Pattern Recognition Approach to Violin Shapes of MIMO database

Thomas Peron1,∗ Francisco A. Rodrigues1, and Luciano da F. Costa21Institute of Mathematics and Computer Science,

University of Sao Paulo, Sao Carlos, SP, Brazil and2Sao Carlos Institute of Physics, University of Sao Paulo, Sao Carlos, SP, Brazil

Since the landmarks established by the Cremonese school in the 16th century, the history of violindesign has been marked by experimentation. While great effort has been invested since the early19th century by the scientific community on researching violin acoustics, substantially less attentionhas been given to the statistical characterization of how the violin shape evolved over time. In thispaper we study the morphology of violins retrieved from the Musical Instrument Museums Online(MIMO) database – the largest freely accessible platform providing information about instrumentsheld in public museums. From the violin images, we derive a set of measurements that reflectrelevant geometrical features of the instruments. The application of Principal Component Analysis(PCA) uncovered similarities between violin makers and their respective copyists, as well as amongluthiers belonging to the same family lineage, in the context of historical narrative. Combined witha time-windowed approach, thin plate splines visualizations revealed that the average violin outlinehas remained mostly stable over time, not adhering to any particular trends of design across differentperiods in music history.

I. INTRODUCTION

Music is among the most abstract of the arts, whichcan be a reason for its widespread appreciation. As such,it becomes specially difficult to identify, compare and dis-cuss its content, properties, quality, and relationships.Compare it with, for instance, painting. In this case, itbecomes possible to devise objective approaches to de-velop all the aforementioned analyses (e.g. how similarto a Brussels griffon is the dog in van Eyck Arnolfini’sportrait?). Used for music production, musical instru-ments have varying properties such as resonances, tessi-ture, harmonic content, and so on. Even though the finalappreciation of a given instrument, as well as its playabil-ity, depends on subjective aspects of human perceptionand cognition, much can be learnt about instruments byapplying the scientific method to them, which involvesmeasuring, analysing and modeling their physical prop-erties [1].

One of the most known classical instruments, whichoften appears in most genres of music, is the violin. Itseems to have been derived from the bowed instrumentsfrom the equestrian cultures in Asia and undergone a longhistory of changes and adaptations, reaching its modernform in Europe, particularly northern Italy, especiallywith Stradivarius and Guarneri [2–4]. In a violin, thesound energy is produced by the friction between the bowand the strings, then reaching the resonance box throughthe bridge, where it is transferred to the surrounding airby the plate vibration. Several elements contribute tothe sound properties and quality of a violin, includingthe employed materials, the bow and strings, and thethree-dimensional shape and dimensions of the body andarm [1]. In particular, the resonances are strongly related

∗ thomaskaue@gmail.com

to the geometric Chladni patterns that are formed in theplate and body of the violin for different fundamentalfrequencies [3, 5].

The history of violin research encompasses many disci-plines related to different aspects of the instrument, suchas vibration analysis [1, 3], material science [6–8], andpsychoacoustics [9, 10]. One of the earliest scientists toperform experiments with the violin was Felix Savart [3].Among his many contributions are the first studies on thevibration modes, including the first reports on applyingthe method by Chladni in order to reveal the nodal andanti-nodal regions in violin plates for different resonancefrequencies. Many other prominent physicists were alsoinvolved with the research of bowed string instruments.For instance, it was Helmholtz who first uncovered thecharacteristic wave form produced by a string under theaction of a bow [1, 3]. His work strongly influenced LordRayleigh, who made several breakthroughs on vibrationalanalysis and laid the foundation for modern experimentaland theoretical acoustics [3]. In the early 20th century,Raman carried out remarkably detailed experiments withviolins and cello strings [3, 11]. Not only the Indian scien-tist shed light on the interplay between many parametersassociated to violin playability (distance of the bow tothe bridge, bow speed, bowing force, etc.), but also elu-cidated some aspects of the so-called “wolf-note” [3, 12].For additional information on the history of violin re-search we refer the reader to [1, 3, 13] and referencestherein.

Probably one of the reasons for the great interest, in-cluding the scientific community, on the violin is thefascination surrounding famous luthiers like Stradivariand Guarneri, who were able to craft remarkable instru-ments. However, despite all the efforts by the mentionedscientists and others in trying to understand the physi-cal properties of the violin, one aspect has received rela-tively little attention along time: the shape of the instru-ment’s body. As remarked by Chitwood [14], the violin

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1600 1650 1700 1750 1800 1850 1900 1950 2000

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FIG. 1. Number of violins over time in the database used inthis paper.

shape has been intentionally ignored in modern acousticresearch, in part due to experimental practicity. Indeed,many of the early experiments with bowed strings wereperformed with the strings isolated from the instruments.

These simplifications regarding the violin shape con-strained more accurate modeling of physical properties ofviolins. Indeed, it remains an unanswered question howmuch shape differences can impact respective acousticalproperties. Another question that naturally arises in thiscontext is how distinguishable instruments fabricated bydifferent luthiers can be. In other words, given a set of vi-olins, is it possible to infer the authorship based solely onfeatures extracted from the body outline? Furthermore,is it possible to infer, based on shape, how a given luthieror school influenced other instrument makers? Recently,a step forward in answering these questions was taken byChitwood [14]. This author carried out a study which, tothe best of our knowledge, is the first to systematicallyanalyze a large dataset of violin shapes. By employingpattern recognition techniques, it was found that the de-sign of violins indeed correlates with the historical timeline. Hierarchical clustering also revealed that famousluthiers tend to group in the same clusters with theirrespective copyists [14].

In this paper, we extend some concepts in the method-ology of [14] in order to morphologically analyze the vi-olin shapes retrieved from another database, namely theMusical Instrument Museums Online (MIMO) [15]. Theinterest in this database resides in the fact that it con-sists in the largest freely accessible platform that providesinformation about instruments held in public museums.Therefore, the availability of the data and the signifi-cance of the collections provide motivation for a thoroughstatistical characterization and systematic comparison ofthe violins held in the museums associated to the MIMOplatform. In order to accomplish this task, we define aset of measures that reflect important geometrical fea-tures of the violin instrument. After defining these mor-phological indicators, the distribution of violins in the at-tribute space is obtained via Principal Component Anal-

ysis (PCA), whereby a significant relationship betweenluthiers and their respective copyists is revealed. More-over, low statistical variability is found among violinsfabricated by luthiers belonging to the same family lin-eage. Interesting results are also reported concerning thecountry of fabrication. In particular, great dispersion inthe PCA space is observed for violins by French makers,whereas instruments fabricated in other countries tendexhibit higher levels of clustering. Time-series analysisof the MIMO database were also carried out in order tocharacterize the evolution of the geometrical features ofviolins across different periods in classical music. In par-ticular, by employing a time-windowed approach alongwith a thin plate splines method [16], we show that theaverage violin shape has remained predominantly stableover time, in agreement with results derived with PCA.

This paper is organized as follows: In Sec. II we de-scribe in detail the MIMO database and the criteria usedto filter the violin images used in our study. The fea-tures to be extracted from the images are also definedin this section. Sec. III is devoted to the discussion onthe statistical similarity of violin parts. Sec. IV showsthe results concerning PCA, followed by Sec. V, whichcontemplates the analysis of the temporal evolution ofviolin measures as well as the time-evolving morphingvia thin-plate splines technique. Finally, our conclusionsare presented in Sec. VI.

II. MATERIALS AND METHODS

This section first describes in detail of the contentdisplayed in the MIMO database as well as the crite-ria used to filter the images. Subsequently, we presentthe methodology proposed for the image processing andfeatures extraction.

A. The MIMO database and information filtering

The Musical Instrument Museums Online (MIMO) [15]is a platform that provides free and easy access to col-lections of musical instruments in important Europeanmuseums. The information provided for each instrumentcomprises, for instance, year of fabrication, place of pro-duction, maker, museum, and images of the instrument.In this paper, we analyze images retrieved by using “vio-lin” as keyword in the MIMO’s website. The respectivelyobtained instruments are filtered so that only images ofnon-rotated instruments are selected. A total of 726 vi-olins out of 1300 instruments was chosen for this work,spanning the interval between 1600-2003.

Some violins in the database have some of the abovementioned fields missing. Figure 1 shows the distribu-tion along time of the 658 violins incorporating year offabriation. The remaining elements either have impre-cise data, e.g “violin fabricated in the 18th century”, orno year is provided at all. Other information, such as in-

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Instrument maker Number of violinsCarl Friedrich Hopf 12

Otto Glass 12Jerome Thibouville 9

Jean-Baptiste Vuillaume 8Carleen Hutchins 7

Others 426Unknown 252

TABLE I. Table showing the most frequent violin makers inthe MIMO database.

Country Number of violinsFrance 233

Germany 232Italy 37

Belgium 17Czech Republic 13

Others 38Unknown 156

TABLE II. Countries of fabrication of violins in the MIMOdatabase.

strument maker and place of production, are also missingsometimes. Tables I and II show the most common coun-tries of fabrication and makers along with the respectivenumber of elements with missing data.

B. Violin Shape Measurements

One of our goals is to comparatively analyze physi-cal properties of violins as estimated from their respec-tive images (see Fig. 2 for an example of contour extrac-tion). To this end, we first need to define, given a violincontour, what measures need to be derived. In partic-ular, good measurements should be able to capture themost relevant geometrical features of the violins, allowinga more objective and precise analysis and comparison.Natural and more direct/intuitive measurement candi-dates include the lengths of parts of violins. Moreover,curvature-based measures also provide a particularly [16]powerful way to represent and characterize shapes: it isinvariant to rotation and translation, it preserves most ofshape information (is invertible up to a rigid-body trans-formation), can be used to detect corners, and is com-patible with human intuition. Curvature is, therefore,also considered in this work. We can thus organize themeasurements adopted here into two major groups: (a)length-based and (b) curvature-based measures.

FIG. 2. Example of contour extraction. (left) Original im-age in gray-scale and (right) extracted contour. Violin imagereprinted with permission by the Musikinstrumentenmuseumsder Universitat Leipzig.

1. Length-Based Measurements and control pointsdetermination

A basic and natural way to describe a violin is bymeans of the length of its parts. Indeed, the length ofan instrument can be a signature of a particular epochor luthier [3, 17]. For instance, violins of the Barroqueperiod are significantly shorter than ones from the 19thcentury, period at which the instruments were lengthenedso as to improve the sound power output in larger con-cert halls [3]. In order to capture such structural changesin the processing of the images, the first set of measure-ments we adopt is based on the length of different violinsections. Figure 3 illustrates in detail the length mea-surements to be used for violin characterization.

One important first step in obtaining length-basedmeasurements consists in defining reference points thatcan be used to organize the overal violin outline. Thesepoints, henceforth called control points, are shown inFig 3 as AL, AR, BL, BR. It turns out that the controlpoints are points in the violin contour with significantlyhigher curvature value than the rest, as exemplified inFig. 4, where we show a typical contour extracted froma violin image.

The values a to f quantify the extension of the up-per, central and lower parts of the instrument; h1 andh2 stand for the width between respective control points;` is the size of the region encompassing the neck, peg-

4

AR

BR

AL

BL

QRQL

FIG. 3. Definition of the measures extracted from the violincontours. Dashed line in the violin contour represents the partwhich encompasses the fingerboard, pegbox, tuning pegs andscroll and which is not considered in the analysis. Symbolsγi (i = 1, ..., 6) denote the contour segments defined by thecontrol points.

box and the scroll; and L corresponds to the total violinlength.

2. Curvature-Based Measurements

Another relevant aspect that differentiates one instru-ment from the other is the shape itself which is, in prin-ciple, a dimensionless property. Therefore, in order tocharacterize geometrical variations in violin contours, wealso calculate the average curvature values si (see Ap-pendix A) for each of the segments γi defined in Fig. 3.

Once the control points (denoted as AL, AR, BL, andBR in Fig. 4) have been identified via the detection ofthe first two (AR and BR) and last two (BL and AL)peaks in the curve of |s| (see Fig. 4(b)), the coefficientsa to f can then be straightforwardly calculated. How-ever, in order to calculate the average curvature valuesof the segments in the violin contour, we need to identifytwo other points, namely QR and QL depicted in Figs. 3

ARBR

BL AL

QR

QL

AR BR BLALQR QL

FIG. 4. (a) Violin contour and (b) its curvature |s| (see Ap-pendix A, Eq. A1). The control points (AR, BR, AL andBL) are depicted in panel (a) with their associated curvaturepeaks in (b). Dashed part in the contour in (a) representsthe fingerboard, pegbox and scroll parts, which are neglectedin the study. Shaded area in (b) corresponds to the curva-ture values associated with the dashed segment in (a). Thecontour used in (a) belongs to a violin by Franz Zucker [18].

and 4. These points mark the position at which the fin-gerboard projects outside of the violin body. Fortunately,as we can see in Fig. 4(b), points QR and QL also havea well defined signature in the curve of |s|, i.e. they cor-respond to the third and fourth peaks along |s|. As itis seen in Fig. 4(b), the contour segment between pointsQR and QL is not considered in the extraction of anymeasurement because of the difficulty to have the tuningpegs in a standard position and scroll shape variations.Thus, in order to not introduce spurious effects, in theanalyses that follow, we discard the curvatures associatedto the dashed segment in the violin contour in Fig. 4(a).

III. DISCUSSION

A. Relationships between violin parts

We start our analysis by characterizing the statisticalsimilarity between different parts of the violins. Moreprecisely, for each image in the database, we extract allthe measurements depicted in Fig. 3 and calculate thecorrelation coefficients between every pair of measure-ment. The resulting correlation map is shown in Fig. 5.These results reveal interesting patterns of relationshipsregarding the instruments structural organization. Inparticular, relevant information can be extracted aboutthe inter-relation between blocks identified in Fig. 5. Forinstance, block A, corresponding to the symmetric pair of

5

A B C D curvatures

A

B

C

D

curv

atur

es

FIG. 5. Correlation map of the measures defined Fig. 3 con-sidering all violins in the database.

measurements a and d, tends to be little correlated withother body features. On the other hand, measurements inblocks B (measurements c and f ) and C (measurements band e) correlate more strongly with the remainder mea-surements. This shows that the central ribs of violinsare a more independent feature of the instrument, withrelative dimensions conserved among instruments irre-spectively of other features. Still regarding blocks B andC, they present moderate correlation one another, creat-ing a larger cluster with two subgroups. This suggeststhat the upper and lower ribs of the instruments are in-trinsically interrelated, defining the main outline of theinstrument body, with the central ribs acting more as akind of detail to the overall shape. However, it is unclearif such phenomena are more a consequence of aestheticor technical constraints.

Of particular interest is the significantly higher valuesof correlation between the neck length ` and blocks Band C. In fact, ` is also strongly related with D (cor-responding to measurements h1 and h2) as well. Such apeculiar characteristic of ` is reflected in the red line inFig. 5 extending along most of the blocks defined by therelative dimensions. As ` corresponds to the length ofthe neck, this result indicates that the neck length workstogether with the other body dimensions for aesthetic,structural or musical reasons. As such, the neck lengthassumes particular importance, defining a reference fromwhich the other dimensions can be derived.

Despite the fact that the measurements belonging togroup A (i.e. a and d) are relatively more independentfrom the rest, the strong correlations values between theother groups of measurements indicate that the other di-mensions preserve an allometric relationship. There isalso a trend towards preserving the aspect ratio (ratiosbetween length and width) of the instruments, as ex-

pressed by dependence between ` with h1 and h2. Inother words, longer necks require larger separations be-tween the left and right sides of the violin, so that thescale is preserved.

The curvatures block in Fig. 5 is formed by the meancurvature values of the segments γi (i = 1, ..., 6) markedin Fig. 3. We have that all measurements in this block areintercorrelated in a surprisingly strong fashion. Proba-bly, this intense interrelationship between the curvaturesis a consequence of a combination of functional, struc-tural and aesthetics constraints.

IV. PRINCIPAL COMPONENT ANALYSISAND CLASSIFICATION OF VIOLINS

In this section we characterize the distribution of vio-lins in the feature space by employing PCA. Importantly,in order to have similar ranges of variations, all measure-ments were normalized prior the application of PCA.

Figure 6 shows the projection of the 728 violins in thespace spanned by the two most significant PCA axes con-sidering all measurements. A great part of the violinensemble gives rise to the large cluster at the origin ofthe coordinate system. Such an agglomeration suggestsa weak geometric variability across the violins belongingto this central group. This is somewhat expected due tothe fact that violin making is based on established in-struments dimension standards [19]. Thus, since manyinstruments are constructed under similar guidelines, itis natural to observe such a clustered structure in thePCA space. Nonetheless, some outliers can still be ob-served in Fig. 6, particularly at significantly high valuesof PCA 1 and/or 2. In these extreme are instrumentswith unusual relative widths, such as violins 4 and 7;and peculiar shapes as is the case of violins 1 and 3.

In order to try identifying some reason accounting forthe dispersion of instruments in Fig. 6, in Fig. 7(a) wevisualize the violins according to the their country of fab-rication. Only instruments containing country informa-tion (572 in total) could be included in this plot, mostof which originate from France, Germany, and Italy, inwhich important makers such as N. Amati, A. Stradi-vari, J. Vuillaume developed their career. In addition,reference production and commerce centers were locatedin such countries, such as Mittenwald in Germany, Mire-court in France, and Cremona in Italy [20, 21]. Inter-estingly, as we can see in Fig. 7(a), all the outliers arefound to belong to these three countries. Instrumentsfrom other countries tended to overlap violins from thethree aforementioned countries, suggesting that Euro-pean violin design and making could have been heavilyinfluenced by trends originated in France, Germany andItaly. It is noteworthy observing that the largest dis-persion of instruments in the PCA space corresponds toFrench violins, suggesting greater diversity of design ex-perimentation or co-existence of more than one dominantset of design rules.

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FIG. 6. PCA projection of the violins considering the whole violin database of this study. Each point in the plot correspondsto an instrument. Violin images reprinted with permission from the Musikinstrumentenmuseums der Universitat Leipzig, Citede la Musique – Philharmonie de Paris, and Germanisches Nationalmuseum.

(a) (b)

FIG. 7. PCA projections of the violins considering (a) only violins with information on country of fabrication, and (d) onlyviolins with data of fabrication data. Each point in the plots correspond to an instrument.

Figure 7(c) depicts the PCA projection annotated ac- cording to the historical periods in classical music. As it

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V1

S1

A2 S5V2

S4

S3

S2

1690 Strad

FIG. 8. PCA projections of the violins considering only violins fabricated by the seven most frequent makers in the database.Each point in the plots correspond to an instrument. Violin images reprinted with permission from the Musikinstrumentenmuse-ums der Universitat Leipzig, Galleria dell’Accademia di Firenze, Cite de la Musique – Philharmonie de Paris, and GermanischesNationalmuseum.

can be seen, the vast majority of the violins in the cen-tral cluster belongs to Baroque, Classical and Romanticperiods. Interestingly, more recent instruments from theImpressionist and Modern periods resulted mostly con-centrated at the core of the central cluster. Figure 7(c)also implies that violins fabricated after the XIX centuryhave geometrical features closely resembling Baroque andClassical instruments. At first sight, this could be under-stood as a surprising result, contrasting to the intenseexperimentation that is characteristic of the modern pe-riod. However, this is more likely a consequence of thefact that a great deal of the music performed at that time,which witnessed wide popularization of music, ultimatelyoriginated from the previous periods.

Finally, Fig. 8 presents the PCA space with the violinsof the eight most frequent violin makers in the MIMOdatabase. A good relationship can be observed betweenthe diverse makers and groups of instrument, showingthat it is indeed meaningful to study the violin datasetin terms of different types of authorial designs. Notice,for instance, the consistence of the group formed by theHopf Family. The Hopf Family is a lineage of violin mak-ers that can be traced back as far as eight generations andthat was initiated by Gaspar Hopf (1650-1711)[21]. He isconsidered one of the founding father of the early luthiertradition in Germany [21]. In the MIMO database, mostof the violins belonging to the Hopf group are labeledsimply as “Hopf”, without mention to any specific fam-

ily member, except for two violins attributed to CarlFriedrich Hopf [22].

It is worth dedicating some attention to the Stradi-vari’s violins, which occupy a central position in thePCA space, strongly overlapping other groups. Addi-tional insights can be gained by observing the resultsof Fig. 8 in the light of important landmarks along thehistory of these famous instruments. For instance, it isknown that between the years 1656–1684, Stradivari wasan apprentice to Nicola Amati’s workshop in Cremona,Italy [17]. In this period (oftentimes referred as Ama-tise period [17]), Stradivari developed violins following“Amati’s grand pattern” [17]. Remarkably, by inspect-ing Fig. 8 one finds that the Stradivari instrument closestto an Amati is a violin dated from 1690 (see Fig. 8). Theother Strads in the PCA space belong to periods laterthan 1690, the year in which Stradivari sought to departfrom Amati’s influence, and began the production of theso-called Long Strads [17].

The Long Strads have significantly larger patterns thanthe traditional styles of the Cremona school, being sodesigned in order to meet other demands of power andtonality [3, 17]. Eventually, Stradivari started producingsmall violins in addition to other sizes. One example isviolin S1 in the lower left part of the cluster in Fig. 8.This violin is significantly smaller than the other instru-ments by Stradivari (compare it, for instance, with S5):S1 has a total length of 53.5 cm, while the other Strads

8

Barroque

Classical

Romantic

Impressionist

FIG. 9. Temporal evolution of violin measures (see Fig. 3 for measures definition). (a) Average central length 〈a〉, (b) averagedistance 〈h1〉 between control points BL and BR, and (c) average curvature 〈s〉 of violin contours. The averages are calculatedby considering time-windows with length ∆t = 20 years (see text for details) and sliding step δt = 1 year.

have lengths varying between 58.8-59.7 cm. This reduc-tion in size was followed by changes in the proportionalmeasurements as indicated in Fig. 3. For instance, S1has a = 0.26, whereas the remaining Strads in Fig. 8have measure a in the range a ∈ [0.21, 0.22].

A further consideration is worth regarding the rela-tionship between Stradivari’s and J. Vuillamme’s instru-ments. Observe that most part of violins by these twomakers resulted nearby one another in the PCA space ofFig. 8. This result is particularly interesting as it con-firms in a quantitative way the well-documented influ-ence of Stradivari on Vuillaume [21]. In fact, not onlythe latter is known to have been a prolific Strad copyist,but he also produced remarkable replicas of other violinsbelonging to the Cremona school. Some of these copieswere hardly distinguishable from the originals, even byhighly skillful violinists such as Niccolo Paganini, whoonce was unable to distinguish between a violin fromthe Cremonese school and its counterfeit crafted by Vuil-laume [21]. The proximity of the points in the PCA thusagrees with the historical narratives of the mentionedluthiers.

Another relevant point in the analysis of violin shapesconcerns the importance of each measurement in pre-dicting characteristics such as country of fabrication andmaker. This task can be addressed by using RandomForest Classifiers [23, 24]. First, we apply this techniqueto the violin data used in the PCA projection visualizedin terms of countries (Fig. 7(a)). By randomly splittingthe data into 70% for training and 30% for testing, we areable to achieve 57% of accuracy in predicting the countryof fabrication, whereas the random case returns 16% ofcorrect assignment. By repeating the procedure for thedata of Fig. 8, we get 44% of accuracy in the predictionof violin makers, over 12% in the random case.

V. TEMPORAL ANALYSIS AND CONTOURMORPHING BY THIN PLATE SPLINES

The PCA projection in Fig. 7(b) reveals the statisticalvariability in the violin design across epochs in classicalmusic history. However, although this result quantifieshow different instruments fabricated in distinct periods

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(a) (b) (c)

FIG. 10. Morphing of a violin fabricated in the year of 1619 to the average shapes of the time windows centered at (a) 1655,(b) 1859, and (c) 1950.

can be, it does not tell us exactly how the violin bodychanged over time. In this section we analyze the evo-lution of the violin shape by employing two approaches,namely (i) the visualization of the time series associatedto the measures defined in Sec. II, and (ii) the study ofthe morphing of contours by using the thin-plate splinesmethod [16].

The oldest violin in the MIMO database goes back to1575, whereas the newest one was fabricated in 2003. Ourgoal here is to investigate changes in the average length-and curvature-based measurements, corresponding to a“characteristic violin”, along the above time spam. Tothis end, we make use of a sliding time window tech-nique. The adoption of windowing is necessary in or-der to provide a prototypic (average) characteristic for atime period and also to reduce high frequency noise inthe time series. First, we set a time window of length∆t = 20 years inside which the quantities of the violinsbelonging to the covered period are averaged. By sub-sequently sliding this time window by δt = 1 year, wegenerate a time series. More specifically, the first ele-ment in the time series associated to, for instance, quan-tity a is calculated by averaging the corresponding val-ues of the violins that were fabricated between the years

y(1)1 = 1575 and y

(1)2 = 1575 + ∆t = 1625. Next, the

second element of the time series is obtained by averag-ing a over the violins belonging to the period betweenyears y(2)

1 = 1575 + δt = 1576 and y(2)2 = y

(2)1 + 1626.

This process is repeated for all possible intervals. Fur-thermore, each time window is labeled with its midpointyear in order to provide a time reference for the averagedvalues. Henceforth, we denote by 〈·〉 averages calculatedover violins that fall in the same time-window.

Figure 9 shows the evolution of some average violinmeasurements obtained according to the methodologydiscussed above. As it can be seen, quantities 〈a〉 and〈h1〉 exhibit slight decreasing and increasing trends, re-spectively. The average curvature 〈s〉, on the other hand,shows oscillatory behavior, but with small amplitudes ofvariation. This supports the conclusions derived withPCA projection in Fig. 7(b) that violin design remainedmostly stable across time, without adhering to partic-ular trends over the different periods in classical musichistory.

Another way to visualize changes in violin shape is bymeans of thin plate splines [16]. Such a method was alsoemployed in [14] so as to depict the morphological de-formations required to transform a representative violin

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outline of a given luthier into a violin shape of anotherviolin maker. Here, we consider the thin plate splinetechnique in order to construct a morphing transforma-tion from a given violin shape to a target contour thatrepresents the average shape of a certain time period.In other words, we seek for the necessary morphologicaldeformation to transform a given violin into the averageshape of a certain epoch. Our methodology works as fol-lows: first, a violin shape from the MIMO database asthe reference contour is selected. Second, for each slid-ing time window, the average control contour points arecalculated and a time evolving set of thin plate splinestransformations is obtained.

In Fig. 10 we show the morphing via thin plate splinesof a violin fabricated in 1619 having as targets three pe-riods, namely the time windows centered at 1655, 1859and 1950. As it can be seen, only a small differenceis observed in the position of the control points. Themost prominent displacement of the grid corresponds tothe year of 1655, where we observe a slight dilation ofthe two most upper control points (see Fig. 10(a)). Inthe Supplemental Material, we provide a video with theanimated version of Fig. 10, which shows the completeevolution of the morphing over the time span coveredby the MIMO database. At first sight, this result seemsto contradict the findings in [14], where a negative cor-relation between time and Linear Discriminant Analy-sis (LDA) coefficients associated to violin shapes was re-ported. However, in the present work, correlation withtime was also observed in the violin time series, but withmoderate amplitudes of variation, which yield small griddeformations in the time averaged contours.

VI. CONCLUSION

The primary goal of this paper was to statisticallycharacterize and study the violins stored in the MIMOdataset. The interest in this platform resides in the factthat it contains the largest freely available online collec-tion of instruments [15]. Aiming at capturing relevantgeometrical features of the violin body, we initiated ourstudy by defining measurements that were divided intotwo groups: length-based and curvature-based.

The analysis of the extracted features by means of PCArevealed interesting relations among violin makers. Inparticular, it was confirmed [14] that famous violin mak-ers tend to cluster with their respective copyists, suchas in the case of Stradivari and Vuillaume. Another in-teresting result concerns the proximity in the PCA spacebetween Stradivari and Amati instruments: it was shownthat the closest Strad to an Amati corresponds to a violincrafted in 1690, which is precisely the year that Stradi-vari changed his design from the “Amati’s grand pattern”to the production of the Long Strads. Moreover, influ-ence between masters and apprentices was reflected inlow statistical variability among violins originated fromthe Hopf family, the only lineage of luthiers represented

in the MIMO database. Such a relationship betweenluthiers, copyists and family lineages was also reportedfor the database studied in [14].

By using thin plate spline analysis, we found that theaverage violin shape changed little along time – a resultcorroborated by the lack of a clustered structure in thePCA as visualized in terms of the period of fabrication.The contour morphing showed, with few exceptions (seethe video in the supplemental material), that small de-formations in the grids are enough to transform standardshaped violins to the average outlines respective to dif-ferent epochs.

All in all, besides providing a quantitative characteri-zation of the violin dataset of MIMO, our analysis alsocomplements previous studies of violin shape contrastedwith historical information as reported in [14]. This pavesthe way to further research encouraging, in the process,more museums to exhibit their collection through onlineplatforms.

ACKNOWLEDGEMENTS

We gratefully acknowledge the MIMO initiative forproviding the images that were used in this study. Wealso greatly acknowledge Daniel Wetterskog (Director ofthe Swedish Museum of Performing Arts), Cecilie Holl-berg (Director of the Galleria dell’Accademia di Firenze),Josef Focht (Director of the Musikinstrumentenmuseumsder Universitat Leipzig), Philippe Provensal (Head ofPress Department, Cite de la Musique – Philharmoniede Paris), and Bianca Slowik (Germanisches National-museum), for kindly authorizing the reprint of the vi-olin images used in Fig. 2, 6 and 8. TP thanks FilipiNascimento, Cesar Comin, and Lucas Assirati for use-ful discussions. TP is further grateful to FAPESP forsponsorship, FAR acknowledges the Leverhulme Trust,CNPq (Grant No. 305940/2010-4) and FAPESP (GrantsNo. 2016/25682-5 and grants 2013/07375-0), and LdaFCthanks CNPq (grant No. 307333/2013-2) and NAP-PRP-USP for sponsorship.

Appendix A: Curvature Definition

Given a planar, regular, parametric curve expressedin terms of Cartesian coordinates γ(t) = (x(t), y(t)), itscurvature is defined [16] as

s = xy − yx(x2 + y2) 3

2, (A1)

where the dots denote the derivatives with respect to pa-rameter t. Informally speaking, the curvature s(t) quan-tifies the rate at which the orientation of the tangentvector changes along γ. This can be intuitively under-stood as an indicator of the local geometric nature ofan arbitrary curve. For instance, regions with constant

11

FIG. 11. Illustration of the concept of curvature (Eq. A1) ofan arbitrary planar curve γ. The curvature value at a point iin γ is given by the inverse of the radius ri of the osculatingcircle at i, i.e si = 1/ri. In the above example, point j hashigher curvature than i.

s = 0 correspond to straight line segments in the originalcurve. On the other hand, if a portion of γ exhibits non-null constant s, then this region corresponds to a perfectcircle. In fact, the value of s at a certain point is given bythe inverse of the radius of the osculating circle to thatpoint [16]. This is exemplified in Fig. 11.

Appendix B: Measurements Normalization

The measurements proposed in the main text for theviolin characterization are divided into two sets: mea-

surements based on the length and width of violin parts(a, b, ..., f ; h1 and h2), and measurements derived fromthe curvature of the segments γi (s1,..., s6). However,as they are defined, these quantities cannot be employedto compare different violins because they clearly dependon the sizes of the original images from which they wereobtained. More specifically, length-based measures aredirectly proportional to the width of the images, whilethe average curvatures scale with the contour size as de-picted in Fig. 12(a).

In the case of the length-based measures, this problemis circumvented by normalizing the measurement valuesby the total violin’s length L, i.e.

a = a

L, (B1)

and analogously for b, ..., f ,h1 and h2.Unfortunately, the normalization in Eq. B1 cannot be

extended to curvature-based measurements. It turns outthat the coefficients si are inversely proportional to thenumber of points Ni in segment γi (see Fig. 12(a)). Tocorrect this effect and obtain non-dimensional measureswe define

˜si = siNi, (B2)

As shown in Fig. 12(b), by doing so, we eliminate thecorrelation between si and the length of the respectivecontour segments. For simplicity’s sake, in the main textwe omit the upper scripts “∼” from the variables nota-tion.

[1] J. Woodhouse, Reports on Progress in Physics 77, 115901(2014).

[2] D. D. Boyden, The History of Violin Playing from itsOrigins to 1761: and its Relationship to the Violin andViolin Music (Oxford University Press, 1990).

[3] C. M. Hutchins, The Journal of the Acoustical Society ofAmerica 73, 1421 (1983).

[4] A. Bachmann, An encyclopedia of the violin (CourierCorporation, 2013).

[5] C. Gough, The European Physical Journal-Special Topics145, 77 (2007).

[6] J. Nagyvary, J. A. DiVerdi, N. L. Owen, and H. D. Tolley,Nature 444, 565 (2006).

[7] J. Nagyvary, R. N. Guillemette, and C. H. Spiegelman,PloS one 4, e4245 (2009).

[8] H.-C. Tai, G.-C. Li, S.-J. Huang, C.-R. Jhu, J.-H. Chung,B. Y. Wang, C.-S. Hsu, B. Brandmair, D.-T. Chung,H. M. Chen, et al., Proceedings of the National Academyof Sciences 114, 27 (2017).

[9] C. Fritz, J. Curtin, J. Poitevineau, H. Borsarello, F.-C.Tao, and T. Ghasarossian, Proceedings of the NationalAcademy of Sciences 111, 7224 (2014).

[10] C. Fritz, J. Curtin, J. Poitevineau, and F.-C. Tao, Pro-ceedings of the National Academy of Sciences 114, 5395(2017).

[11] C. V. Raman, Proc. Indian Assoc. for the Cultivation ofScience 6, 19 (1920).

[12] C. V. Raman, The London, Edinburgh, and DublinPhilosophical Magazine and Journal of Science 32, 391(1916).

[13] M. Katz, The violin: a research and information guide(Routledge, 2006).

[14] D. H. Chitwood, PLoS One 9, e109229 (2014).[15] http://www.mimo-international.com/.[16] L. d. F. Costa and R. M. Cesar Jr, Shape analysis

and classification: theory and practice (CRC Press, Inc.,2001).

[17] W. H. Hill, A. F. Hill, and A. E. Hill, Antonio Stradivari,His Life & Work (1644-1737), Vol. 140 (1963) p. 281.

[18] http://www.mimo-international.com/MIMO/doc/IFD/OAI_ULEI_M0004501.

[19] C. Johnson, R. Courtnall, and Y. Menuhin, The Art ofViolin Making (Robert Hale Ltd, London, 1999).

[20] G. Hart, The violin: its famous makers and their imita-tors (Dulau, 1909).

[21] H. R. Haweis, Old violins and violin lore (London: W.Reeves, 1979).

[22] http://www.mimo-international.com/MIMO/doc/IFD/OAI_ULEI_M0003038/violine, http://www.mimo-international.com/MIMO/doc/IFD/OAI_

12

2000 4000 6000 8000 10000

Ni

0.00

0.01

0.02

0.03

0.04

0.05

s i

2000 4000 6000 8000 10000

Ni

0

25

50

75

100

125

150

175

200

s iNi

FIG. 12. Normalization of curvature-based measurements. (a) Scatter plot showing si as function of the number of points Ni

in a contour segment γi. (b) Dependence of the normalized averaged curvature (Eq. B2) on Ni.

ULEI_M0004375/violine.[23] L. Breiman, Machine learning 45, 5 (2001).[24] G. James, D. Witten, T. Hastie, and R. Tibshirani, An

introduction to statistical learning, Vol. 112 (Springer,2013).