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.