[math-fun] Fourier analysis and music
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?
Perhaps wavelets are what you want. -- Gene On Sunday, March 1, 2020, 9:59:24 AM PST, Henry Baker <hbaker1@pipeline.com> wrote: 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?
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?
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
I believe the term used was Cepstral analysis. Unfortunately, most of the good properties of Fourier transforms did not exist with Cepstral transforms.
On Mar 2, 2020, at 2:11 PM, rcs@xmission.com wrote:
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?
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (4)
-
Eugene Salamin -
Henry Baker -
rcs＠xmission.com -
Tom Knight