[sdiy] Fixed filter bank questions

mskala at ansuz.sooke.bc.ca mskala at ansuz.sooke.bc.ca
Thu May 9 04:33:31 CEST 2019

On Wed, 8 May 2019, Ben Bradley wrote:
> I spend a few minutes thinking about this, and what you'll have
> musically is when a note "hits a frequency" of a filter, not only will
> a fundamental be emphasized, but also its even and multiple of 3
> harmonics as well. Other notes would have one or a very few harmonics

This is probably not a real problem for two reasons:  one, the filters
will not be so narrow as to make such an effect noticeable, and two, the
filters will be arranged in an exponential series (1, 2, 4, 8, 16) whereas
harmonics are arranged in an arithmetic series (1, 2, 3, 4, 5).  These two
factors mean any issue of "lining up" can only practically exist across
two or maybe at most three filters; higher-frequency filters will all pass
multiple harmonics anyway.

The bandwidth of each filter is in proportion to its frequency and doubles
with each octave, while the spacing between harmonics does not.  You do
NOT get one filter passing harmonic 2, one passing harmonic 4, one passing
harmonic 6, one passing harmonic 8, and so on.  Instead it's more like one
filter passes harmonic 2, if it's exactly half-octaves with a root-2 ratio
then next filter basically passes harmonic 3 because 3 is close to 2.818,
the filter at the octave passes harmonic 4 mostly and a bit of 3 and 5,
one another octave up passes roughly everything from harmonics 6 to 10...
and if the fundamental were not aligned exactly it wouldn't make much
difference because even the lowest filter is pretty wide and overlaps with
its neighbours on either side.

So trying to make the filters be in an exponential series that doesn't
include octaves, to avoid "lining up" with harmonics of notes, is unlikely
to be a real issue.  Nonetheless, if you want to aim for it (possibly to
avoid lining up with octave-doubling musical material?), it might make
some make sense to use a ratio of 1.5348..., which is 2^phi where phi is
the Golden Ratio 0.6180...  That gives a fraction of an octave that is in
some sense maximally far away from being a rational number, so a repeating
pattern of octaves (or any other just musical intervals) from one filter
centre frequency will be as far as possible away from hitting any other
filter centre frequencies.  Some root of 1.5348 (square root, cube root,
etc) if you want filters more closely spaced.

A comb filter might a different story because they have peaks and nulls
spaced in an arithmetic series, not exponential; so a note at an exact
multiple of the comb frequency could have all its harmonics pass, or none
of its harmonics pass, and that would make a difference.  But that's a
very different kind of filter from what would be in the kind of analog
filter bank we're talking about here, and if you want to avoid this
effect, basically all you can do is not use comb filters.

In my Leapfrog VCF module design I played with the poles and zeroes to get
the peaks and nulls at harmonically-related frequencies in the hope that
it might have some musically relevant effect.  If you put the peak at the
edge of the cutoff slope right on top of the fundamental of a note, then
the second and third harmonics are notched out; and there's a small peak
in the passband at 2/3 of that fundamental frequency.  Having those
harmonic relationships is why I call it a "musical near-elliptic" filter
instead of just an elliptic filter.  But honestly, I think it's unlikely
that the difference between doing that and not doing it is really audible
in normal use.

Matthew Skala
mskala at ansuz.sooke.bc.ca                 People before tribes.

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