Natural Science Forum / Physics / Acoustics / June 2005

i'm not Ethan, but this isn't exactly what i meant to be bringing up.  i
sure hope that no one here ignores phase values for different drivers in a
loudspeaker or between different loudspeakers in a stereo or multichannel
system.  i think everybody here agrees that applying some all-pass filter to
your left speaker but not to your right will change something that we can
hear.

my issue is about whether we can hear the difference between two harmonic or
quasi-periodic musical notes that are played one right after another and are
identical except the later has different slowly varying functions of time
for the phase component of all of the harmonics than for the former.  even
though Ghost complained about this representation, the two tones are:

           N
   x(t) = SUM{ r_n(t) * cos(n*w0*t + phi_n(t)) }
          n=1

and

           N
   y(t) = SUM{ r_n(t) * cos(n*w0*t + theta_n(t)) }
          n=1

w0 is the fundamental frequency of the tone, w0 would be 2*pi*440 Hz if the
note is the A just above middle C.

r_n(t) ( >= 0 ) is the time-variant magnitude envelope for the nth harmonic
for both x(t) and y(t).  phi_n(t) is the time-variant phase for the nth
harmonic for x(t), theta_n(t) is the time-variant phase for the nth harmonic
for y(t).  all are slowly moving functions of time (i.e. they are
bandlimited to much less than w0).

x(t) is identical to y(t) *except* that the phase functions for the
harmonics, phi_n(t) and theta_n(t) are not equal to each other (but they are
still both slowly moving functions of time, otherwise x(t) and y(t) would
not be nearly periodic).  the magnitude of each harmonic, r_n(t), is the
same for x(t) and y(t).

i think what Ethan is saying (please correct me if i am misinterpreting,
Ethan) is that theta_n(t) can be different than phi_n(t) and we will not
hear that difference.  (assuming that theta_n and phi_n are approximately
equally bandlimited.)

in article ZoSdnUaS55qcmirfRVn-sQ@giganews.com, Ethan Winer at ethanw at
ethanwiner dot com wrote on 06/20/2005 14:12:

pretty close.  it's wavetable synthesis (not to be confused with the PCM
sampling that E-mu kwyboards or the zillion of "wavetable" soundcard chips
do) or sometimes called "group additive synthesis" in the lit.  the
restriction that wavetable synthesis has that additive synthesis does not is
that all of the partials or overtones of wavetable synthesis must be very
nearly harmonic, very nearly an integer multiple of some common fundamental
whereas additive synthesis can have non-harmonic partials.  wavetable
synthesis might not work so good for bells and such (except if you could
group the partials into two or three harmonic groups, then two or three
simultaneous wavetable synths could do such a bell).  for normal wavetable
synthesis, the deviation of a partial's instantaneous frequency from the
exact harmonic value is equal to the time derivative of the phase function
(that's another reason i am unhappy that someone would publish a zillion
papers all beginning with the assumption that phase just does not matter).

the promise of additive synthesis was that it could realistically recreate
an arbitrary sound because it would accurately (to some finite degree)
recreate each sinusoidal partial component of that given sound.  in the case
of many musical tones (after the attack transient) it is known that the
partials are all _pretty_much_ harmonic (not perfectly harmonic, e.g. the
piano has its higher harmonics progressively sharper and sharper).  if the
partials are nearly exactly harmonic, wavetable synthesis is sufficient (and
computationally cheaper) and the small deviations of some partial from its
exact harmonic value can be dealt with by slowly changing the phase.  but if
you throw the phase away to start with, it's kinda hard to do that.

how about N phase knobs, one for each harmonic?

my issue is, we got here a synthesis method.  sure the synthesizer could be
used solely to generate cool, but unnatural sounds, and that's fine.
anything goes w.r.t. phase.  but often, a measure of merit of a synthesis
method is how realistically it can generate a sound of some existing (and
usually acoustic) musical instrument.  a good example of such is the
"physical modeling synthesis".

wavetable synthesis is cheap.  much cheaper than physical modeling or
additive synthesis.  a little more expensive than 2-operator Chowning FM
synthesis.  but even if it's cheap, wavetable synthesis can generate
waveforms that match the waveforms generated by additive synthesis *but*
*only* for the case where all of the partials (or those with any decent
amount of energy) have frequencies that are very nearly harmonic, that have
frequencies that are very nearly integer multiples of some common
fundamental frequency.

so for this class of musical tones or notes, we try to use wavetable
synthesis to (re)create some tones, there will be some data reduction for
the representation of those slowly moving envelopes and phase functions
(just as there would be for additive synthesis), and we wish to minimize or
at least constrain the effect of that data reduction on how (bad) the note
sounds.  one way to reduce the data is to simply toss it all away.  if we
toss away the envelopes r_n(t) (set them to zero or to some totally
unrelated functions of time), that would definitely have a noticeable effect
on the sound of the tone, so we wouldn't do that!

but it is a lot easier for some to do the same to the phase (with no
justification offered) and i'm still wondering how free we are to do that if
we're trying to persuasively emulate another musical instrument.

Bert,

drivers <

Phase shift in a crossover can be audible, but only for frequencies output
by both drivers around the crossover frequency. And then, what's audible is
the change in frequency response, not the phase shift itself. But was Robert
even talking about loudspeakers?

--Ethan