It's a totally naive laymans approach
I hope the formatting stays in place.
The feedback delay in the loop folds the signal back
so we have periods of a comb filter.
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Now we want to fill the period densly with impulses:
First bad idea is to place a first impulse exactly in the middle
that would be a ratio for the allpass delay of 0.5 in respect to the
comb filter.
It means that the second next impulse falls on the period.
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The next idea is to place the impulse so that after the second cycle
it exactly fills the free space between the first pulse and the period
like this,
exactly in the middle between the first impulse and the period:
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this means we need a ratio "a" for the allpass delay in respect to the
combfilter loop that fulfills:
2a - 1 = a/2
Where 1 is the period of the combfilter.
Alternativly, to place it on the other side, we need
2a - 1 = 1 - a/2;
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This gives ratios of 0.5. 0.66667 and 0.8
These are bad ratios since they have very small common multiples with
the loop period.
So we detune them slightly so they are never in synch with the loop
period or each other.
That was my very naive approach, and surprisingly it worked.
The next idea is to place the second impulse not in the middle of the
free space
but in a golden ratio in respect to the first impulse
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2a - 1 = a*0.618...
or
N*a mod 1 = a*0.618..
or if you prefer the exact solution:
a = (1 + SQRT(5)) / ( SQRT(5)*N + N - 2)
wich is ~ 0.723607 and the same as 1/ (1+ 0.382...) or 1/ (N + 0.382)
where N is the number of impulses, that means instead of placing the 2nd
impulse on a*0.618
we can also place the 3rd, 4th etc for shorter AP diffusors.
(And again we can also fill the other side of the first impulse with
0.839643
And the solution for N = 1 is 2.618.. and we can use the reciprocal
0.381 to place a first impusle)
The pattern this gives for 0.72.. is both regular but evenly distributed
so that each pulse
falls an a free space, just like on a Fibonaccy flower pattern each
petal falls an a free space,
forever.
(I have only estimated the first few periods manually, and it appeared
like that
Its hard to identify in the impulse response since I test a loop with 3
APs )
The regularity is a bad thing, but the even distribution seems like a
good thing (?).
I assume it doesn't even make a huge difference to using 0.618.. for a
ratio though it seemed to sound better.
(And if you use 0.618, what do you use for the other APs?)
So it's not the solution I am looking for but interesting never the less.
I believe that instant and well distributed echo density is a desired
property
and I assume that the more noise like the response is as a time series
the better it works also in the frequency/phase domain.
For instance you can make noise loops with randomizing all phases by FFT
in circular convolution
that sound very reverberated.
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