When the

delay-line lengths need to be varied in real time, or
interactively in a GUI, it is convenient to choose each delay-line
length

as an integer power of a distinct

prime number
[

457]:

where we call

the ``multiplicity'' of the prime

. With
this choice, the delay-line lengths are always coprime (no factors in
common other than

), and yet we can lengthen or shorten each delay
line individually (by factors of

) without affecting the mutually
prime property.
Suppose we are initially given desired delay-line lengths

arranged in ascending order so that

Then good prime-power approximations

can be expected using
the prime numbers in their natural order:

Since

(for any
logarithmic base), an optimal (in some sense) choice of prime
multiplicity

is

round

where

is the desired length in samples. That is,

can be
simply obtained by

*rounding*
to the
nearest integer (max 1). The prime-power delay-line length
approximation is then of course

and the multiplicative approximation error is bounded by

(when

).
This prime-power length scheme is used to keep 16 delay lines both
variable and mutually prime in

Faust's

`reverb_designer.dsp`
programming example (via the function

`prime_power_delays` in

`effect.lib`).

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