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Re: Tritone paradox
Here are some ideas on the application of the pitch algorithm of
Terhardt, Stoll & Seewann (1982) [hereafter TSS82] to the perception
of Shepard tones, which I prefer to call octave-complex tones or OCTs
(a term borrowed from David Butler's recent book -- or did someone
else use it first?). I will explain how the results of such an
analysis may be used to explain Deutsch's and Repp's "tritone
First, let's get one thing straight! TSS82 measure pitch on a
ONE-DIMENSIONAL scale. In their approach, it's not chroma that
counts. It's a SPECIFIC pitch, or chroma PLUS register.
Terhardt et al. (1986) investigated the pitch of OCTs by means of the
standard psychoacoustic technique of adjusting the frequency of a
successively presented pure comparison tone until its pitch is the
same as that of the OCT. For OCTs with flat amplitude envelopes over
almost the entire audio range, Terhardt et al. (1986) found that the
distribution of all pitches of all 12 OCTs in the chromatic scale has
a maximum near 300 pitch-units (that is, the pitch of a 300-Hz pure
tone). Note that this implies a _specific octave register_ for the
pitch of an OCT.
Earlier, Pollack (1978) had found that the pitch properties of OCTs
are almost unaffected by quite large changes in spectral envelope --
a result that is consistent with calculations according to TSS82. It
follows that the above "300 pitch-units" result applies,
approximately, not only to the flat spectral envelopes of Terhardt et
al. (1986) but also to the bell-shaped spectral envelopes of Shepard
and of Deutsch.
In the tritone paradox experiment, two OCTs are presented a tritone
apart, and a listener indicates which of the two is higher in pitch.
In Terhardt's approach, their task is to compare two specific
pitches, where pitch is measured on a one-dimensional scale. The
model of TSS82 may be used to predict which of the two pitches is
higher. It's that simple! (Given that the chroma of each OCT is
obvious, all the model needs to predict is the octave register of
Setting aside the TSS82 model for the moment, the "tritone paradox"
effects described by Deutsch and Repp may probably be accounted for
simply by repeating the pure-tone pitch-matching experiment (for all
12 OCTs) of Terhardt et al (1986) -- but this time, comparing results
for different language or dialect groups. Data analysis: first, look
for a significant effect of language on the shape and center of the
pitch distribution of all 12 OCTs. If there is an effect, proceed to
make specific predictions regarding which pairs of OCTs should rise
and which should fall for a given listener or group, on the basis of
the experimental pitch-match data. Then compare those predictions
with experimental data from the "tritone paradox" experiment.
If this idea works, then the "paradox" will evaporate. It will remain
to explain why language or dialect affects the octave register of the
main pitch of an OCT. Again, TSS82 can help. But I would like to
postpone that story until a later episode (there's a halloween party
Pollack, Irwin (1978). Decoupling of auditory pitch and stimulus
frequency: The Shepard demonstration revisited. JASA, 63, 202-206.
Terhardt, E., Stoll, G., & Seewann, M. (1982b). Algorithm for
extraction of pitch and pitch salience from complex tonal signals.
Journal of the Acoustical Society of America, 71, 679-688.
Terhardt, E., Stoll., G., Schermbach, R., & Parncutt, R. (1986).
Tonhoehenmehrdeutigkeit, Tonverwandschaft und Identifikation von
Sukzessivintervallen (Pitch ambiguity, successive harmonic
relationship, and melodic interval recognition). Acustica, 61, 57-66.