## 2 Materials and Methods
### Statistical Analysis Methods
The centroids of the clusters were considered the target point, thus, with reference to a generic cluster j of historic buildings having centroid CjL(x\({}_{Cj}\), \(y\)Cj\({}_{Cj}\), \(z\)Cj\({}_{rCj}\), \(s\)Cj) the Euclidean distance dij of a generic i-th contemporary building having parameters (HWi, HLi, EVi, EWi, BFOi) was computed as follows:  
\[d_{ij}=\sqrt{\left(HW_{i}-x_{Cj}\right)^{2}+\left(HL_{i}-y_{Cj}\right)^{2}+ \left(EV_{i}-z_{Cj}\right)^{2}+\left(EW_{i}-r_{Cj}\right)^{2}+\left(BFO_{i}-s _{Cj}\right)^{2}} \tag{1}\]  
\(d_{ij}\) was considered a proximity index quantifying the level of architectural consistency of each contemporary building with the traditional formal features identified through the clustering procedure. The procedure expressed by Equation (1) was applied also to compute the distances \(d_{k}\) of every \(k\)-th historic building belonging to cluster \(j\) from its centroid:  
\[d_{k}=\sqrt{\left(HW_{k}-x_{Cj}\right)^{2}+\left(HL_{k}-y_{Cj}\right)^{2}+ \left(EV_{k}-z_{Cj}\right)^{2}+\left(EW_{k}-r_{Cj}\right)^{2}+\left(BFO_{k}-s _{Cj}\right)^{2}} \tag{2}\]The distribution of \(d_{k}\) was considered to assess resonance or dissonance of contemporary buildings: in particular, the property of asssonance was acknowledged to a building having dij smaller than a threshold defined as a percentile of \(d_{k}\). Then the distance of a building from the nearest cluster was defined as follows: \(d_{i,\min}=\min(d_{ij})\), with j varying from 1 to the number of clusters. \(d_{i,\min}\) was used as a more synthetic parameter to define the properties of \"assonance\" or \"dissonance\" of contemporary farm buildings with the historic-typological characters of the local context. Moreover, a qualitative analysis was performed considering every \(d_{ij}\) value. Figure 3: Diagram of the methodological process. A detailed analytical comparison of the features of the \(i\)-th contemporary building with those expressed by the centroids of historic building clusters was carried out by computing the differences in terms of single parameters, as follows:  
\[\Delta HW_{ij}=HW_{i}-x_{j};\ \Delta HL_{ij}=HL_{i}-y_{cj};\ \Delta EV_{ij}=EV_{i}-z_{cj};\ \Delta EW_{ij}=EW_{i}-r_{cj};\ \Delta BFO_{ij}= BFO_{i}-s_{cj}\]  
Then, we decided to identify a range of proximity of each parameter to the average values of each historic cluster. The 70 th percentiles of the absolute values of the differences \(\Delta HW_{k}\), \(\Delta HL_{k}\), \(\Delta EV_{k}\), \(\Delta EW_{k}\), \(\Delta BFO_{k}\) between each parameter of every \(k\)-th historic building belonging to cluster \(j\) and the corresponding coordinate of its centroid were adopted as significance boundaries for that cluster. Respectively named these boundaries \(\overline{x_{j}}\), \(\overline{y_{j}}\), \(\overline{z_{j}}\), \(\overline{r_{j}}\), \(\overline{s_{j}}\), the following indicators were defined:  
\[\delta HW_{ij} =\left|\Delta HW_{ij}\right|-\overline{x_{j}}\] \[\delta HL_{ij} =\left|\Delta HL_{ij}\right|-\overline{y_{j}}\] \[\delta EV_{ij} =\left|\Delta EV_{ij}\right|-\overline{z_{j}}\] \[\delta EW_{ij} =\left|\Delta EW_{ij}\right|-\overline{r_{j}}\] \[\delta BFO_{ij} =\left|\Delta BFO_{ij}\right|-\overline{s_{j}}\]  
Therefore, a positive value of one of these indicators means that the corresponding parameter of the contemporary building under study lies outside the proximity interval defined for the average parameter of the cluster. In this case, the greater the indicator, the farther the parameter from the historic reference value. On the contrary, a negative value means that the parameter falls within the 70 th percentile interval around the historic mean value. In this case, the greater the absolute value of the indicator, the nearer the parameter to the historic average.