Page 12 - Summer 2010

Fig. 4. Algorithm for producing pitch circularity employed by Deutsch.
show the progression of the relative amplitudes of Harmonics 1 and 2, Harmonics 3 and 4, and Harmonics 5 and 6, as F0 moves upward from the ‘tonic’ of the scale.
See text for details. Reprinted from Deutsch, Dooley, and Henthorn.
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The graphs
Fig. 5. Subjects were presented with pairs of tones created using the algorithm by Deutsch,16 and they judged whether each tone pair rose or fell in pitch. The graph plots the percentages of judgments based on pitch class proximity, as a function of
distance between the tones within a pair along the pitch class circle. Adapted from
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Deutsch, Dooley, and Henthorn.
to generate circular banks of tones by systematically varying
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tude. Then for the tone with F0 a semitone lower, the odd harmonics are reduced in amplitude relative to the even ones, so raising the perceived height of the tone. Then for the tone with F0 another semitone lower, the odd harmonics are fur- ther reduced in amplitude, so raising the perceived height of the tone to a greater extent. We continue moving down the octave in semitone steps, reducing the amplitudes of the odd- numbered harmonics further with each step, until for the lowest F0 the odd-numbered harmonics no longer contribute to perceived height. The tone with the lowest F0 is therefore heard as displaced up an octave, and so as higher in pitch than the tone with the highest F0—and pitch circularity is thereby obtained.
After some trial and error, I settled on the parameters shown in Fig. 4. Complex tones consisting of the first six har- monics were employed, and the amplitudes of the odd-num- bered harmonics were reduced by 3.5 dB for each semitone step down the scale; therefore for the tone with lowest F0 the odd harmonics were 38.5 dB lower than the even ones. To achieve this pattern for harmonic pairs 1 and 2, and harmon- ic pairs 3 and 4, the even numbered harmonics were at a con- sistently high amplitude, while the odd numbered harmonics decreased in amplitude as F0 descended. To obtain the same pattern of relationship for harmonic pairs 5 and 6, harmonic 5 was consistently low in amplitude while harmonic 6 increased in amplitude as the scale descended.
In a formal experiment to determine whether such a bank of tones—hereafter referred to as a scale—would indeed be perceived as circular, my colleagues Trevor Henthorn, Kevin Dooley and I created two such scales;15 for one scale the lowest F0 was A4 and for the other the lowest F0 was F#4. (For want of a better word, we refer to the tone with the lowest F0 as the tonic of the scale) For each scale, each tone was paired with every other tone, both as the first and the second tone of a pair, and subjects were asked to judge for each pair whether it rose or fell in pitch.
We found that judgments of these tones were over- whelmingly determined by proximity along the pitch class
We can begin with a bank of twelve tones, each of which consists of the first six components of a harmonic series, with F0s varying over an octave in semitone steps. For the tone with highest F0 the odd and even harmonics are equal in ampli-
the relative amplitudes of the odd and even harmonics.
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