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The Ando-Beranek’s model, a linear version of Ando’s subjective preference theory, obtained by the authors in a recent work, was combined with Barron revised theory. An optimal volume region for each reverberation time was obtained for classical music in symphony orchestra concert halls. The obtained relation was tested with good agreement with the top rated halls reported by Beranek and other halls with reported anomalies.

Obtaining a criterion based on objective measures to assess the quality of a concert hall is one of the fundamental aims of room acoustics. However, recently, it has been recognized that

There is no unique and definitive design parameter that can be said to correspond with perceived hall acoustic quality. However, different authors have tried to find models that combine the studied acoustic parameters with the aim of obtaining an objective ranking that correlates with the subjective ranking.

Otherwise, it is very important for designers to have a model that enables the prediction of subjective response from the objective measurements.

One of the most important references for determining optimal values from psycho-acoustic studies can be found in the works by Ando [

Beranek added two independent parameters to those of Ando: bass ratio (BR) and the surface diffusion index (SDI). These parameters were used with a modification of Ando’s theory to obtain an objective hall ranking method [

On the other hand, room acoustics is a well-established branch of acoustical engineering having its own standard. The ISO standard proposes that the acoustics of a hall can be described with just a few numbers obtained by spatially averaging over several measured seats. In the current decade, that standard has been criticized on many stages: the algorithms to compute the parameters are imprecise, the applied frequency range is too narrow, and a single omnidirectional source is a poor representation of the dozens of sound sources present in a real orchestra [

In previous studies [

In [_{mid}, LFC_{E}_{4}, _{late} and IACC_{L}_{3} (the last two parameters are used in LEV computation).

As shown in extensive objective acoustic measurements by Barron and Lee [_{late}:

Traditional statistics is strong in devising ways of describing data and inferring distributional parameters from sample. But, the obtained statistical models can be spurious or have a poor validity. A way to show validity of the model is to apply it to another data set. Albeit our model was computed using mean values of room parameters, in a recent work, we used Ando-Beranek’s model to evaluate quality in a hall by using quality maps [

In this work, we combine the results obtained in [

In this section we introduce the Ando-Beranek’s model inspired by the work of these two authors [

In Beranek [_{t} Ando’s functions. We have introduced [_{AB} of the orthogonal factors obtained in previous works [_{a} of _{t} functions (_{AB} was named Ando-Beranek’s model).

The general problem to be solved is the following: how to find the coefficients in:
_{mid} + _{E4} + _{a})

In [_{AB} = −1_{mid} + 2_{E4} − 0

With regard to our subjective hall assessment [_{AB} parameter was:
_{AB}

This model and the quality criterion for symphony orchestra concert halls, supported by our experimental data, was used to mapping quality in a hall with satisfactory results [

At this point we must review the knowledge about volume in design room acoustic [

The first step in acoustic design is to choose a reverberation time adequate to hall’s use. After considering the volume and area following Sabine’s theory, the next level of detail includes specific reflections. Certain 1st-order reflectors arrive at the listener from above. A low ceiling tends to promote low reverberance, lack of envelopment and a generally inadequate increase of loudness (as it directs sound into the absorbing audience). So, an important factor becomes Height/Width ratio or equivalently EDT/RT ratio [

In the early 60s, Leo Beranek postulated the importance of Initial Time Delay Gap, and this often led to arrays of small reflectors suspended below the ceiling. These allowed the ceiling to be higher, in order to sustain reverberance, but the reverberation was still addressed as a simply function of volume and area, or the number of seats.

Designs based on this approach resulted in several wide fan-shaped and oval halls with overhead reflectors. Some halls designed using these rules showed problems. The analysis of what these halls are missing is related to the loudness of the lateral sound, both early and late lateral sound. The importance of lateral reflections was advanced by Barron and Marshall. Note here that these arguments can be used as a support of the quality criterion 6 because our expression involves RT, LFC and LEV.

These properties depend on constructional data [

shape of the room;

volume of the room;

number of seats and their arrangement;

materials of walls, ceiling, floor, seats,

While the reverberation time is determined by factors (b), (d) and not significantly by (a), the room shape influences strongly the number, directions, delays and strengths of the early reflections received at a given position or seat. The strength of the direct sound depends on the distances to be covered, and also on the arrangement of the audience. In specific reference to the volume, large room volume is necessary to get an adequate reverberance and a good spaciousness [

Beranek’s design procedure use strength G to determine audience area and, combined with reverberation, establish the hall’s volume. In his procedure, it may go unnoticed reverb efficiency related to Height/Width ratio and the assumption that about 75% of the total sound absorbtion in a hall is contributed by the audience and orchestra [_{mid} ϵ [3.5, 5] dB.

V-RT region following Beranek’s design procedure. Concert Halls from his 2006 book are shown [

To evaluate the model of Ando-Beranek’s and its optimum interval from the point of view of the acoustical design of concert halls for symphony orchestra music halls, we are going to consider some basic approximations from room acoustics. First of all, we consider the classical expression of the Sabine’s reverberation time:

In the quality relationship (6), some data from the room RT, LFC, IACCL, are considered design parameters. From the quality criterion we find that _{l}_{ate} satisfy the inequalities:

We can use now the Barron’s model [_{l}_{ate}) will be given by the Equation (1). From the quality Equation (6), we can get an interval of variation for _{l}_{ate} that we consider as optimum [_{late;min}, _{late;max}] [_{min}. This distance is where the _{l}_{ate} will be maximum and, therefore, it will allow us to deduce the corresponding volume through the formula:

According to the Barron’s model, the variation of _{l}_{ate} with distance is very slow (−0.174/_{late;min}, _{late;max}] is always 3.51 dB wide. This supposes that in case the minimum distance condition is fulfilled, the maximum accepted distance is:
_{max} ≈ 20·

As a numerical example, we now consider as design values the following ones: RT = 2.5 s, LFC = 0.2 and IACCL = 0.15. These values give an optimum interval for the late strength: _{late} = [1.01, 4.52]. If we consider a minimum distance of 8 m, we found that the room volume is 15,544 m^{3}. Furthermore, we have guaranteed the optimum interval of late energy up to a 50 m distance to the source, which is clearly enough.

Taking as design parameters LFC, IACCL and _{min}, we can represent the Equation (3) for different values of reverberation time T. This representation allows us to validate the obtained result by applying it to well-known halls.

_{min} = 7 m). In a first sight, we can appreciate that the combination of the Barron’s model with the quality Equation (6), gives an expression which satisfies the following limits:

This means that out of the fitted interval of reverberation time, it is not meaningful to design rooms for classical music according to this result, because it would have a very little volume.

There are three groups of halls in _{late}. Finally, we included a group of large halls with regrettable reputation [_{late}) is low. As we can see in

V-RT region following Ando-Beranek’s formula.

In this paper, we test Ando-Beranek’s model of sound quality for symphony orchestra concert halls obtained by the authors in a previous work [

Although traditional statistics is strong in devising ways of describing data and inferring distributional parameters from sample, the obtained statistical models can be spurious or have a poor validity. A way to show validity of the model is to apply it to another data set. A previous test of our model was presented in [

In this paper, we have combined the Ando-Beranek’s model with the Barron’s theory. This has allowed us to obtain a formula that combines RT, LFC, IACCL3 and the Volume of the room. Using the minimum distance from the center of the scenario to the first line of seats as a reference, we have obtained expression (3). This expression can be represented (in

This work has been financially supported by FEDER funds and by the Ministry of Science and Technology with references Nos. BIA2003-09306, BIA2008-05485 and BIA 2012-36896.

Authors are part of the Research Group in Virtual Acoustics (GIAV-UPV-UVEG) (

The authors declare no conflicts of interest.