A little further down the waveform rabbit hole: I revisited Biminiroad’s morphing version of Robert Syrett’s 7 Waves VCO, and experimented a little with the order in which the waveforms are placed. Robert’s original setup had them organized in terms of all, odd, and even harmonics – with the inclusion of some new neighbouring waveforms invented by Don Tillman. I’ve rearranged them so as to provide the smoothest transitions – at least to my ears – when morphing between the waveforms. I‘ve placed the square wave last, not only because it provides better transitions for the sawtooth, but also because it does indeed (subjectively) seem to be louder, even though the sawtooth has greater overtone content. It seems that Jerobeam Fenderson was right in his explanation that it is the amount of (abrupt) change that catches our attention – the square has two vertical edges while the sawtooth only has one.
As I mentioned yesterday I was intrigued by the placement of the sawtooth between the triangle and square waves in the new Ableton Wavetable synth, and I spent a little time today exploring the results of changing the order of the oscillator shapes in one of the Audulus µVCOs. In the case of crossfading between oscillators, that placement, along with shifting the phase of the sawtooth to match the edge of the square, results in the octave harmonic of the sawtooth appearing before the fundamental when shifting between the sawtooth and the square. With the Wavetable synth the interpolations don’t appear to be the same as would occur when fading between set waveforms and I guess this is what opened up the possibility of arranging them in that order. When simply crossfading between waveforms a (co)sine–triangle–square–sawtooth ordering with all the waves sharing the same phase provides the smoothest transitions.
While a change from sine to triangle is fairly subtle, the square and sawtooth waves are more tricky when attempting to morph smoothly between waveforms. It’s an issue that Jerobeam Fenderson touches on in the third of his Oscilloscope Music tutorials. He explains the discrepancy between perceived loudness and the actual amplitude of the waveform in terms of our not hearing positions, but rather changes in positions – referring to the explanation of DC-offset in his second tutorial. The discrepancy might also be explained in terms of the higher overtone content of the sawtooth and square waves and the increased sensitivity of our hearing in the ranges in which the overtones of mid to lower tones appear.
In its current state there’s no easy way to implement wavetables in Audulus, but taking a look at Brian Funk’s tutorial/walkthrough/exploration of the Wavetable synth that will be included in Ableton Live 10, I was struck by how much has been achieved in Audulus recently on the waveshaping front – a lot of it thanks to the interest and efforts of Robert Syrett. Making wavetable waveforms without using wavetables, as he puts it.
A detail that intrigued me with the Ableton Wavetable setup was the order of the basic oscillator waves as sine–triangle–sawtooth–square, with the phase of the sawtooth shifted 90 degrees to match the edge of the square instead of the peak of the triangle, as one might expect. In the Audulus morphing oscillator included in the µCollection the sawtooth comes last and shares the same phase as the other waves.1
Something to take a closer look at and experiment with.