Fractionation of pitch intervals: an axiomatic study testing monotonicity, commutativity, and multiplicativity in musicians and non-musicians.

Direct ratio scaling (e.g., magnitude estimation; Stevens, 1956, American Journal of Psychology) is a widely used approach in psychophysics resting on the assumption that participants are able to refer to sensations with numbers that are valid on a ratio scale. Only recently, the necessary conditions of commutativity and multiplicativity have been formulated (Narens, 1996, Journal of Mathematical Psychology) and tested empirically, e.g., for the sensation of loudness. The present investigation evaluated these properties for the ratio production of pitch intervals. Musically trained (n = 10) and untrained (n = 11) participants adjusted intervals defined by two ascending pure-tone frequencies to given fractions (1/3, 1/2, 2/3), starting either from a 12 or 17-semitone standard pitch interval. The results show that the axioms of commutativity and multiplicativity held for most of the participants, irrespective of musical training. Furthermore, all participants produced larger frequency intervals in response to larger ratio numbers used in the instructions (monotonicity), but only musically trained participants were sensitive to the size of the standard interval (thus producing strictly increasing magnitudes). Overall, the results indicate that pitch intervals are ratio-scalable. However, restrictions must be made, especially for non-musical listeners, and when an octave is exceeded.