To formally determine the degree to which instruments are separable from each other based on body shape, linear discriminants were used to predict instrument class ( Table 1 ). Allocation of instrument types reveals that most violins, cellos, and double basses are distinguishable from each other. However, only a fraction of violas (27.8%) are correctly predicted as such, and a majority (62.7%) is wrongly predicted to be violins. The data corroborates common knowledge that, although on average larger than violins, violas are often nearly identical in their shape to them. Nonetheless, a small subset of violas exhibit distinguishable shapes ( Fig. 2C ; Table 1 ). Historically, the viola shape and size is non-standardized, and a variety of new shapes, to accommodate playability and reduce injury in players with instruments that are too large (e.g., see the Pellegrina viola, Fig. 1A ), are currently being designed [19] .

A) Scatterplot of the separation by linear discriminants (LDs) 1 and 2, providing 66.4% and 30.8% of total instrument separation, respectively. Vertical line indicates the LD1 values separating all double basses from other instrument types. B) Histogram of LD2 values (30.8% of separation), which largely separate violins and cellos. No cellos have LD1 values less than the LD1 value of the left line, and no violins have LD1 values greater than the LD1 value of the right line. C) Histogram of LD3 values (2.8% of separation), which differentiate some violas from violins and cellos. The indicated tail of the viola LD3 distribution does not include violins or cellos. Note: for both panels B) and C) , double basses are not shown to better focus on violin, viola, and cello distributions. D) Pairwise thin plate splines, using grids to show the deformations necessary to transform reference instrument outlines (vertical) into targets (horizontal). Mean outlines of instruments are overlaid and colored to indicate type. Violins, teal; violas, magenta; cellos, burnt orange; double bases, lavender.

To determine the extent that different instrument types are distinguishable from each other, a Linear Discriminant Analysis (LDA), maximizing separation of instrument shapes based on harmonic coefficients, was performed ( Fig. 2A–C ; Dataset S3 ). The first linear discriminant (LD1) explains 66.4% of instrument type separation and mainly differentiates doubles basses, which qualitatively have a distinct and highly variable shape, from other types ( Fig. 2A ). LD2, explaining 30.8% of separation, mainly separates violin shape from that of cellos ( Fig. 2B ) and LD3 explains only 2.8% of separation by type, and is capable of only distinguishing some violas from violins and cellos ( Fig. 2C ). Thin plate splines, which deform a grid so that a reference shape matches a target, can be used to qualitatively analyze the shape characteristics unique to each type ( Fig. 2D ). Violas are wider in the lower bout than violins, whereas in cellos the center bout is displaced more distally and the upper bout narrowed. The shape of double basses is immensely different from other types, with a much wider lower bout, tapered shoulders, and a distally displaced center bout.

Schelleng has noted that the problem of scaling instrument size to accommodate different ranges is theoretically possible by maintaining all dimensions and using identical materials [11] . Practically, this is impossible. Because the average human dimensions do not change relative to instrument type, simply building a larger violin-to-scale instead of existing viola, cello, and double bass shapes would significantly impact playability, be limited by player stamina, and potentially increase player injury. This is particularly true for the viola, which ideally would be a larger size and played between the legs to accommodate its range, but because of the tradition of playing on the shoulder, is scaled inappropriately, sometimes compromising tone quality [19] .

>9,000 body outlines of violins, violas, cellos, and double basses were obtained from iconography collected from various sources through cozio.com (Tarisio Auctions). Instruments not belonging to the violin family, such as members of the lira da braccio and viola da gamba families, and experimental and oddly shaped instruments, were not included in the analysis ( Fig. 1A ). An Elliptical Fourier Descriptor analysis was used to measure the shape of violin family members ( Figs. 1B, S 1 ) [13] – [18] and Principal Component Analysis (PCA) performed to visualize patterns of variance ( Fig. 1C ; Datasets S1 , S2 ). The resulting “eigenviolins” describe shape variations among instrument body outlines. Together, the first four principal components (PCs) describe approximately 77.6% of the measured shape variance ( Figs. 1C, S 2 ). PC2, which describes a pattern of shape variance related to the ratios in width of the upper and lower bouts and the proximal-distal placement of the center bout ( Fig. 1C ) separates instrument types by their range (i.e., violins and violas have lower PC2 values and cellos and doubles basses higher PC2 values) ( Fig. 1D ). Double basses, which have viola da gamba-esque tapering shoulders and c-shaped center bouts, show separation from other instruments types by PC1 (describing instrument width) and PC3 (which describes the shallowness of the center bout) ( Fig. 1D–E ).

The evolution of violin shape over time

The 16th century was an innovative time in the evolution of Western string instrument shape. As previously mentioned, the violin family likely arose from luthiers in Brescia, using elements from popular string instruments of the 1500s (including the rebec, vielle, and the lira da braccio) (Fig. 1A) [1]–[4]. Although violin design is always improving and changing, the overall features, and especially the body shape we associate with violins today, arose as early as the mid-16th century. Has the shape of violins remained stagnant since this time, or like other complex morphological phenomena, has it been evolving over the course of four centuries?

To answer this question, I consider only violin outlines for the remainder of this study, which with >7,000 samples, dominates the dataset relative to other instrument types. This dataset is derived from the iconography of auction houses, and so therefore encompasses the most highly desirable violins, but also those of historical importance. An advantage of studying instrument outlines is the increased sampling it allows. For example, works studying the material, physical, or psycho-acoustical properties of Cermonese instruments are often limited to a few or handful of instruments because of the preciousness of the material [20]–[26]. Shape, as derived from photos in this work, presents no such sampling limitations.

It is helpful to understand the structure of the dataset with respect to luthier, date of manufacture, and locale, which generally follows the history of violin making in Europe (Fig. 3) [27]–[29]. Cermonese instruments dominate the dataset, but only until around 1750, after which other Italian schools of violin making rise, including Milan, Naples, Venice, and Turin, as well as those outside Italy, such as Paris and London (Fig. 3A–C). Instruments from these cities are often associated with distinct periods of history. Some of this structure is due to the fact that only a handful of luthiers often contribute to a city's output (Fig. 3D). MANOVA modeling was used to determine the significance of luthier, year, and city covariates in explaining harmonics coefficients of violin outlines. The final model included luthier and year as significant explanatory variables (Table 2; Dataset S4). Country was not significant, but only if luthier is the first factor, reflecting the unbalanced design and dependence of factors (i.e., luthiers mostly come from a single city and each city is composed of a small number of luthiers). For these reasons, I chose to ignore city in this particular model and focus on luthiers and time.

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larger image TIFF original image Download: Figure 3. The historical and geographic context of luthiers and their violins. A) Geography of violin production in Europe. Overlaid on a map, circle color and location indicates cities of production and size is proportional to the violin output represented in this dataset. B) Stacked histrogram of violin production by year. Colors indicate city of manufacture. C) Same data as in B), but scaled to show the proportional output of each city by year. D) Output of prolific luthiers (with>45 violins in the dataset) over historical time. Points correspond to violins and the year of production, colored by the city of production. Luthiers are organized temporally, by the mean year of their violins represented in the dataset. Cremona, red; Naples, blue; Milan, green; Paris, purple; Venice, orange; Turin, yellow; Mantua, brown; Florence, pink; London, grey; other cities, black. https://doi.org/10.1371/journal.pone.0109229.g003

Linear Discriminant Analysis (LDA) was used to separate luthiers by the shape attributes that most distinguish them (Fig. 4A; Datasets S5, S6). A more important question is what factors influence the shapes of violins that most distinguish their makers. The most obvious factor, available in this dataset, is time. Are the shapes that distinguish individual makers under the control of a higher influence, such as their place in history? To answer this question, linear discriminants for the outlines of violins produced by each luthier were averaged and correlated with the averaged age of instruments for each maker (Fig. 4B; Dataset S5). After multiple test adjustment, only three LDs were significantly correlated with year, and LD1 (explaining 9.4% of separation between >400 luthiers) exhibited exceptional correlation with time (rho = −0.61, p = 1.04×10−38) (Fig. 4C).

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larger image TIFF original image Download: Figure 4. Linear discriminants of luthier correlated with time. A) Percent separation contributed by linear discriminants (LDs) 1–85, separating violin outlines by luthier. Averaged LDs by luthier were correlated with the mean year of manufacture for luthiers. Those LDs significantly correlated with time are indicated in red. B) Scatterplot of Spearman's rho (x-axis) and -log 10 p-values (y-axis) for averaged LD values by luthier correlated with mean year of luthier production. Circles, labels, and red indicate significant LD correlation with time. Note that LD1 (9.4% separation) is exceptionally correlated with time. C) Scatterplot showing correlation of averaged LD1 values for luthiers with average year of luthier violin production. rho and p values and indicated. All p values shown in this figure are multiple test adjusted across LDs to control false discovery rate (FDR) using the Benjamini-Hochberg (BH) method. https://doi.org/10.1371/journal.pone.0109229.g004

Visualizing LD1 values of individual violins (Fig. 5A) and prolific luthiers (defined by>45 violins) (Fig. 5B) over time, and comparing with historical accounts of violin making, can offer insights into why this particular shape attribute is temporally modulated. Much of the correlation of LD1 with time seems to be attributable to extreme values before ∼1650 and after ∼1800. Instruments made before 1650 have exceptionally high LD1 values and are almost exclusively derived from Brescian luthiers (e.g., Giovanni Paolo Maggini, Fig. 5), representing the first violins. Interestingly, the instruments of Eugenio Degani and his son Giulio Degani at the beginning of the 20th century have anachronistically high LD1 values, perhaps suggesting the Brescian school influenced them. The first luthier to innovate a violin with the opposite extreme of LD1 values was Antonio Stradivari of the Cremonese school. This is noteworthy for two reasons: 1) low LD1 values uniquely define A. Stradivari from his contemporaries, which is important for the identification of violins from this period because of their desirability, and 2) the documented influence of A. Stradivari on subsequent luthiers provides a hypothesis that the low LD1 values in violins after ∼1800 may arise from imitation.

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larger image TIFF original image Download: Figure 5. The contributions of luthiers to the correlation of violin shape attributes with time. A) Scatterplot showing individual violins with LD1 values (9.4%) plotted against year. Colors indicate violins produced by select luthiers. B) Similar to A), showing boxplots of LD1 values of violins produced by prolithic luthiers. Luthiers are arranged temporally along the x-axis by the average year of the violins they produced. See text for details for the relationships of luthiers to each other and known copying of violin design. Purple, Giovanni Maggini; red, Antonio Stradivari; yellow, Nicolas Lupot; orange, Jean-Baptiste Vuillaume; blue, Eugenio Degani; green, Giulio Degani; black, other. https://doi.org/10.1371/journal.pone.0109229.g005

Two of the most famous luthiers that began the trend of low LD1 values after 1800 are the known Stradivari copyists Nicolas Lupot and Jean-Baptiste Vuillaume of Paris (Fig. 5). Hart, in his The Violin: Its Famous Makers and Their Imitators [27], not only declares N. Lupot the “French Stradivarius” but says, “Stradivari was his idol, and from the fact already mentioned, that he is very rarely found to have followed any other model than that of Stradivari, he would seem to have been aware of his own peculiar fitness for the great master's design.” The violins of J.B. Vuillaume may even have been more influential than Lupot in disseminating the Stradivari shape attribute around the world. His purposeful imitation of Stradivari was profit-driven [28]:

Of all the great Italian masters of violin-making, Stradivari was always his ideal, and by constant study, and cultivation of his own rare natural powers of observation, he acquired such an intimate knowledge and judgment of Stardivari's work in every detail, that he might almost be said to be better acquainted with the maker's instruments than the master himself. Vuillaume soon found the sale of violins, issued as new works without any semblance of antiquity, an unprofitable undertaking, and, recognizing the growing demand in all parts of the world for instruments resembling the great works of Cremona, he determined to apply his great skill as a workman, and his extraordinary familiarity with Stradivari's models, to the construction of faithful copies of the great maker's works. This was the foundation of his success, for the modern copies found a ready sale, and orders poured in upon Vuillaume from all parts of the world.

Although LD1 is exceptionally correlated with time, it still represents only a fraction (9.4%) of the total separation of violin shape by luthier (Fig. 4). Using linear discriminants to predict luthier is a particularly relevant way to detect imitation while using all available separation (Table 3; Dataset S7). Of the prolific luthiers, Giovanni Paolo Maggini (the only early Brescian in this group) is one of the most discernable, with 78.6% of his instruments being correctly reassigned to him. Instruments by Eugenio Degani, with the anachronistically high LD1 values (Fig. 5) are also relatively distinguishable at 63.5% correctly reallocated violins. Nicolas Lupot, the known copyist previously mentioned, has one of the lowest correct reallocation rates (19.5%). But the lowest reassignment among prolific luthiers is Vincenzo Trusiano Panormo, with only 8.2% correctly reallocated instruments. Hart describes both N. Lupot and V.T. Panormo as the “faithful copyists” of A. Stradivari [27]:

Panormo and Lupot share the palm as the faithful copyists of the great Cremonese master. Neither appears to have attempted to create a model of his own; their sole aim was to imitate to their utmost the various patterns of Stradivarius, Guarnerius, and Amati, but they principally confined themselves to those of Stradivarius.

An analysis of instrument shape by luthier indicates that specific shape attributes are highly correlated with time (Figs. 4–5). Detailed analysis of the discernibility of shapes from different luthiers and historical accounts suggest widespread copying (Table 3; Dataset S7), particularly of A. Stradivari, contributing to the temporal structure of shape variance and the evolution of the modern violin’s outline.