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J Acoust Soc Am. 2008 Jul;124(1):589-97. doi: 10.1121/1.2931957.

Pitch circularity from tones comprising full harmonic series.

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Department of Psychology, University of California, San Diego, La Jolla, California 92093, USA.


This paper describes an algorithm for producing pitch circularity using tones that each comprise a full harmonic series, and reports an experiment that demonstrates such circularity. Banks of 12 tones (i.e., scales) were created, with F0 varying in semitone steps. For each scale, as F0 descended, the amplitudes of the odd-numbered harmonics were reduced relative to the even-numbered ones by 3.5 dB for each semitone step. In consequence, the tone with the lowest F0 was heard as though displaced up an octave. In an experiment employing two such scales, all possible ordered tone pairs from each scale were presented, making 132 ordered tone pairs for each scale. Sixteen subjects judged for each tone pair whether the second tone was higher or lower than the first. The data derived from these pairwise comparisons were subjected to Kruskal's nonmetric multidimensional scaling, and excellent circularities were obtained. Individual differences in the subjects' judgments were also explored. The findings support the argument that musical pitch should be characterized as varying along two dimensions: the monotonic dimension of pitch height and the circular dimension of pitch class.

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