[music-dsp] audibility of phase shifts to higher harmonics

Bill Schottstaedt bil at ccrma.Stanford.EDU
Fri Oct 8 09:04:02 EDT 2010

If you vary the initial phases to minimize the peak amplitude,
the minimum peak version sounds "raspier" (if there are
enough harmonics and so on).  I thought this might make a
difference if used in FM as the modulating wave (mentioned
in https://ccrma.stanford.edu/software/snd/snd/fm.html),
but the result was disappointing.  That got me interested
in what is the minimum peak of (say) n equal amplitude
harmonics.  Horner and Beauchamp wrote a paper about 
this in the 90's, and I've followed up with


My initial guess was that you could get below the square root
of n, and that's holding up so far.  For my aged ears, if I
want to hear the difference, n has to be about 60, but surely
younger ears could hear it at lower n.  In the limit (say 65k
harmonics), the all cosine wave is a click, and the minimized
peak wave sounds like a seriously damaged attempt at white
noise.  Two things have always amazed me in this area: how
similar the waveforms sound though they look completely
different, and how much cancellation you can get in the best
case.  (By the way, if you simply randomize the initial phases
you can around n^.6 or n^.7 -- good, but no where near the

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