).

-pete