Reverb Part 1—“Freeverb”

Thursday, June 05, 2025

#audio #dsp #maxmsp #plugin #reverb #schroeder

All posts in this series: Part 1, Part 2

I started work on a reverb plugin in C++/JUCE back in 2023, and I've slowly added to it since. Today, I'm going to discuss how algorithmic (i.e., delay-based) reverbs work using the “Freeverb” algorithm that's available in that plugin. I've included a Max/MSP version of the reverb, and in a later post, I will discuss getting our hands dirty with the code in C++/JUCE, as well as a number of other cool algorithms. Let's get started!

How Reverb Works

There are two main categories of digital reverb—convolution and algorithmic—and today we will be discussing algorithmic reverb. An algorithmic reverb is based on a network of delays or “echoes,” with individual delays roughly corresponding to the sound reflections from individual walls or objects, and the network of connections between them simulating the way a reflection from one surface may bounce off many others.

Sean Costello of Valhalla DSP has a series of posts giving helpful resources on reverb design, including influential papers, online resources, and books, and you can find the type of algorithm we'll discuss today among the papers he mentions.

Feedforward and Feedback Delays

In the most basic form of delay, we take a copy of the incoming audio signal, delay it by a certain amount, and combine it with the original signal, usually with the option to adjust the proportions of original and delayed copies. To contrast with the other types of delay, this is often referred to as a “feedforward” delay—the delayed copy is “fed forward” and recombined with the original without any “feedback,” which we will discuss in the next section.

In the diagram below, $x (n)$ represents the incoming signal; $z^{- M}$ is the delay block (with $M$ representing the amount of delay in samples); ^[1] $b_{0}$ and $b_{M}$ represent the amount of gain applied to the original and delayed copies respectively; ^[2] the $+$ symbol represents summing the two copies; and $y (n)$ is the output signal.

block diagram of a feedforward delay/comb filter — Feedforward delay/comb filter (diagram from Julius O. Smith)

Next, we have a “feedback” delay. Note how in the diagram below, the incoming signal is summed with the fed-back output of the delay before entering the delay; the gain $- a_{M}$ applied to the fed-back output of the delay is negative; and the output of the whole delay $y (n)$ is taken from that sum of the input and feedback before it enters the delay.

block diagram of a feedback delay/comb filter — Feedback delay/comb filter (diagram from Julius O. Smith)

Allpass Filters

Below I have the amplitude curves from feedforward and feedback delays, caused by constructive and destructive interference. The repeated notches in the feedforward delay frequency plot are caused by destructive interference; adding the delay to the original signal causes certain frequencies to cancel out. In the feedback example, as the sound feeds back on itself, certain frequencies constructively interfere—they reinforce each other and create the “spikes” in the amplitude response. Both of these delay configurations are also referred to as “comb filters,” due to the shape of the spikes and notches.

feedforward comb filter amplitude plot — Feedforward comb filter amplitude plot (diagram from Julius O. Smith)

negative feedback comb filter amplitude plot — Negative feedback comb filter amplitude plot (diagram from Julius O. Smith)

Notice how the feedforward and feedback amplitude responses are the inverse of each other. Having a negative gain in the feedback loop of a feedback delay is necessary for the peaks to align with where the troughs would be in a feedforward delay that has the same delay time. As shown here, if the feedback delay's gain were positive, the peaks would be at 0.2, 0.4, etc., instead of 0.1, 0.3, etc.

This alignment is important for our next type of filter: the “allpass” filter. As shown below, we combine a feedforward delay with a feedback delay that has a negative gain on the feedback. As a result, our peaks and troughs in the frequency response cancel out. This kind of delay configuration has a “flat” frequency response; all frequencies pass through at the same loudness.

block diagram of an allpass filter — Allpass filter (diagram from Julius O. Smith)

In contrast to the frequency response, our time response is much more complex. An allpass filter delays the different frequencies in the signal by different amounts. As a result, a single impulse or “click” going through it would become multiplied and “smeared” in time, giving the effect of multiple clicks piling on top of each other, a very useful effect for creating a smooth- and dense-sounding reverb.

Now let's put these all together to make a reverb!

The Classic Schroeder Reverberator

Series Allpasses

Manfred Schroeder has two papers from 1961 that introduce the idea of allpasses and using them for reverberators.

In “Colorless” Artificial Reverberation, ^[3] Schroeder and co-author B.F. Logan note six features that they seek from a delay-based reverberator:

There should be a flat frequency response
Normal modes (i.e., resonant emphases on specific frequencies) “must overlap and cover the entire audio frequency range”
Different frequencies should decay at the same or similar rate
There should be high echo density within a short period after an impulse
Echoes must be non-periodic (i.e., no “flutter echoes”)
“Periodic or comb-like” frequency responses must be avoided

Below, I have an 808 drum machine clap played through a feedback comb filter. The delay is set at successively shorter lengths, from 50ms down to 5ms. Notice how at longer delay times, there is a “fluttering” tremolo effect. This is similar to the echoes with two walls directly facing each other, as in a long, narrow hall. At shorter lengths, there is an audible tone created by the repeated echoes. Both of these features are undesirable in a reverb effect.

In “Colorless” Artificial Reverberation, Schroeder and Logan propose a chain of allpass filters as an answer to this issue. The allpass filter is already better than the feedback comb at creating smooth-sounding results, and to improve on this further, the authors suggest “incommensurate” delay times. In other words, the delay times do not easily divide into each other, preventing the echoes from one allpass from lining up with another. The result of the incommensurate delay times is something more like the random tapping of rain, rather than any periodic rhythm.

On this algorithm, Sean Costello comments that

The resulting reverberator structure is a bit hard to tune, as both the reverb attack and decay are controlled by the feedforward/feedback allpass coefficient, and is less “general purpose” than the structure discussed in Schoeder’s next paper.

As Costello then mentions, some reverbs such as the Eventide Blackhole make artistic use of the weirdness of this structure. However, it would be helpful to have a more general-purpose algorithm.

Below I have a synth riff: first the dry signal, then a mix of the dry signal with a “wet” signal through 8 series allpasses. For the allpass example, I increase the feedforward/feedback parameter from about 0.1 up to 1, then back to 0.1 again.

As mentioned, this doesn't sound as natural as we'd like, as well as not being super flexible. It's a start though, and our next version will sound much nicer.

Parallel Combs into Series Allpasses

In Natural Sounding Artificial Reverberation, ^[4] Schroeder proposes the class of algorithm shown below. Feedback comb filters create a nice exponential amplitude decay (unlike the allpass chain in which, as mentioned above, both attack and decay depend on the same parameter). In response to the problem of “coloration” in the frequency spectrum, Schroeder notes that

extreme response irregularities are imperceptible when the density of peaks and valleys on the frequency scale is high enough

In other words, if we combine several feedback comb filters in parallel, the “comb teeth” in their frequency spectra will tightly “interlock,” creating what sounds like a flat frequency response.

The next problem after the feedback comb's frequency response is the “flutter” effect. Schroeder gives the goal of around 1000 echoes per second to make a “natural”-sounding reverb. Assuming 4 parallel comb filters with delays that don't easily divide into each other, but are in the neighborhood of 40ms (25 echoes per second), we get only about 100 echoes per second. Assuming (as Schroeder does) that a single allpass multiplies the number of echoes by about 3, two series allpasses after 4 parallel combs is enough to meet the minimum requirement.

DSP block diagram of 8 filtered-feedback comb filters summed in parallel, and feeding into four allpass filters in series — The “Freeverb” Schroeder reverberator (diagram from Julius O. Smith)

The combination of parallel feedback combs into series allpasses (or sometimes vice versa) is often referred to as a “Schroeder reverb.” The “Freeverb” algorithm shown above is one popular Schroeder reverb. The delays are given in samples, with the floating point values representing the delay feedback coefficients. In the combs, there are two values, and as far as I can tell, the first is the feedback coefficient, and the second is the lowpass filtering (more details in a moment).

Below I have first the dry synth riff, followed by the riff through the “Freeverb” algorithm on my algorithmic reverb plugin. This time it sounds much more natural, and if you play around with the Max patch (see below) or my plugin, there's much more flexibility.

One additional feature of this particular version is that there is a first-order (i.e., 6dB/octave) low-pass filter in the feedback loop of the combs. This causes higher frequencies to decay faster. While Schroeder and Logan seek to make all frequencies decay at an equal rate, the most important aspect seems to be to prevent individual frequency bands from ringing and standing out. Reverberations in the physical world decay more quickly at higher frequencies, due (at least in my understanding) to the sound absorbency of building materials, as well as the fact that the atmosphere can be approximated with a low-pass filter.

Some final design notes: First, we have the stereo spread. The right channel has a slightly longer delay time than the left, with a default value of 23 samples added to each delay value, according to Julius O. Smith. This simulates the effect of different physical environments to the right and left of the listener. Second, in my JUCE implementation I have a low-frequency oscillator (LFO) slowly modulating the first and third allpass filter delay times longer and shorter, giving a chorusing effect and making the sound richer.

If you want to play with this reverb algorithm in Max/MSP, you can use the patch below:

“Freeverb” Max patch. Copy code and select New From Clipboard in Max.

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-----------end_max5_patcher-----------

Final Notes

That's all for today! My plan is to do a series of posts, each covering a class of reverb algorithms. For the next one, I'll be writing about rings of allpass filters—this is relevant to, for example, Jon Dattorro's popular 1997 algorithm that appears to be based on a plate reverb from Lexicon's 224 and 480 reverb units. Until next time!

This notation comes from the idea of the Z-transform. ↩︎
Note that the triangle symbol is the standard DSP symbol for amplification/an amplifier. Also note that $b$ and $a$ are often used for feedforward/feedback coefficients, although it seems to be somewhat inconsistent which is which. ↩︎
M. R. Schroeder and B. F. Logan, “‘Colorless’ artificial reverberation,” IRE Transactions on Audio, vol. AU-9, no. 6, pp. 209–214, Nov. 1961, doi: 10.1109/TAU.1961.1166351. ↩︎
M. R. Schroeder, “Natural sounding artificial reverberation,” in Audio Engineering Society Convention 13, Audio Engineering Society, 1961. Accessed: Dec. 29, 2024. [Online]. Available: https://www.aes.org/e-lib/download.cfm?ID=343 ↩︎