A consequence of using the standard paths was periodic oscillations of the variables between their maximum and minimum values. (The overall periodicity of the resulting sound was far more complex than the periodicity of any one path, but, nonetheless, the resulting regularity was noticeable.) Since periodicity is (in information theoretical terms [9]) a greater degree of redundancy than needed to convey a message, my interests as a composer led me to look for ways in which it was minimized.
Polynomials are used in Herbert Brün's SAWDUST system in the implementation of the VARY algorithm. I decided to try to use them in TrikTraks by specifying equally spaced zero-crossings, then scaling them to their specified limits. The values are loaded into a table, and interpolating wave table lookup algorithms are used to extract values with the desired frequency and phase. This procedure allows for a polynomial to occur with a variable number of periods over the requested duration.
From TrikTraks, a polynomial path is specified in the following way, here given for only one segment:
# TrikTraks: input file for polynomials 200 100 poly 5 3 0 # samples 30000 -30000 poly 7 2.5 0 # amplitude
The range is given first (maximum and minimum values), then the type of path (``poly''), followed by the degree of the polynomial to be generated, the number of periods over the sound's duration, and the initial phase of the polynomial. Polynomials can be requested of degree 3 to degree 10.
Here is a plot of the paths for both the duration and the amplitudes, for the data given above:
Please note that the above are the paths for one segment of a waveform, and that a waveform can have up to 64 segments, and thus there can be up to 128 distinct paths.