Interesting.... is there a relationship between Chip-z and a Constant-Q ? Could this be used to compute an inverse Contant-Q?

It's not obvious to me but I don't have a deep knowledge of the Chirp-z transform. And it's quite possible someone might find some tricky way to use the Chirp-z transform for that job in the future.

What did occur to me after this thread is the possibility of processing specific frequency bands with the Chirp-Z. For example if we only want EQ 4kHz to 8kHz, (for the sake of the argument approximating a constant Q equalizer). A real filter or equalizer will affect the spectrum outside of the 4kHz to 8kHz region, so you would expect some "non ideal" behaviours by chopping the spectrum. If we added some guard bands to allow a transition say 3kHz to 9kHz it might help. However if we need to add guard bands 2kHz to 16kHz then we are already stretching the window to cover most of the spectrum and the whole idea of focusing on a smaller region is lost.

I see the value of a FFT/IFFT-like transform that accomodates the way musical notes are spread in the spectrum

There's already a constant Q algorithm. It's quite slow but people have invented fast versions. It's used for *analysing* signals. (AFIK) The problem is there's no inverse like a true transform-pair but people have worked on approximate inverses. The other avenue is filter banks covering unequal spans.

One of the problems with FFT processing is messing with the signal in the frequency domain in a way which is non-minimum phase. For example if we make an equalizer and just change the magnitudes the results can be quite unnatural.

Another problem is we are making modifications to *points* what happens between the points can be vastly different to the points. If you look at this curve fit (not and FFT) you can see oscillations between the points, which can get large in some instances. Changing relatively far way points can cause large changes to the shape,

A good example of such bad behaviours is the Equalizers built into sound cards. You can decrease bands above 3kHz and the treble increases!