As to the critical case of “cross-dimensional commutativity,” they found all four participants studied in that condition (two using multiples 2 and 3, and two using proportions 50% and 75% see their Table B.2) to exhibit cross-dimensional commutativity (re Eq. The experimental paradigm they employed addresses what they call “cross-dimensional commutativity” in that loudness of 1-kHz tones is mapped into the loudness of 2-kHz tones, thereby involving two different physical dimensions f and g, but no change between sense modalities the sensation with respect to which the judgments/productions are made is always loudness. ( 2010) also tested selected cases of their theory, though on a relatively small number of participants each.
In addition to formulating a theory of cross-modal psychophysics, Luce et al.
His thoroughly formulated theory basically states that in order for magnitude estimates (as well as magnitude productions, or cross-modality matches) to be valid, two empirically testable axioms have to hold: (1) commutativity (Narens’ Axiom 4) and (2) multiplicativity (Narens’ Axiom 9).Ĭommutativity means (in a simplified notation compared to Narens, 1996) that given x is a stimulus intensity produced in a magnitude production trial, and p and q are positive numbers, adjusting a physical stimulus to have p times the subjective magnitude as some reference intensity (thereby resulting in x p), and subsequently, starting from that outcome, to produce another stimulus intensity q times as strong subjectively (i.e., x p,q) should result in the same stimulus intensity as performing the two operations in the reverse order, i.e.: However, it was not until Louis Narens’ ( 1996) seminal paper that a comprehensive theory of magnitude scaling, not just cross-modality matching, based on axiomatic measurement theory emerged. The first formal conceptualizations of cross-modality matching (Krantz, 1972 Krantz et al., 1971) contrasted a “mapping theory” of cross-modality matching with the eventually adapted “relation theory” (Shepard, 1981) emphasizing the role of sensation ratios rather than the subjective magnitude of single stimuli. Of course, this reasoning relies on what has been pejoratively labelled a “curve-fitting” approach, as opposed to a mathematically formulated psychophysical theory. It is further stipulated that if cross-modality matches pass this test, a common underlying scale of sensory magnitude – independent of modality – may be assumed. In theory, the power function exponent of the matching function should simply be the ratio of the two unimodal power functions which, in the relatively few direct empirical tests of this proposition, has been found (e.g., loudness and line length: Collins & Gescheider, 1989 Hellman & Meiselman, 1990 brightness and loudness: Stevens & Marks, 1965 Walsh & Browman, 1978: pain and handgrip strength: Gracely et al., 1978). This suggests that both loudness and brightness sensations may be measured on a common ratio scale of subjective intensity, but cautions against interpreting the numbers involved at face value.Ĭross-modality matching assumes a pivotal role in Stevens’ ( 1975) “New Psychophysics.” By matching sensations on one modality, for example, loudness, to equal sensation magnitudes on another modality, for example, brightness, the resulting cross-modality matching function provides an empirical consistency check for the independently determined unimodal psychophysical power functions obtained via magnitude estimation.
Multiplicativity, by contrast, i.e., comparing the outcome of a ×1×6 adjustment with ×2×3 sequences, irrespective of order, was violated in 17 of 40 tests, or at least once for all but six participants. checking whether ×1, ×2, and ×3 adjustments are strictly ordered, was evaluated on the same data set and found to hold. When this property was individually evaluated for each of 20 participants and for two possible directions, i.e., starting out with a noise burst or a luminous patch, only seven of the 40 tests indicated a statistically significant violation of cross-modal commutativity. Cross-modal commutativity was tested by comparing a set of successive ×2×3 productions with a set of ×3×2 productions. In their fundamental paper, Luce, Steingrimsson, and Narens ( 2010, Psychological Review, 117, 1247-1258) proposed that ratio productions constituting a generalization of cross-modality matching may be represented on a single scale of subjective intensity, if they meet “cross-dimensional commutativity.” The present experiment is the first to test this axiom by making truly cross-modal adjustments of the type: “Make the sound three times as loud as the light appears bright!” Twenty participants repeatedly adjusted the level of a burst of noise to result in the desired sensation ratio (e.g., to be three times as intense) compared to the brightness emanating from a grayscale square, and vice versa.
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