# Bow moment of inertia affects bouncing point and rotational motion during sautillé stroke performance by violinists

Víctor Salvador Castrillo, Frédéric Ablitzer, Claudia Fritz

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Víctor Salvador Castrillo, Frédéric Ablitzer, Claudia Fritz. Bow moment of inertia affects bouncing point and rotational motion during sautillé stroke performance by violinists. International Symposium on Music and Room Acoustics, May 2025, New Orleans, United States. pp.035018, 10.1121/2.0002115. hal-05291258

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# Bow moment of inertia affects bouncing point and rotational motion during sautillé stroke performance by violinists

Víctor Salvador Castrillo ; Frédéric Ablitzer; Claudia Fritz

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Proc. Mtgs. Acoust. 58, 035018 (2025)

https://doi.org/10.1121/2.0002115

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# Bow moment of inertia affects bouncing point and rotational motion during sautillé stroke performance by violinists

**Víctor Salvador Castrillo**

Lutheries-Acoustique-Musique, Institut Jean le Rond d'Alembert, Paris, Ile de France, 75005, FRANCE;
victor.salvador_castrillo@sorbonne-universite.fr; vicsalcas@gmail.com

**Frédéric Ablitzer**

Le Mans Universite, Le Mans, Pays de la Loire, FRANCE; frederic.ablitzer@univ-lemans.fr

**Claudia Fritz**

Institut Jean le Rond d'Alembert, Paris, Ile-de-France, FRANCE; fritz@lam.jussieu.fr

The moment of inertia of a violin bow is a mechanical property to which violinists appear particularly sensitive, especially during bouncing bow strokes. In this study, we explored how the bow's moment of inertia influences the performance of one of these bowing techniques: sautillé. Sixteen violinists, including professionals, students, and skilled amateurs, participated in the experiment using a bow with an adjustable moment of inertia. They were invited to perform sautillé strokes on the same violin under varying experimental conditions, including string (A and G), dynamics (forte and piano), and bow inertia (low and high). The rhythm was kept constant across participants using a metronome. Bowing parameters were measured using optical motion capture and analyzed statistically with linear mixed models to evaluate the effects of the experimental conditions. The results indicate that increasing the moment of inertia significantly reduces the amplitude of rotational oscillations around two axes perpendicular to the bow's length. Additionally, the bouncing point shifts significantly toward the tip as the moment of inertia increases. These findings emphasize the role of the moment of inertia in the bow's dynamics during bouncing strokes and illustrate how bow design influences violin performance.
# 1. INTRODUCTION

The art of playing the violin requires a good mastering of all the different bow strokes. In Western violin technique, some strokes involve a bouncing movement with the bow jumping off the string. One of them is *sautillé*, in which the bow bounces off the string while moving in alternating directions (down-bow for one note, up-bow for the next). The violinist controls this horizontal movement (parallel to the bow hair), while the vertical bouncing motion (perpendicular to both the bow hair and the string) is primarily generated by the bow stick itself, with minimal intervention from the player [1]. However, the stroke may fail if it is attempted at a wrong place, which violin pedagogue Lucien Capet [2] fixed at the third, fourth and fifth eighths of the bow depending on the desired speed (Fig. 1).

<figure><img src="image_7.png" /><figcaption>**Figure 1: Best divisions for the performance of sautillé bow stroke, from [2].**</figcaption></figure>

A previous study on the perception of violin bow mass distribution [3] revealed that when playing with bows with different moment of inertia, some participants perceived a change in the dynamic behavior of the bow during bouncing strokes. Some comments referred to the bows as having “less bounce”, being “less precise” or being “more difficult to make bounce”. Others also referred to a need to change the way they were controlling the bow to produce the desired musical action or bowing stroke, including variations in bowing parameters such as the distance between the hair-string contact point and the frog. This parameter seems to be perceived as an inherent property of the bow, with participants describing it as “the bouncing point”. For instance, some bows would have a higher bouncing point (closer to the tip), while others would have a lower one (closer to the frog).

By tracking the movement of bows with contrasting properties, we may be able to identify differences in how the bow is manipulated by the musician due to changes in its mechanical properties. To this end, in this paper we describe an experiment consisting in a motion capture test conducted with a group of 16 violinists. We aimed to compare the bowing parameters used by participants while performing the *sautillé* stroke with an experimental violin bow. This bow allows for adjustments in mass distribution, enabling contrasting moments of inertia along a single stick. The objective was to analyze how variations in the moment of inertia influence bowing parameters in this type of bouncing stroke.

# 2. MATERIALS

Let us first present the key elements used during these experiments: the violin, the bow, and the motion capture system.

## A. VIOLIN

All participants played on the same violin, which was unfamiliar to all of them. It is a contemporary copy of the Gibson ex-Huberman Stradivarius of 1713, made by luthier Georges Stoppani in 2018 for the Bilbao Project [4]. The E string was a Thomastik Infeld’s Vision Titanium Solo, while the other strings were Pirastro Evah Pirazzi Gold (with G-gold string).

We chose to use a single violin rather than each player’s personal instrument for several reasons. First, using only one instrument eliminates variability in construction parameters and string characteristics, which could have influenced bow behavior. This approach ensures that variations in the bowing parameters primarily stem from changes in the bow’s moment of inertia and the player’s interaction with the bow. Additionally, there was a technical constraint: the motion capture system, as described below, requires reflective markers
to be attached to what is considered a rigid body, an object in which the distances between any two points remain constant over time. Had each participant used a different violin, we would have needed to redefine the rigid body for every new instrument due to potential geometric differences. This would have significantly increased the duration of the test, as participants would have had to bring their instruments to the motion capture room in advance. Furthermore, slight variations in each rigid body could have introduced an additional level of uncertainty in the calculation of bowing parameters.

## B. BOW

The bow used by the participants was the same experimental bow described in [3], shown in Fig. 2. This bow features a stick made from low-density pernambuco wood (0.878 g/cm³), with a total mass of 52.2 g (including the hair and all components). A standard weight of 60 g was achieved by adding small masses at various positions along the stick. Neodymium magnets were used as additional weights and could be attached in three magnetized locations: at the screw, at the tip, or on a clamp positioned on the upper side of the stick. This clamp was consistently positioned 15 cm from the pivot point, defined as the spot where the violinist places their thumb (the point of the frog in contact with the stick closest to the tip). For further technical details, see [5].

<figure><img src="image_8.png" /><figcaption>**Figure 2:** Experimental bow with added masses. Top: close-up of the magnets. Bottom: participants' view of the bow, with the tip and frog magnets covered with paper and a lightweight piece of foam covering the magnetic clamp.</figcaption></figure>

Five different mass configurations were designed to create both small and large variations in the bow's center of mass and moment of inertia, while keeping the total mass constant. In this study, we focus on two specific configurations, labeled A and C, which represent the greatest variation in moment of inertia while maintaining identical total mass and center of mass position. We made this choice because we believe the moment of inertia is a key factor influencing the bow's behavior during bouncing strokes. The properties of these configurations are shown in Table 1. Although both configurations use the same stick, their properties differ significantly. Therefore, we will refer to them as two distinct bows, namely bow A and bow C. For further details on how bow properties were measured, see [5].

<table><caption>**Table 1:** Mass configurations and mechanical properties of the two bows (without reflective markers). $\Delta m_t$, $\Delta m_c$, $\Delta m_f$: added masses at the tip (64.5 cm from the pivot point), clamp (15 cm) and frog (–8.6 cm); $m_{bow}$: total bow mass; $x_{CM}$: distance from the center of mass to the pivot point (where the violinist puts their thumb); $I$: moment of inertia with respect to the pivot point; $\{\Delta x_{CM}, \Delta I\}$: variation of mechanical properties compared to the reference bow A, expressed as percentages. The mechanical properties of the bow without masses are as follow: $m_{bow}$ = 52.2 g, $I$ = 4.75 gm², $x_{CM}$ = 20.4 cm.</caption><thead><tr><th>Bow</th><th>Δm<sub>t</sub> (g)</th><th>Δm<sub>c</sub> (g)</th><th>Δm<sub>f</sub> (g)</th><th>m<sub>bow</sub> (g)</th><th>x<sub>CM</sub> (cm)</th><th>Δx<sub>CM</sub> (%)</th><th>I (gm<sup>2</sup>)</th><th>ΔI (%)</th></tr></thead><tbody><tr><td>A</td><td>+0.14</td><td>+6.88</td><td>+0.77</td><td>59.99</td><td>19.5</td><td></td><td>4.95</td><td></td></tr><tr><td>C</td><td>+1.75</td><td>+1.91</td><td>+4.27</td><td>60.13</td><td>19.5</td><td>0.0</td><td>5.52</td><td>+11.6</td></tr></tbody></table>
### C. MOTION CAPTURE: ROOM, CAMERAS AND MARKERS

The experiments were conducted in a motion capture room at our laboratory in Paris. The room's walls are acoustically insulated with foam panels, making it relatively non-reverberant. It is equipped with an OptiTrack motion capture system with nine Flex 13 cameras, operating at 120 Hz. The cameras are positioned around the room, capturing a defined volume within it.

The positions of the bow and violin were tracked using spherical reflective markers, attached to various parts of each structure with reusable adhesive putty that would not damage the wood varnish. The motion capture system was calibrated according to the manufacturer's instructions, yielding a mean ray error of 0.57 mm. This value indicate that the marker position could deviate by approximately 0.57 mm based on the camera system's triangulation process. The mean wand error was equal to 0.12 mm, evaluated by the software as "excellent".

## 3. METHODS

In this section, we first outline the steps taken to maintain consistent hair tension across participants, followed by a description of the marker configurations on the bow and violin used to calculate the bowing parameters, which are presented at the end of the section.

### A. HAIR TENSION AND ROSIN

The bow hair was set to the same tension for all participants and tests. To achieve this, we applied a method described by Ablitzer et al. [6], which is based on the relationship between the hair-stick distance and hair tension: as the hair tension increases, the bow stick deforms, moving further away from the hair. To standardize the tension, we placed a laser-cut cylinder with a diameter of 8 mm between the untightened hair and the stick of the bow, at a distance of 28 cm from the frog-hair attachment point. The hair tension was then gradually increased until the cylinder fell off, which occurred consistently when the hair-stick distance reached approximately 8 mm, corresponding to a specific hair tension.

For this specific hair-stick distance, we estimated the hair tension three times, obtaining an average value of 36.2 ± 0.4 N. This estimation was made using the software PAFI [7], which uses a finite element model of the stick that incorporates its elastic properties and geometry, determined through measurements (e.g., the Young's modulus, estimated at 20.0 ± 0.2 GPa). This method is explained in [5].

Additionally, before each session, the violin strings were cleaned with a cotton cloth. To ensure a consistent amount of rosin to the bow hair, we applied four strokes of rosin (Pirastro Gold) on the hair, twice in each direction, and then removed the excess with two gentle shakes.

### B. MARKER CONFIGURATIONS FOR THE MODELS OF THE HAIR AND THE STRING

The methodology used to track the violin and bow in the motion capture room is largely based on the work of Schoonderwaldt and Demoucron on extracting bowing parameters from violin performances [8]. We used software similar to theirs (OptiTrack), which also supports the use of kinematic models (mathematical representations that describe motion by focusing on geometric relationships between components). The software also allows for the definition of rigid bodies, meaning elements where the distance between their components remains constant (i.e., they do not deform).

To enable the motion capture system to identify these rigid bodies, we placed a set of reflective markers at various points on both elements, as shown in Fig. 3. Once a rigid body is registered, if a marker is removed or lost during the motion capture session, its position is calculated based on the remaining markers. We used this feature during the rigid body identification phase by placing a set of six markers that would later become virtual, meaning they would not be physically present during the actual test. These virtual markers
were essential for estimating bowing parameters, as they were placed at key locations on the strings and bow hair: two at the endpoints of the bow hair, two at the endpoints of the vibrating A string (i.e., the bridge and nut contact points), and two at the endpoints of the vibrating G string. Accordingly, the virtual A and G strings were defined as the segments connecting the respective marker pairs at their endpoints. Similarly, the virtual hair was defined as the segment connecting the markers at its endpoints, which were positioned at the midpoint of the hair strand's width.

Regarding the marker configuration, we optimized their placement to minimize any impact on the mechanical properties of the bow. For instance, we used three smaller markers at the tip to keep the moment of inertia as close to the original values as possible while ensuring accurate tracking of this part of the bow. For the violin, we replicated exactly the marker placements used by Schoonderwaldt and Demoucron [8].

<figure><img src="image_9.png" /><figcaption>**Figure 3: Markers placed at different points of the violin and the bow. The markers on the strings and on the hair were used just for calibration, and removed for the test.**</figcaption></figure>
## Influence of the markers on the bow's mechanical properties

Due to the addition of the reflective markers to the bow, the total mass and the mechanical properties given by mass distribution are inexorably modified. To know to which extent this variation was significant, we remeasured these properties on the bow with markers to make sure that it remained within desired values. They are presented in Table 2.

<table><caption>Table 2: Mechanical properties of the bows used in the motion capture test (with reflective markers). $m_{bow}$: total bow mass; $x_{CM}$: distance from the center of mass to the pivot point; $I$: moment of inertia with respect to the pivot point; $\Delta x_{CM}$, $\Delta I$: variation of mechanical properties compared to the reference bow A, expressed as percentages.</caption><thead><tr><th>Bow</th><th>$m_{bow}$ (g)</th><th>$x_{CM}$ (cm)</th><th>$\Delta x_{CM}$ (%)</th><th>$I$ (gm<sup>2</sup>)</th><th>$\Delta I$ (%)</th></tr></thead><tbody><tr><td>A</td><td>61.19</td><td>19.7</td><td></td><td>4.98</td><td></td></tr><tr><td>C</td><td>61.42</td><td>19.6</td><td>-0.5</td><td>5.48</td><td>+10.0</td></tr></tbody></table>

When we compare the results of this measurement presented in Table 2 to those in Table 1, we observe that they are very similar. This supports the hypothesis that the added markers do not significantly affect the mechanical properties of the bow. However, there is a detail that should not go unnoticed: the moment of inertia of bow C with markers is $5.48 \pm 0.03$ gm$^2$, which is slightly lower than the value without markers (5.52 gm$^2$). This value is based on three measurements, making the difference significant (in absence of repeated measures of the bow without markers). This is surprising, as it is theoretically impossible: the moment of inertia should only increase when mass is added at positions other than the axis of rotation around which this parameter is calculated. We hypothesize that this discrepancy is due to uncertainties in the calculation of the moment of inertia (refer to [5] for details on how this property was estimated).

## C. DEFINITION OF THE BOWING PARAMETERS

The bowing parameters are defined as the six degrees of freedom of the bow relevant to violin playing, comprising three translational and three rotational components. These parameters were estimated geometrically based on the positions of virtual markers placed at the endpoints of the hair and the A or G strings, along with additional real markers positioned on the violin and the bow. Once again, our approach was heavily influenced by the work of Schoonderwaldt and Demoucron [8]. For further details on how all bowing parameters were calculated, refer to [5].

The three translational parameters are depicted in Fig. 4. The virtual hair is defined as the segment connecting markers F and T, and the virtual string as the segment connecting markers B and N. The translational bowing parameters are defined as follows:

* Position of the string on the hair $x_S$: distance between the closest point of the virtual hair to the virtual string $P'$ and the marker at the frog F.

* Position of the hair on the string $y_H$: distance between the closest point of the virtual string to the virtual hair P and the marker at the bridge B.

* Hair-string minimum distance $z_{HS}$: minimum distance between the virtual hair and the virtual string. It is defined as positive when the bow moves away from the violin. Due to the deflection of both real elements during sautillé, this distance can become negative.

The three rotational bowing parameters are associated with the corresponding translational parameters, and describe rotations around their respective axes. These angles are illustrated in Fig. 5 and are defined as follows:
<figure><img src="image_10.png" /><figcaption>**Figure 4:** Translational bowing parameters and geometric relations for their calculations. Redrawn from Figure 4(a) of "Extraction of bowing parameters from violin performance combining motion capture and sensors", Schoonderwaldt et al., JASA, Vol. 126, No. 5, November 2009, p. 2699 [8].</figcaption></figure>

* Tilt $\psi$: angle of rotation of the bow around its longitudinal axis $\hat{x}_H$.

* Inclination $\vartheta$: angle of the bow around the longitudinal axis of the string $\hat{y}_S$.

* Skewness $\phi$: angle of deviation between the longitudinal direction of the hair and its orthogonal alignment to the string, i.e. rotation around $\hat{z}_{HS}$.

<figure><img src="image_11.png" /><figcaption>**Figure 5:** Rotational bowing parameters. From left to right: tilt $\psi$, inclination $\vartheta$ and skewness $\phi$. All the angles indicated in the figure are positive.</figcaption></figure>

# 4. EXPERIMENTAL PROCEDURE

In this section, we present the procedure of the motion capture test as well as an overview of the data to be analyzed. Finally, we outline the statistical model used to assess the effect of the bow.

In the experiment, a set of 16 violinists were asked to perform a specific technical skill, the sautillé stroke. The test was conducted with two bows with contrasting moments of inertia, A and C, on two open strings, A and G, and at two dynamic levels, piano and forte. A metronome set at 120 beats per minute was used, with the same volume for all participants. They were allowed to select their own piano and forte dynamics, under no other constraint than the volume of the metronome.

## A. PARTICIPANTS

A total of 16 participants took part in the experiment, four of whom had participated in the previous experiment presented in [3]. The group consisted of six students from the Conservatoire à Rayonnement
Régional de Boulogne-Billancourt, six professional musicians with music diplomas, and four skilled amateur musicians. Each participant was offered a €15 gift card for their participation, as well as some chocolate. All participants voluntarily took part in the test, with each session lasting approximately one hour. They were informed of their rights, and all the data were anonymized. Participants were given time to rest and could quit at any time.

## B. TEST PROCEDURE

Each violinist was instructed to find the point where they felt the greatest comfort performing the sautillé stroke. The test protocol for each combination of bow, dynamic, and string was as follows. First, the participant placed the bow on the string at the location marked by the metallic clamp fixed to the stick. Then, at the given rhythm, they were asked to play four notes per beat, equivalent to 8 notes per second. Maintaining this pace, they were invited to find what they perceive as the most comfortable bouncing point of the bow. They were instructed to adjust the bow along the stick axis to locate the position where they perceived the most comfortable bounce, without restrictions regarding the bow's distance to the bridge or how high the bow should bounce. Once they found the bouncing point, they were instructed to sustain the bouncing at that position for at least five seconds (equivalent to 10 metronome beats). From now on, we will refer to this task as the “sautillé finding process”.

A preliminary test, which considered initiating the movement from both the tip of the bow and the clamp, showed no significant differences in the final bouncing point relative to the starting position. Therefore, we chose the clamp as the starting point for convenience, as it serves as a clear visual reference.

The sautillé finding process was explained to each participant, with the author demonstrating the procedure using the violin and bow. Demonstrating how the test should be performed proved to be a much more effective method for helping participants understand the procedure, significantly reducing the time needed for the explanation. At the same time, participants were informed about the position they needed to maintain: seated and facing the wall where the music stand was placed. This position prevented the motion capture system from losing markers due to excessive bow height. All participants were aware that they would test two different bows, but no information was given about the difference between them.

For each combination of bow, dynamic level, and string, the sautillé finding process was repeated three times, resulting in a total of 24 trials per participant.

## C. DESCRIPTION OF THE DATA TO BE ANALYZED

For the analysis of the effect of the bow on the performance of sautillé, we will consider the average bowing parameters over the last three seconds of the sautillé finding processes, which would correspond to the coordinates of the preferred bouncing point in the space of the player's bowing parameters.

Since the frequency of the sautillé stroke was fixed at 8 notes per second (imposed by the metronome), any change in the amplitude of a given bowing parameter implies a corresponding change in its speed. For periodic motions at a constant frequency, the RMS (root mean square) speed is directly related to the amplitude of the motion. As the stroke frequency remained consistent across all conditions and participants, RMS speed serves as an indirect measure of motion amplitude. Therefore, we also analyzed the RMS speed of each bowing parameter, computed as:

$$ \text{RMS}(\dot{P}) = \sqrt{\frac{1}{N} \sum_{i} \dot{P}_{i}^{2}}, \quad \dot{P}_{i} = (P_{i} - P_{i-1})f_{s}, $$

where P is any bowing parameter, $P_i$ is the bowing parameter at the instant i, $\dot{P}_i$ is the rate of change of P between two consecutive samples and $f_s$ is the sampling rate of the motion capture system (120 Hz).
## D. DATASET ARCHITECTURE AND REGRESSION MODEL

For the handling of the data and the statistical model, we used `pandas` and `statsmodels` Python toolboxes. For each bowing parameter, the data were organized into tables with the following column structure: "Musician", "Bow", "String", "Dynamics", "Measurement", and a final column containing the average (or RMS) value of the bowing parameter.

By performing a statistical analysis across all participants, we can assess whether there were significant variations in the bowing parameters due to the change in bow's moment of inertia. We observed that each musician used different bowing parameters when performing the *sautillé* stroke, resulting in non-negligible between-subject variability. As a result, considering each measured parameter as independent from the musician that produced it could lead to inaccuracies in the significance of the effect of the bow. For this reason, we chose to use linear mixed models (LMM) because they allow us to model between-subject differences.

We analyzed the average over the last 3 seconds of all six bowing parameters, as well as the RMS speed, as dependent variables. To assess the significance of the effect of the bow, we fitted a separate LMM for each parameter, using restricted maximum likelihood estimation (REML). The columns of each table are modeled as follows: "Musician" is considered as a random effect with 16 categorical levels, each corresponding to a participant. The three main factors "Bow", "String" and "Dynamics" are treated as fixed effects, each with two categorical levels: A and C for "Bow", A and G for "String" and *forte* and *piano* for "Dynamics". The column "Measurements" represents the index of each of the three repeated measurements. The last column, containing the average or RMS speed of all bowing parameters over the last 3 seconds of the *sautillé* finding process, denotes the measured response variable. For the Bow-String-Dynamics design $i \in \{1, ..., 8\}$, the musician $j \in \{1, ..., 16\}$ and the measurement $k \in \{1, 2, 3\}$, the model takes the following form:

$$P_{ijk} = (\beta_0 + u_{0j}) + \beta_1(\text{Bow})_i + \beta_2(\text{Dynamics})_i + \beta_3(\text{String})_i + \varepsilon_{ijk} \quad (1)$$

$$u_{0j} \sim N(0, \sigma_{u_0}^2), \quad \varepsilon_{ijk} \sim N(0, \sigma^2),$$

where $P_{ijk}$ is the average or RMS speed of any bowing parameter. The terms [Bow, Dynamics, String] take values equal to zero or one depending on the level of each factor. The coefficient $\beta_0$ represents the fitted value for the parameter $P$ under the baseline conditions (bow A, *forte*, A string). The rest of the $\beta$-coefficients represent the contributions of the fixed effects to the value of the output parameter $P$.

Let us focus now on the random effects, which in our case are the participants of the study. The term added to the intercept $u_{0j}$ captures the deviation of each subject's baseline from the overall population average (represented by $\beta_0$), and it is modeled as a normal distribution with variance $\sigma_{u_0}^2$. The residuals $\varepsilon_{ijk}$, i.e. the difference between the predicted and the measured values, are also assumed to follow a normal distribution. We will refer to this LMM as "random-intercepts-only model", because it only accounts for random effects on the intercept of the linear regression. It does not account for between-subject variability in the slopes, that is, it assumes that the effect of the fixed effects is the same for all subjects, regardless of their individual intercepts.

To capture the variability in the effect of the bow between musicians, we used a variant of the LMM that accounts for the between-subject variance in the slope of the bow effect:

$$P_{ijk} = (\beta_0 + u_{0j}) + (\beta_1 + u_{1j})(\text{Bow})_i + \beta_2(\text{Dynamics})_i + \beta_3(\text{String})_i + \varepsilon_{ijk} \quad (2)$$

$$u_{1j} \sim N(0, \sigma_{u_1}^2).$$

This model, with random intercept ($\beta_0 + u_{0j}$) and a random slope in the bow term ($\beta_1 + u_{1j}$), also accounts for individual differences in the effect of the bow. We will refer to this LMM as "random-intercepts-and-slopes model". The estimated term $u_{1j}$ informs us about how different the effect of the bow is on different subjects, and it is assumed also to be normal with variance $\sigma_{u_1}^2$. The covariance between the distributions
of $u_{0j}$ (random intercept) and $u_{1j}$ (random slope) indicates whether, and to what extent, they vary together across groups.

While adding terms can improve model fit, it also increases complexity and the risk of overfitting. For this reason, we only added a random slope term for the bow effect when this term significantly improved model fit in comparison to the random-intercepts-only model. To test whether this was the case, we compared the random-intercepts-only model, Eq. (1), with the random-intercepts-and-slopes model, Eq. (2), using the Bayesian information criterion (BIC), a widely accepted method for model selection.

## 5. RESULTS AND DISCUSSION

In this section, we do not present the LMM regression results for all bowing parameters; interested readers may refer to [5] for the complete analysis. Instead, we focus only on those parameters for which the bow's moment of inertia appears to influence their values. These selected parameters, along with their corresponding LMM results, are summarized in Table 3. Among all the analyses conducted (12 in total: 6 averages and 6 RMS speeds), the only case in which the random-intercepts-and-slopes model provided a better fit than the random-intercepts-only model, according to the BIC criterion, was for the $x_S$ parameter, both in its average value and RMS speed. However, the LMM results for the $x_S$ parameter's RMS speed are not shown in Table 3 due to the absence of a significant bow effect.

Whereas for most cases the bowing parameters are significantly influenced by the string and dynamics factors, only four of them are affected by the bow: the average distance between the bowing point and the frog ($x_S$), the average hair-string distance ($z_{HS}$), and the RMS angular velocities for inclination ($\vartheta$) and skewness ($\phi$). These effects are interpreted below:

* For the distance between the bowing point and the frog $x_S$, on average, bow $C$ is estimated to be placed one centimeter closer to the tip ($\beta_1 = 10.2$ mm) compared to bow $A$. From the negative covariance of the random effects (intercept and slope), we infer that this displacement due to the increased moment of inertia of the bow is greater for players whose baseline $x_S$ is lower, that is, for those whose bouncing point with bow $A$ lies closer to the frog. Conversely, this displacement is smaller for players whose bouncing point is closer to the tip with bow $A$ (i.e., higher baseline $x_S$). Although the bow effect is significant ($p$-value < 0.05), it is small when compared to the effects of the string or the dynamics, as shown in Fig. 6. Specifically, playing in *piano* would shift the bouncing point towards the tip about 2.7 cm ($\beta_2 = 26.7$ mm), compared to playing in *forte*. Finally, playing *sautillé* on the open $A$ string would also shift the bouncing point of the bow about 1.7 cm towards the tip on average ($\beta_3 = 16.9$ mm), compared to playing *sautillé* on the open $G$ string.

* The average $z_{HS}$ represents the mean distance between the virtual hair and the virtual string, which can be negative due to the deflection of both real elements during *sautillé*, specifically when the virtual hair spends more time below the virtual string than above it. When playing in *forte* or on the open $G$ string, the estimated average $z_{HS}$ is lowered about 1 mm in each case. Regarding the bow effect, the model suggests that this parameter is also lowered by 0.2 mm when playing with bow $C$ compared to bow $A$. However, this statistically significant difference is very small compared to the effect of the string or dynamic level. We may be tempted to conclude that bow $C$ spends more time in contact with the string than bow $A$. However, due to the uncertainty of the motion capture system, assessing contact between the real elements based solely on negative values may be inaccurate. More notably, this effect could imply that bow $C$ bounces at a lower altitude than bow $A$, which would imply a reduced amplitude in the oscillation of the bow's inclination during the *sautillé* stroke. And it is the case, as discussed here below.

* Finally, for both inclination and skewness angles, the RMS speed of these parameters is significantly reduced when playing with bow $C$ compared to bow $A$. The difference in these speeds also change
<table><caption>**Table 3:** LMM regression results in the technical test, for the bowing parameters for which the moment of inertia of the bow seems to be a determining factor (p-value< 0.05): the average distance between the bowing point and the frog (x<sub>S</sub>), the average hair-string distance (z<sub>HS</sub>), and the RMS angular velocities for inclination (ϑ) and skewness (φ). The baseline [Bow, Dynamics, String] = [0, 0, 0] corresponds to playing with bow A in forte on the A string.</caption><thead><tr><th></th><th></th><th>Coef.</th><th>Std.Err.</th><th>z</th><th>p > |z|</th><th>[0.025</th><th>0.975]</th></tr></thead><tbody><tr><td rowspan="7">Average x<sub>S</sub> (mm)</td><td>Intercept (β<sub>0</sub>)</td><td>246.1</td><td>8.6</td><td>-</td><td>-</td><td>229.1</td><td>263.0</td></tr><tr><td>Bow (β<sub>1</sub>)</td><td>10.2</td><td>5.2</td><td>1.97</td><td><strong>0.048</strong></td><td>0.1</td><td>20.3</td></tr><tr><td>Dynamics (β<sub>2</sub>)</td><td>26.7</td><td>2.1</td><td>12.46</td><td><strong>0.000</strong></td><td>22.5</td><td>30.9</td></tr><tr><td>String (β<sub>3</sub>)</td><td>-16.9</td><td>2.1</td><td>-7.88</td><td><strong>0.000</strong></td><td>-21.1</td><td>-12.7</td></tr><tr><td>Subject intercept var (σ<sup>2</sup><sub>u0</sub>)</td><td>1120.5</td><td>20.7</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Cov(u<sub>0</sub>, u<sub>1</sub>)</td><td>-378.2</td><td>10.2</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Subject slope var (σ<sup>2</sup><sub>u1</sub>)</td><td>354.1</td><td>7.6</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td rowspan="7">Average z<sub>HS</sub> (mm)</td><td>Intercept (β<sub>0</sub>)</td><td>-0.7</td><td>0.2</td><td>-</td><td>-</td><td>-1.0</td><td>-0.4</td></tr><tr><td>Bow (β<sub>1</sub>)</td><td>-0.2</td><td>0.1</td><td>-2.96</td><td><strong>0.003</strong></td><td>-0.3</td><td>-0.1</td></tr><tr><td>Dynamics (β<sub>2</sub>)</td><td>1.2</td><td>0.1</td><td>17.48</td><td><strong>0.000</strong></td><td>1.1</td><td>1.3</td></tr><tr><td>String (β<sub>3</sub>)</td><td>-0.8</td><td>0.1</td><td>-11.90</td><td><strong>0.000</strong></td><td>-0.9</td><td>-0.7</td></tr><tr><td>Subject intercept var (σ<sup>2</sup><sub>u0</sub>)</td><td>0.3</td><td>0.2</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Intercept (β<sub>0</sub>)</td><td>40.2</td><td>2.7</td><td>-</td><td>-</td><td>34.9</td><td>45.6</td></tr><tr><td rowspan="5">RMS inclination ϑ (deg/s)</td><td>Bow (β<sub>1</sub>)</td><td>-1.6</td><td>0.7</td><td>-2.31</td><td><strong>0.021</strong></td><td>-3.0</td><td>-0.2</td></tr><tr><td>Dynamics (β<sub>2</sub>)</td><td>-19.5</td><td>0.7</td><td>-27.83</td><td><strong>0.000</strong></td><td>-20.8</td><td>-18.1</td></tr><tr><td>String (β<sub>3</sub>)</td><td>8.9</td><td>0.7</td><td>12.79</td><td><strong>0.000</strong></td><td>7.6</td><td>10.3</td></tr><tr><td>Subject intercept var (σ<sup>2</sup><sub>u0</sub>)</td><td>111.3</td><td>6.2</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Intercept (β<sub>0</sub>)</td><td>23.9</td><td>1.7</td><td>-</td><td>-</td><td>20.6</td><td>27.3</td></tr><tr><td rowspan="5">RMS skewness ϕ (deg/s)</td><td>Bow (β<sub>1</sub>)</td><td>-1.8</td><td>0.6</td><td>-2.85</td><td><strong>0.004</strong></td><td>-3.0</td><td>-0.6</td></tr><tr><td>Dynamics (β<sub>2</sub>)</td><td>-12.7</td><td>0.6</td><td>-20.24</td><td><strong>0.000</strong></td><td>-13.9</td><td>-11.4</td></tr><tr><td>String (β<sub>3</sub>)</td><td>4.2</td><td>0.6</td><td>6.80</td><td><strong>0.000</strong></td><td>3.0</td><td>5.5</td></tr><tr><td>Subject intercept var (σ<sup>2</sup><sub>u0</sub>)</td><td>41.0</td><td>2.6</td><td>-</td><td>-</td><td>-</td><td>-</td></tr></tbody></table>

significantly depending on the string (higher on the G string for both cases) and on the dynamic level (higher in *forte*). Once again, the effect of the change in the bow's moment of inertia is smaller compared to the effect of the string and the dynamic level. However, this effect suggests that a higher moment of inertia leads to a slower oscillation of the bow's inclination (ϑ) and skewness (φ). Since the frequency is fixed by the metronome, this results in a smaller amplitude of the movement. Oscillations in the ϑ and φ rotations involve pivoting the bow around two orthogonal axes perpendicular to its longitudinal axis (along which the mass distribution was changed). The reduction in the amplitude of this motion in bow C is expected, as the increase in the bow's moment of inertia directly hinders this kind of oscillations.

In summary, bow C, with a greater moment of inertia, seems to “have” a bouncing point placed slightly closer to the tip and lower RMS angular velocities in both inclination and skewness compared to bow A. While these effects are statistically significant, they are less pronounced than the influence of string and dynamics, which have a greater impact on the overall bowing behavior.

The observed displacement of the bouncing point towards the tip is small, averaging about 1 cm (sig-
<figure><img src="image_12.png" /><figcaption>**Figure 6:** Comparison between the effects of the bow, the string and the dynamics on the $x_S$ position of the bow. The vertical axis represents the difference, in absolute value, between the averages of $x_S$ for each level: in green, the difference between the measured values: $\bar{x}_S(\text{Bow C}) - \bar{x}_S(\text{Bow A})$, $\bar{x}_S(\text{piano}) - \bar{x}_S(\text{forte})$ and $\bar{x}_S(\text{A string})$ – $\bar{x}_S(\text{G string})$; in purple, the $\beta_i$ coefficients estimated with the linear mixed model (first row of Table 3), in absolute value. The error bars represent 95% confidence intervals.</figcaption></figure>

nificant). Although this average displacement may seem small when compared to the total length of the bow hair (about 65 cm), it is a parameter to which some musicians seem particularly sensitive, as reflected by their comments describing changes in the bouncing point (during a previous study on the perception of violin bow mass distribution [3]).

In addition to the displacement of the bouncing point towards the tip, the RMS angular velocity in both inclination and skewness when playing bouncing strokes with bow C is estimated to be between 4% (inclination) and 8% (skewness) lower than with bow A. This reduction suggests a corresponding decrease in the amplitude of oscillations in both inclination and skewness during bouncing strokes. This outcome is not surprising, as these rotations involve pivoting the bow around two axes perpendicular to its longitudinal axis, along which the mass distribution has been altered. The moment of inertia of bow C, calculated around both the skewness and inclination axis, is higher than for bow A, which naturally hinders these rotational oscillations.

## 6. CONCLUSION

In this paper we presented an experiment aimed at investigating the influence of the bow's moment of inertia on the sautillé bouncing stroke as performed by musicians. We used an experimental bow whose moment of inertia could be adjusted by adding small masses to different parts of the stick. Two distributions were considered, with contrasting moments of inertia but fixed total mass and center of mass. The bow and the violin were tracked using motion capture technology, and the movement of the bow was characterized by its six degrees of freedom relevant to violin playing.

A total of 16 violinists participated in the test. The experiment involved inviting the violinists to find the so-called “bouncing point” of the bow on the open G and A strings, both in *forte* and *piano* dynamics. Results show that the moment of inertia of the bow has an important influence not only on the position of the bouncing point along the longitudinal axis of the bow, but also on the amplitude of its rotations around axes perpendicular to that axis. However, although the effects of the bow's moment of inertia are statistically significant, they appear to be less substantial than the effects of the dynamic level or the string on which the sautillé stroke is performed.
While this study focused solely on the influence of the moment of inertia on the bowing parameters of the sautillé bouncing stroke, similar analyses could be conducted with respect to the center of mass position. This parameter could also be critical for bouncing strokes, as changes in the torque exerted by the weight of the bow may alter its dynamic behavior during bouncing strokes. Additionally, similar studies could be conducted on other bouncing techniques, such as spiccato or ricochet, to further explore the relationship between the bow's mechanical properties and performance.

## 7. ACKNOWLEDGEMENTS

We are grateful to all the participants for their availability and engagement in the test. Special thanks go to Coline Girard and Waithika Sinnathamby for their invaluable support during the motion capture sessions, and to David Poirier-Quinot for his expert technical assistance.

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