A mathematical exploration of the mystery of loudness adaptation

Loudness adaptation, or the decrease in perceived loudness of a steady, prolonged tone is rather a mysterious phenomenon. When measured by one technique (utilizing both ears), loudness of an extended tone will decrease by as much as 35 decibels; when measured by another technique (using only a single ear), loudness does not adapt at all regardless of how long the tone persists. The mystery is even more intriguing. When loudness adaptation does occur, the fractional reduction in the loudness of a tone (adaptation dB/sound level of extended tone dB SL) provides a good measure of the Stevens exponent, n, for loudness, an exponent which depends on sound frequency. When we analyze mathematically the two methods for measuring loudness adaptation, the reason for the apparent difference in adaptation emerges. Moreover, we derive the approximate equation showing that n equals fractional adaptation, and a method for improving the derivation of n from adaptation data. These results, derived mathematically, compare well with measured data, both our own and those obtained from the literature.