source file: mills2.txt Date: Mon, 25 Nov 1996 11:43:47 -0800 Subject: McLaren on Fourier analysis From: Ben Denckla > The problem with this way of thinking about music > and sound is that the method you're using (Fourier > analysis) is incapable of ever giving you *anything* > BUT a set of perfectly harmonic frequencies. If you > Fourier-analyze a set of inharmonic frequencies, your > output will be a set of perfectly harmonic frequencies-- > albeit an infinite number of 'em. I am not familiar with the works that Mr. McLaren is cricizing, but I feel that there are is some mathematical imprecision here. There are various types of Fourier analysis. It seems that the type of analysis referred to here is Fourier series analysis. McLaren claims that "If you Fourier-analyze a set of inharmonic frequencies, your output will be ..." This is a somewhat nonsensical statement in that you cannot perform a Fourier series analysis on a non-periodic waveform. The technique is only *defined* for periodic waveforms. So McLaren's general point, that Fourier series analysis "limits the types of sounds which can be usefully analyzed" has the right spirit, but is in detail wrong. In fact Fourier series analysis limits the types of sounds which can be analyzed at all, usefully or otherwise. My point is that there are no spurious results to the analysis, since it is only defined for periodic signals. Periodic signals are those that can be expressed as a superposition of harmonically related signals. Other types of Fourier analysis can deal with aperiodic signals. For instance, if you take a long-term FFT of a gong sound, you may be able to see its inharmonic structure quite nicely. The tradeoff that you have made here is that you have very little time resolution with a long-term FFT, so you do not know much about how the spectrum dynamically evolves. Ben Denckla Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Tue, 26 Nov 1996 00:15 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA07577; Tue, 26 Nov 1996 00:16:53 +0100 Received: from eartha.mills.edu by ns (smtpxd); id XA07644 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id PAA08814; Mon, 25 Nov 1996 15:16:50 -0800 Date: Mon, 25 Nov 1996 15:16:50 -0800 Message-Id: <199611251528_MC1-C6D-45C1@compuserve.com> Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu