[sdiy] Better waveforms of our nature

Tue Oct 18 16:53:21 CEST 2016

Thanks all, it answers my question (and more).
    -m

On Tue, Oct 18, 2016 at 4:05 PM, Olivier Gillet <ol.gillet at gmail.com> wrote:

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