### 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 :

#### 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]:
round

where is the desired length in samples. That is, can be
simply obtained by *rounding*to the nearest integer (max 1). The prime-power delay-line length approximation is then of course

`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