The acoustics of 16th-century Jesuit churches in Andalusia: Latin cross plans following the model of Il Gesù

The acoustics of 16th-century Jesuit churches in Andalusia: Latin cross plans following the model of Il Gesù J. Le´ on * , E. Alberdi , P. Bustamante , M. Galindo Instituto Universitario de Arquitectura Y Ciencias de La Construcci´ on (IUACC), Escuela T´ ecnica Superior de Arquitectura, Universidad de Sevilla. Av, Reina Mercedes 2, 41012 Seville, Spain ARTICLE INFO Keywords: Worship acoustics Acoustic simulation Acoustic heritage Jesuit architecture ABSTRACT Following the Council of Trent, profound changes were observed in ecclesiastical architectural space as a result of the new spirit infused into the Catholic Church by the Counter-Reformation. The Jesuit mother church, Il Gesù, contributed to the consolidation of the model of the single nave church, eliminating the aisles and reducing the size of the transept. This architectural model was disseminated in step with the expansion of the Society. The churches of Santa Catalina (Cordoba), La Anunciaci´ on (Seville), and La Encarnaci´ on (Marchena, Seville) provide emblematic examples of the model adopted by the Society of Jesus in the second half of the 16th century in Andalusia. The typological, geometric, and material similarities displayed by these churches enable a common analysis of the sound field, to be carried out through in-situ acoustic measurements and simulations. The study of different source positions in similar models enables the determination of the optimal source and receiver configurations. It was found that the position on the main altar, coinciding with the position of the priest in the Tridentine rite, does not facilitate the perception of the sound field, except in the positions in the high tribunes. When the source is under the dome of the transept, the sound field evaluation is more favourable. Conversely, a comparative analysis of churches considered to be ’twins’, St Catalina and La Anunciaci´ on, has shown the impact that the interior cladding of these spaces has on the perception of their sound field. 1. Introduction Following the Council of Trent (1545–1563) the importance of preaching in the Catholic Church was made evident through the formalisation of a Counter-Reformation ecclesiastical model. The singlenave plan was established as the most suitable model, diverging from compositional principles based on aesthetic considerations. In his book Instructiones Fabricae et Supellectilis Ecclesiasticae (1577) St Charles Borromeo, Archbishop of Milan [1], applied the Tridentine decree to the problem of architecture, deeming churches with a Latin cross plan and a single nave to be the most suitable, in contrast to the centralised plans of Renaissance architects, viewed by Borromeo as typical of pagan temples. The church of San Fedele (Tibaldi, 1569–1579), built in Milan for the Jesuits at the request of Borromeo, was adapted to the needs of the Tridentine liturgical rite and applied the principles set out by Borromeo. The mother church of Il Gesù in Rome (Vignola, 1568–1584) consolidates this model. In this case, the large single nave is flanked by side chapels, eliminating the side aisles and reducing the size of the transept, according to the Tridentine liturgical rite. In addition, the raising of the presbytery further improved the visibility of the high altar for the faithful. Vallery-Radot [2] produced an inventory of more than 300 plans of Jesuit buildings from 1580 to 1690. This typology is identified in more than half of the plans analysed. Following its foundation in 1569, the Jesuit order expanded rapidly in Europe and America during the 16th and 17th centuries. As a result of this expansion, many churches were built according to the Jesuit Modo Nostro, following the model of the church of Il Gesù, but adapted to the architectural characteristics of each region. In the case of Spain [3], the predominant model was the Latin cross plan with side chapels and a straight headwall, which were adapted to local building traditions in each case. After the Council of Trent, the importance of preaching led to an interest in the acoustic quality of sacred spaces. As Howard highlights [4], understanding the knowledge of acoustics in the 16th century is crucial. Vitruvius adopted the Aristotelian theories of sound propagation as concentric waves, a concept that persisted throughout the 16th century. In this context, prominent Jesuits such as Biancani and Bettini studied acoustic phenomena. Biancani [5] analysed the phenomenon of * Corresponding author. E-mail addresses: [email protected] (J. Le´ on), [email protected] (E. Alberdi), [email protected] (P. Bustamante), [email protected] (M. Galindo). Contents lists available at ScienceDirect Applied Acoustics journal homepage: www.elsevier.com/locate/apacoust https://doi.org/10.1016/j.apacoust.2025.110774 Received 16 January 2025; Received in revised form 24 March 2025; Accepted 23 April 2025 Applied Acoustics 238 (2025) 110774 Available online 8 May 2025 0003-682X/© 2025 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license ( http://creativecommons.org/licenses/bync-nd/4.0/ ). the echo, demonstrating the interest of Jesuit mathematicians in the study of acoustic phenomena, also highlighted by Briatore [6]. The concern about the most suitable method for covering churches, either with vaults or with wooden ceilings, was debated in Italy during the Renaissance. Howard [4] and Barbieri [7] point out that, in the memorandum of the church of San Francesco della Vigna, Francesco Zorzi proposed vaults for the choir and a wooden ceiling for the main nave, considering the vaulted ceiling to be more appropriate for singing. Widespread preoccupation for the acoustic aspects of churches led the Jesuits to reflect on the most appropriate way to cover them. Navarro [8] analyses three Jesuit documents which show the concerns expressed by the Jesuit order regarding this issue. In the specific case of the first Jesuit church in Madrid (Spain) dedicated to the Holy Name of Jesus and designed by Father Bustamante, now disappeared, documents show that it was resolved with a wooden roof, reflecting the documented opinions in the excellent acoustic conditions of the church by those who knew the building. While the advantages of wooden ceilings were praised, as seen in Ceballos [9], Farnese and Vignola defended the use of the barrel vault to cover the church of Il Gesù, which was under construction. This went against the criterion defended by the Jesuits, supported by Father Trist´ an, who favoured the use of wooden ceilings to allow preaching to be heard with perfect clarity. On this issue, the analysis by Martellotta [10], based on the data of the initial acoustic study of the church of Il Gesù [11], evaluates the acoustics of the different proposals for the church ceiling. A model with a flat wooden ceiling at different heights, with and without a dome, was studied against the model built with a barrel vault. Martellotta concluded that the wooden ceiling was more suitable than the barrel vault. However, the variations obtained were not as significant as might be expected. The results show that the chosen ceiling height had a greater influence on the acoustics of the church than the construction typology selected. In relation to the acoustics of Jesuit churches that correspond to the typology of Il Gesù, research has been carried out on several churches, albeit within larger study samples, together with churches of other historical periods and/or typologies [12]. Among the churches studied by different researchers is the church of S. Roque (Lisbon, Portugal (16th-17th c.)) studied by Carvalho [13] alongside a group of 41 Portuguese churches of different styles. Engel [14] includes the church of St. Peter and St. Paul (Krakow, Poland (16th-17th c.)) in a group of 14 Polish churches while Diandreras [15] analyses three Jesuit churches from a large sample of Peruvian colonial churches - the Society in Arequipa, Ayacucho and Cusco (Peru, 17th c.) - which follow the model of Il Gesù (52). The studies by Abadía [16] analyse and evaluate the acoustics of Jesuit churches in the city of Cordoba (Argentina). Three of these churches (Sta. Catalina, S. Isidro Labrador and La Compa˜ nía (17th-18th century)), follow Jesuit architectural models deriving from the Spanish ones, but mainly influenced by local decorative and constructive aspects, featuring unique elements like the wooden vaulted ceiling of the church of La Compa˜ nía [17]. The churches of Sta. Catalina and San Isidro Labrador, despite their relatively modest volume (2991.3 and 4434.2 m 3 ), display reverberation times at medium frequencies (3.32, 6.17s) which exceed the optimum, as documented by Abadía. In the case of the church of La Compa˜ nía, which has a larger volume (8236.5 m 3 ), the values obtained for T 30m are lower (3.14s) due to the presence of the wooden ceiling in the nave of the church. The research introduces a novelty study of the sound field in a group of churches that adhere to the principles proclaimed by the Council of Trent applied by the Jesuit Order. The selected sample comprises churches with common features in terms of geographical location (Andalusia (Spain)), construction dates (second half of the 16 century), dimensions, construction systems, materials and even the same Jesuit architects and supervisors. These three churches follow the model established by the Jesuits for their mother church, Il Gesù, featuring a Latin cross plan. These similarities allow us to carry out a joint analysis of the characterisation of the sound field for three source positions, the main altar, under the transept dome, and the choir. The analysis, based on in situ measurements and acoustic models, demonstrates how a greater proportion of altarpiece surfaces and furnishings have a significant impact on the perception of the sound field, particularly in terms of reverberation, musical clarity and speech intelligibility. Although the Jesuits were aware that wooden ceilings improved the acoustics of their enclosures, other factors led to the preference for vaults and domes as the main system. Comparing these three churches with other Jesuit churches of the same period, volume and Latin cross plan typology, it becomes clear how the choice of wooden roofs would have meant a reduction in reverberation times and enhanced clarity. Finally, the analysis allows us to establish that the position of the speaker on the main altar, as prescribed by the Tridentine rite, is acoustically less favourable compared to positioning the speaker under the transept dome. Furthermore, this study aims to analyse different positions of sources and receivers inside the churches in order to determine how these positions affect the acoustics of these enclosures and also stating the locations of the participants in religious events and concerts for which these spaces are also used. 2. Comparative analysis of the sample The Jesuit province of Hispania was founded in 1547 and divided into four provinces in 1554 (Aragon, Castilla, Toledo and Andalucía). In the province of Andalusia, Father Bartolom´ e de Bustamante, who during the Second General Congregation (Rome 1565) had helped draft brief rules on how to construct Jesuit buildings and churches, was commissioned by Francisco de Borja to draw up the plans for the buildings being founded in the province [18]. The three churches selected are some of the colleges established by the Jesuits in the second half of the 16th century (Fig. 1). Their designs, supervised by Bustamante, follow the Latin cross plan and offer similar construction characteristics. The church of the college of Santa Catalina (SC) (1564–1589) in Cordoba belongs to the first foundation of the Society in Andalusia and became the prototype for the following constructions. The Church of La Anunciaci´ on (AN) (Seville, 1565–1579) was constructed for the Jesuits’ Casa Profesa in Seville and is regarded as a “twin” of SC, given its architectural plan and dimensions. Once Bustamante’s plans had been drawn up, they were adapted and built by Hern´ an Ruiz el Joven. Both churches, with a single nave without side chapels, are covered with vaulted ceilings in the nave, transept and main chapel, and a hemispherical dome in the transept. Although the churches follow the same construction pattern, SC has two bays in the nave, while AN has three. As Ceballos [9] points out, the proportions of AN are more harmonious. In SC, the altarpiece of San Ignacio, directly connecting the nave of the original church with the chapel of the Sagrario was eliminated in 1853. As Fig. 1 shows, a secondary volume (567 m 3 ) was added to the initial volume of the church, resulting in a final volume of 11,926 m 3 (Table 1). The third church selected for the exhibition is the Colegio de la Encarnaci´ on (EN) (1566–1588) in Marchena (Seville). As with the two previous churches, its initial designs were the work of Bustamante, assisted in this case by Martín de Gainza, master builder of Seville Cathedral. At a later stage, Hern´ an Ruiz El Joven took over and participated in the construction of the church [9]. The plan of the church, with a single nave and three vaulted bays before the transept, follows a similar layout to those of AN and SC. However, in EN there are side chapels on either side of the main nave. These chapels, of the same depth as the transept arms, follow the same pattern as that used by the Society. Following the expulsion of the Jesuits in 1767, the chapels were closed off, as evidenced by the plan by Fray Antonio Ramírez of 1780. This remained in place until the recent renovation of the church (1998–2002). In this last intervention, the platform of the main altar was expanded, and is located further forward than in the original design. In J. Le´ on et al. Applied Acoustics 238 (2025) 110774 2 Fig. 1. Sample churches. Geometrical parameters. Position of sources and receivers in-situ measurement. J. Le´ on et al. Applied Acoustics 238 (2025) 110774 3 addition, this church has a balcony on the right arm of the transept that was used as a tribune by the Dukes of Arcos, offering a privileged vantage point for the events that took place in the church. Although the three churches are contemporaneous and of similar conception, EN, while maintaining a similar longitudinal axis dimension to AN and SC, has a smaller volume, due to the lower heights of the naves and the lantern (Table 1). From materials perspective, the three churches exhibit notable similarities. Painted mortar is predominant, covering the largest areas of cladding in all three cases (39.2–59.7 %), as well as marble, mainly present in the floors (13.4–17.6 %) or the wooden benches and the altarpieces (Table 6). As Ceballos [9] points out, these three churches (AN, SC, EN) are the purest examples of the Jesuit church typology in Spain in the first half of the 16th century, repeating the same construction pattern and appropriating a Mannerist-Trento style, reflecting the sober spirit of the early Society of Jesus. 3. Methods The study of the sound field of the sample churches combines in-situ acoustic measurements of the enclosures and simulations corresponding to the different positions of sources and receivers commonly used in the enclosures. The three-dimensional acoustic models of the churches have been analysed with CATT-Acoustic software [19], featuring two different calculation engines, CATT-Acoustic v.9.1b and CATT TUCT v2.0b (The Universal Cone Tracer). The software application used is suitable for both predicting the acoustic parameter values and generating the binaural impulse responses [20,21], providing an accurate representation of the acoustic behaviour in these spaces. Graphic design and modelling software SketchUp v.14, which allows the geometry to be exported to CATT-Acoustic has been used to reproduce the 3D model of the interior of the three churches. No details smaller than 0.50 m were considered given the simplification required by the complex interior geometry of these spaces [21]. The models were subsequently calibrated in order to reproduce the sound field inside the churches with the strict application of the parameters obtained from in-situ measurements. The acoustic characteristics of the materials, such as the absorption and diffusion of the different interior surfaces, play a decisive role in this process, over and above any previous geometric simplifications [22]. Different source and receiver positions have been analysed based on the initial calibrated models, and considering different scenarios with empty and occupied churches and varying the positions of the existing furniture, particularly the pews used to seat the public, following configurations commonly used in the sample churches. 3.1. Acoustic parameters The sound field of churches has been evaluated on the basis of the objective parameters defined in ISO 3382-1 [23]. The parameter T 30 (s) has been considered in the analysis of the physical reverberation of these spaces. However, when evaluating the perceived reverberation, EDT (s) (Early Decay Time) was analysed to provide information on the initial speed of the decay curve. T 30 was used, together with all other parameters, to adjust the initial models that reproduce the sound field of churches. Sound strenght, parameter G (dB), allows us to evaluate the subjective sound level and uniformity of the sound field. The assessment of perceived sound clarity is based on three parameters, C 80 (dB) Musical Clarity Index, T S (ms) Center Time, and D 50 Definition, which measure the ratio between the initial and late energy reaching the listener. The apparent source width from J LF (Just Noticeable Level Fluctuation) and IACC E (Early Interaural Cross-Correlation Index) corresponds to the first part of the impulse response. The results are represented in two ways: firstly, by frequency averaging in octave bands (125–4000 Hz) and then by single number averaged values over the frequency range indicated in Table A.1 of Annex A of ISO 3382-1. In addition to the above parameters, the STI (Speech Transmission Index) parameter according to IEC 6026816 [24], is used to evaluate speech intelligibility, classifying the results obtained on a standardised scale. 3.2. In-situ measurement The acoustic parameters that define the sound field were calculated from the impulse responses (IR) of the rooms. For this purpose, in-situ measurements were carried out following the criteria indicated in the standards [23,24] with the churches empty of public. The sources were placed in the usual positions of the speakers in S1 (main altar), S4 (side altar), and for the music, S2 (dome of the transept) and S3 (choir). The receivers distributed in the audience area included different positions according to the most frequent uses of each of the churches. The acoustic parameters were obtained at each point where a receiver was placed through sweeps of sinusoidal signals, where the frequency increases exponentially with time. The environmental conditions of the churches were measured and taken into account for the calculation, obtaining the temperature, with an accuracy range of ±0.1 ◦C and the relative humidity with a range of ±0.5 %, at receiver position 1. Table 2 shows the environmental parameters for each church, as well as the number of receivers studied for each source, as numbered in Fig. 1. IRIS [25], as integrated hardware and software system for measuring environmental acoustics, was used for the in-situ measurements. IRIS software [25] was used to generate the signal, record the response, and analyse the results, connecting the processor computer to the rest of the measurement chain via the MOTU 4PREHYBRID hybrid audio interface. The Omnipower 4292 Type 4292-L dodecahedral sound source, located 1.50 m above ground level, reproduces the excitation signal, which is amplified by a B&K 2734 power amplifier. The Core Sound TetraMic microphone, 1.20 m above the ground and aligned in the source-receiver direction, captures the 3D impulse responses. It utilizes a tetrahedral microphone array, comprising four capsules arranged in the shape of a tetrahedron. This configuration enables the capture of the sound field from all directions. Once the measurements were taken, in the postprocessing phase, IRIS placed microphones in a figure of 8 to analyse the first 2 ms of the impulse response obtained in the measurement in order to determine the directional parameters (J LF ). The software used identifies the direct sound and aligns the null position of the virtual side microphone in the same direction as that of the analysis of the initial 2 ms of the recorded impulse response. According to the software manual instructions, the parameters pertaining to sound intensity (G) are obtained by calibrating the sound intensity within a semi-anechoic chamber. The frequency range and duration of the sweep are designed to ensure that the signal-to-noise ratio exceeds 45 dB in all octave bands subjected to analysis, spanning a range from 125 to 4000 Hz. Head Acoustic’s HMS III torso simulator was used to evaluate the signals with the EASERA software [26]. Background noise was measured with a PCE430-Class I sound level meter, averaged over 5 min. The results were obtained for all frequencies in the range of 8 to 16000 Hz, expressing the BNC (Balance Noise Criterion) value taken as a representative value in Table 2. Table 1 Geometrical Characteristics. SC AN EN MAIN DIMENSIONS (m)  Longitudinal axis 45.28 47.25 45.28 Transverse axis 24.25 26.06 19.67 Main nave height 19.51 19.82 15.81 Height under lantern 32.07 34.25 20.64 Dome diameter 11.59 14.09 10.17 SURFACES (m 2 ) Surface. Ground floor 653.05 760.27 669.81 Surface. Choir floor 125.83 166.39 113.06 VOLUME (m 3 )11,926 13,681 7,674 J. Le´ on et al. Applied Acoustics 238 (2025) 110774 4 3.3. Model calibration As indicated above, the acoustic models which reproduce the interior of the churches, made with SketchUp v.14 and imported into CATTAcoustic v.9.1b, required calibration in order to adjust their sound behaviour to the results obtained in the in-situ measurement. TUCT algorithm 1 is used for closed rooms facilitating short calculation and basic auralisation with ‘max split-up order 0 ′ . The number of beams and the truncation time of each model are set manually, and the results obtained were verified to be acceptable (Table 4). The initial models are calibrated using a simplified iterative procedure that allows T 30 to be calculated and compared with the T 30 values obtained from insitu measurements. To perform this process, it is necessary to assign absorption and scattering coefficients to the materials of the inner envelope, based on literature sources (Table 6). The iterative calculation procedure is repeated to adjust those absorption and scattering coefficients values that present a greater uncertainty and for which bibliographic data is unavailable. Table 6 shows the material considered. Once the model has been adjusted using this simplified process, another calculation is carried out using algorithm 2, which improves the stabilisation of the simulated results. The results obtained for all selected acoustic parameters are compared with those obtained from the acoustic measurements, considering the initial models’ conditions of humidity, temperature, background noise and occupation similar to those existing during in-situ measurements [21,27]. According to Bork [28] simulations are considered adequate when the differences between measured and simulated values are between one and two times the Just Notifiable Difference (JND) value for each parameter, applying the JND values seen in ISO 3382-1 [23]. The coincidence between in-situ and simulated values with differences of less than 1 JND for the T 30 parameter (less than 5 % of the measured values for each octave band) is considered a suitable calibration criterion [29]. For the rest of the parameters, the calibration of the model for values below 2 JND was considered adequate [30]. Martellotta [31] suggests that, for musical clarity and sharpness in church spaces, denoted by the parameters C 80 and T s , in particularly reverberant spaces, the value of 1 JND indicated in the ISO 3382-1 [23] standard can be modified, considering 1.5 dB for C 80 and 8.5 % of the reference value for T s . The comparison of the results obtained in-situ and from the initial model (empty churches) is graphically represented in terms of acoustic parameters in octave bands. 3.4. Spatial configurations Based on the validated models for each church, the different source positions, receiver positions and audience distributions were reproduced, considering both the empty and occupied church hypotheses. In total, 22 models were produced: 6 models each for SC and EN (3 source positions) and 10 for AN (4 source positions). The positions of sources and receivers correspond to those shown in Fig. 1 for SC and EN. In the case of AN, by keeping the stairs leading up to the main altar at the head of the church, as originally designed, a large space was freed up under the dome of the transept, allowing a raised platform to be introduced for concerts and the side altar to be used for small liturgical events. Fig. 2 shows the audience arrangements in AN for S2 and S4, introducing new receiver positions and audience areas in the raised platform arrangement. In the case of AN, Rico et al. [32] presented a proposal for the acoustic rehabilitation of the church, analysing two source positions, Table 2 In-situ measurements. Sources and receivers analysed per . R S1 S2 S3 S4 T (◦C), H r (%) BNC (dBA) SC 21 1–17, 19–21 1–18 1,2,13,15,19 –26.0, 46.3 20.7 AN 25 1–25 1–22, 24–25 1–12, 23–25 1–12, 16, 18, 22–25 18.0, 59.6 32.1 EN 23 1–21, 23 1–13,15–18,20,21,23 1–7,19–22 –15.6, 52.7 21.6 Source. R: total model receivers. X-Y: all numbered receivers in the range are considered. Environmental parameters Fig. 2. Different configurations of sources and receivers in La Anunciaci´ on church (AN). a) Photograph of the S2 position without the platform, and b) S2 position with the platform. J. Le´ on et al. Applied Acoustics 238 (2025) 110774 5 one in the transept under the dome, coinciding with S2, and the other under the choir. However, the intervention proposed an alteration to the finishes on the side walls of the church, and installing panels to form an acoustic shell on the platform, obstructing the direct view of the main altar. These suggestions were not taken into account in this study. Table 3 shows the total number of receivers for the acoustic models (R) and those considered for the different source positions (S1 to S4). For the AN case with the source under the dome (S2), the receivers analysed in the model introduced by the platform are highlighted in red. Table 4 details the calculation conditions for the three churches in the sample. The environmental parameters (temperature, humidity and background noise) obtained in in-situ measurements were considered in the models and the absorption coefficients of the pews were modified in the scenarios with the church occupied (0.23, 0.37, 0.83, 0.99, 0.98, 0.98) following the model established by Martellotta et al. [31]. The models made with CATT Acoustic represent the interior of the three churches (Fig. 3) in a simplified way (SC (1834 planes), AN (1144 planes), EN (2288 planes)) so that the number of planes used in the models depends on the degree of complexity of the envelope of the indoor space. 4. Results and discussion 4.1. In-situ measurement Table 5 shows the results of the in-situ measurements, spectrally and spatially averaged for each acoustic parameter. The results are categorized by church classification (SC, AN, EN) and source positions (S1 to S4). For each acoustic parameter, the lowest value is highlighted in red and the highest value in blue. In the case of EN, the results obtained with the source located at the main altar (S1) and under the transept dome (S2) are given for the receiver R23, located on a high rostrum (see Fig. 1). This position, known as the count’s balcony, is analysed separately. The results obtained from this location are indicated in grey in the table. Fig. 4 illustrates the single-number averaged values of reverberation times (T 30m ) and musical clarity (C 80m ) versus volume for a group of Jesuit Latin cross churches studied by other researchers [13,14,15,16,17] and including SC, AN and EN in the graphs. All values correspond to in-situ measurements laid down in ISO 3382 [23] at various times, except for Carvalho (SR), whose research predates the standard. In the volume range of between 5,000 and 15,000 m 3 , SC presents T 30m values close to the churches in its range. For AN and EN the reverberation times were 1.5–2.0s above the most representative group. The C 80m results for SC, AN and EN were in the central zone of the values seen in similar churches. The results obtained for CPC show low T 30m and high C 80m with respect to the sample. This church is the only one in the sample with a vaulted wooden ceiling. As Abadía [17] states, especially at low frequencies, the wood of the vault surface, given its large size, absorbs at these frequencies, considerably decreasing the reverberation time, emphasising the role of cladding in acoustically qualifying these spaces. Moreover, Alberdi [33] analyses a group of 27 churches from the same historical period with a Latin cross plan, including Jesuit churches and churches of other religious orders. The values of T 30m are in line with the average for this typology. In the case of EN, the values of C 80 are in line with the average value of this sample, while SC and AN yield more favourable values, as can also be seen in the graph in Fig. 4. As expected, the comparative analysis of the three churches (SC, AN, EN) shows similar reverberation times (T 30m ), regardless of source position. In Fig. 5, in the graphs showing the results for reverberation (T 30 and EDT) expressed by frequency in octave band, it is observed that SC and AN, with very similar dimensions (volume variation 14.7 %, ground floor area 16.4 %, and length 4.3 %) present differences of around 1.25s at all frequencies, with the exception of 4000 Hz (0.5s) for T 30 . When considering the materiality of the three churches, painted mortar represents the largest proportion of surface area in the interior finishes in all three cases, while other materials such as marble, altarpieces and wooden pews can be found in significant proportions, accounting for more than 70 % of the total (75.6 % SC, 80.6 % AN, 73.3 % EN) (Table 6), highlighting the similarity between geometries and finishes. The proportions of these four materials show differences of between 5 and 7.3 % when assessed jointly, although it is observed that for marble the differences are small, with variations of between 3–4.2 %. Furthermore, when altarpieces and pews, wooden elements with similar absorption coefficients at low and medium frequencies, are evaluated together they show greater variations (4.5–9.1 %), especially when comparing SC and AN. In short, although their geometries are very similar, particularly in the case of SC and AN, the proportion of the materials which form part of the furniture show notable differences. The significance of the materials constituting the interior of churches is highlighted by an examination of the reverberation phenomenon as elucidated by the Sabine [34] equation. T60 =0.161*V A(1) Considering the mean absorptions at frequencies of 500 and 1000 Hz, calculated from the CATT Acoustic models, the value of the absorption coefficient α is 0.0890, 0.0870, and 0.0605 in SC, AN, and EN, respectively. In turn, the value of the sound absorption area (A) depends on the absorption coefficient ( α ) and the interior cladding on the surfaces of each church. It should thus be noted that SC, with a lower volume, has a higher value of A (521.7 m 2 ) than AN (484.3 m 2 ), which, applying Sabine, results in lower reverberation times in SC, as confirmed with insitu tests. In EN, the relationship between volume and sound absorption area (A) (285.4 m 2 ) brings the reverberation time results closer to AN. The EDT analysis in all churches shows no variations of more than 1 JND regardless of the position of the source (S1 to S4) (Table 5). However, when analysing the sample as a whole, although this parameter depends on the first reflections where the position of the source can have some impact, the differences in the JND values for SC and AN for the sources are below 1 JND. In the case of EN, the differences are more Table 3 Source positions analysed in the simulations. Receptors analysed per source. R: total model receivers. X-Y: all numbered receivers in the range are considered. The receivers with the source on the raised platform model are underlined. R S1 S2 S3 S4 SC 21 1–21 1–17, 20, 21 1–10, 13–15, 19–21 – AN 37 1–25 1, 3, 5, 7, 9, 10–20,21,22, 24–29 9–18, 24, 25, 30–37 1–14, 19–25 1–8, 19, 20, 23 EN 23 1–21, 23 1–18, 20, 21, 23 1–8, 13, 15–22 – Table 4 Calculation conditions. Background noise (dBA) 125 250 500 1000 2000 4000 SC 10.4 18.2 21.8 24.1 22.9 16.1 AN 36.8 42.7 47.2 57.0 63.2 70.3 EN 11.9 17.8 16.8 22.2 25.4 17.2 Calculation conditions Calculation algorithm 2 Number of rays 130,000 (SC-AN); 100,000 (EN) Echogram / impulse response (ms) 6000 (SC, AN); 5500 (EN) Air density 1.20 kg/m 3 Air absorption Activated Diffraction Activated J. Le´ on et al. Applied Acoustics 238 (2025) 110774 6 appreciable, especially when comparing S2 and S3 (1.79 JND), while the results could be expected to be more similar between S1 and S2, which are placed closer together (Table 5 and Fig. 5). In contrast, when the source is under the dome (S2), the perception of the sound level is higher in SC and AN. In EN this occurs when the source is positioned in the choir (S3) (Table 5). In all hypotheses the results obtained are related to the source-receiver distance. In the case of SC and AN, the average source-receiver distances are lower when the source is under the dome, while the same is true for EN when the source is in the choir, taking into account the receivers considered in each scenario as shown in Table 2. Sound clarity, evaluated from the C 80 plots in Fig. 5 and the results expressed in Table 5 (C 80 , D 50 , T s ), indicates better values in SC and AN with the source located under the dome (S2) than in the choir position (S3). In the case of AN, the values obtained for C 80 are very similar for S1, S2, and S3, with differences of below 1 JND. However, for the sidealtar source position (S4), the results for all the clarity parameters worsen, although these values are close to 1 JND. When considering the positioning of the source (S4), it is essential to acknowledge that as the receivers under study are situated below the dome (Table 2, Fig. 2) they do not cover all of the church nave. Additionally, the distance between source and receiver is smaller than that observed in the other models. In SC, the differences in sound clarity are more noticeable between the different source positions, with differences above 1 JND (1.8 JND S2S3). However, in EN the behaviour for sound clarity is discrepant, as the source position in the choir (S3) is more favourable, with differences of above 2 JND compared to other positions. In EN, the height of the main nave of the church is lower than that of SC and AN (4 m lower), and so is that of the choir (2.4 m lower), which brings the source (S3) closer Fig. 3. CATT Acoustic 3D models. SC (Santa Catalina), AN (La Anunciaci´ on), EN (La Encarnaci´ on). Table 5 In-situ measurement. Single number of spatially averaged values for different source positions. JND value for each acoustic parameter. (*): lowest value of the acoustic parameter. (**): highest value of the acoustic parameter. Underlined: results R23 (EN). Source EDT m (s) T 30m (s) T sm (ms) G m (dB) C 80m (dB) D 50m J LFm STI IACC E SC S1 3.95 3.89 297.18 8.85 −5.65 0.15 0.15 0.33 0.33 S2 3.93* 3.88* 257.06* 10.14 −3.61** 0.28 0.18 0.39** 0.37** S3 4.11 3.90 318.21 8.73* −6.34 0.11* 0.09* 0.29 0.29* AN S1 5.25** 5.15 370.71 10.25 −5.97 0.19 0.17 0.28* 0.36 S2 5.17 5.17** 339.65 11.21 −4.85 0.26** 0.17 0.33 0.37** S3 5.07 5.15 334.96 10.74 −4.98 0.17 0.17 0.28* 0.24 S4 5.23 4.89 373.70 10.47 −6.45 0.18 0.18 0.29 0.31 EN S1 4.78 4.93 374.38** 12.24 −7.31* 0.12 0.19** 0.30 0.30 S2 4.93 4.95 364.73 12.62 −6.75 0.14 0.20** 0.31 0.33 S3 4.50 4.88 312.22 13.88** −4.61 0.22 0.18 0.37 0.31 EN_R23 S1 3.94 4.93 256.39 11.17 −3.07 0.23 0.30 0.38 0.22 S2 4.05 4.93 277.03 11.10 −3.57 0.16 0.27 0.38 0.35 JND –5 % 5 % 10 1.00 1.50 0.05 0.05 –0.075 Fig. 4. Jesuit Churches Latin cross plan. Reverberation (T 30m ) and clarity (C 80m ). Single number values. Main source altar. J. Le´ on et al. Applied Acoustics 238 (2025) 110774 7 to the plane of the audience, improving the clarity parameters obtained. The musical clarity results are conditioned by the source-receiver distance. According to this criterion, the best results are obtained with lower average distances. In the cases of SC and AN, the average distances with the source under the dome (S2) are 13.58 and 12.80 m, respectively, whereas when the source is placed in the choir (S3) these distances are 26.75 and 20.73 m. In EN, the average distance for S2 is 17.09 m, while in the choir (S3) it is 13.96 m, which in this case yields better musical clarity results with the source located in the choir. In churches, the presence of elements such as chapels, columns, apses and lateral naves can contribute to attenuate the early reflected energy and increase it with source-receiver distance [35,36,37,38,39]. In the sample churches, studying the spatial distribution of sound strength (G) in relation to source-receiver distance, values within 1 or 2 JND are obtained with respect to the Barron and Lee model [40] (Fig. 6), but with greater attenuations than predicted by this model, as expected. In order to quantify them, the value of the μ parameter has been calculated based on the μ -model [41]. The value of the μ parameter obtained for the churches are 0.14 (SC), 0.22 (AN) and 0.20 (EN), higher values than those indicated by Zamarre˜ no et al. [41] for Gothic-Mudejar Spanish style churches (0.09–0.16, average 0.13). These churches, in addition to having a smaller average volume, feature a wooden roof. Among the three churches studied, only SC obtains a μ value within the range of the Gothic-Mudejar churches, as well as a T 30 value close to them, coinciding with a higher percentage of wooden elements (altarpieces and pews), as evaluated in the reverberation times analysis. Furthermore, we can observe that for the three churches the points closest to the altar show possible sound concentrations caused by the domes, with results exceeding those of the theoretical models. The enveloping sensation, evaluated from J LF , shows markedly homogeneous behaviour for AN and EN. In SC, the results obtained for the position of the source in the choir (S3) show lower values, with a difference of 1.80 JND compared to S2. The receivers analysed for this source position (S2, SC) are all located under the dome of the transept and not in the main nave, so that the fraction of lateral energy received is lower than for the receivers located in the nave. In AN and EN, the number of receivers included in the measurement for this source position covers the entire nave (Fig. 1, Table 2). In EN, the results obtained for the position of the Dukes of Arcos tribune (R23, S1, S2) show a lower perceived reverberation, with EDT differences of around 3.5 JND compared to the average for all the receivers analysed (Table 5). Better clarity is observed with differences of more than 8 JND for T s , while better clarity and definition with differences of more than 2 JND are found for C 80 and D 50 . A separate comparison of the results of R23 and the other ground floor receivers shows that only R18 and R19, positioned halfway from the source, show similar results. Compared to the ground floor receivers located at a similar distance (R3), the results for the transept rostrum position (R23) are much more favourable, as seen in the assessment of musical clarity, where C 80 obtains average values of −7.32 dB for R3. Therefore, as Alberdi [41] points out, receiver positions in tribunes above the plane of the audience yield a better subjective acoustic sensation, especially with the source on the main altar. In SC, placing the source under the dome (S2) results in improved speech intelligibility and musical clarity. This is accompanied by lower values of T s and higher values of C 80 , D 50 and STI. In the case of AN, Fig. 5. T 30 , EDT, C 80 and J LF spatially averaged versus frequency octave band and errors bars (standard error). J. Le´ on et al. Applied Acoustics 238 (2025) 110774 8 while it can be argued that the optimal results are achieved when the source is positioned below the dome, these are not significantly superior to those obtained when the source is situated in the choir (S3) or the main altar (S1). For EN, which is smaller and features side chapels off the main nave, the optimal position is that of the choir (S3). 4.2. Calibration assessment To validate the models, the empty church has been considered taking into account the conditions of the on-site measurement and the position of the source in the main altar (S1), which is the most representative of Table 6 Main surfaces used in the simulation: Materials, surface (%), absorption (up) and scattering (down) coefficients. Material Church Area (%) 125 Hz 250 Hz 500 Hz 1 kHz 2 kHz 4 kHz Painted mortar (A) * SC 40.0 0.130 0.090 0.055 0.043 0.050 0.043 AN 59.7 0.083 0.073 0.055 0.050 0.050 0.020 EN 39.2 0.080 0.070 0.050 0.030 0.010 0.010 Plaster (A) [42] SC/AN/EN 17.4/3.2/2.7 0.02 0.02 0.03 0.03 0.04 0.05 Facing brick (B) [42] EN 14.5 0.03 0.03 0.03 0.04 0.05 0.07 Limestone (B) [42] EN 3.2 0.02 0.02 0.03 0.04 0.05 0.05 Limestone-mortar (A) [42] EN 2.4 0.02 0.03 0.03 0.04 0.05 0.06 Marble (A) [42] SC/AN/EN 14.6/13.4/17.6 0.01 0.01 0.01 0.02 0.02 0.02 Ceramic tiles (A) [42] EN 0.6 0.02 0.02 0.03 0.03 0.04 0.05 Wooden decking (A) [42] AN 3.1 0.18 0.12 0.10 0.09 0.08 0.07 Altarpieces (C) [21] SC/AN/EN 12.3/7.2/5.5 0.12 0.12 0.15 0.15 0.18 0.18 Canvas (A) [43] AN 1.6 0.01 0.01 0.01 0.10 0.20 0.45 Glass (A) [42] SC/AN/EN 0.8/1.8/0.3 0.1 0.05 0.04 0.03 0.03 0.03 Wooden pews (C) [44] SC/AN/EN 8.7/4.7/11.0 0.04 0.12 0.17 0.22 0.23 0.24 Lightly upholstered chair (C) [42] AN 0.5 0.44 0.56 0.67 0.74 0.83 0.87 Wooden door (B) [42] SC/AN/EN 2.5/2.4/1.6 0.14 0.10 0.06 0.08 0.10 0.10 Carpet (B) [42] SC/AN 1.3/8.5 0.07 0.31 0.49 0.81 0.66 0.54 Cotton curtain (C) [42] SC/AN 0.8/2.0 0.30 0.45 0.65 0.56 0.59 0.71 Velvet curtain (C) [42] EN 0.4 0.05 0.12 0.35 0.45 0.38 0.36 Air conditioning duct (C) [45] SC 1.5 0.16 0.39 0.91 1.01 1.01 1.01 Organ (C) [46] AN 0.30 0.12 0.14 0.16 0.16 0.16 0.16 Wooden lattice (C) [42] (semitransparent) EN 0.7 0.27 0.23 0.22 0.15 0.10 0.07 * Adapted from the experimental results in the tuning process. Scattering coefficients by frequency in octave bands (125–4000 Hz) [30]. A (Smooth: 0.12/0.13/0.14/0.15/0.16/0.17). B (Moderately irregular: 0.20/0.25/0.360/0.35/0.40/0.45). C (Irregular: 0.30/0.40/0.50/0.60/0.70/0.80). 4.3. Acoustic evaluation of models. Fig. 6. Sound strength (G). Main altar source (S1). Values averaged to a single number for the different models. Barron and Lees model and μ -model, vs sourcereceiver distance. J. Le´ on et al. Applied Acoustics 238 (2025) 110774 9