Re: Pitch of a complex tone

[Malcolm Slaney <malcolm@xxxxxxxxxxxx>]

At 6:28 AM -0800 2/4/98, Alexander Galembo wrote:
>I would appreciate to be informed about any publications providng a doubt
>in this phase independence (if exist).

I think the first were Flanagon and Guttman (1960).  I'm not sure if they
described their observations as a phase change, but they are.  Pierce, a
few years ago, redid the experiments.

A description of this stimulai that Richard Duda wrote for the Apple
Hearing Demo Reel is appended to the end of my note.

-- Malcolm


This animation was produced in conjunction with Richard Duda of the
Department of Electrical Engineering at San Jose State University during
the Summer of 1989. Thanks to Richard Duda for both the audio examples and
the explanation that follows and to John Pierce for calling this
experiment to our attention.

Researchers in psychoacoustics have long looked to cochlear models to
explain the perception of musical pitch [Small70]. Many experiments
have made it clear that the auditory system has more than one mechanism for
pitch estimation. In one of these experiments, Flanagan and
Guttman used short-duration impulse trains to investigate two different
mechanisms for matching periodic sounds, one based on spectrum and
one based on pulse rate [Flanagan60]. They used two different impulse
trains, one having one pulse per period of the fundamental, the other
having four pulses per period, every fourth pulse being negative . These
signals have the interesting property that they have the same power
spectrum, which seems to suggest that they should have the same pitch. The
standard conclusion, however, was that below 150 pulses per
second the trains "matched" if they had the same pulse rate; they "matched"
on spectrum only when the fundamental frequency was above about
200 Hz.

[Pierce89] modified this experiment by replacing the pulses by tone
bursts=F3short periods of a 4,800-Hz sine wave modulated by a raised-cosine
Hamming window. In essence, he used Flanagan and Guttman's pulses to
amplitude modulate a steady high-frequency carrier. His purpose in
doing this was to narrow the spectrum, keeping the large response of the
basilar membrane near one place (the 4,800-Hz place), regardless of
pulse rate.

To be more specific, Pierce used the three signal "patterns" shown below.
All have the same burst duration, which is one-eighth of a pattern
period. Pattern a has four bursts in a pattern period. Pattern b has the
same burst rate or pulse rate, but every fourth burst is inverted in phase.
Thus, the fundamental frequency of b is a factor of four or two octaves
lower than that of a. Pattern c has only one burst per pattern period,
and thus has the same period as b; in fact, it can be shown that b and c
have the same power spectrum. Thus, a and b sound alike at low
pulse rates where pulse-rate is dominant, and b and c sound alike at high
pulse rates where spectrum is dominant. Pierce observed that the ear
matches a and b for pattern frequencies below 75 Hz, and matches b and c
for pattern frequencies above 300 Hz. He found the interval
between 75 and 300 Hz to be ambiguous, the b pattern being described as
sounding inharmonic.



       Pierce's tone bursts. Patterns a and b have the same pulse rate
frequency, while b and c have the same power spectrum. Here the
       test sounds are shown with one cycle per burst.

To see if and how these results are reflected in correlograms, a similar
set of tone burst signals were generated. The only difference between our
signals and Pierce's signals was due to differences in the digital sampling
rate used. To get a Fourier spectrum with minimum spectral splatter,
Pierce imposed two requirements:

              1)The tone-burst frequency fb was set at half the Nyquist
rate. Where Pierce's 19,200-Hz sampling rate led to fb =3D 4,800 Hz,
              our 16,000-Hz sampling rate forced fb down to 4,000 Hz.

              2)Each burst had to contain an exact integral number n of
cycles. This number, n, is a major parameter for the experiments,
              ranging from 1 to 128. If the pattern period is T, then to
obtain exactly n cycles of frequency fb in time T/8 requires that fb T/8 =3D
              n, so that T =3D 8n/fb .



Thus, to obtain the same spectral characteristics, we had to use different
numerical values for the tone-burst frequency fb and the corresponding
pattern period T. The table shown below is our version of Table I in
Pierce's paper.



A set of eight test signals was generated according to this scheme. Each
test signal consists of a sequence of the a, b and c patterns, each
pattern lasting 1.024 seconds. This time interval was chosen to get an
exact integer number of bursts, ranging from 4 for Case 1c to 2000 for
Cases 8a and 8b.