On smoothness under parameter changes

Is your synthesizer a mathematical function?

At least it can be considered in such terms. Each setting of all its parameters represents a point in parameter space. The output signal depends on the parameter settings. Assuming the parameters remain fixed over time, the generated audio signal may also be considered as a point in another space. In order to relate these output sequences to perceptually more relevant terms, signal descriptors (e.g. the fundamental frequency, amplitude, spectral centroid, flux) are applied to the output signal.

Now, in order to assess how smoothly the sound changes as one turns any of the knobs that controls some synthesis parameter, the first step is to relate the amount of change in the signal descriptors to the distance in parameter space. The distance in parameter space corresponds to the angle the knob is turned. Let us call this distance Δc. It is trickier to define suitable distance metrics in the space of audio signal sequences, but why not use a signal descriptor φ which itself varies over time and take its time average ⟨φ⟩. The difference Δφ between two such time averages as the synthesizer is run at two different points in parameter space may be taken as the distance metric.

A smooth function has derivatives of all orders. Therefore the smoothness of a synthesis parameter may be described in terms of a derivative of the function that maps points in parameter space to points in the space of signal descriptors. This derivative may be defined as the limit of Δφ/Δc as Δc approaches 0. It makes a significant difference whether a pitch control of an oscillator has been designed with a linear or exponential response. But abrupt changes, corresponding to a discontinuous derivative, will be even more conspicuous when they occur.

Whereas the derivative is about the smoothness locally at each point in parameter space, another way to look at parameter smoothness is to measure the total variation of a signal descriptor as the synthesis parameter goes from one setting to another. As a compromise, the interval over which the total variation is measured may be made really small, so that a local variation can be measured instead over an interval of a parameter.

Is this really useful for anything?

Short answer: Don't expect too much. But seriously, whether we like it or not, science progresses in part by taking vague concepts and making them crisper, by making them quantifiable. "Smoothness" under parameter changes is precisely such a vague concept that can be defined in ways that make it measurable. Such a smoothness diagnostic may be useful in the design of synthesis models and their parameter mappings, as well as perhaps for introducing and testing hypotheses about the perceptual discrimination of similar synthesized sounds.

The paper was presented as a poster at the joint SMAC/SMC conference.

Total variation

The total variation of a real valued function f on an interval I is defined as

taking the supremum over all possible partitions of I = [p₀, p_N]. Notably, if the function is (continuously) differentiable, the total variation becomes

but f does not have to be differentiable, and the total variation may be unbounded.

Sometimes the function itself may be evaluated at any point of the interval, although its derivative either does not exist or is far too complicated to deal with. Then the total variation may be estimated by sampling the function at several points and checking whether or not it converges to some limit as the mesh gets finer. If it doesn't, the curve may be a fractal, so its fractal dimension can be estimated from the procedure.

The length of a fractal curve is a function of the scale of measurement. As the scale of measurement ε varies, the measured length N varies according to N ~ ε ^−D, where D is the fractal dimension. The common procedure then is to fit a double logarithmic plot of N against ε and finding the slope. However, it would be a grave mistake to blindly accept any automatically calculated slope without checking the error of the fit.

Estimating the total variation at several arbitrary sampling resolutions can be inefficient, unless a clever trick is used. Suppose we begin with a fine resolution with uniform distance Δ = x_i - x_i-1 > 0 between the points. Then it is easy to obtain the total variation for subdivisions by nΔ, for n = 1, 2, … just by skipping so many points. Even better, one can take averages

so as to obtain estimates that do not depend (as much) on the particular chosen sample points.  

A somewhat related concept is arc length, which is, conceptually, the length of a string superposed on the graph of the function (assuming the function is continuous). The total variation is smaller than the arc length. For the straight line y = kx, 0 < x < t, the total variation squared is V² = (kt)² as compared to the arc length squared which is t²+(kt)². Now suppose the function is monotonous over the interval under consideration. Then, if the function is deformed so as to become more curved, only the arc length will increase while the total variation remains the same. For example, if f_n(x) = xⁿ, 0 ≤ x ≤ 1 and n = 1, 2, ..., then the arc length approaches 2 as n increases, whereas the total variation remains 1.