*EDIT*: I just realised that the demo you're talking about doesn't include the "early reflections"! So that demo only includes echoes which have gone through the main feedback delays at least once.
You can tune the diffuser to have an almost-instant onset. I can't remember what I did last time, but at a guess, having the largest diffuser stage increase by a factor of N (number of channels) instead of 2 might do it.
But also, if you're playing acoustic drums in a big space like a concert hall (instead of a long one like a staircase), the first echoes coming back from the walls are actually a bit delayed (1 foot ~= 1ms, at the speed of sound). So if your nearest wall is 10ft away, the first wall-echoes will come 10-20ms after the initial direct sound.
If you mix in some of the diffused signal directly, it bridges the gap between the initial sound and the feedback echoes: https://signalsmith-audio.co.uk/writing/2021/lets-write-a-re...
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You can tune the diffuser to have an almost-instant onset. I can't remember what I did last time, but at a guess, having the largest diffuser stage increase by a factor of N (number of channels) instead of 2 might do it.
But also, if you're playing acoustic drums in a big space like a concert hall (instead of a long one like a staircase), the first echoes coming back from the walls are actually a bit delayed (1 foot ~= 1ms, at the speed of sound). So if your nearest wall is 10ft away, the first wall-echoes will come 10-20ms after the initial direct sound.