Re: [tuning] Approx GCD's are Phase Modulated JI Intervals

Graham said:
> I don't know what a Hilbert transform is. Fortunately, I don't think
> Helmholtz did either. Wikipedia says even Hilbert didn't come up with
> it until 1905.
> //snip
> That formula does depend on the amplitudes being equal. When they
> aren't you have to decide how to define the frequency of the modulated
> signal. You can get a result identical to a combination of frequency
> and amplitude modulation. I did it with phasors. After all this time
> I don't remember exactly what a phasor is, but it does make the
> calculation simple to understand. However Helmholtz did it he got the
> same result as me.

Turning a sine wave into a phasor involves the Hilbert transform. The
Hilbert transform was invented as a way of generalizing the "phasor"
concept to all waveforms. All the Hilbert transform really does is
create a linear +90-deg phase shift across the board, but this can be
used to generate what they call the "analytic signal" from the
original, which has a lot of uses. A phasor is the analytic
representation of a real sinusoid.

A phasor is e^(jwt), and a sine wave is cos(wt). As you remember,
e^(jwt) = cos(wt) + jsin(wt). So the way you generate a "phasor" from
a single sinusoid (e.g. turn cos(wt) into e^(jwt)) mathematically is
to:

1) come up with a 90 degree phase-shifted version of it (this is what
the Hilbert transform actually is)
2) make that the imaginary part of the signal, and make the original
the real part
3) The resulting waveform is now called the "analytic representation"
of the signal and you can do all kinds of fun but usually meaningless
instantaneous frequency/amplitude stuff with it

Sorry for using such confusing terminology, hope that clears it up a
bit. What Helmholtz did was turn the constituent sine waves into
phasors and then derive the instantaneous frequency that way. In doing
so, he used the mathematical concept that was later generalized by the
Hilbert transform, hence my later comments.

> > 2) The Hilbert transform isn't a causal function, so the ear can't
> > really be performing a real-time 90-degree phaseshift on the incoming
> > signal. It would have to go forward in time to figure out what's
> > coming next. Some of these limitations could be solved by using a
> > windowed Hilbert transform, but to assume that the auditory system is
> > actually doing that is complete speculation.
>
> Say what?

To actually convert a signal into the analytic equivalent in real
life, the brain would have to "read into the future" and figure out
what's coming next. A possible around this is to just use a "windowed"
Hilbert transform (similar to how you can use a "windowed" Fourier
transform), but to assume that this is actually happening (and hence
leads to this particular "wobble" artifact) is just speculation.

> > 3) I have, empirically speaking, never personally heard this "wobble"
> > in pitch, although I can certainly convince myself of it if I try.
>
> So what pitch do you hear when two sine waves beat? I think we know
> you get a certain fuzziness when intervals are slightly out of tune.
> Maybe that's part vibrato as well as tremolo. We're talking about
> small differences in frequency.

I hear fuzziness even when hearing a single sine wave played by
itself. If you try, you can sort of "detune" the sine wave with your
mind (and hear it as an upper harmonic of a phantom undertone and all
kinds of fun stuff).

But when two sine waves beat, I hear roughly the mean of the two
pitches. Perhaps there is some very slight vibrato and I'm just not
hearing it because it's too slight. Keep in mind that this would also
mean that we would hear pitch changes whenever instruments change in
volume, or wherever auditory envelopes occur.

My point was that the instantaneous frequency showing a bit of FM
going on doesn't actually mean that it translates over to perception
that way. It sounds good in theory, but all that that is really
showing you is how you'd have to apply FM and AM to a single sinusoid
at the mean frequency to get back the original complex waveform. Does
440:445 count as "simple" or "complex"?

-Mike