Direct Estimation of Multidimensional Tonal Dissimilarity

Abstract

Magnitude estimates were obtained for the dissimilarity of pure tones varying in frequency and intensity. In Expt. 1, a spatial embedding for a set of seven tones was determined by assuming the magnitude estimates represented distances in a two-dimensional space whose coordinate axes were pitch and loudness. Embeddings were determined for both Euclidean and "city-block" distance functions. A linear relation was consistently observed between the projections of the tones on dimensions of pitch and loudness and their values on Stevens's mel and sone scales, with the city-block projections usually yielding higher correlations. In Expt. 2, 11 tones were used and the magnitude estimates of dissimilarity were compared to the tightly constrained "monotone" distances derived from the rank-order properties of the dissimilarities. The agreement observed between the magnitude estimates and the monotone predicted distances strongly supports a distance interpretation for the dissimilarities.

Multidimensional scaling of color difference in the pigeon

“…A two-dimensional spatial configuration was determined from the average ranks, using the iterative technique described by Carvellas and Schneider (1972). The loci of the stimuli in this two-dimensional Euclidean space are represented by the squares in Fig.…”

The dimensions of tonal experience: A nonmetric multidimensional scaling approach

“…In that case, the value of stress (an index of the degree of nonmonotonicity) will be high and the obtained configuration will not accurately reflect the true configuration (Young, 1970). The high stress values found in both Carvellas and Schneider (1972) and Chipman and Carey (1975) indicate that this may, indeed, have been the case. Consequently, it would seem reasonable to employ a scaling technique that would not require that subjects generate numbers to represent dissimilarities.…”

“…In fact, two previous studies have indicated that only two dimensions may be necessary to represent tonal dissimilarity. Carvellas and Schneider (1972) found that a two-dimensional auditory space with loudness and pitch as dimensions was able to account for magnitude estimates of the dissimilarity of pure tones varying in frequency and intensity. Chipman and Carey (1975) also found that a twodimensional auditory space could account for magnitude estimates of the dissimilarity of narrow-band noises varying in intensity and center frequency.…”

“…In fact, it is usually difficult to discriminate between the Euclidean metric and the so-called city-block metric (e.g., Behrman & Brown, 1968;Carvellas & Schneider, 1972). In the city-block metric, the distance between points in a space is the sum of their differences along the component dimensions.…”

An interval scale of saturation for the pigeon

“…Accordingly, the ranks were averaged across the two experiments to arrive at a representative rank order of color difference for the 55 pairs of 630-nm lights that differed in colorimetric purity. This average rank order served as input to a nonmetric scaling program (Carvellas & Schneider, 1972) which determined the best one-dimensional representation for the 11 stimuli. Stress (Kruskal, 1964) for this configuration was 6070.…”