**A Theory of "Ratios"**
Birnbaum (1978) theorized that judgments of "ratios" of psychophysical
magnitudes such as loudness (Birnbaum & Elmasian, 1977) satisfy
the following equation,
R_{ij} = *a*e^{(si - sj)} + *b*,
where R_{ij} is the judged "ratio" of sensation of stimulus *i* to stimulus *j*;
s_{i} and s_{j} are the subjective values of the stimuli; e^{x} is the exponential
function; *a* and *b* are constants.

**References**
Birnbaum, M. H. (1978). Differences and ratios in psychological measurement.
In N.J. Castellan & F. Restle, (Eds.), *Cognitive Theory, Vol. 3.*
Hillsdale, N.J.: Lawrence Erlbaum Associates, p. 33-74.
Birnbaum, M. H., & Elmasian, R. (1977). Loudness ratios and differences
involve the same psychophysical operation. *Perception & Psychophysics,
22,*383-391.
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