In the 1960s & 70s, there was chatter about variations of the Fourier Transform. One was called SCEPTRUM; there were others. Rich ---- Quoting Henry Baker <hbaker1@pipeline.com>:
This problem has been bothering me ever since I learned about Fourier transforms and the frequency domain as an undergraduate who loves music.
Unlike in electrical engineering and physics, where frequency is a *linear* scale, and an entire bounded spectrum can be translated up/down in frequency using mixing converters, 12-tone music uses a *log* frequency scale, and I'm not aware of any simple Fourier formulae that work on a log frequency scale.
The ultraviolet catastrophe shows that the amount of information encoded in higher musical octaves grows enormously, so any invertible Fourier translation of standard 12-tone scales is going to have to throw away this extra information.
Is anyone here aware of any interesting papers or theorems which involve *linear* time and *logarithmic* frequency? Or perhaps there are Fourier-like theorems that involve *logarithmic* time and *logarithmic* frequency? Or *exponential* time and *logarithmic* frequency?
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