## 4 Discussion  
By characterising the construction of each of the towers in this study, the mechanical properties of the structural materials in the walls could be estimated, thus providing information on E, as well as on the masonry's compressive strength. These data are essential when elaborating numerical  
Figure 21: Comparison between (a) the slenderness (\(\lambda\)) and the estimated natural frequency; and (b) slenderness and measured natural frequency. Green square: cube-shaped; blue circle: cylindrical; orange diamond: Agicampe tower (oval); purple triangle: horseshoe. models to assess their stability and vulnerability, especially against horizontal loads [39]. The MQI method provides an initial approximate value for these parameters: although based on quantitative and qualitative analysis of different variables associated with the walls, it returns a wide range of values for E that need to be detailed and reduced according to the specific conditions of the structures under assessment. To do so, a correlation can be established between the natural frequency obtained from ambient vibration testing and the average values obtained for E using MQI (Figure 22). The graphs exclude towers built with rammed earth walls, which were not included in the MQI method. The resulting curve represents the optimal adjustment between E and the natural frequency of the structure for each type of tower. Dispersion from the curve indicates a need to adjust the value of E, bearing in mind the average value established in respect of the minimum and maximum values included in Table 3, from which the values of E are adjusted for the towers that have higher dispersion: Campanillas and Cantal. In the case of Campanillas, the value changes from 1556 to 1828 MPa, and for Cantal, it varies from 1583 to 1859 MPa. Once the value of E has been adjusted, it is possible to fine-tune the curve and thus obtain an R\({}^{2}\) value of 0.88. Such adjustment is made coherently with the actual characteristics of the masonry walls in the towers. As far as the numerical analysis of masonry structures is concerned, establishing a value for the structure's global elastic modulus is particularly relevant for an accurate assessment of its overall mechanical behaviour and as a basic starting-point to study its seismic vulnerability and develop non-linear analysis using finite element models [40]. Furthermore, slenderness is one of the key parameters involved in the resistance of historical structures against horizontal actions such as earthquake [41]. This study has shown the relevance of slenderness when estimating the natural frequency to be expected in these defensive structures. Slenderness can be easily worked out from the results of ambient vibration testing and can be compared with the curve proposed in Figure 21b. In addition, depending on the percentage of dispersion from the curve, the relevance of any structural damage present in the wall can be estimated and thus its general bracing against horizontal loads assessed. Our results are sustained by other studies that propose an empirical formula to obtain the fundamental vibration frequency of historic masonry towers [15]. In addition, also significant is how slenderness can be used to estimate the number of levels in some of the towers. Our analysis shows that medieval towers of Nasrid origin (13 th-15 th century) used for agricultural purposes have a slenderness between 1.8 and 2.0. Based on that value, it appears that towers with distinctly lower slenderness had no upper level, possibly, as in the case of Unique and Esteril towers, because they were pruned by Castilian conquerors to prevent them being used for defence in the event of a military uprising. On the other hand, some watchtowers from the same period would have a slenderness around 2.3-2.5, a value that serves as a reference to estimate the original height of the Guadalmansa, Virfuuela or Jaral towers.