"Optimum" Room Dimensions
The standard modal approach for designing a room with good acoustics is to create as many different resonances as possible, and to spread them as evenly as possible across the frequency spectrum, as discussed in the Handbook for Sound Engineers, Chapter 3. There is even a complicated "Bonello Criterion" to evaluate the spread. The lowest resonance is determined by the largest dimension of the room. (Technically there is also a resonance at zero Hz for all rooms, but this is generally not considered a true resonance). In general, the lower the better for the first resonant frequency, because this region is where the frequency response is most variable. Bigger rooms also reduce the spacing between resonances. The limiting factor here is usually cost. For a 19-foot long room the first resonance is about 30 Hz. Every harmonic of this frequency (60, 90, 120, etc.) is also a resonance. The width and height of the room each give rise to another series of resonances. These are the primary "axial" resonances, involving reflections from two opposing surfaces. Additional resonances are created by reflections that ricochet off four different surfaces. These "tangential" resonances are generally weaker, because energy is lost at each reflection. Finally there are "oblique" resonances which ricochet off all six surfaces. Each resonance gives rise to a "mode" with a characteristic spatial pressure variation. The mathematical description of a mode is given in the physics section, and some graphical examples are illustrated below. To spread these resonances as uniformly as possible, various ratios between the room height, width, and length have been proposed. Three such sets from the Handbook for Sound Engineers are shown in the table below.
See the link for the table....
According to the modal design theory, the worst possible room shape is a cube. The next worst is a room where all dimensions are multiples of the height. A pretty horrible example is a room 8-ft high, 16-ft wide, and 16 ft long. The resonances for an optimum room (design #3, width =1.60 x height and length = 2.33 x height) and for the latter horrible example illustrate the difference in the resonances [10.7 kb]. The two rooms have the same total volume. The horizontal frequency scale varies from 0 to 200 Hz. Each vertical line represents a resonance. There are three tiers of lines; the highest tier represents axial modes, the middle tier tangential, and the lower tier oblique. The resonances for the horrible room are less dense, because many occur at exactly the same frequency, and there is a fairly large gap between the second and third resonances, at about 60 Hz. The blue, green, and red lines represent resonances related to the room length, width, and height, respectively.