# [sdiy] Stretched harmonic synthesis

cheater cheater cheater00social at gmail.com
Mon Feb 21 11:53:05 CET 2022

```On Mon, Feb 21, 2022 at 4:51 AM Ian Fritz <ijfritz at comcast.net> wrote:
>
> See, eg, Fletcher and Rossing, Sect. 2.18. Pretty well-known stuff.
>
> Ian

Thanks Ian. Why not post this to the group?

I wasn't aware of this book. Great resource, thanks!

The equation 2.66 for the stiff string (on page 64) reads:

u d^2y/dt^2 = T d^2y/dx^2 - ESK^2 d^4y/dx^4

Notice that y is a function of two variables, x and t, and we
differentiate over both of them. The equation is a PDE, a partial
differential equation. In print it tips you off that the d is written
as "curly d" which is rounded off, unlike a normal d. This PDE is not
separable because the term on the left (the acceleration of the
string) is balanced out by its curvature (T d^2y/dx^2) as well as
stiffness (the rest). They are again "competing for the same finite
resource". So it cannot be turned into an ODE.

My original mail was started because I posted about the equation you
pointed me to, on Julius Smith's site, and we talked about solving it
with use of delay lines.

https://ccrma.stanford.edu/~jos/pasp/Dispersive_1D_Wave_Equation.html

For those who are unaware how the equation relates to the one I
excerpted out of the book: Smith uses Newton notation, while the book
above uses Leibniz notation.

https://en.wikipedia.org/wiki/Notation_for_differentiation

y with two dots above it is the same as d^2y/dt^2 (each dot means
differentiating once over time).
y with two ticks is the same as d^2y/dx^2 (each tick means
differentiating once over x).

Calculating the curvature and stiffness requires knowing the state of
every point of the vibrating string. They need to be stored somewhere
and Smith has a section there on using a delay line to store those
values. This results in well known Karplus-Strong synthesis. That was
a couple days back. The discussion is under the subject "Dispersive
Karplus-Strong". So the question is how to solve the problem without
delay lines, and for that, it cannot be a PDE which cannot be
separated, and I'm trying to establish what it can be. It can be an
ODE, but maybe it can be some other form of differential equation as
well. Delay lines are undesirable because K-S oscillators aren't quite
the same as integrator core oscillators.

(I remember a similar discussion about partial differential equations
happening on this list about 15 years ago... I'm sure David remembers
it too... fun times)

> > On Feb 20, 2022, at 5:31 PM, cheater cheater via Synth-diy <synth-diy at synth-diy.org> wrote:
> >
> > ﻿Tom, I know what Matthew's saying, but my question hasn't been
> > answered or even touched upon. First people were talking about PDEs
> > instead of ODEs. Then they were talking about finite difference
> > 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
> >
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>

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