<div dir="ltr">Thanks all, it answers my question (and more).<div> -m</div></div><div class="gmail_extra"><br><div class="gmail_quote">On Tue, Oct 18, 2016 at 4:05 PM, Olivier Gillet <span dir="ltr"><<a href="mailto:ol.gillet@gmail.com" target="_blank">ol.gillet@gmail.com</a>></span> wrote:<br><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex"><span class="im HOEnZb">Yes. For periodic signals, differentiation is equivalent to<br>
multiplying the amplitude of a harmonic by its rank ; and integration<br>
equivalent to dividing the amplitude of a harmonic by its rank. Up to<br>
a normalization constant and ignoring phase shifts.<br>
<br>
<a href="https://en.wikipedia.org/wiki/Fourier_transform#Differentiation" rel="noreferrer" target="_blank">https://en.wikipedia.org/wiki/<wbr>Fourier_transform#<wbr>Differentiation</a><br>
<br>
That's why the "integral" waveforms have a tamer spectrum.<br>
<br>
Note that there is an entire continuum between the "mellow" and<br>
"bright" columns - that can be computed in the time-domain with<br>
fractional derivation/integration, or in the frequency domain by<br>
scaling the harmonics according to n^alpha, alpha between 0 and 1.<br>
<br>
On Tue, Oct 18, 2016 at 3:29 PM, Matthias Puech<br>
<<a href="mailto:matthias.puech@gmail.com">matthias.puech@gmail.com</a>> wrote:<br>
</span><div class="HOEnZb"><div class="h5">> Hello,<br>
> Thank you Donald for the interesting read!<br>
> I've been wondering: is there a known relationship between an arbitrary<br>
> waveform's spectrum and its integral/derivative's? Are all integral<br>
> waveforms "mellow" versions of their derivatives?<br>
><br>
> I've asked recently on another DSP mailing list but did not get any answer.<br>
> It might be very simple maths...<br>
> -m<br>
><br>
> On Mon, Oct 17, 2016 at 10:45 PM, David G Dixon <<a href="mailto:dixon@mail.ubc.ca">dixon@mail.ubc.ca</a>> wrote:<br>
>><br>
>> The Rubicon (Intellijel eurorack) has double-frequency saws as a standard<br>
>> output. It also has a "sigmoid" wave (a saw put through a sine-shaper<br>
>> instead of a triangle), and a double-frequency sigmoid. I don't know of<br>
>> any<br>
>> other "commercial" VCOs with these waveforms. Another nice one is the<br>
>> "zigzag" wave, which is really just the sum of triangle and square. This<br>
>> waveform is the control signal for the core comparator in my VCO designs,<br>
>> and I just brought it out through a buffer to an output jack.<br>
>><br>
>> One thing I'm wondering about lately is the Minimoog "sawtooth" which<br>
>> looks<br>
>> more like a shark-fin. What does the spectrum look like for that<br>
>> waveform?<br>
>><br>
>> > -----Original Message-----<br>
>> > From: Synth-diy [mailto:<a href="mailto:synth-diy-bounces@dropmix.xs4all.nl">synth-diy-bounces@<wbr>dropmix.xs4all.nl</a>]<br>
>> > On Behalf Of David Moylan<br>
>> > Sent: Monday, October 17, 2016 11:33 AM<br>
>> > To: <a href="mailto:mskala@ansuz.sooke.bc.ca">mskala@ansuz.sooke.bc.ca</a><br>
>> > Cc: <a href="mailto:synth-diy@dropmix.xs4all.nl">synth-diy@dropmix.xs4all.nl</a><br>
>> > Subject: Re: [sdiy] Better waveforms of our nature<br>
>> ><br>
>> > Ah, yeah, got confused about which was at twice the<br>
>> > frequency. I do think this method would cover a lot of the<br>
>> > same ground as the bright/even harmonics block in Don's<br>
>> > chart. And could be fairly simple to add a x2 saw to a<br>
>> > typical saw vco.<br>
>> ><br>
>> > That would still leave the mellow/even and mellow/all.<br>
>> > Mellow/all looks a lot like a full wave rectified sine (which<br>
>> > would automatically be at double frequency of the input<br>
>> > sine). Similar mixing as you described with sine would cover<br>
>> > most of mellow/even range. For mellow/all (FWR<br>
>> > estimate) the oscillator could just be retuned or octave<br>
>> > switched if available.<br>
>> ><br>
>> > Of course, this is all theoretical; haven't tried it. Looks<br>
>> > like denominator of rectified sine is (4n^2 - 1). So not<br>
>> > mathematically equivalent but ballpark and again, low parts<br>
>> > count to provide this wave in analog hardware. Here's a<br>
>> > table of harmonic divisors scaled against<br>
>> > n=1 value to get relative divisors. So, the rectified sine<br>
>> > would be mellower then the parabolic wave as the amplitude of<br>
>> > the harmonics is dropping faster. More specifically, 80% of<br>
>> > amplitude at n=2 and approaching 75% of amplitude as n increases.<br>
>> ><br>
>> > n2 | 4n^2 - 1<br>
>> > 1 1 1<br>
>> > 2 4 5<br>
>> > 3 9 11.6<br>
>> > 4 16 21<br>
>> > 5 25 33<br>
>> > 6 36 47<br>
>> > 7 49 65<br>
>> > 8 64 85<br>
>> ><br>
>> > I still think it would be fun to experiment with given its<br>
>> > simplicity.<br>
>> > Come to think of it, since the difference in harmonic<br>
>> > amplitudes is only in the range of 75-80%, you could just mix<br>
>> > a bit more of the FWR wave to compensate and have a fairly<br>
>> > small error.<br>
>> ><br>
>> > Dave<br>
>> ><br>
>> > On 10/17/2016 08:25 PM, <a href="mailto:mskala@ansuz.sooke.bc.ca">mskala@ansuz.sooke.bc.ca</a> wrote:<br>
>> > > On Mon, 17 Oct 2016, David Moylan wrote:<br>
>> > >> Not quite. A sine wave only represents one<br>
>> > harmonic/overtone. So if<br>
>> > >> you add one an octave up you're just adding that single<br>
>> > harmonic, no<br>
>> > >> other even harmonics. You can build it up with multiple sines (if<br>
>> > >> you have extra<br>
>> > ><br>
>> > > I said "adding a sine wave to a traditional oscillator at twice the<br>
>> > > frequency" and meant that the traditional oscillator would be<br>
>> > > something like a sawtooth - so the sine wave provides the<br>
>> > fundamental<br>
>> > > and the sawtooth at twice the fundamental frequency<br>
>> > provides all even harmonics.<br>
>> > ><br>
>> ><br>
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