# [sdiy] Stretched harmonic synthesis

rburnett at richieburnett.co.uk rburnett at richieburnett.co.uk
Sun Feb 20 13:46:06 CET 2022

```Cheater, if you think about it you must store at least two states in
order for an oscillator to *keep going*!  Otherwise when the sinewave
goes through zero it would just stop right there because there wouldn't
be any energy stored anywhere else to keep it going.  That's why
resonant systems have to be 2nd order at minimum.  As long as there are
two states, and the energy moves between them, then the system can keep
oscillating.

These two states can be voltages across components, currents through
them, integrals, differentials, or previous outputs in the case of a
digital filter, or even things like potential energy and kinetic energy
in the case of a mass oscillating on the end of a spring.

-Richie,

On 2022-02-20 11:41, Mike Bryant wrote:
> 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
>
> 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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>>
>
>
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```