[sdiy] Fourier Analysis Question

[ASSI]

Hi Tom,

On Saturday 18 December 2010, Tom Wiltshire wrote:
> In part, I want to separate out the parts of the piano waveform
> I *can* analyse with FFT and model with wavetable synthesis (e.g. all
> the harmonic bit) and see what and how much is left over.

You will find that there really isn't all that much periodicity in a piano 
sample.  That doesn't mean you couldn't produce a convincing wavetable (PPG 
had done it decades ago) of a piano, but it won't play like a piano.  In a 
narrow range and for certain styles of playing it can be very convincing 
even without adding inharmonic partials, however.

I have some spectral representations of wavetables up on my site if you want 
to have a look.

http://Synth.Stromeko.net/Tuning.html

Not to discourage you from doing your research, but in some respect you've 
picked the most difficult instrument to start with...

> If I can see
> and hear the enharmonic part alone, it might give me clues as to how to
> model that element of sounds.

The relation of partials to fundamental is inherently non-linear in a piano.  
There is a component that comes from string stiffness that makes the upper 
partials progressively sharper and another one from string mass 
(distribution) that would make the partials flat if it wasn't for the first 
component winning out.  Still the sharpness of the upper partials is 
significantly modified by the second component.  The fundamental in a single 
piano bass string actually has several modes with slightly different 
frequencies that transfer energy to each other.  The upper registers are two 
or three strings that also have coupled modes between them.  They aren't 
tuned to exactly the same fundamental either as that would create a very 
dull sound and short sustain.

> There really isn't any difference in result between having a database of
> partial sets (a frequency-domain representation) and having a wavetable
> (a time domain representation).

While that is true on a very high level, keep in mind that the theory that 
links the two is only really valid for quite restricted classes of signals.  
Having a finite Fourier spectrum in hand really requires to have an infinite 
time-domain signal with strict periodicity, for instance.  Now, you can bend 
the rules quite a bit in practise as everyone knows, but the nice guarantees 
of a linear transform -- that it doesn't matter in which order and in which 
domain you do certain operations, for instance -- falls apart very quickly.

For short-time spectral analysis you need to be aware that higher resolution 
on the time scale means lower resolution on the frequency axis and vice-
versa.  The literature is full of papers that use properties of special 
signal classes to get around this fundamental limitation, but don't expect 
anything as easily accessible as an FFT.


Regards,
Achim.
-- 
+<[Q+ Matrix-12 WAVE#46+305 Neuron microQkb Andromeda XTk Blofeld]>+

SD adaptation for Waldorf microQ V2.22R2:
http://Synth.Stromeko.net/Downloads.html#WaldorfSDada