Hi Ryan-

The sines and cosines that get used to build up a waveform in
Fourier analysis are the "basis functions" of the Fourier
transform. It is possible to decompose signals using many
different types of bases. The Fourier basis (sines and cosines,
harmonically related if the signal is of finite extent) has
some nice mathematical properties that make the decomposition
(and recomposition) simpler, mathematically, than it is with
many other bases. But that simplicity doesn't make the
Fourier basis "right" for all applications.

In your case, you want to use as "basis functions" the signals
played by standard instruments. These are much more complicated
than the sines and cosines in a Fourier basis. Besides the
fact that the sustained waveform from an instrument playing
a note has a non-sinusoidal shape, notes are transient (they
start and stop in time) and also dynamic (their pitch, volume,
and timbre vary in time, e.g., due to
tremolo, vibrato, etc.). Although it is mathematically possible
to represent signals with such dynamic, transient structure via
a Fourier transform, I don't think a Fourier decomposition
is well-suited to your problem.

One approach is to actually take samples of the instruments
you'll use, playing all the notes available, and use them (with
various durations) directly as your basis. This would be the most
accurate approach, but the calculations you'd need to do to find
the expansion coefficients (i.e., the score!) would probably
be extremely difficult computationally, and probably not
well-defined (the basis is likely neither complete nor
orthogonal). You'd be doing something like additive synthesis,
but with a much bigger basis than is usually used! Looking
up some of the math associated with additive synthesis might
provide you with some leads.

A possible option that has the potential to be more computationally
tractible would be to use some kind of wavelet or other
time-scale or time-frequency transform rather
than a Fourier transform. Very roughly speaking, you can
think of such a transform as breaking up a signal into
*localized* pulses, i.e., notes! That is, where a Fourier
transform represents a signal as a sum of "eternal" sines
and cosines of specific frequencies, a time-frequency transform
breaks up the signal into separate parts that are localized both in
frequency *and* time. You might be able to find some way to
project a wavelet or other time-frequency transform of the sound
you are interested in onto the transforms of sounds from the
instruments you have available; this would give you the notes
and volumes needed to most closely match the desired signal.
This won't make any fundamental problems with the incompleteness
or redundancy of your basis (choice of instruments & notes) go
away, but use of such transforms might provide methods of
approximation that make the problem more tractable computationally.

A google search on "wavelets" and "music" will probably get you
started. This wavelet FAQ might also help:

http://www.math.ucdavis.edu/~saito/c...avelet_faq.pdf

Here's a review article on time-frequency analysis of sounds
from musical instruments---your basis functions, so to speak:

http://epubs.siam.org/sam-bin/getfil...cles/38228.pdf

If you want to learn more about Fourier expansions from
a musical point of view, see:

http://ccrma.stanford.edu/~jos/mdft/

Here's a reference that turned up in my own quick googling using
"time scale transform music" that may provide a starting point
for thinking along these lines, if you can find a copy:

Kronland-Martinet R., Grossmann A. "Application of time-frequency and
time-scale methods to the analysis, synthesis and transformation of
natural sounds." in "Representations of Musical Signals", C. Roads,
G. De Poli, A. Picciali Eds, MIT Press, october 1990.

Interlibrary loan may help you here!

A similar search using "time frequency transform music" turned up
"Musical Transformations using the Modification of Time-Frequency
Images" in a 1993 issue of *Computer Music Journal*:

http://mitpress.mit.edu/catalog/item...d=6768&ttype=6

This is just from some quick googling and these are probably not
the best or most recent references that may be relevant. Wavelet
and time-frequency analysis is now very mature and there are
entire textbooks and monographs on these topics. Good
luck with this.

Peace,
Tom Loredo

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