On 12/06/2016 8:04 PM, Andy Farnell wrote:
Great to follow this Ross, even with my weak powers of math its informative.
My powers of math are still pretty weak, but I've been spending time at the gym lately ;)
I did some experiments with Bezier after being hugely inspired by the sounds Jagannathan Sampath got with his DIN synth. (http://dinisnoise.org/) Jag told me that he had a cute method for matching the endpoints of the segment (you can see in the code), and listening, sounds seem to be alias free, but we never could arrive at a proof of that. Now I am revisiting that territory for another reason and wondering about the properties of easily computed polynomials again.
My less-than-stellar understanding is that at the breakpoints, higher-order continuity determines the bandwidth of the harmonics induced by the discontinuity (This is related to the BLIT, BLEP, etc story discussed here many times). Each additional matched derivative gives you an extra 6db of roll-off. Which would stand to reason, since in the limit (i.e. infinite matched derivatives in a periodic waveform) you'd get a sinusoid. Or you could bandlimit the breakpoints using some other scheme (e.g. oversampling).
As to the exact impact the order of the polynomial has on bandwidth, aside from at the breakpoints, I'm not sure. But taking a stab at it: a polynomial of order n will have at-most n zero-crossings -- that might allow for a rough estimate of the maximum bandwidth. As for the minimum bandwidth: it doesn't take that many terms to get a sine wave with 100db-accuracy (consider the error term in the Taylor series).
It may well be that the harmonic content in the DIN synth comes mainly from controlling (dis)continuity at the breakpoints rather than the spectrum of the polynomial curve per-se.
Cheers, Ross. _______________________________________________ dupswapdrop: music-dsp mailing list music-dsp@music.columbia.edu https://lists.columbia.edu/mailman/listinfo/music-dsp
