Subject:Re: Tritone paradoxFrom:at <parncuttSOUND.MUSIC.MCGILL.CA>Date:Fri, 29 Oct 1993 22:19:55 EDTHere 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 paradox" findings. 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 each pitch.) 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 going on). Richard Parncutt References 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.

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