# Acoustics in Baroque Catholic Church Spaces

^{1}

^{2}

^{*}

*Acoustics*

**2024**,

*6*(4), 911-932; https://doi.org/10.3390/acoustics6040051 (registering DOI)

## Abstract

**:**

## 1. Introduction

^{3}for the sample of Baroque churches, is smaller than in previous architectural styles. When comparing the reverberation time against the architectural style, a decrease can be observed in relation to its predecessors, as reflected by Carvalho in the Portuguese churches.

^{2}) between the acoustic parameters of reverberation time and musical clarity and the geometrical parameters of the churches (maximum height and volume), it was ascertained that in all cases r

^{2}values were lower and non-significant in the churches of Peru than in Portugal. The investigation concludes that this could be due mainly to the different construction procedures and materials typically used in churches in Peru (Figure 9).

## 2. Methods

#### 2.1. Selection of the Study Sample

^{3}are studied, so that churches with higher volumes, such as Saint Peter’s Basilica in Rome, are not included in the study as they display different acoustic behaviours. Based on the criteria detailed, a total of 66 Catholic churches were selected in 8 countries—Spain, Portugal, Italy, Germany, Poland, Peru, Argentina and Brazil.

#### 2.2. Acoustic Parameters Analysed

_{30}values and the perceived reverberation from the early decay time (EDT), which is more closely linked to the subjective sensation, and the perceived sound clarity, through the musical clarity (C

_{80}), the definition (D

_{50}) and the central time (T

_{S}), considering the single-number frequency averaged values, calculated according to ISO 3382-1:2009 [39], as shown in table A.1 of Annex A. Other researchers have used a similar methodology to analyse the acoustics in churches [8,11,12,16,17,32,33,34,40,41,42], although mixing churches of different architectural styles, without the homogenising factor of being Catholic churches and from the Baroque period. Desarnaulds and Carvalho studied 63 baroque churches but mixed Catholic, Protestant and Orthodox confessions. Only Meyer [43] found a logarithmic correlation between reverberation time and volume. Cirillo and Martellotta [44] also found a correlation between reverberation time and volume for Romanesque churches and significant linear correlations (r

^{2}> 0.8) between the mean values of the parameters T

_{30}, D

_{50}, C

_{80}and T

_{S}, but not with the geometrical parameters. They also found non-linear correlations between C

_{80}, D

_{50}and T

_{S}as a function of reverberation time. A lack of published data made it impossible to analyse the parameters per receiver, while the subjective level of sound, apparent source width and listener envelopment could not be analysed either. Cirillo and Martellotta [44] presented a more detailed analysis by presenting the values of the acoustic parameters measured at each reception point for Romanesque. This analysis of the individual position results showed that EDT, G, C

_{80}, D

_{50}, T

_{S}and RASTI measured in each church proved to be related to source–receiver distance. The use of the single-number mean value could introduce considerable dispersion in the parameter values depending on the different source–receiver distances involved, even though, as mentioned above, the source position is common in all cases and the receivers are always in the listening area. To mitigate this possible dispersion, we will also analyse, where possible, depending on the number of churches and available acoustic parameters, the single-number frequency averaged values by typology, where the drop in acoustic parameters in relation to distance is common to all of the churches in the group [45,46]. In addition, more homogeneous samples will be analysed within each subgroup where possible (Latin cross and single-nave), based on geometrical and acoustic data, in order to obtain better approximations when determining the acoustic dependencies of the selected sample.

## 3. Results

#### 3.1. Typological Analysis of the Sample

_{30m}and EDT

_{m}) and the clarity based on musical clarity (C

_{80m}) (see Table 3). The 25th percentile is shown at the bottom of the box, followed by the 50th percentile, and, finally, the 75th percentile. The black line inside the graph marks the median, and the whiskers (error bars) below and above the box mark the 10th and 90th percentiles. The points at the ends of the graphs are the 5th and 95th percentiles.

_{30}and EDT, is less homogeneous for the Latin cross plan, with a better distribution observed for the Basilica typology. However, the highest reverberation times are observed in the basilica churches, while the single-nave churches, with the lowest volume range, have the lowest reverberation times. Furthermore, the reverberation results are similar for churches with central, Greek cross and Latin cross floor plans, although there is a greater variability of results for churches with Latin cross floor plans. The graph corresponding to T

_{30}includes the average values of the optimum reverberation time by typology, with consideration given to the optimum values defined by Beranek [48] for religious music (T

_{opt}= 0.55 log (V) − 0.14). In all cases, the median obtained for all typologies is found to be significantly above the values defined by Beranek as optimal. In the case of the behaviour of the sample for musical clarity (C

_{80m}), especially for the Latin cross and single-nave typologies, greater uniformity is observed in the results of the acoustics measurements. The single, central and Greek nave models display the best results, while the basilica plan churches provide the worst results. A comparison of the results obtained for musical clarity with the scale established by Marshall [49], which evaluates the suitability of music according to the value of C

_{80}(3), indicates that churches with a central plan, Greek cross and single nave are generally conducive to organ music.

#### 3.2. Statistical Correlation with Geometrical Variables

^{2}, excluding correlations where R

^{2}< 0.5. The significant p values have also been calculated.

_{30}increases while the volume decreases, this should be discarded.

_{30}presented a weak correlation (R

^{2}) as a function of the S and V/S or the L and V/S (0.58, 0.50). As expected, the reverberation increases as the S and V/S or the L and V/S increase.

#### 3.3. Statistical Correlation Between Acoustic Parameters

_{30}-EDT (R

^{2}= 0.73) and T

_{S}(ms)-EDT (R

^{2}= 0.60), as shown in (Equations (1) and (2)):

^{2}≥ 0.5) and better results are obtained for the individual parameters.

_{30}can be obtained from the rest of the parameters in three of the six combinations possible, EDT in four, D

_{50}in three, T

_{S}in one and C

_{80}in no single combination. All of these are of the surface type described in Table 4.

_{S}appears as a common variable for determining the rest of the parameters. This parameter, closely related to the fine structure of the energy, presents a great spatial variability. The importance of the chosen variables should also be noted, since in the estimation of T

_{30}and EDT, the three parameters involved are the same, albeit with significant differences in the value of R

^{2}.

#### 3.4. Statistical Correlation with Geometrical Variables by Typology

^{2}≥ 0.65 and exceeding a significant p level are presented.

#### 3.4.1. Latin Cross Typology (Geometrical Variables)

^{3}, (II) 10,000 m

^{3}≤ V ≤ 20,000 m

^{3}and (III) V > 20,000 m

^{3}. Only correlations with R

^{2}≥ 0.75 exceeding a significant p level are presented.

_{30}and EDT only in interval (II) (seven cases). As expected, the increase in variables increases functions. The best correlations, the geometrical variable, the coefficients of the equation and their determination coefficient are all presented in Table 7.

#### 3.4.2. Basilica Typology

_{30}and T

_{S}. The best correlations, surface type, coefficients of the equation and their determination coefficient are presented in Table 8.

#### 3.5. Statistical Correlation Between Acoustic Parameters by Typology

^{2}≥ 0.65 are presented.

#### 3.5.1. Latin Cross Typology (Acoustics Parameters)

_{30}-EDT and EDT-T

_{30}). The correlations between two individual acoustic parameters, equation type, the coefficients of the equation and their coefficient of determination are presented in Table 9.

_{30}and EDT (Table 10).

^{3}, (II) 10,000 m

^{3}≤ V ≤ 20,000 m

^{3}and (III) V > 20,000 m

^{3}. Only correlations with R

^{2}≥ 0.75 exceeding a significant p level are presented.

_{30}and EDT combinations and in interval (II) (five combinations for T

_{30}, EDT, D

_{50}and T

_{S}, none for C

_{80}) (Table 11).

#### 3.5.2. Single-Nave Typology

_{50}correlation obtained with a single variable.

^{3}and (II) V ≥ 2500 m

^{3}. Only correlations with R

^{2}≥ 0.75 exceeding a significant p level are presented. For brevity, when analysing dependencies between two or three acoustic parameters, only the best correlation of the two or nine possible combinations is presented. The best correlations between two acoustic parameters, surface type, coefficients of the equation and their determination coefficient are presented in Table 14.

## 4. Discussion

^{2}> 0.65 exceeding the significant p level are analysed in Latin cross and basilica typologies. Overall, the dependence of acoustic parameters on geometric parameters remains low. The study of more homogeneous samples finds some dependencies with parameters related to reverberation in Latin cross churches with a volume of between 10,000 and 20,000 m

^{3}(R

^{2}> 0.75) In general, the inclusion of two geometrical parameters does not significantly improve (or worsen) the correlations obtained with a single variable.

_{80}(Equations (1) and (2) and Table 5). A typological analysis was carried out on Latin cross and single-nave typologies and the results are presented in Table 9, Table 10, Table 12 and Table 13. Again, the inclusion of two acoustic parameters as variables does not improve the correlations obtained with a single variable, nor does it yield any correlations. The study of a more homogeneous sample, in volume intervals, for Latin cross and single-nave churches found high 2D correlations, as shown in Table 11 and Table 14. Again, no significant 3D correlations are found in these intervals.

## 5. Conclusions

_{30m}parameter, and the initial reverberation (EDT

_{m}) and the clarity of the sound was analysed using musical clarity (C

_{80m}), definition (D

_{50m}) and central time (T

_{sm}).

_{30m}) with an average value of 5.17 s, while the single-nave churches are at the opposite end of the scale with an average volume more than seven times lower and an average T

_{30m}of 2.81 s. The proposal made during the Council of Trent, based on the Latin cross plan model developed from the Il Gesú model in Rome, or central plans, resolved as central or Greek cross plans, have similar average reverberation time values (4.39 Latin cross; 4.10 s central; and 3.79 s Greek cross), even though the average volume of the Latin cross churches (20,670 m

^{3}) is practically double the average volume of the central churches (10,983 m

^{3}) or Greek cross churches (11,308 m

^{3}). For the Latin cross plans, corresponding to a more diverse sample, a higher variability of results is also observed. However, the use of central floor plans (central and Greek cross) in smaller churches during the Baroque period improves musical clarity compared to the Latin cross typology, with average values for C

_{80m}of −3.85 and −4.83 dB, respectively, compared to −7.69 dB for Latin cross churches. The central configurations (central and Greek cross), with volumes three times higher than the single-nave floors, have similar musical clarity, with an average value of C

_{80m}for single-nave churches of −3.70 dB. Comparing the sample means for musical clarity in the two highest volume typologies (Latin cross (−7.79 dB) and basilica (−7.73 dB)), the results are similar. In general, the central plan and Greek cross configurations are those with the most suitable average C

_{80m}values when related to the average volume of the sample of each typology.

^{2}≥ 0.5 exceeding the significant p level) was found between the values of the acoustic parameters and one of the geometrical variables for the whole sample due to the different typologies studied. The same can be said when the study was extended to two geometrical variables. When the linear and non-linear regressions studied were obtained between a pair of acoustic parameters or one acoustic parameter with another two of those studied, correlations with R

^{2}≥ 0.61 are obtained for all the acoustic parameters with the exception of C

_{80m}.

^{2}≥ 0.65 exceeding the significant p level are presented.

_{30}and EDT) are found. In the case of more homogeneous samples, churches with volumes of between 10.000 and 20.000 cubic metres, 2D correlations are found with acoustic variables for all the acoustic parameters with the exception of C

_{80m}(R

^{2}≥ 0.75).

_{30}.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**(

**a**) Floor plan of the church of Santa Maria in Campitelli. (

**b**) Floor plan of the church of San Andrea al Quirinale (Rome).

**Figure 2.**Reverberation times by frequency. Obermarchtal (OB), Ottobeuren (OM), Ochsenhausen (OH), Rot an der Rot (RT) and Ebersmunster (EM) [7]. (

**a**) Obermachtal Church [Mythos Schwäbische Alb]. (

**b**) Ottobeuren Church [delso.photo, License CC-BY-SA].

**Figure 3.**Carvalho. Measurements of RASTI speech transmission index [9]. (

**a**) RASTI, with standard error for different architectural styles. (

**b**) RASTI index based on distance for different periods. Regression lines. Both graphs show the Baroque period in red.

**Figure 4.**Martellotta. Graphs showing EDT at mid frequencies (500–1000 Hz) [12]. (

**a**) Basilicas of Santa Maria Maggiore (SMM) and Saint Paul Outside the Walls (SPX). (

**b**) Basilicas of Saint John Lateran (SJL) and Saint Peter’s (SPB).

**Figure 5.**Caniato. Church of San Lorenzo in Turin [18]. (

**a**) Floor plan and positions of sources (S) and receivers (1 to 16) of the measurements. (

**b**) Comparison of average reverberation time for source S1 and the values obtained for each receiver.

**Figure 6.**Martellotta. Crypt of Cádiz Cathedral [23]. (

**a**) Floor plan of the crypt. Location of sources (S) and receivers (R). (

**b**) Bayesian analysis. Source S1 Receiver R2. (

**c**) Schema of flutter echoes. Source S2. Receiver S11.

**Figure 7.**Alberdi. Church of San Luis de los Franceses [14]. Cross section. Directional intensity maps. (

**a**) 10 ms after direct sound (

**b**) 20 to 120 ms.

**Figure 8.**Queiroz-de-Sant’Ana. Igreja Nossa Senhora do Rosário de São Benedito [31]. Floor plan showing position of sources (S1 and S2) and receivers (1 to 8). Graph showing reverberation time T30 for source position S1.

**Figure 9.**Jiménez-Dianderas and Carvalho [34]. Baroque churches. Church of San Francisco, Arequipa (Peru).

**Figure 11.**Box diagrams for different acoustic single-number parameters (T

_{30m}, EDT, C

_{80m}) and typologies. The 5th (down) and 95th (up) dot percentile. The 10th and 90th error bars. Box with 25th, 50th, 75th percentile lines and median bold line. Optimal reverberation time (T

_{opt}) by Beranek.

Country (Cases) | Researcher (Churches) | Churches |
---|---|---|

Spain (6) | Segura et al. (1) | St. Jaume (1550–1582 *, B, V) (Valencia) |

Planells et al. (1) | Santa María de Elche (1672–1784, LC, B) (Murcia) | |

Alberdi et al. (3) | Santa María Magdalena (1694–1709, LC, B), San Luis de los Franceses (1699–1731, C, D), San Telmo (1721–1724, SN, V) (Sevilla) | |

Alvarez-Morales et al. (1) | Cádiz Cathedral (1722–1838, LC, V) (Cádiz) | |

Portugal (8) | Carvalho (8) | Estrela (1779–1790, LC, V), St Roque (1590–1619, LC, V) (Lisboa). Mosterio de Bustelo (1633, LC, V) (Penafiel), Dos Clérigos (1732, C, D) (Oporto), Misericordia (1554–1574, SN, V) (Évora), Monastery Tibaes (17th–18th c, LC, V) (Braga), Matriz (16th–17th c., LC, W) (Vila do Bispo), St. Lourenzo (17th c., SN, V) (Azeitao) |

Italy (15) | Cirillo et al. (6) | Santi Luce e Martina (1634–1664, GC, D) Santa Agnese in Agone (1652–1657, GC, D), Santa Croce in Gerusalem (1740–1758 *, B, V) (Rome), San Lorenzo, Superga (1634–1680, C, D) (Turín), San Martino (1747–1775, LC, V) (Martina Franca) |

Magrini et al. (4) | N.S. Consolazione (1684–1706, B, V) (S.M. Vigne (1642 *, B, V), Annunziata (1615 *, B, V), St Cosma and Damiano (1684 *, B, V), (Genoa) | |

Ricciardi (3) | St. Luca (1634–1664, GC, D), St Croce and St. Camillo (1667–1695, GC, D), Gesú (16th c., GC, D) (Genoa) | |

Martellotta (2) | St Giovanni in Laterano (1646–1649 *, B, W), St. Maria Maggiore (1740–1758 *, B, W) (Roma) | |

Germany (2) | Lottermoser (2) | Monastery of Weingarten (1715–1724, LC, V), Monastery of Ottobeauren (1748, LC, V) |

Poland (2) | Engel et al. (2) | Reformati Fathers (18th c., SN, V) (Wieliczka), St. Peter and St. Paul (1597–1619, LC, V) (Krakow) |

Peru (28) | Jimenez Dianderas (28) | Nuestra Señora de la Merced (1647, LC, V), Convento San Francisco (16th c., LC, V), Santa Rosa de Santa María (1747, SN, V), Santo Domingo (1677–1680, LC, V), La Compañía de Jesús (1654–1698, LC, V) (Arequipa), Compañía de Jesús (1645, LC, V), Pampa San Agustín (16th c., SN, W) San Juan de Dios (1627, SN, V), Santa Ana (1748, LC, V), Santa María Magdalena (1558, LC, W), Santo Domingo (1548–1649, LC, V) (Ayacucho), Belén (1650, SN, V), Cathedral (1559–1659, B, V) Compañía de Jesús (1654–1671, LC, V), Merced (1651–1659, B, V), San Antonio Abad (1678–1699, SN, W), San Francisco (1651, B, V), San Pedro (1688–1699, LC, V), San Sebastián (17th c., SN, W), Santa Catalina (1651–1654, SN, V), Santa Teresa (1673–1676, SN, V), Santiago (17th c., SN, W), Santo Domingo (1680, B, V), Virgen de la Almudena (1698, LC, V), (Cusco), Carmen de la Legua (17th c., SN, V) San Francisco El Grande (17th c., B, V) (Lima), San Francisco de Asís (1677–1696, B, V), San Pedro (16th c., LC, V) (Puno) |

Argentina (4) | Abadía Succi, L. (4) | Santa Catalina (1754, LC, V), Alta Gracia (1767, LC, V), La Compañía de Jesús (1645, LC, W) (Córdoba), San Isidro Labrador (1729, LC, V) (Jesús María) |

Brazil (1) | Queiroz (1) | Nossa Senhora do Rosário de São Benedito (18th c., LC, V) (Paraty) |

Latin Cross (LC) | Central (C) | Greek Cross (GC) | Single Nave (SN) | Basilical (B) | |
---|---|---|---|---|---|

No. churches | 27 | 4 | 5 | 16 | 14 |

Length (m) | 25–89 | 30–59 | 25–66 | 21–52 (83) | 23–126 |

Surface area (m^{2}) | 321–2970 | 273–650 | 140–1580 | 174–597 (1870) | 241–5800 |

Vol. (103 m^{3}) | 1.9–130 | 4.8–22 | 2.4–25.6 | 0.7–8.5 | 9.1–120 |

(LC) | (C) | (GC) | (SN) | (B) | Total | |
---|---|---|---|---|---|---|

T_{30m} | 27 | 4 | 5 | 16 | 14 | 66 |

EDT_{m} | 20 | 3 | 5 | 12 | 10 | 50 |

C_{80m} | 24 | 4 | 5 | 16 | 14 | 63 |

D_{50m} | 22 | 2 | 3 | 15 | 10 | 52 |

T_{Sm} | 20 | 2 | 3 | 14 | 7 | 46 |

Surface Type | Equation |
---|---|

Plane | $z={z}_{0}+ax+by$ |

Paraboloid | $z={z}_{0}+ax+by+c{x}^{2}+d{y}^{2}$ |

Gaussian | $z=a{e}^{-0.5\left[{\left({\displaystyle \frac{x-{x}_{0}}{b}}\right)}^{2}+{\left({\displaystyle \frac{y-{y}_{0}}{c}}\right)}^{2}\right]}$ |

Lorentzian | $z={\displaystyle \frac{a}{\left[1+{\left(\frac{x-{x}_{0}}{b}\right)}^{2}\right]\left[1+{\left(\frac{y-{y}_{0}}{c}\right)}^{2}\right]}}$ |

**Table 5.**Best correlations between one acoustic parameter and another two parameters, the coefficients of the surface type and their determination coefficient.

Parameter | Surface Type | Variable | Coefficients |
---|---|---|---|

T_{30} | Paraboloid | EDT, T_{S} | ${z}_{0}=0.9387,a=0.7267,b=-0.0011,$ $c=0.0214,d=2.2279\xb7{10}^{-7},{R}^{2}=0.78$ |

EDT | Gaussian | T_{30}, T_{S} | ${z}_{0}=11.3568,{y}_{0}=722.9645,a=9.1560$ $b=6.7779,c=533.7480,{R}^{2}=0.87$ |

D_{50} | Paraboloid | C_{80}, T_{S} | ${z}_{0}=0.3533,a=0.0179,b=-0.0005,$ $c=0.0005,d=4.4410\xb7{10}^{-7},{R}^{2}=0.67$ |

T_{S} | Plane | T_{30}, EDT | ${z}_{0}=24.7522,a=-26.8540$ $b=116.2150,{R}^{2}=0.61$ |

**Table 6.**Best correlations between one acoustic parameter and a geometrical variable, the coefficients of the equation and their determination coefficient.

Parameter | Variable | Coefficients |
---|---|---|

EDT | V/S | ${y}_{0}=-72.6052$ $a=15.8495$ $b=-1.1003$ $c=0.0255$ ${R}^{2}=0.997$ |

**Table 7.**Best correlations between one acoustic parameter and another two geometrical parameters and the coefficients of the surface type and their determination coefficient. Latin Cross typology.

Parameter | Surface Type | Variable | Coefficients |
---|---|---|---|

T_{30} | Paraboloid | V, V/S | ${z}_{0}=8.2704,a=0.0009,b=-1.7042,$ $c=-3.1090\xb7{10}^{-8},d=0.0633,{R}^{2}=0.91$ |

EDT | Paraboloid | S, V/S | ${z}_{0}=37.3904,a=-0.0039,b=-4.5453,$ $c=-2.3165\xb7{10}^{-6},d=0.1575,{R}^{2}=0.99$ |

**Table 8.**Best correlations between one acoustic parameter and another two geometrical parameters and the coefficients of the surface type and their determination coefficient. Basilica typology.

Parameter | Surface Type | Variable | Coefficients |
---|---|---|---|

T_{30} | Paraboloid | V, V/S | ${z}_{0}=-4.2176,a=-3.0026\xb7{10}^{-5},b=1.3185,$ $c=6.0367\xb7{10}^{-10},d=-0.0429\xb7{10}^{-6},{R}^{2}=0.66$ |

T_{S} | Paraboloid | S, V/S | ${z}_{0}=2597.8576,a=-10.0082,b=913.1179$ $c=0.0031,d=-36.7648,{R}^{2}=0.98$ |

**Table 9.**Best correlations between pair of acoustic parameters, the coefficients of the equation type and their determination coefficient.

Parameter | Variable | Equation Type | Coefficients |
---|---|---|---|

T_{30} | EDT | Polynomial (Cubic) | ${y}_{0}=-1.0622,\text{}a=1.8296,\text{}b=-0.2249,\text{}$ $c=0.0171,\text{}{R}^{2}=0.98$ |

EDT | T_{30} | Logarithm (3rd Order) | ${y}_{0}=4.5769,\text{}a=-7.3837,\text{}b=6.9055,\text{}$ $c=-1.2362,\text{}{R}^{2}=0.97$ |

**Table 10.**Best correlations between one acoustic parameter and another two ones, the coefficients of the surface type and their determination coefficient.

Parameter | Surface Type | Variable | Coefficients |
---|---|---|---|

T_{30} | Paraboloid | EDT, T_{S} | ${z}_{0}=8.3135,\text{}a=0.0001,\text{}b=-0.2326,\text{}$ $c=-6.9291\xb7{10}^{-10},\text{}d=0.0022,\text{}{R}^{2}=0.74$ |

EDT | Gaussian | T_{30}, T_{S} | ${z}_{0}=12.5575,\text{}a=9.5165\xb7{10}^{-5},\text{}b=-0.3647,\text{}$ $c=-6.6241\xb7{10}^{-10},\text{}d=0.0033,\text{}{R}^{2}=0.84$ |

**Table 11.**Best correlations between two acoustic parameters, volume interval, equation type, the coefficients of the equation and their determination coefficient. Latin Cross typology.

Parameter | Variable | V (m^{3}) | Equation Type | Coefficients |
---|---|---|---|---|

T_{30} | EDT | <10^{4} | Polynomial (Linear) | ${y}_{0}=-0.2682,a=1.0417,{R}^{2}=0.98$ |

EDT | T_{30} | <10^{4} | Polynomial (Linear) | ${y}_{0}=0.3183,a=0.9418,{R}^{2}=0.98$ |

T_{30} | EDT | 10^{4} ≤ V ≤ 2∙10^{4} | Polynomial (Linear) | ${y}_{0}=-0.2237,a=1.0001,{R}^{2}=0.996$ |

EDT | T_{30} | 10^{4} ≤ V ≤ 2∙10^{4} | Polynomial (Linear) | ${y}_{0}=0.2415,a=0.9955,{R}^{2}=0.996$ |

D_{50} | T_{S} | 10^{4} ≤ V ≤ 2∙10^{4} | Polynomial (cubic) | ${y}_{0}=-0.0781,a=0.0027,$ $b=-8.8432\xb7{10}^{-6},$ $c=8.1714\xb7{10}^{-9},{R}^{2}=0.87$ |

T_{S} | D_{50} | 10^{4} ≤ V ≤ 2∙10^{4} | Polynomial (inverse 3rd Order) | ${y}_{0}=4998.4231,a=-1742.0565,$ $b=205.0782,c=-7.5684,{R}^{2}=0.86$ |

**Table 12.**Correlation between two individual acoustic parameters, the coefficients of the equation and their determination coefficient.

Parameter | Variable | Coefficients |
---|---|---|

D_{50} | C_{80} | ${y}_{0}=0.4106,a=0.0818$ $b=0.0069$ $c=0.0002,{R}^{2}=0.95$ |

**Table 13.**Best correlations between two individual acoustic parameters, the coefficients of the surface type and their determination coefficient.

Parameter | Surface Type | Variable | Coefficients |
---|---|---|---|

D_{50} | Paraboloid | EDT, C_{80} | ${z}_{0}=0.4929,a=-0.1359,b=0.0407,$ $c=0.0277,d=0.0014,{R}^{2}=0.94$ |

T_{S} | Paraboloid | EDT, D_{50} | ${z}_{0}=453.1379,a=-160.5536,b=-1728.3538$ $c=45.1449,d=4446.0818,{R}^{2}=0.93$ |

**Table 14.**Best correlations between two acoustic parameters, volume interval, equation type, the coefficients of the equation and their determination coefficient. Single-Nave typology.

Parameter | Variable | V (m3) | Equation Type | Coefficients |
---|---|---|---|---|

T_{30} | C_{80} | <2500 | Pseudo-Voigt, 5-Parameter | ${y}_{0}=3.7705,{x}_{0}=-0.0625,a=-2.0094$ $b=3.1063,c=0.8045,{R}^{2}=0.97$ |

C_{80} | D_{50} | <2500 | Polynomial (cubic) | ${y}_{0}=-21.2656,a=213.7030,$ $b=-833.4227$ $c=1145.2392,a=0.9418,{R}^{2}=0.95$ |

T_{S} | C_{80} | <2500 | Polynomial (cubic) | ${y}_{0}=123.0558,a=-4.5771,$ $b=15.2778$ $c=3.6348,a=0.9418,{R}^{2}=0.85$ |

EDT | T_{30} | V ≤ 2500 | Modified Gaussian, 5-Parameter | ${y}_{0}=-3366.9459,{x}_{0}=4.3420,$ $a=3371.9446$ $b=3371.9446,c=1.0133,{R}^{2}=0.90$ |

T_{30} | C_{80} | V ≤ 2500 | Polynomial (cubic) | ${y}_{0}=2.5834,a=-1.3545,b=-0.5072,$ $c=-0.0453\xb7{10}^{-9},{R}^{2}=0.85$ |

D_{50} | C_{80} | V ≤ 2500 | Modified Gaussian, 5-Parameter | ${y}_{0}=1.7601,{x}_{0}=-7.5000,a=-1.6503$ $b=22.4887,c=1.0007,{R}^{2}=0.96$ |

T_{S} | EDT | V ≤ 2500 | Polynomial (inverse 3rd Order) | ${y}_{0}=1303.1102,a=-5643.5922,$ $b=9145.4647,c=-4674.0578,$ ${R}^{2}=0.92$ |

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## Share and Cite

**MDPI and ACS Style**

Alberdi, E.; Galindo, M.; León-Rodríguez, A.L.; León, J.
Acoustics in Baroque Catholic Church Spaces. *Acoustics* **2024**, *6*, 911-932.
https://doi.org/10.3390/acoustics6040051

**AMA Style**

Alberdi E, Galindo M, León-Rodríguez AL, León J.
Acoustics in Baroque Catholic Church Spaces. *Acoustics*. 2024; 6(4):911-932.
https://doi.org/10.3390/acoustics6040051

**Chicago/Turabian Style**

Alberdi, Enedina, Miguel Galindo, Angel L. León-Rodríguez, and Jesús León.
2024. "Acoustics in Baroque Catholic Church Spaces" *Acoustics* 6, no. 4: 911-932.
https://doi.org/10.3390/acoustics6040051