Fig. 7.14). To realize these conditions simultaneously, geometrically uneven surfaces for the side walls of the concert hall have been proposed (Ando and Sakamoto, 1988).

3.3 Theory of Subjective Preferences for Sound Fields

We will now put these results into theory. Because preference involves a limited number of orthogonal acoustic factors that are implicit in the sound signals arriving at both ears (Table 3.1), the scale value of any one-dimensional subjective response may be expressed as a function of these factors

A linear scale value for subjective preference can be derived using the law of comparative judgment or paired-comparisons tests (Thurstone, 1927; Mosteller, 1951; Gullikson, 1956; Torgerson, 1958) for both groups of subjects as well as for individuals (Ando, 1998). Through a series of experiments, it has been verified that four objective acoustic factors act independently of the scale value when two of the four factors are varied simultaneously, as indicated in Table 3.3. Results obtained in a series of experiments that used different source signals indicate that the four units of scale appear to be almost constant, so we can add individual scale values to obtain the total scale value (Ando, 1983),

where Si , i = 1, 2, 3, 4 is the scale value obtained relative to each objective factor. Equation (3.6) represents a four-dimensional continuum.

The dependence of the scale values on each objective factor is shown graphically in Fig. 3.5. From the nature of the scale value, it is convenient to set its value to zero at the most preferred conditions, as shown in this figure. Scale values of subjective preference obtained from other experimental series that used different music programs, yield similar results when each factor is normalized by its most preferred value. The following common formula is given:

Table 3.3 Subjective preference tests examining independent effects changing two of four orthogonal factors of the sound field

1Unpublished (see Ando 1998).

Fig. 3.5 Scale values of subjective preference obtained for simulated sound fields in an anechoic chamber, as a function of four normalized orthogonal factors of the sound field. Different symbols indicate scale values obtained from different source signals (Ando, 1985). (a) As a function of listening level, LL. The most preferred listening level, [LL]p = 0 dB. (b) As a function of first reflection time t1/[ t1]p. (c) As a function of later reverberation time Tsub/[Tsub]p. (d) As a function of interaural correlation magnitude IACC. The most preferred values [ t1]p and [Tsub]p are calculated by Equations (3.3) and (3.4), respectively. Even if different signals are used, consistency of scale values as a function of the normalized factor is maintained, fitting a single curve

where xi is the normalized factor and the values of αi are weighting coefficients as listed in Table 3.4. If αι is close to zero, then a lesser contribution of the factor xi on subjective preference is signified.

Table 3.4 Four orthogonal factors of the sound field, and its weighting coefficients αi in Equation (3.7), which was obtained with a number of subjects

The factor x1 is given by the sound pressure level difference, measured by the A-weighted network, so that,

P and [P]p being, respectively, the sound pressure or listening level (LL) present at a specific seat and the most preferred sound pressure that may be assumed at a particular seat position in the room under investigation:

The values of [ t1]p and [Tsub]p may be calculated using Equations (3.1) and (3.4), respectively.

The scale value of preference has been formulated approximately in terms of the 3/2 power of the normalized factor, expressed in terms of the logarithm of the normalized factors, x1, x2, and x3. The remarkable fact is that the spatial binaural factor x4 = IACC is expressed in terms of the 3/2 power of its real values, indicating a greater contribution than those of the temporal parameters. The scale values are not greatly changed in the neighborhood of the most preferred conditions, but decrease rapidly outside this range. Since the experiments were conducted to find the optimal conditions, this theory remains valid in the range of preferred conditions tested for the four orthogonal factors. When t1 and Tsub are fixed near their preferred conditions, for example, the scale value of subjective preference calculated by Equation (3.6) for the LL and the IACC is demonstrated in Fig. 3.6. Agreement between cal-

Fig. 3.6 Scale values of subjective preference for the sound field with music motif A as a function of the listening level (LL) and as a parameter of the IACC (Ando and Morioka, 1981). (_ _ _ _): Calculated values based on Equation (3.6) taking the two factors into consideration; (____): Measured values

culated values and observed ones are satisfactory, so that the independence of the listening level LL and the IACC on the scale value is achieved (Ando and Morioka, 1981). The same is true for the other two factors (Ando, 1985).

3.4Evaluation of Boston Symphony Hall Based on Temporal and Spatial Factors

As a typical example, we will consider the quality of the sound field at each seating position in an existing concert hall using the Symphony Hall in Boston, Massachusetts, as a model. Suppose that a single source is located at center stage, 1.2 m above the stage floor. Ear positions are receiving points situated at a height of 1.1 m above floor level. The 30 earliest reflections with their amplitudes, delay times, and directions of arrival at the listeners are taken into account using the image method.

Contour lines of the total scale value of preference, calculated for music motif B are shown in Fig. 3.7. The left plot (a) demonstrates the effects of the reflections from the sides on the stage in their original shape configuration. Adding angled reflecting sidewalls on the sides of the stage, as in right plot (b), may produce decreasing values of the IACC for substantial portions of the audience area, which increases the total preference value at each seat (compare left and right plots). In this calculation, reverberation time is assumed to be 1.8 s throughout the hall and the most preferred listening level, [LL]p = 20 log[P]p in Equation (3.10), is set for a point on the center line 20 m from the source position.

Fig. 3.7 Predicted effects of modified room dimensions on listener satisfaction. An example of calculating scale values with the four orthogonal factors using Equations (3.6) through (3.11). (a) Contour lines of the total scale value for Boston Symphony Hall, with original side walls on the stage. (b) Contour lines of the total scale values for the side walls optimized

In order to further test the subjective preference theory, subjective preference judgments in another existing hall (Uhara Hall in Kobe, Japan) were performed using paired-comparison tests (PCT) at each set of seats. Here source locations on the stage were varied. The theory of subjective preference with orthogonal factors was reconfirmed (Sato et al., 1997; Ando, 1998). It was also shown that the theory holds at the condition of τ IACC = 0, i.e., a source at its optimal location at center stage. For other, off-center positions, preference values decrease.