# [sdiy] Stretched harmonic synthesis

Mike Bryant mbryant at futurehorizons.com
Sun Feb 20 12:41:03 CET 2022

```You can use a delay line to store individual states, or an integrator to store sums of previous states.  It's just the same values saved in a different manner.  For any sinusoidal generation using differential equations you need at least two integrators or delay elements, i.e. solving a d2y/d2x function.

-----Original Message-----
From: cheater cheater [mailto:cheater00social at gmail.com]
Sent: 20 February 2022 07:33
To: Mike Bryant
Cc: synth-diy at synth-diy.org
Subject: Re: [sdiy] Stretched harmonic synthesis

Perhaps, but I'm asking about differential equations that can be solved without a _delay line_. Those are necessary for solving PDFs because they depend on two variables. For example, d^2y/dt^2 + K =
d^2y/dx^2 + e. The delay line is necessary to store the states of the system along the X axis. Meanwhile all that's necessary to store the state across time is time itself, you don't need to keep a value around after you've used it, after that it only exists in the past.

On Sun, Feb 20, 2022 at 12:53 AM Mike Bryant <mbryant at futurehorizons.com> wrote:
>
> Any differential equation solver needs to integrate.
>
>
> -----Original Message-----
> From: cheater cheater [mailto:cheater00social at gmail.com]
> Sent: 19 February 2022 23:50
> To: Mike Bryant
> Cc: synth-diy at synth-diy.org
> Subject: Re: [sdiy] Stretched harmonic synthesis
>
>
> On Sat, Feb 19, 2022 at 11:24 PM Mike Bryant <mbryant at futurehorizons.com> wrote:
> >
> > All differential equations require a delay to solve them - either
> > z-1 in digital or s-1 in analogue (often an integrator but can be a
> > sample and hold)
> >
> > -----Original Message-----
> > From: Synth-diy [mailto:synth-diy-bounces at synth-diy.org] On Behalf
> > Of cheater cheater via Synth-diy
> > Sent: 19 February 2022 20:57
> > To: synth-diy
> > Subject: Re: [sdiy] Stretched harmonic synthesis
> >
> > I was wondering if anyone knows any ODEs which generate a signal with harmonics which are progressively stretched? Meaning higher partials are further apart.
> >
> > Or alternatively progressively contracted - higher partials are closer together.
> >
> > Are ODEs the only kind of differential equation that can be solved without a delay line?
> >
> > Thanks
> >
> >
> > On Fri, Feb 18, 2022 at 6:07 PM cheater cheater <cheater00social at gmail.com> wrote:
> > >
> > > Hi, I was wondering if anyone ever tried Karplus-Strong synthesis
> > > with an all-pass filter (APF) in the feedback, in order to obtain
> > > stretched tuning, and what your results were.
> > >
> > > https://ccrma.stanford.edu/~jos/pasp/Dispersive_1D_Wave_Equation.h
> > > tm
> > > l
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>

```