[sdiy] Better waveforms of our nature

[Olivier Gillet]

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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