Modes in small rooms often lead to uneven frequency responses and extended sound decays. In critical listening spaces this causes unwanted coloration effects, which can be detrimental to the sound quality. The problem arises at low frequencies, because of the relatively low modal density. Many methods and optimum room ratios have been suggested over the years to minimize coloration. Essentially these methods try to avoid degenerate modes, where modal frequencies all fall within a small bandwidth, and also bandwidths with absences of modes. The assumption being that as music is played in the rooms, the absence or boosting of certain tonal elements will detract from the audio quality. The starting point for previous methods to determine room dimensions is usually the equation defining the eigen-frequencies within a rigid rectangular enclosure, shown in the following equation.

Where nx, ny and nz are integers and Lx, Ly and Lz the length, width and height of the room. Often the best dimensions are given in terms of ratios. Previous methods for determining room ratios differ, however, in how they utilize Eq. (1).

A New Approach to Automatically Sizing Critical Listening Rooms

Bolt produced design charts to determine good room ratios. His method investigated the average modal spacing to try and achieve evenly spaced modes; the assumption being that if the modal frequencies are evenly spaced, then the flattest possible frequency response is achieved. Gilford discussed a loose methodology whereby the modal frequencies are calculated and listed. An acoustician then looks for groupings and absences assuming a modal bandwidth of about 20Hz. This is a cumbersome process to undertake by hand, but this type of iterative search is easily accomplished with modern computers using numerical optimization techniques and it is the basis of the Direct Modal Optimization approach we will describe. Louden calculated the modal distribution for 125 combinations of room ratios at a spacing of 0.1 and published a list of preferred dimensions based on a single figure of merit. The figure of merit used to judge room ratios is the standard deviation of the inter-mode spacing, so again this is a regime to achieve evenly spaced modes. This type of quantized search can limit the potential solutions found. Bonello developed a heuristic criterion based on the fact that the modal density should never decrease when going from one third-octave band to the next highest band. He also makes the empirical assumption that five or more modes in a bandwidth mask the effects of coincident modes. The method we will discuss in a series of issues uses a theoretical model, which although not perfect, is a more accurate model of low frequency room behavior than Eq. (1). The new method acts directly on the modal response of the room, rather than the mode spacing, which is one more level removed from the actual signals received by the listener. Below we compare various aspects of the Direct Modal Optimization method with Conventional Approaches. In the next issue we describe the method.

Theoretical model
Conventional Approaches calculate the eigenfrequency positions, using Eq. (1) for rigid, rectangular rooms.

Direct Modal Optimization offers a more accurate calculation of the actual modal response including the effects of absorption on eigenfrequency position, weighting of axial, tangential and oblique modes and mode bandwidth.

Criteria
Conventional Approaches try to achieve evenly spaced modes, which is a step away from the actual modal response. Criteria are heuristic or empirical rather than fundamentally based on the modal response.

Direct Modal Optimization minimizes the standard deviation of the actual modal response perceived by a listener.

Effect of Absorption on Modal Position
Conventional Approaches do not consider absorption. If the criterion requires grouping into third-octave bands, modes occurring on or near the border of adjacent bands, may be incorrectly assigned, since they may be shifted by absorption.

Direct Modal Optimization includes absorption in the calculation of the modal response.

Effect of Absorption on Modal Weighting
Conventional Approaches do not calculate modal weighting and typically modes are weighted heuristically.

Direct Modal Optimization includes the effect of absorption on the weighting of axial, tangential and oblique modes in the calculation of the modal response.

Effect of Absorption on Modal Bandwidth
Conventional Approaches only calculate the position of the eigenfrequencies.

Direct Modal Optimization directly includes the modal bandwidth in the calculation of the modal response.

Modal Overlap
Conventional Approaches do not address modal overlap.

Direct Modal Optimization includes the modal overlap of adjacent modes directly in the modal calculation.

Sensitivity
Conventional Approaches empirically assign a 20 Hz modal separation as being perceptible.

Direct Modal Optimization automatically searches for the flattest modal response (lowest standard deviation) and monitors the worst response as well. Experiments suggest a difference limen of greater than 0.3.

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