Re: [music-dsp] a family of simple polynomial windows and waveforms

[robert bristow-johnson]

 Original Message 

Subject: Re: [music-dsp] a family of simple polynomial windows and waveforms

From: "Ross Bencina" 

Date: Mon, June 13, 2016 12:25 am

To: music-dsp@music.columbia.edu

--



>> many hours of integration by parts
there's gotta be easier ways of doing it (like Euler's with binomial).
anyway, i haven't quite groked Charles Z's thing quite yet.


--
�


r b-j � � � � � � � � �r...@audioimagination.com
�


"Imagination is more important than knowledge."
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Re: [music-dsp] a family of simple polynomial windows and waveforms

[Ross Bencina]

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.





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