### The SPINC function

The Spectral Pitch INCrement function defines an(other) aurally adequate distortion of the frequency scale [97] [104] p. 261-264. The formula

 F(f) = 1414arctan[f/(1414 Hz)] spinc

by definition assigns to any frequency a particular value of the SPINC variable F(f) such that F(f) grows monotonically with f. The unit of the SPINC variable, by definition, is: 1 spinc. For low frequencies, i.e., below about 500 Hz, there is no distortion at all: The SPINC variable is nearly proportional to frequency, and in this range even its numerical value is (nearly) equal to that of the corresponding frequency (when measured in Hz). For higher frequencies, the graph of the function gets flatter and flatter, such that a progressive frequency compression is achieved.

The essential difference between SPINC scaling and logarithmic scaling of frequency is, that for the former the low-frequency range is linear. This, indeed, is aurally adequate, as the logarithmic scale, while being largely appropriate at high frequencies, inadequately overstretches the low-frequency region of the hearing range.

So the SPINC function resembles the Bark function, and indeed the former is intended to serve essentially the same purposes as the latter. The difference between the SPINC function and the Bark function - and the benefit of the former - is, that the former is more "economical" at high frequencies, i.e., beyond 2 kHz. That is to say: While for frequencies below 2 kHz 1 Bark (corresponding to one critical band) includes one and the same number of spincs, beyond 2 kHz the number of spincs included in 1 Bark gets smaller when frequency grows.

This characteristic of the SPINC function is of particular interest with regard to subdivision of the range of audible frequencies into discrete intervals, i.e., for models of the peripheral auditory system, and for technical audio-signal processing. In such a system, the range of audible frequencies is properly subdivided into steps each of which comprises the same number of spincs. One can be fairly sure that such kind of discretization of the frequency scale is aurally adequate, i.e., that no aurally relevant information is missed, provided that the interval size (measured in spincs) is chosen small enough. The latter condition, of course, should not be understood in the trivial sense. Naturally, if the intervals of subdivision are chosen strictly "small enough", any type of scaling will do. The crucial point is, that with using the SPINC scale one can expect the number of intervals required to be minimal.

The benefit of SPINC scaling in the above respect becomes intuitively apparent when one notices that 1 spinc approximately corresponds to one JND of frequency (of a sine tone). In fact, I have deduced the above formula of the SPINC function from an analytical representation of the JND as a function of frequency:

 DfD(f) = 1+0.5(f/kHz)2 Hz

Here DfD denotes the JND of frequency. This formula was chosen as an approximation to experimental data from several authors [97].

Obviously, the idea to discretize the frequency continuum into intervals equal to the JND of frequency so strongly suggests itself that it actually is far from new. However, in contemporary models and systems of the aforementioned kind it does not appear to have been taken advantage of. Rather, the Bark scale appears to dominate.

We smart scientists should indeed more often consider that an idea does not necessarily have to be wrong or useless just because it is obvious, plausible, and simple.

Author: Ernst Terhardt terhardt@ei.tum.de - Feb. 20, 2000

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