Hi Brian

Saw your second post and a couple of extra poitns:

FFT bin sizes depend on the sample length, and are

fixed in size. E.g. for sample length of 4096

= just less than 10th of a second at 44100 samples

per sec, the freq. bins are 10.77 hz apart.

So the FFT is more accurate for higher frequencies than

for lower ones. It is indeed a harmonic series, but a

harmonic series based on the frequency of 10.77 hz, rather

than on the frequency of the instrumetn being sampled.

As I've just posted, but you may not have seen yet,

one can improve on this accuracy by frequency

interpolation.

As regards amplitude, if one uses the complex FFT,

and discards the imaginary

part, then the real part may fluctuate wildly, as

it will depend on the phase.

One should instead find the length of the complex

vector (sqrt of sum of squares), as you mention.

Or use a real instead of complex fft (same result,

but faster algorithm, if one is willing to discard

the phase information).

This gives reasonably stable amplitudes I find in my

tests, finding the amplitudes of the random frequencies

of an 0.1 second clip with accuracy of often less than

a decibel, but with the occasional wild value out by about 2 or 3 decibels,

e.g. finding it as 24 decibels instead of 27.

It's not so good as the pitch determination.

Perhaps there are ways of interpolating for the

amplitude as well as the pitch?

The angle of the complex vector gives the phase information.

Real and imaginary parts are two components of a vector,

and from that one can recover an angle, which tells one

where in the cycle the particular sine wave component

is at the beginning of the clip.

As for hearing partials in musical timbres, it is

interesting to speculate about which came first,

the FFT analysis, or hearing partials.

Did we lean to recognise partials as a

result of FFT analysis or was that something we

could do anyway?

I think one can demonstrate historically that hearing

partials came first, as for instance bell makers

in C17 listened to partials in bell sounds

and measured them, in order to help with making

of church bells.

I wonder what is the earliest record of someone

hearing partials in a musical sound?

Robert