# [sdiy] Stretched harmonic synthesis

cheater cheater cheater00social at gmail.com
Mon Feb 21 01:29:00 CET 2022

```Tom, I know what Matthew's saying, but my question hasn't been
equations needing a unit difference term in their solution (z^-1)
which is an artifact of using *difference* equations instead of ODEs
which are *differential* equations. They use infinitesimal
differences, and when they're being solved, there is no unit delay to
speak of in the solutions. There hasn't been one email yet about ODEs
in this thread. Some people were trying to tell me what ODEs are but
they were not correct in what they were saying because they confused
differential equations with difference equations. I was trying not to
point this out since it felt impolite but I also feel I should mention
I studied for a mathematics degree at the university since some are
trying to teach me (thanks) what I already know, but are teaching
incorrectly (no thanks).

The closest we got to a discussion was Matthew's email when he tried
reasoning in terms of a solution's degrees of freedom. Saying that a
simple triangle core oscillator would have a single degree of freedom,
meanwhile something with two separately controllable tones would have
two degrees of freedom. It is not clear to me why an oscillator with a
particular stretch has to have more than one degree of freedom. It
doesn't. Similarly having a triangle vs sawtooth oscillator gives you
a different set of harmonic amplitudes, but they aren't a new degree
of freedom. Even though a subsequent low pass or high pass filter can
dynamically alter these amplitudes in real time, even an equation
describing both the oscillator and the filter at the same time will
still be an ODE.

We know that DSF synthesis (see 3.5 here
https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.60.4437 )
only uses five simple sinewave oscillators and some basic algebra, so
that's five ODEs. Alternatively you could say it's a separable PDE
which can be broken down to five ODEs with one free variable each. DSF
allows you to have frequency spread that's constant across the
spectrum (rather than progressive or better said monotonic). So we've
learned two things. The first is that you can achieve something
similar to what I'm looking for without an infinite amount of state
variables, and therefore without delay memory. The second is that
there are differential equations that can be solved without delay
lines that are not ODEs. Separable PDEs are one such example. But I
still wonder about other sorts of differential equations that can be
solved without delay lines.

So here we have three degrees of freedom - one is phi, one is beta,
and one is N (according to the paper): phi is the base frequency, beta
is the partial spread, and N is the amount of partials you want total.
Yet this system can be solved by reducing it to ODEs. What makes it
different than a drum head which cannot be reduced to two ODEs? The
state of the two variables doesn't depend on the other and vice versa.
When there's such a circular relationship, I believe (but don't quote
me on that) that it is difficult to separate the PDE. In the drum
head, the variables are cross-dependent because both modes share the
same piece of drum head they're trying to stretch, so they're both
competing for the same resource, and this in turn balances out the
Newtonian mechanics of the vibrating particles: the more tension, the
more acceleration back to origin. So we have a system where each of
two separate modes influences the tension, and then the tension
influences the modes back. It would be about one million times more
listen to your elders when they're telling you how to milk a duck
correctly".

TIL https://en.wiktionary.org/wiki/don%27t_teach_your_grandmother_how_to_milk_ducks
:-)

On Sun, Feb 20, 2022 at 10:42 PM Tom Wiltshire <tom at electricdruid.net> wrote:
>
>
>
> > On 20 Feb 2022, at 20:51, cheater cheater via Synth-diy <synth-diy at synth-diy.org> wrote:
> >
> > On Sun, Feb 20, 2022 at 8:24 PM <mskala at northcoastsynthesis.com> wrote:
> >>
> >> On Sun, 20 Feb 2022, cheater cheater wrote:
> >>> system at those points in a delay line), let me ask my original
> >>> question again.
> >>>
> >>> I was wondering if anyone knows any ODEs which generate a signal with
> >>> harmonics which are progressively stretched? Meaning higher partials
> >>> are further apart.
> >>
> >> Asking it again isn't going to make the truthful answer simpler, but if
> >> you refuse to pay attention to the details that several people have given
> >> you, then it's best to go with the answer being "no."  And that's because
> >> such equations do not exist - not just because nobody knows about them.
> >>
> >> --
> >> Matthew Skala
> >> North Coast Synthesis Ltd.
> >
> > Damn, do you always get angry on the internet?
>
> He's not getting angry, Cheater. He's just pointing out to you that your question has already been answered, by Richie, by Mike, by others too. Asking it again won't alter reality, and just makes it look like you're not really listening to what you've been told thus far, which makes people less likely to want to help you.
>
> Tom

```