# A Statistical Approach to Violin Evaluation
## 1 Introduction  
Assessing the quality of a violin is an extremely difficult matter. There are no objective descriptors that can separate a \"good\" violin from a \"great\" one and yet some instruments can reach prices up in the millions. Even professional musicians have trouble distinguishing old from new violins, and their preference does not necessarily correlate with the price of the instrument [1, 2, 3]. At the end of the day, it comes down to the subjective preference of the player. Having an objective measure to support the evaluation of an instrument could help instrument makers and musicians in the search for the perfect match between violinist and violin. The bridge admittance could be such a simple objective tool to qualify an instrument. This is the frequency response function (FRF) measured at the bridge on the plane perpendicular to the strings. The measurement is performed by means of an instrumented hammer and an accelerometer, located at opposites on the bridge. It is a good estimator of the vibrational behavior of the instrument soundbox from whence the sound of the violin comes from. It is fast to perform and the equipment needed is relatively cheap, at least for a scientific lab if not for the luthier's workshop [4, 5]. FRFs have in fact been used for a long time as descriptors of the violin in the literature [6, 7]. It has even been stated, without hard evidence though, that the quality of the violin correlates with some spectral features of the bridge admittance. In particular, the presence of the so-called bridge hill has been hailed as a mark of quality. According to [7], this bridge hill is \"a hump [sic] between 2 and 3 kHz in their frequency responses\". Despite the importance of this feature, the lack of a scientifically grounded definition makes its use in research questionable. Our objective in this article is to produce physically informed and objectively measurable features from the bridge admittance that allow one to compare, and maybe eventually discriminate, good violins from great ones. At the very least, have a standard way of comparing them beyond the mere visual inspection, which seems to be the preferred way in the state of the art [5]. To compare different FRFs one needs to understand their physical provenance. Woodhouse [8] divides the signal into three separate regions. In the low range, the influence of the individual modes of the instrument is clearly visible in the signal as peaks. Eventually, as the frequency of the modes becomes higher, the damping factor makes the peaks blend together and a clear identification of the modes is not possible: these are the transitional modes. At high frequency, in fact, the mode overlap becomes greater, and even small differences cause a great variability of the mode shapes between instruments. Although the modes still determine the behavior of the FRF, analytical approaches to model it becomes progressively less effective [9]. Different instruments have different resonance frequencies. These differences are due to material and geometric variation between the instruments [10]. Therefore, a direct comparison in the frequency domain is not immediately applicable. If two instruments have the same FRFs but shifted in a few Hertz, does that make them more similar than a pair of FRFs where the modes are at the same frequency but their amplitude is completely different? Developing a way to answer those kinds of questions is the aim of this article. Following the separation highlighted by [PERSON], one can consider different models for each frequency range of the FRF. Modal approaches are revealed as powerful reconstructions of the individual (also denoted as \"signature\") and transitional modes. Indeed, the bridge can be interpreted as a filter coupling strings and violin soundbox such that its admittance can be seen as the filter transfer function [11]. Accordingly, the fundamental properties of modes, i.e., frequency, amplitude, and damping ratio, can be used to define the poles and zeros of such filter.