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# Optimizing the acoustics of small recital rooms

15 March 2021

A room’s length-to-width ratio is crucial in determining the smoothness of its frequency response.

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The 88 notes of a grand piano range in wavelength from about 8 centimeters to about 13 meters. If the piano is in an enclosed room, some frequencies within that range will be in resonance with one or more of the room’s dimensions; those frequencies are supported by the room resonances while other frequencies in between are not. Your ability to hear the low notes of an étude as Frédéric Chopin intended will depend crucially on where your head is in relation to a resonant mode’s nearest peak or valley.

The presence of modes in a recital room is unavoidable. Their cumulative effect, however, can be mitigated by contriving to distribute them in frequency as evenly as possible—that is, to create the smoothest possible frequency response. How to do that is the subject of a new paper by Jens Holger Rindel of Odeon, a Danish company that develops room acoustics software.

In a large recital room, reverberation influences the perception of sound. But, Rindel argues, when a room’s volume is around 300 m3 or smaller, sound perception shifts to the frequency domain. Thus the smoothness of the frequency response is particularly important in small recital rooms.

Rindel’s principal criterion for acoustic smoothness is the frequency spacing index (FSI). Developed in 1945 by Richard Bolt, the FSI embodies the variance of frequency intervals between the highest relevant low-frequency mode n and all the other low-frequency modes from 1 to n − 1. Low values of the FSI correspond to smooth-sounding rooms. Both the FSI and the modes are readily calculated. In his analysis, Rindel considered rectangular rooms up to a maximum volume of 300 m3 and with a range of values of two ratios: room length to room width, l/w, and room width to room height, w/h. For such rooms, a value of n = 25 is appropriate.

Surprisingly, the FSI depended only weakly on w/h. Far more important was l/w. The lowest FSI, 1.33, corresponded to a l/w of 1.21. The highest value of the FSI, 3.91, corresponded to a l/w of 2.0—that is, to a room shaped like a double cube. To achieve optimal acoustics, Rindel recommends small recital rooms be built with a l/w between 1.15 and 1.45 and with a w/h that exceeds 1.1. (J. H. Rindel, JASA Express Lett. 1, 021601, 2021.)

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