# Echo Analysis in Iberian Bullfighting Arenas Through Objective Parameters and Acoustic Simulation 1
## 4 Results and Discussion
### Reverberation Times  
Reverberation time is the main parameter describing the acoustics of a room. In closed enclosures, its value does not vary practically with distance, but in open enclosures, it can vary and present non-linear energy decay curves [7; 8; 20; 22]; for this reason, a detailed study of its behaviour is presented. Table 4 shows the spatial average results of the \(T_{30}\) reverberation times in the spaces studied, for the sound source S1 and for all sound sources at 1000 Hz, and for S1 and all sound sources with the average of 500 and 1000 Hz octave bands. Table 4 highlights the longest reverberation time values in the Las Ventas and Campo Pequeno bullrings, the former due to its large size and the latter because it has a roof. Regarding the latter, there is very little difference between the \(T_{30}\) values with the oculus closed (CO) and partially open (PO). In the Ronda and Seville bullrings, the reverberation time values are similar, although slightly longer in Seville. The shortest \(T_{30}\) values correspond to the Italica amphitheatre and are of the same order of magnitude as in the Segobriga amphitheatre [7]: \(T_{30\text{mid}}\) for the sound source at the centre of the conic is 1.3 s.  
Figure 11: Floor plan of the Italica amphitheatre in Seville, showing the sound sources (S) and the receptors (R) in the cavea and the arena. (+) Topographic references. \begin{table}
\begin{tabular}{l l l l l l l l l l l l l l l l l l} \hline \hline
**Sound Sources** & **R1** & **R2** & **R3** & **R4** & **R5** & **R6** & **R7** & **R8** & **R9** & **R10** & **R11** & **R12** & **R13** & **R14** & **R15** & **R16** & **R17** \\ \hline S1: & North Minor axis & x & o & o & o & x & o & o & o & o & o & o & o & o & o & o & x & \\ S2: & South Minor axis & x & o & o & o & x & o & o & o & o & o & o & o & o & o & o & x & x \\ S3: & East Major axis & x & o & o & o & x & o & o & o & o & o & o & o & o & o & x & x \\ \(\circ\) & Combination to be carried out in the cavea. & & & & & & & & & & & & & & & & & & & \\ \(\mathbf{x}\) & Combination with both the source and receiver in the arena. & & & & & & & & & & & & & & & & & & & & \\  & Location of the sound source. & & & & & & & & & & & & & & & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Source–receiver combinations measured in the Italica amphitheatre. All recorded impulse responses show an impulse response-to-noise ratio, INR \(>\) 45 dB in each octave band. The behaviour of the energy decay curves is fundamentally of two types: cliff and plateau [27]. In the first case, the abrupt decrease occurs in the first 5 decibels of the decay, which enables linear drops for the calculation of \(T_{30}\) to be obtained, and justifies the high percentage of IRs in accordance with the standard. The linearity of the energy decay curves has been evaluated by using the degree of curvature \(C\), and the degree of non-linearity \(\xi\) for the \(T_{30}\) parameter, defined in Annex B of the ISO 3382-2 standard [17]. In the Ronda bullring, 74.7% of the 288 values (48 source-receiver combinations and 6 octave bands) meet the linearity criteria of 0-5% and 0-5%, respectively, for both parameters. Moreover, 8.7% of the values exceed the linearity threshold, set at 10% for C and 10% for \(\xi\), which renders the \(T_{30}\) results corresponding to these decay curves as questionable. The most notable non-linearity occurs in the extreme bands of 125 and 4000 Hz. In the Seville bullring, the linearity of the decay curves is determined by the results obtained from 450 recordings (15 reception points, 6 octave bands, and 5 sound-source positions), showing that 66.8% meet the linearity criterion \(C\) of 0-5% and \(\xi_{\rm T30}\) of 0-5%; 17.6% fall within the range of doubtful values \(C\) of 5-10% and \(\xi_{\rm T30}\) of 5-10%; and 15.6% fail to comply with the linearity criterion \(C>10\%\) and \(\xi_{\rm T30}>10\%\).