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Comments and Conjectures on Roughness for a Sunday Morning

1 August 1999

        The question on sensory dissonance from Leman and Sethares reminds me
of the thesis work on roughness that Jian-Yu Lin did in my lab in 1995.

        Jian-Yu studied roughness produced by both amplitude modulation of
sine tones and and beats of sine tones. He studied center frequencies from 70
Hz to 2000 Hz. He found, like everybody else, that as the center frequency
increases, the modulation rate or beat rate that produces maximum roughness
also increases. Beyond that, we ran into two kinds of troubles.

Theoretical trouble: Like others, we imagine that roughness is caused by
fluctuations of excitation in the auditory system. This involves three
components: (1) The spectral components must excite the same auditory filter.
This is the critical-band connection with roughness noted by everybody. (2)
There is a temporal modulation transfer function. The TMTF is a lowpass filter
on fluctuation rates because the auditory system cannot follow very rapid
fluctuations. (3) There is a "speeding factor" whereby increased fluctuation
rate leads to increased sense of roughness. The need for the speeding factor
is clear because the other two factors favor small separations between the
spectral components. Without the speeding factor, one would expect maximum
roughness at the lowest possible modulation frequencies or beat rates. Because
of all these factors, we do not expect, a priori, any simple relationship
between maximum roughness and critical band.

Experimental trouble: We found it difficult to get consistent judgements of
maximum roughness over time. We tend to think that roughness may be
multidimensional, perhaps related to the tradeoffs between the three factors
above. In the end, we asked our listeners a different question. We asked them
to find the highest modulation frequency or beat rate that produced a large
sense of fluctuations. This question was based on our observation that as the
modulation frequency increases, there comes a point where listeners experience
a rather rapid falloff in perceived fluctuation strength. We got stable
results with this question. We think that it is a better question than maximum

Asking listeners to find this maximum modulation rate or beat rate has the
additional advantage of eliminating the speeding factor. Instead of asking
listeners to maximize a perceptual quantity (roughness) this new question asks
listeners to find the limiting characteristic. Therefore model calculations
intended to explain the results do not need to include the speeding factor. It
is enough to consider auditory filtering and TMTF.

Now to the question from Leman and Sethares: When Jian-Yu did the experiment
in this way, he found the 70 Hz limit suggested by Zwicker and Fastl. As the
center frequency of the components increases (frequency of the carrier for AM
or mean frequency of a beating pair) the
"highest-modulation-rate-before-a-dramatic-decrease-in-fluctuation" increases
until about 1000 Hz, and there it tends to saturate at a value of about 70 Hz.

More trouble: We think that it is important to study both AM and beats. They
sound rather similar, and any good model ought to be able to deal with them
both. For the same fluctuation rate, the AM signal has a bandwidth that is
twice as large as the beating sines, and that is where the model problems
begin. In the end, we were not satisfied with our ability to find a model that
would explain both. Therefore, this work appears only in a an ASA abstract,
JASA volume 97, page 3275.

Conjecture about the 70 Hz limit: This conjecture must be seen as heretical
because it is outside the bounds of any modeling concept for roughness,
including our own. There are also no data to support it. Nevertheless, here is
is: As the modulation frequency or beat rate grows to become as large as 70
Hz, the listener starts to hear the fluctuation rate as a low pitch. Maybe
it's like the missing fundamental effect or maybe it's a difference tone.
Whichever it is, the presence of this low pitch tends to homogenize the
auditory sensation and lead to a dramatic decrease in perceived roughness.
That is why one does not see maximum roughness fluctuation rates greater
than 70 Hz.

Bottom line: If this conjecture is correct, then it means that there is a
70-Hz boundary. Studies of critical bands and roughness really ought not to go
there because the rules suddenly change at fluctuation rates that high. That,
in turn, would limit roughness studies to center frequencies less than 1000 Hz.

Best wishes,

Bill H.