I am reading about reverb algorithms and I came across the concept of nested allpass filters.

Here's an illustration of a regular allpass filter:

And here's a nested allpass structure:

source: http://gdsp.hf.ntnu.no/lessons/6/33/

I used those illustrations as well as H14.all.pass from the audio examples as the basis. But I am not sure I am doing this right at all....

Here's a vanilla allpass I made:
vanilla_allpass~.pd

And here's a nested allpass using the above abstraction:
nested_allpass~.pd

Is this a proper nested allpass? If not, where am I going wrong? As you might guess, I don't fully understand [rzero_rev~] and [rpole~].
Cheers

@Ice-Ice Well, I was hoping that if I followed that same tutorial I might be able to remedy some of my ignorance of low-level filter construction.

Nope!

But maybe what I made is still useful to you. This is untested because I honestly don't understand what I should be testing for. The examples on that tutorial sound more like feedback delays than all pass filters to me, but that just may be more of my ignorance. That said, my gut tells me that you can't just mashup H14 and this filter diagram--they're slightly different instances of the same thing.
allpass~.pd

Edit: Oooh, I think I see what's going on!
H14 implements the APF using [rzero-rev~] fed into [rpole~]. From help:

    [rzero-rev~] implements y[n] = -a * x[n] + x[n-1]
    [rpole~] implements y[n] = x[n] + a * y[n-1]

Since the first equation provides input to the second, substitute x[n] in the second with the output equation of the first:

    y[n] = -a * x[n] + x[n-1] + a * y[n-1]

Now make an equation for that tutorial's diagram:

    y[n] = -g * x[n] + x[n-M1] + g * y[n-M1]

The first term represents the upper path, the second term is the middle path through the delay, and the last term is the feedback going through the same delay!

So H14 is the same only if a = g and M1 = 1.

Hi, I have a deep section on filter theory in my tutorial, and how to build some reverbs with it and comb filters. It is part of the ELSE download!

Screenshot 2026-01-13 at 19.48.07.png

Also, note that [rev1~] uses a series of allpass filters in Vanilla, but not a nested one. Give me a minute and I can try and make nested ones.

@Ice-Ice by the way, please don't use illegal arguments in inlet/outlet objects, it's not a good practice.

what you need for reverbs is not a 1st or 2nd order allpass filter, but an allpass reverberator with a delay time, so forget about H14. Here's a proper allpass unit from my tutorial

This is where the example is and I'm sending another screenshot as the one before isn't showing somehow

from the referenced nested allpass structure, this is what I make of it...

see nested-allpass.pd

here's the CSound triple nested example from that tutorial translated to Pd using objects from ELSE.

see
triple-nested-allpass.pd

Here's a revised version, there were minor mistakes...

see:
triple-nested-allpass-2.pd