### Choice of Delay Lengths

Following Schroeder's original insight, the delay line lengths in an
FDN ( in Fig.3.10) are typically chosen to be *mutually
prime*. That is, their prime factorizations contain no common
factors. This rule maximizes the number of samples that the lossless
reverberator prototype must be run before the impulse response
repeats.

The delay lengths should be chosen to ensure a
*sufficiently high mode density* in all frequency bands. An
insufficient mode density can be heard as ``ringing tones'' or an
uneven amplitude modulation in the late reverberation impulse
response.

#### Mean Free Path

A rough guide to the average delay-line length is the ``mean free
path'' in the desired reverberant environment. The *mean free
path* is defined as the average distance a ray of sound travels before
it encounters an obstacle and reflects. An approximate value for the
mean free path, due to Sabine, an early pioneer of statistical room
acoustics, is

*diffuse field*assumption,

*i.e.*, that plane waves are traveling randomly in all directions [349,47] (see §3.2.1 for a simple construction). Normally, late reverberation satisfies this assumption well, away from open doors and windows, provided the room is not too ``dead''. Regarding each delay line as a mean-free-path delay, the average can be set to the mean free path by equating

*diffuse*, especially at high frequencies. In a diffuse reflection, a single incident plane wave reflects in many directions at once.

#### Mode Density Requirement

A guide for the *sum* of the delay-line lengths is the desired
*mode density*. The sum of delay-line lengths in a lossless
FDN is simply the *order* of the system :

Since the order of a system equals the number of poles, we have that is the number of poles on the unit circle in the lossless prototype. If the modes were uniformly distributed, the mode density would be modes per Hz. Schroeder [417] suggests that, for a reverberation time of 1 second, a mode density of 0.15 modes per Hz is adequate. Since the mode widths are inversely proportional to reverberation time, the mode density for a reverberation time of 2 seconds should be 0.3 modes per Hz, etc. In summary, for a sufficient mode density in the frequency domain, Schroeder's formula is

#### Prime Power Delay-Line Lengths

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]:

Suppose we are initially given desired delay-line lengths arranged in ascending order so that

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

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`).

**Next Section:**

Achieving Desired Reverberation Times

**Previous Section:**

Choice of Lossless Feedback Matrix