Ambiguity and information in musical-tone structures

Previous studies of the relative pitch of intervals composed of Shepard tones (Shepard, 1964) revealed a sinusoidal structure of the directional distortion. Deutsch and coworkers found a similar sinusoidal structure when tritones alone were presented to listeners (Deutsch, 1987). In the first study, the tritone paradox was studied among listeners in Florida. The typical sinusoidal response function appeared for averaged data as well as for tones generated in two different octave ranges, consistent with earlier work (Deutsch, 1991; 1994). The degree of context sensitivity of tritones composed of Shepard tones and Deutsch tones is discussed in terms of what we call a whole tone context. This idea is explored in the second study. The second study measures the relative distribution and rate of new musical information growth in melodic flows in pre-20th and 20th century diatonic styles, and compares these with 20th century melodies in whole tone and pantonal compositional settings. Three measures of complexity were used in these analyses: (1) metric or probabilistic entropy, hM, is derived from the melodic sequence as a Markov process yielding a logarithmic quantifier of the dispension of statistical weight in its asymptotic distribution; (2) topological entropy, hT, quantifies the logarithmic rate of growth of new pitch pattern subsequences along the melodic line; (3) hT - hM = 0 in a uniformly random system so that deviation from zero indicates nonuniformity in the melodic line, i.e., [(hT - hM) < 0]1 < [(hT - hM) < 0]2 says the former makes more motions on fewer melodic elements. Analyses of 34 compositions indicated that the metric entropy, hM, alone is not distinguishing, consistent with the findings of Boon and Decroly (1990). However, the topological entropy, hT, quantified differences within a composer's work, and the ratio hT - hM distinguished among composers. The findings suggest that complexity measured by the difference (ratio) in entropy computations is lower in 20th century styles in which the tritone is part of the interval vocabulary (e.g., Bartok and Schoenberg) than in 19th or 20th century styles (e.g., Mozart and Prokofiev) in which functional harmony dominates the tonal picture and the tritone is considered an altered fourth or fifth. The metric and topological entropies considered together suggest a technique for quantifiable characterization of differences between the works of composers studied, including those of the 20th century.