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A Mathematical Exploration of the Mystery of Loudness Adaptation

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Abstract

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.

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Abbreviations

φ :

Sound pressure level expressed as a multiple of a subject’s own threshold level; φ is also known as “sensation level” or SL. May also be expressed as sound energy per unit area per unit time, again as a multiple of threshold level. φ may also be expressed as “sound pressure level” or SPL, although we shall not do so in this paper

dB SL:

Decibels sensation level; 10 log10 φ

SDLB:

Simultaneous dichotic loudness balance

A :

Sound level of a tone introduced into the control ear in an SDLB procedure after adjustment by the subject to match the loudness of a tone in the other ear. Expressed in decibels sensation level (dB SL). A=10log 10 φ a

B :

Sound level of a sustained tone introduced into the adapting ear. Expressed in decibels sensation level (dB SL). B=10log 10 φ b

γ :

Parameter that governs the degree of adaptation

m :

Slope of the regression line obtained when decibels of adaptation are plotted against sound level of the sustained tone in decibels sensation level (dB SL)

n :

Loudness exponent that appears in the power law of sensation

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  1. Institute of Biomaterials & Biomedical Engineering, Departments of Physiology and Physics, University of Toronto, 164 College Street, Toronto, Ontario, Canada, M5S 3G9

    Kenneth H. Norwich

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Correspondence to Kenneth H. Norwich.

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Norwich, K.H. A Mathematical Exploration of the Mystery of Loudness Adaptation. Bull. Math. Biol. 72, 298–313 (2010). https://doi.org/10.1007/s11538-009-9447-1

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