Room acoustical parameters: A factor analysis approach

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Applied Acoustics 70 (2009) 97–109 www.elsevier.com/locate/apacoust

Room acoustical parameters: A factor analysis approach S. Cerda´ a,b,*, A. Gime´nez b,c, J. Romero b,c, R. Cibria´n d, J.L. Miralles e b

a Applied Mathematics Department, Universitat Polite`cnica de Vale`ncia, Camino de Vera s/n, 46022 Valencia, Spain Grup dAcu´stica Arquitecto`nica, Ambiental i Industrial, Universitat Polite`cnica de Vale`ncia, Camino de Vera s/n, 46022 Valencia, Spain c Applied Physics Department, Universitat Polite`cnica de Vale`ncia, Camino de Vera s/n, 46022 Vale`ncia, Spain d Facultad de Medicina, Universidad de Valencia, Blasco Iba´n˜ez s/n, Valencia, Spain e Facultad de Psicologia, Universidad de Valencia, Blasco Iban˜ez s/n. Valencia, Spain

Received 4 January 2007; received in revised form 30 August 2007; accepted 6 January 2008 Available online 4 March 2008

Abstract In this study, we determined the most representative acoustical parameters for halls intended for verbal or music audition. Our study was carried out in nine halls of different shapes and designed for different uses. We measured the impulse response at a great number of points (many more than the minimum required by the ISO 3382 norm). From a physical viewpoint, all halls are enclosed three-dimensional areas. Our work hypothesis is that objective (measurable) acoustical parameters, or a combination of such parameters, must provide the acoustical information specific to each hall and must make it possible to grade each hall. Factor analysis was used to obtain these grading parameters and the considerable number of measurements we determined guaranteed the application of this type of analysis. The convergence provides corroboration of the main correlations between parameters. A group of orthogonal parameters was thus obtained, made up of three factors that group the parameters used by different outstanding researchers. These factors provide a clear acoustical interpretation. We have termed the first of these ‘‘intelligibility” as it contains intelligibility parameters; the second is associated with spaciousness; the third and last parameter has been termed ‘‘strength” as it is a linear combination of the parameters that measure the amplification (G) and the bass ratio (BR). The optimal scores of these factors for different uses of halls make it possible to grade any hall, independently of its shape, for its corresponding use. Ó 2008 Elsevier Ltd. All rights reserved. PACS: 43.55.Gx Keywords: Room acoustics; Correlations; Orthogonality; Factor analysis

1. Introduction The considerable number of parameters proposed by different researchers to determine acoustical quality in concert halls has brought about different ways of tackling such study, depending on the basic attributes to be considered. The Gottingen school [1–3] contemplates three parameters: the interaural cross-correlation index (IACC), reverberation time (RT), and clarity (C). Yamamoto and Suzuki [4], following the techniques of this school, measure clarity (C), strength (G), and space with the IACC index. Marshall and *

Corresponding author. E-mail address: [email protected] (S. Cerda´).

0003-682X/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.apacoust.2008.01.001

Barron [5,16] mainly opt for space impression measured by the lateral force (LF) factor. More recently, Bradley–Soulodre [17,18] and Barron [14] have studied the influence of late energy on the lateral impression from late lateral energy level, (GLL), and late lateral energy fraction (LLF). However, of all the researchers’ proposals we could mention, Ando’s proposal [6] is the most outstanding, because of all the parameters mentioned in the previous paragraph, only four are statistically independent, i.e., any variation in each one does not affect the rest. These are the interaural crosscorrelation index (IACC), early decay time (EDT), strength factor (G), and the time between the first direct sound and the first reflection (ITDG). More recently, Beranek [7] has confirmed that there are two parameters more than those

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indicated by Ando that are also statistically independent, bass ratio (BR) and the surface diffusion index (SDI). These ‘‘independent” parameters can provide overall grading for concert halls. The work plan proposed by Beranek [7] is the following: (1) Identify the orthogonal acoustical parameters that are necessary to judge the acoustical quality of a hall. (2) Choose the best scores for each hall. (3) Determine the specific weight adequate for each one of said parameters to obtain a good correspondence with the subjective response of the audience. This study uses factor analysis to perform the first two points of the procedure. Our first hypothesis was that all halls could be studied simply as enclosed 3D areas, which is why we opted for such a varied choice of halls. The acoustical parameters of each hall encompassed all the acoustical information. As each parameter was defined to represent a physical or subjective quality, the components obtained were not expected to be arbitrary, but rather to provide relevant information and make it possible to reduce the number of parameters. First, the main features of the halls are presented and a code is established to identify the main use of the hall. Then the parameters of the halls, their definition, and their acoustical interpretation are given. The procedure and experimental equipment are described in Section 2. Finally, the study and discussion of the results provided by statistical analysis are set out in Section 3. The main correlations between parameters are given in the first subsection and the second subsection presents the factor analysis. Three principal components are obtained and explained in the last section. The study closes with the main conclusions obtained. 1.1. The halls that were studied This study includes different halls located in Valencia (Spain) which aim to provide correct verbal or musical audition. As mentioned above, we did not choose halls of a similar typology, size, age, or use which is in keeping with our work hypothesis that specifies that the parameters of

the halls must provide all the acoustical information. Each hall has been given a number and a letter that indicate the hall’s number and the main activity carried out there. Halls with the prefix C are mainly concert halls, prefix T refers to theater halls, and prefix S to conference and multiuse halls. Table 1 provides a list of their main features (reverberation time was measured from the response of an unoccupied hall and the mean of octaves of 500 Hz and 1 kHz). The most important features are listed below: Conference halls (400–3000 m3)  Assembly hall of the College of Industrial Engineers of the Polytechnic University of Valencia (S1) [16 measures].  Auditorium of the Polytechnic University of Valencia (S2) [24 measures].  Auditorium of the Polytechnic City of Innovation in Valencia (6G-UPV) (S3) [40 measures]. Concert halls (3000–8000 m3)  Ribarroja Auditorium (C1) [39 measures].  Benaguasil Auditorium (C2) [31 measures].  Torrent Auditorium (C3) [48 measures]. Theater halls (5000–6000 m3)  La Banda Primitiva de Lliria theatre (T1) [81 measures].  La Unio´ Musical de Lliria theatre (T2) [45 measures].  El Teatro Principal de Valencia theatre (T3) [53 measures]. Halls S1, S2, S3, and C1 are rectangular but with considerable differences: Halls S1 and C1, though of a similar typology (rectangular), differ in audience capacity (142 and 783), lateral covering materials (carpeting and wood), and volumes 434 and 7830 m3 . The volume/audience ratio is between 3 and 10 m3 , respectively. Hall S1 is used for conferences, congresses, and recitals given by soloists; on the other hand, hall C1 is used for theatrical plays, opera, dance, and concerts.

Table 1 Main typological characteristics of the halls that were studied: Id. (identification of the hall): prefix C is mainly for concerts, prefix T for theatrical performances, and prefix S for conferences and multiuse, year of opening, RT reverberation time, V/seat (volume per seat) Hall

Id.

Year

Seating capacity

Measures

V ðm3 Þ

TRmid ðsÞ

Assembly Hall CIE, PUV Auditorium, PUV Auditorium of the Polytechnic City of Innovation, PUV Ribarroja Auditorium Benaguacil Auditorium Torrent Auditorium La Banda Primitiva de Lliria Theatre La Unio´ Musical de Lliria Theatre Teatro Principal de Valencia Theatre

S1 S2 S3

1978 (2000) 1978 2000

142 385 475 (380 + 95)

16 24 40 (30 + 10)

434 2700 3266

0.68 1.3 1.51

3 7 6.9

C1 C2 C3 T1 T2 T3

1994 1960 1997 1951 (1992) 1951 (1992) 1832 (1991)

783 509 (413 + 96) 606 (478 + 128) 967 (602 + 145 + 142 + 78) 967 (602 + 145 + 142 + 78) 1224 (460 + 226 + 156 + 232 + 150)

39 31(22 + 9) 48(34 + 14) 81 (40 + 11 + 18 + 12) 45 (24 + 15 + 3 + 3) 53 (36 + 6 + 11)

7830 3480 6430 5314 6287 5911

1.79 2.25 1.87 1.35 1.43 1.5

10 6.9 10.6 5.5 7.3 4.9

V=seat

S. Cerda´ et al. / Applied Acoustics 70 (2009) 97–109

Hall S2 is rectangular but its side corridor makes it asymmetrical with regard to the central axis of the stage, and hall S3 is rectangular with a dress circle. The interior panelling in both cases is wooden. The volumes are 2700 and 3266 m3 and the number of seats – 385 and 475 – provides an equivalent volume/audience ratio, which is 6.9 and 7 m3 . They are both used for similar purposes: conferences, congresses, and soloist musical concerts; chamber orchestras and choirs. Halls C2, T1, and T2 are fan-shaped with volumes between 3480 and 6287 m3 and number of seats between 509 and 967. The interior decoration and overlay in the Benaguasil Auditorium (smooth walls) differs from the Unio´ Musical and Banda Primitiva Theatres which approach proscene theatres. The Torrent Auditorium (C3) has an irregular hexagonal shape (fan-shaped + an inverted fan-shape) with an audience capacity of 606 people; the volume/audience ratio is 10.6 m3 . This hall is used for conferences, congresses, all types of concerts, opera, and dance. The Teatro Principal de Valencia theatre (Hall T3) was opened in 1832 and is in the style of Baroque theatres; it is horseshoe-shaped and has boxes on different floors; there are 1224 seats and the volume/audience ratio is 4.9 m3 . 1.2. Studied parameters The first step in this study was to choose the objective parameters and their calculations. The parameters studied, grouped according to main subjective sensations [19,20,7, 6,16] were:    

Energy parameters: G, C50, C80, Ts. Reverberation parameters: TRmid , EDTmid , BR, Br. Intelligibility parameters: STI, RASTI, %ALcons . Spatial parameters: IACCE , LFE , LFCE .

These parameters are associated with the main subjective qualities of the halls:  Transparency: with regard to the audition of music, transparency refers to the perception of separate tones in time and instruments played simultaneously.  Reverberation represents the degree of vivacity of the hall.  Intelligibility, this parameter is essential for verbal audition and quantifies verbal comprehension.  Space sensation quantifies the sensation of feeling enveloped by sound, giving the impression of a small hall, and being close to the source of sound.

operating at the same power level and located in an anechoic chamber. The equation is R1 2 p ðtÞdt ; dB ð1Þ G ¼ 10 log R 01 2 pA ðtÞdt 0 where pA ðtÞ is the free-field sound pressure level at a distance of 10 m [7]. We are used direct sound as reference. Comparison with properly calibrated measurements have demonstrated that this method will normally provide too high G values at low frequencies due to insufficient window length, fairly good G values at mid frequencies (500– 1000 Hz), and too low G values at high frequencies due to the influence of the immediate surroundings of the transducers [8]. We have worked with [7]: 1 Gmid ¼ ðG500 Hz þ G1 kHz Þ 2 The clarity factors, C50 and C80 are given by [9] Rx 2 p ðtÞdt 0 C x ¼ 10 log R 1 ; dB p2 ðtÞdt x

ð2Þ

ð3Þ

We worked with the averages given by [9] Hz þ 0:25  C 150kHz þ 0:35  C 250kHz C 50 ¼ 0:15  C 500 50

C 80

þ 0:25  C 450kHz 1 Hz ¼ ðC 500 þ C 180kHz þ C 280kHz Þ 3 80

Center time Ts is calculated as follows [32]: R1 2 tp_ ðtÞdt T s ¼ R01 2 p ðtÞdt 0

ð4Þ ð5Þ

ð6Þ

We have worked with center time at 1 kHz band 1.2.2. Reverberation parameters Reverberation times EDT, T30, were calculated as follow. EDT is the 60 dB decay time calculated by a line fit to the portion of the decay curve between 0 and 10 dB. T30 is the 60 dB decay time calculated by a line fit to the portion of the decay curve between 5 and 35 dB. We worked with mid values, the bass ratio (BR) and brilliance (Br) as follows [7]: 1 TRmid ¼ ðTR30500 Hz þ TR301 kHz Þ 2 1 EDTmid ¼ ðEDT500 Hz þ EDT1 kHz Þ 2 BR ¼ Br ¼

1.2.1. Energy parameters Strength factor G is a measure of the sound pressure level at a point in a hall, with an omni-directional source on stage, minus the sound pressure level that would be measured at a distance of 10 m from the same sound source

99

TR30125 Hz þ TR30250 Hz TR30500 Hz þ TR301 kHz

TR302 kHz þ TR304 kHz TR30500 Hz þ TR301 kHz

ð7Þ ð8Þ ð9Þ ð10Þ

1.2.3. Intelligibility parameters We have worked with STI and RASTI from the original versions [10] which are based on weighted sums of modulation transfer function (MTF) values. STI is calculated as

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the weighted sum of modulation transfer indices MTI, one for each octave frequency band from 125 Hz through 8 kHz (where each MTI value is derived from MTF values over 14 different modulation frequencies) taking into account auditory effects according to IEC 60268-16. The RASTI is calculated as the weighted sum of MTI’s over the 500 and 2000 Hz octave bands, where the MTI values are derived from MTF values over 4 and 5 different modulation frequencies respectively. %ALcons [11], were determined by the Farrell Becker empirical formula [10]: %ALcons ¼ 170:5045  e5:419ðSTIÞ

ð11Þ

1.2.4. Spatial parameters The interaural cross-correlation coefficient IACC, introduced by Schroeder [2] and Ando [6] as the maximum within the delay time interval s < 1 ms of the cross-correlation function IACCFð Þ between the pressures pL ðtÞ and pR ðtÞ at the left and right ear, respectively, of a real or dummy head, calculated over a time window t2  t1 R t2

pL ðtÞpR ðt þ sÞdt t IACCFðsÞ ¼ R t2 1 Rt 1=2 2 ð t1 pL ðtÞdt  t12 p2R ðtÞdtÞ

ð12Þ

Following Okano et al. [12], we worked with early IACC as 1 Hz IACCE3 ¼ ðIACC500 þ IACC1EkHz þ IACC2EkHz Þ E 3 The early lateral energy fraction is R td 2 p8 ðtÞdt LF ¼ Rtted p2 ðtÞdt 0

ð13Þ

ð14Þ

where p8 ðtÞ is the impulse response measured with a figureof-eight microphone with one of its nulls pointed towards the source. The time limits are td , the direct time, and te , the early time. LFE is normally used for music application, with direct time 5 ms and early time 80 ms [5]. We worked with average: 1 LFE4 ¼ ðLF125 Hz þ LF250 Hz þ LF500 Hz LF1 kHz Þ 4 And the early lateral energy fraction cosine is R td pðtÞ  pL ðtÞdt LFC ¼ te R td p2 ðtÞdt 0

2. Procedure and experimental equipment The experimental methodology was that set out by the ISO-3382 [21] and IEC 60268 [22]. The equipment was made up of a laptop PC with a professional sound card (Vxpocket v2). The two microphones used were G.R.A.S. Type 40 AK (sensitivity at 250 Hz 50 mV/Pa, frequency response (dB): 3.15 Hz–20 kHz, upper limit of dynamic range (3% Distortion): 164 dB, re. 20 Pa, lower limit dynamic range: 14 dB, re. 20 Pa); and their corresponding supply source was GRAS 12AA and preamplifiers G.R.A.S Type 26AK (frequency range: 2 Hz–20 kHz, noise: A-weight: <2.5 V). We used the multi pattern capacitor microphone AT4050/CM5 (frequency response: 20–20,000 Hz, sensitivity: 15.8 mV, polar patterns: cardioids, omni directional, figure-of-eight) and the corresponding phantom supply source to determine the spatiality parameters. Finally, the binaural measures were carried out with the Head acoustic HMS III.0 (transmission range: 3 Hz–20 kHz, 3 dB/+0.1 dB; dynamic range: typ. >118 dB, max SPL 145 dB), binaural head (HEAD Acoustics). The emission system was made up of power amplifier M-1000 (power output level RL = 4O: 520 W + 520 W) and the dodecahedral source was Dodecahedral loudspeakerDO12 (rated power 600 W, sound power > 120 dB, frequency range: 80 Hz–6.3 kHz, directivity: nearly spherical). We used the WinMLS program for measuring and analysis. This program gives the acoustical parameters of impulse response in accordance with the ISO 3382 norm and other recent parameters which were not included in the norm such as LG, LFC or strength (G) using direct sound as a reference. The impulse response of the halls was obtained by sinusoidal logarithmic sweep tests in view of the advantages this type of signal has over others. The ISO 3382 norm has been followed when working with the adequate signal/noise ratio. All measures were determined in unoccupied halls and the source was situated in the center of the stage. 3. Analysis of the results and discussion

ð15Þ 3.1. Statistical analysis

ð16Þ

where the impulse responses are the same as those for LF. This provides an approximation of a weighting of lateral reflections according to the cosine of the angle of incidence, which is thought to be better correlated with the subjective impression than the cosine-squared weighting of the LF [8]. We work with the average: 1 LFCE4 ¼ ðLFC125 Hz þ LFC250 Hz þ LFC500 Hz LFC1 kHz Þ 4 ð17Þ

If we follow Beranek’s proposal for establishing a system for classifying halls using acoustical parameters, the first step is to identify the statistically independent parameters. Numerous authors have already carried out this process and as a result, different schools use different groups of independent parameters. Here we propose two work designs: we feel the first is the traditional design, in which correlations between parameters are sought out and the mathematical models are established that have a relevant correlation index. The second design, which must include the first, promises to provide more acoustical information and is a data reduction process by means of factor analysis. All statistical analyses require a suitable number of data to

S. Cerda´ et al. / Applied Acoustics 70 (2009) 97–109

have statistical value. Thus we have determined numerous measurements in each hall, many more than the ISO 3382 establishes as the minimum. The statistical analysis was carried out using the SPSS 13.0 program [23]. The mean scores in each hall and for each parameter are presented in Table 2.

become hall parameters). On the other hand, if we obtain formulas that are independent of the hall, these ratios will be of general validity. This would mean that there would be a mathematical relation between the correlated variables. Only three intra-hall correlations appear and are shown in Table 4, and the only one of them that maintains an independent hall relation is LFE4 with LFCE4. Notice that this relation also appears between mean scores in spite of its being a spatiality parameter [26]. The following section deals with this relation.  Overall correlations: We include all the measurements for overall correlations. This is the process for determining independent parameters, or orthogonal parameters as termed by Beranek. Table 5 shows our results; the

3.1.1. Correlation analysis To establish correlations we considered the following possibilities:  Correlations between mean scores: With regard to the acoustical characteristics of a hall, mean scores of parameters are used. Technical descriptions of the halls provide this type of overall information, or mean scores by frequencies if greater detail is required. This type of analysis presents a problem, because the mean scores cannot be represented as there are parameters that vary greatly in one hall [24,25]. Our study was carried out on nine halls and the results we calculated are shown in Table 3. We would like to point out that the three spatiality parameters are interrelated, as are the intelligibility and clarity parameters.  Intra-hall correlations: These kinds of correlations would make it possible to obtain calculation formulas from another known parameter for the same hall. If these expressions are dependent on the hall, they provide limited interest (unless the emerging coefficients have a relevant physical significance, in which case they would

101

Table 4 Intra-hall correlations between the mean acoustical parameters in the halls that were analysed Id.

LFE4  LFCE4

C 50avg  C 80avg

C 80avg  T s1 kHz

S1 S2 S3 C1 C2 C3 T1 T2 T3

0.95 0.95 0.78 0.94 0.85 0.93 0.88 0.95 0.90

0.91 0.90 0.90 0.82 0.82 0.79 0.72 0.82 0.77

0.94 0.93 0.94 0.87 0.84 0.92 0.58 0.51 0.73

Table 2 Mean scores of the acoustical parameters in all the halls Id.

RTmid

EDTmid

C 50avg

C 80avg

T s1 kHz

STI

RASTI

ALcons

BR

Br

LFE4

LFCE4

Gmid

IACCE3

C1 C2 C3 S1 S2 S3 T1 T2 T3

1.79 2.25 1.87 0.68 1.30 1.51 1.35 1.43 1.50

1.63 1.85 1.87 0.56 1.12 1.22 1.10 1.35 1.51

0.45 1.25 0.97 3.61 0.32 0.68 1.65 0.90 2.55

1.81 1.02 0.92 7.68 2.75 2.18 4.44 3.24 3.86

103 123 106 44 81 87 64 73 64

0.55 0.55 0.50 0.70 0.59 0.58 0.61 0.58 0.62

0.53 0.50 0.50 0.70 0.57 0.55 0.60 0.57 0.60

8.9 8.9 11.3 3.8 7 7.4 6.2 7.4 6.2

1.08 0.78 1.22 0.79 0.85 0.76 1.33 1.21 1.06

0.91 0.74 0.90 1.47 1.06 0.88 0.84 0.77 0.85

0.19 0.18 0.19 0.25 0.24 0.22 0.16 0.17 0.14

0.25 0.25 0.25 0.29 0.29 0.27 0.23 0.25 0.19

9.72 9.13 4.91 10.89 9.51 8.19 3.60 4.78 2.33

0.41 0.33 0.32 0.29 0.33 0.28 0.35 0.32 0.49

Table 3 Statistically significant correlations between the mean acoustical parameters in all the halls

LFCE4 Gmid T s1 kHz STI RASTI C 80avg ALcons BR Br RTmid EDTmid IACCE3

LFE4

LFCE4

0.94 0.82

0.81

Gmid

C 50avg

0.92 0.88 0.76 0.93 0.84

T s1 kHz

STI

RASTI

C 80avg

ALcons

Br

RTmid

0.84 0.89

0.79 0.75

0.95

0.87 0.92 0.86

0.96 0.98

0.91

0.93 0.84

0.76 0.88 0.90

0.77 0.91 0.88

0.75 0.71 0.79 0.82

0.80

102

S. Cerda´ et al. / Applied Acoustics 70 (2009) 97–109

Table 5 Statistically significant overall correlations between mean acoustical parameters in all the halls: correlations with r value (Pearson’s coefficient) higher in module than 0.6 are expressed in bold characters

two shaded areas correspond to the spatiality and speech parameters. Absolute value correlations of over 0.6 are given in bold characters. Parameters that show a correlation of under 0.6 of absolute value are deemed independent parameters. Correlations that appear between mean scores are marked with an asterisk. As can be seen, the overall correlations coincide in most cases with the correlations between mean scores. In general, they are higher between mean scores.

and the other that of music. And the relation between RTmid and EDTmid . On the other hand, for the diffuse field mid is achieved in ms [27]. The the linear relation T s1 kHz ¼ RT 13:8 equations we obtained for the halls are: (1) C 50avg ¼ 2:2 þ 0:91  C 80avg (0.86), (2) EDTmid ¼ 0:886  RT mid (0.98), mid (0.99). (3) T s1 kHz ¼ RT 18:9 4. Factor analysis

3.1.2. Spatiality parameters This section analyses spatiality parameters. The three parameters analysed, LFE4 , LFCE4 and IACCE3 show correlations. Barron proposes that LFC ¼ 1:5  LF, in a diffuse field, and that there are theoretical reasons to expect a LFC ¼ 1k ð1  IACCÞ relation [14]. We obtained: (1) LFCE4 ¼ 1:24  LFE4 with a correlation of 0.98. (2) LFCE4 ¼ 0:377  0:371  IACCE3 . with a correlation of 0.72. This would mean a k = 2.7. In diffuse field, k ¼ 3 [27]. The best relation is corroborated between LFCE4 and IACCE3, both related to the subjective response [14,7]. 3.2. Speech parameters It is known that intelligibility in halls is characterised by the noise signal and reverberation time ratio [28]. Although there are several parameters that study intelligibility in a hall, all of them are related among themselves and with reverberation time by analytical formulas, valid for small halls, and as a first approach for large halls [29]. This has been corroborated by our results. It may be interesting to note the existing correlation between C 50avg and C 80avg , as both parameters measure clarity, but one that of speech

The main applications of factor analysis are the reduction in the number of variables and the detection of a structure in the relation between variables. The factor analysis we carried out consists of extracting the principal components, by analyzing the correlations matrix, for values over one. We completed the process by rotating the factors by means of the varimax procedure. Thus we obtained three Table 6 Grouping of the parameters studied in three factors in accordance with the reduction of variables method (factor analysis) with varimax rotation

LFE4 LFCE4 Gmid RTmid EDTmid C 50avg C 80avg T s1 kHz IACCE3 BR Br STI RASTI ALcons

1

2

3

0.035 0.107 0.010 0.861 0.752 0.857 0.967 0.926 0.201 0.084 0.584 0.952 0.971 0.922

0.834 0.878 0.470 0.239 0.407 0.336 0.024 0.137 0.755 0.522 0.517 0.008 0.025 0.031

0.169 0.307 0.768 0.180 0.230 0.086 0.009 0.178 0.385 0.695 0.236 0.125 0.029 0.084

The parameters integrated in each factor and with their correlation coefficient are marked in bold characters.

S. Cerda´ et al. / Applied Acoustics 70 (2009) 97–109

factors that explain 83% of the total variance. Table 6 shows the components we obtained in accordance with the parameters studied and their correlations. (1) Factor 1: Reverberation–intelligibility–clarity factor. This factor encompasses all intelligibility and clarity parameters. In the previous section, we commented that they were all related among themselves. Therefore, it is not surprising that a factor that encom-

103

passes them should emerge. As we previously drew attention to the dependence of these parameters with RT, we can see what the relation is between Factor 1 and RTmid . Factor 1 ¼ 3:94  2:57  RTmid ð0:93Þ

ð18Þ

Fig. 1 shows the mean scores of Factor 1 and RTmid for each hall: they are arranged according to the score of Factor 1, from high to low. This result suggests that Factor 1 is

2.50 2.00

Factor 1 and RT (s)

1.50 1.00 0.50

0.00 S1

T1

T3

S2

T2

S3

C1

C3

C2

-0.50 -1.00

-1.50 -2.00

Factor 1

RTmid

Fig. 1. Mean RTmid and Factor 1 scores associated with intelligibility obtained with the factor method, for each of the halls. The order of the halls is that of decreasing values of Factor 1.

id

3.00000

C1 C2 C3 S1

2.00000

S2 S3 T1 T2

FACTOR 1

1.00000

T3 Fit line for Total

0.00000

-1.00000

-2.00000 R Sq Linear = 0.774

-3.00000

0.50

1.00

1.50

2.00

2.50

RTmid Fig. 2. Linear regression between the values of Factor 1 vs RTmid (r ¼ 0:93).

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104

essentially determined by the reverberation time in the hall and this result is confirmed by the statistical analysis of linear regression with all the parameters, as the most significant coefficient is that of RTmid and the excellent correlation between Factor 1 and RTmid shown in Fig. 2. (2) Factor 2: Spatiality–subjectivity factor. The spatiality parameters appear grouped in this factor thus LFE4,

LFCE4, IACCE3. In order to analyze the weight of each one of the spatiality parameters in Factor 2 we have made a linear regression between this Factor and the three previous parameters. The result is Factor 2 ¼ 0:77  2:92  LFE4  6:28  LFCE4 þ 3:73  IACCE3 ð0:98Þ:

ð19Þ

1.5

Factor 2 and BQ I

1

0.5

0 T3

C2

C1

T1

T2

C3

S3

S2

S1

-0.5

-1

Factor 2

BQI

Fig. 3. Mean BQI and Factor 2 values associated with spatiality obtained with the factor method for each of the halls. The order of the halls is that of decreasing values of Factor 2.

id

4.00000

C1 C2 C3 S1 S2

2.00000

S3 T1 T2

FACTOR 2

T3 Fit line for Total 0.00000

-2.00000

-4.00000

R Sq Linear = 0.739

0.20

0.40

0.60

0.80

BQI Fig. 4. Linear regression between values of Factor 2 vs BQI (r ¼ 0:86).

S. Cerda´ et al. / Applied Acoustics 70 (2009) 97–109

This equation can be reduced to a simpler expression if we use the relations we encountered in the section on correlations between spatiality parameters: Factor 2 ¼ 4:45  6:93  BQI

ð20Þ

105

BQI being Beranek’s binaural clarity index [7]. Fig. 3 shows the mean values of Factor 2 and BQI for each hall. They are arranged according to Factor 2 values, from high to low.This result suggests that Factor 2 corresponds essen-

id

3.00000

C1 C2 C3 S1

2.00000

S2 S3 T1 T2

FACTOR 1 (s)

1.00000

T3

0.00000

-1.00000

-2.00000

-3.00000

0.00

10.00

20.00

30.00

dist (m) Fig. 5. Factor 1 vs distance.

id

4.00000

C1 C2 C3 S1 S2

2.00000

S3 T1 T2

FACTOR 2

T3 0.00000

-2.00000

-4.00000

0.00

10.00

20.00

dist (m) Fig. 6. Factor 2 vs distance.

30.00

S. Cerda´ et al. / Applied Acoustics 70 (2009) 97–109

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tially to the BQI parameter (as shows Fig. 4). Therefore, it is a spatiality–subjectivity factor, in accordance with Beranek’s interpretation of BQI [7]. (3) Factor 3: Strength factor. Parameters Gmid and BR are involved in this Factor. The first measures the strength or amplification of the hall. This parameter particularly depends on the position [14,7,30]. BR

measures the strength in low frequencies and it is near constant in the halls. These parameters are a reference of the quality of a hall, as intuitively one wants a concert hall that amplifies sound and is rich in basses frequencies [7]. However, none of these parameters correlates with the subjective ranking obtained by Beranek [7], as spatiality parameters do. Nonethe-

id

3.00000

C1 C2 C3 S1 S2

2.00000

S3 T1 T2 T3

FACTOR 3

1.00000

0.00000

-1.00000

-2.00000

0.00

10.00

20.00

30.00

dist (m) Fig. 7. Factor 3 vs distance.

95% IC REGR factor score 1 for analysis 1

3

2

1

0

-1

-2 C1

C2

C3

S1

S2

S3

T1

T2

id

Fig. 8. Confidence interval of 95% of the mean for Factor 1.

T3

S. Cerda´ et al. / Applied Acoustics 70 (2009) 97–109

less, as they appear in the same factor it must be supposed that they have something in common and that they may be related to a general acoustical quality in halls. The regression lines between Factor 3 and Gmid and BR are shown in the following equation: Factor 3 ¼ 1:19 þ 0:17  Gmid  2:21  BR ð0:98Þ

would receive the best grading according to the scoring criterion of Ando–Beranek [6,7]. 4.0.1. Orthogonality Following Beranek’s hall analysis process [7]:

ð21Þ

As it shows, the dependence of Factor 3 on Gmid and BR is opposite, BR being weightier and negative which provides the richness in basses/low frequencies. Taking into account that the mean Gmid values are almost an order of magnitude greater than those of BR, Factor 3 could be interpreted as a parameter that seeks a balance of strength in a hall, neither excessive amplification nor excessive intensification of basses frequencies. Analysis of the behaviour of the three factors with distance proves interesting (Figs. 5–7). Factor 1, which is in essence RTmid , shows little variability (Fig. 8) and no dependence on distance. Factor 2, a spatiality factor, shows a slight dependence on position and thus on distance, and consequently greater variability (Fig. 9). On the other hand, Factor 3 (Fig. 7) shows a clear decrease in dependence on distance. The shape of dependence of Factor 3 on distance is similar to Gmid [13] as BR is near constant. The optimal scores for Gmid and BR are Gmid 2 ½3:5; 5 and BR 2 ½1:1; 1:45[7]. These scores provide an optimal interval of Factor 3 [1.41; 0.39]. Table 7 shows the Gmid , BR and Factor 3 scores, Factor 3 is arranged from lower to higher scores. As can be seen, the first four halls, the T1, T2, and T3 theatres and the C3 hall, are within the optimal interval. These halls also correspond to those that

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1. Identify the orthogonal acoustical parameters that are necessary to judge the acoustical quality of a hall. 2. Choose the optimal scores for each hall. The factor analysis gives: 1. The group of orthogonal parameters comes from RTmid , QDI, Factor 3 (Gmid , BR) as shown in Table 8.

Table 7 Gmid , BR, and ‘‘strength” factor scores for the halls Id.

Gmid

BR

Factor 3

T1 T2 C3 T3 C1 S3 S2 C2 S1

3.60 4.78 4.91 2.33 9.72 8.19 9.51 9.13 10.89

1.33 1.21 1.22 1.06 1.08 0.76 0.85 0.78 0.79

1.18 0.79 0.52 0.40 0.51 0.76 0.83 1.01 1.41

The scores in bold characters of F3 are in the optimal interval of this factor.

95% IC REGR factor score 2 for analysis 1

3.00000

2.00000

1.00000

0.00000

-1.00000

-2.00000

-3.00000 C1

C2

C3

S1

S2

S3

T1

T2

id Fig. 9. Confidence interval of 95% of the mean for Factor 2.

T3

S. Cerda´ et al. / Applied Acoustics 70 (2009) 97–109

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Table 8 Correlations between Factors and RTmid , BQI, Gmid and BR

RTmid BQI Gmid BR Factor 1 Factor 2 Factor 3

RTmid

BQI

Gmid

– 0.01 0.33 0.14 0.88 0.23 0.04

– 0.06 0.15 0.14 0.86 0.00

– 0.59 0.01 0.07 0.90

BR

Factor 1 Factor 2 Factor 3

parameter that seeks the balance of strength in a hall, neither excessive amplification nor excessive intensification of basses frequencies. Acknowledgements

– 0.08 – 0.16 0.00 0.86 0.00

– 0.00

–

2. The optimal RTmid scores (depending on the kind of music) and of the QDI are studied in detail in [7] and the optimal values of Factor 3 are 1:42 < Factor 3 < 0:39

ð22Þ

5. Conclusions This study proposes an orthogonal group of parameters for characterising halls independently of the purpose for which they were designed. The group of orthogonal parameters comes from RTmid , QDI, and a linear combination of (Gmid , BR), which we have termed strength factor. This group of parameters is compatible with the orthogonal groups mentioned in the literature [31]. This means our work hypothesis is corroborated: the properties that measure acoustical parameters must be physical properties of halls, like enclosed three-dimensional areas. The choice of halls designed for different purposes should not present a problem when coming to conclusions from the acoustical parameters analysed. Therefore, determining many measures in each hall, at different positions, will guarantee that we will obtain the relevant acoustical information by means of statistical analysis. This hypothesis and work methodology has also let us obtain correlations between parameters that confirm those already known [7,31] and some hypotheses [15]. Although some authors criticise the use of mean scores of parameters in hall analysis [24,25], we found that the main overall correlations coincide with the correlations of mean scores; the latter present greater correlation coefficients. We found that very few intra-hall correlations emerge, the most noteworthy being the relation between LF and LFC which is independent of the hall. The existence of relations that are repeated between halls but that do not provide the same equation in each one of them makes us wonder if regression coefficients can contribute relevant hall information. The optimal intervals of these parameters were obtained from the measurements determined in a considerable number of halls provided by [7]. The optimal scores of the Strength factor for the design of music halls are in the [1.42, 0.39] interval, determined by the optimal scores of Gmid and BR proposed in [7]. Taking into account that the mean scores of Gmid are almost one order of magnitude greater than those of BR, Factor 3 could be interpreted as a

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