Matt Barber brbrofsvl at gmail.com
Wed Jul 2 20:51:47 CEST 2008

```On Wed, Jul 2, 2008 at 1:26 PM, Mathieu Bouchard <matju at artengine.ca> wrote:
> On Tue, 1 Jul 2008, Matt Barber wrote:
>
>> Any ideas??
>
> Just drop the idea of matching more than two sample points. It's what makes
> [tabread4~] miss the opportunity to be C1, but it's also in exchange for
> pretty much nothing. Well, maybe it's not nothing, but I still have no clue
> about what's the point of matching x[t-1] and x[t+2] for a curve that will
> only be used from x[t+0] to x[t+1]. When you get to x[t+2], one
> almost-arbitrarily different cubic has just passed, and you're entering
> another almost-arbitrarily different cubic, so the cubic used between x[t+0]
> and x[t+1] seems irrelevant at x[t+2].
>
> Perhaps you have a totally different way of explaining it that would show
> that two adjacent cubic's continuations into each other are closely related
> and have special meaning, but if you do, please speak up cause I don't see
> any of it.
>

Nope, I don't have any explanation; I'm just a kid! =o)

Seriously though, I tend to agree with you -- this should explain my
unease about searching for every polynomial possibility with a certain
number of points.  I want to help out as much as I can, but I just
don't want to be the one to close a door on an option.  I am only
qualified to deliver some of the formulae and maybe do some of the
programming, but I don't pack the mathematical guns to do the kinds of
analytical work Chuck has been doing.

On the other hand, doesn't [tabread4~]'s Lagrange interpolator have a
continuous 2nd derivative while the [tabread4c~] Hermite one does not?
I don't know what that would mean spectrally, if anything.  It's the
"almost" in the "almost-arbitrary" curves you mention that I don't
know how to gauge -- intuitively one could imagine that the more
pieces of its surrounding environment are matched, the better the
interpolation, but I certainly wouldn't put money on that argument.
The other side of the coin is, is it optimal that any points at all
should be matched?  I could imagine the existence of interpolations
which deliver all kinds of spectral benefits but which aren't
constrained to pass through any particular sample value...

Also fairly intuitively one could imagine that some of this is
signal-dependent, which would seem to stress the need for careful
analysis and benchmarking.  Curiosity would seem to demand testing a
few options to feel out a direction before getting rid of any, testing
some absurd options to confirm absurdity.  Of course, there's more
than a strong possibility that this wheel has already been invented
several times over and that the answers have been thoroughly
established somewhere.  The thing that keeps bothering me is that the
smart people who designed Pd and Csound both arrived at the Lagrange
interpolation, and those who designed SC3 arrived at the Hermite, and
I am certainly not qualified to second-guess any of them.  I can do
the algebra, though.  =oD

Matt

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