Publications

Abstract

Firstly, theories are reviewed on the explanation of tonal consonance as the singular nature of tone intervals with frequency ratios corresponding with small integer numbers. An evaluation of these explanations in the light of some experimental studies supports the hypothesis, as promoted by von Helmholtz, that the difference between consonant and dissonant intervals is related to beats of adjacent partials. This relation was studied more fully by experiments in which subjects had to judge simple-tone intervals as a function of test frequency and interval width. The results may be considered as a modification of von Helmholtz's conception and indicate that, as a function of frequency, the transition range between consonant and dissonant intervals is related to critical bandwidth. Simple-tone intervals are evaluated as consonant for frequency differences exceeding this bandwidth. whereas the most dissonant intervals correspond with frequency differences of about a quarter of this bandwidth. On the base of these results, some properties of consonant intervals consisting of complex tones are explained. To answer the question whether critical bandwidth also plays a rôle in music, the chords of two compositions (parts of a trio sonata of J. S. Bach and of a string quartet of A. Dvorák) were analyzed by computing interval distributions as a function of frequency and number of harmonics taken into account. The results strongly suggest that, indeed, critical bandwidth plays an important rôle in music: for a number of harmonics representative for musical instruments, the "density" of simultaneous partials alters as a function of frequency in the same way as critical bandwidth does.

Abstract

An introductory study of the perception of stochastically specified events is reported. The initial problem was to determine whether the perceiver can split visual input data of this kind into random and determined components. The inability of subjects to do so with the stimulus material used (a filmlike sequence of dot patterns), led to the more general question of how subjects code this kind of visual material. To meet the difficulty of defining the subjects' responses, two experiments were designed. In both, patterns were presented as a rapid sequence of dots on a screen. The patterns were more or less disturbed by “noise,” i.e. the dots did not appear exactly at their proper places. In the first experiment the response was a rating on a semantic scale, in the second an identification from among a set of alternative patterns. The results of these experiments give some insight in the coding systems adopted by the subjects. First, noise appears to be detrimental to pattern recognition, especially to patterns with little spread. Second, this shows connections with the factors obtained from analysis of the semantic ratings, e.g. easily disturbed patterns show a large drop in the semantic regularity factor, when only a little noise is added.

Abstract

As a preliminary to further research on musical consonance an explanatory investigation was made on the different modes of judgment of musical intervals. This was done by way of a semantic differential. Subjects rated 23 intervals against 10 scales. In a factor analysis three factors appeared: pitch, evaluation and fusion. The relation between these factors and some physical characteristics has been investigated. The scale consonant-dissonant showed to be purely evaluative (in opposition to Stumpf's theory). This evaluative connotation is not in accordance with the musicological meaning of consonance. Suggestions to account for this difference have been given.