On the functional form of partial masking functions in psychoacoustics

Partial masking of pure tones is often investigated in terms of matching functions which record the intensity φ(x, n) of an unmasked tone which matches, in perceived loudness, a tone of intensity x embedded in a flat, broadband noise of intensity n. Empirically, the following property of “shift invariance” is observed to hold for these matching functions: φ(λx, λ^θn) = λφ(x, n), for any λ > 0, and some θ < 1. In the context of the model $\text{φ(x, n) = F}{}_0\text{[}\text{g}{}_0\text{(x)}\text{(h}{}_0\text{(x) + k}{}_0\text{(n))}\text{]}$ which expresses the idea that the effect of a noise mask is to control the gain of a pure tone signal in a multiplicative fashion, shift invariance proves to be an extremely powerful theoretical constraint. Specifically, we show that only two parametric families of functions (F₀, g₀, h₀, k₀) are possible candidates for interpolating empirical data. These two parametric families are given by the following expressions: $\text{φ}{}_1\text{(x, n) = A(}\text{x}{}^{\mathrm{\alpha }}\text{(x}{}^{\mathrm{\alpha \prime }}\text{+ Kn}{}^{\mathrm{<mtext>\alpha \prime </mtext><mtext>\theta </mtext>}}\text{)}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>(\alpha \; ?\; \alpha \prime )</mtext>}}$, $\text{φ}{}_2\text{(x, n) = A[x}{}^{\mathrm{\alpha }}\text{(x}{}^{\mathrm{\alpha \prime }}\text{? Kn}{}^{\mathrm{<mtext>\alpha \prime </mtext><mtext>\theta </mtext>}}\text{)]}{}^{\mathrm{<mtext>1</mtext><mtext>(\alpha \; +\; \alpha \prime )</mtext>}}$. Both of these expressions are in good agreement with a large array of partial masking data.