Lionel Schouveiler
email: lionel.schouveiler@irphe.univ-mrs.fr

Experiments on fall of open tori

Keywords: Fluid-solid interaction, Falling torus, Fluttering mode, Path instability

Fall of open tori horizontally dropped in quiescent water is experimentally investigated. The fall is controlled by the Galileo number, the torus aspect ratio and the solid to uid density ratio. The dependence of the descent velocity and of the drag coecient on the control parameters is rst analyzed. As the control parameters are varied, vertical translation, zigzagging and spiraling descent are observed. These dierent descent modes are characterized and a phase diagram of the falling tori are presented as function of the three control parameters.

Introduction

At the end of the 17th century, Newton performed experiments to investigate the air resistance and reported in The Principia that bodies "did not always fall directly down, but sometimes uttered a little in the air and waved to and fro as they were descending". Almost two centuries later, Maxwell [START_REF] Maxwell | On a particular case of the descent of a heavy body in a resisting medium[END_REF] observed that "when a slip of paper falls through the air, its motion, though undecided and wavering at rst, sometimes becomes regular". This variety of motions of bodies descending through a uid, that everybody can experience by observing fall of snowakes, hailstones or leaves, has excited the curiosity of researchers for a long time. Studies have mainly concerned bodies with symmetries as spheres, disks or cylinders, for which a review can be found in [START_REF] Ern | Wake-induced oscillatory paths of bodies freely rising or falling in uids[END_REF]. For example, successive experimental works of Willmarth et al. [START_REF] Willmarth | Steady and unsteady motions and wakes of freely falling disks[END_REF], Stringham et al. [START_REF] Stringham | The behavior of large particles falling in quiescent liquids[END_REF], Field et al. [START_REF] Field | Chaotic dynamics of falling disks[END_REF] and Zhong et al. [START_REF] Zhong | Experimental study of freely falling thin disks: Transition from planar zigzag to spiral[END_REF] have provided a comprehensive phase diagram for falling thin disks which reveals, depending on the physical parameters (uid viscosity and density, disk diameter and density), dierent motions as steady vertical descent, tumbling, chaotic motion and periodic utter. Flutter can be spiraling, planar zigzagging or a zigzag motion in a plane precessing around a vertical axis. In these studies, the disk thickness is implicitly assumed to be small enough to have no eect, but numerical simulations conducted by Auguste et al. [START_REF] Auguste | Falling styles of disks[END_REF] have revealed dierences, when compared with zero thickness disks, even for diameter-to-thickness ratios as large as 50. The various descent modes are mainly driven by uid loads induced by the vorticity produced at the body surface and shed in its wake [START_REF] Ern | Wake-induced oscillatory paths of bodies freely rising or falling in uids[END_REF]. The knowledge of the ow around the body xed in a ow can therefore provide insight at least on the transition from vertical translation to nonstraight paths. For this reason, the studies devoted to the ow around open tori, and more generally rings, xed in a uniform ow are reviewed, then, the works devoted to falling open rings are presented. Unless otherwise stated, the cited studies concern rings held normal to a uniform ow or horizontally released in an initially quiescent uid. The ow behind a torus held in a uniform ow is controled by the torus aspect ratio, dened as the ratio of the torus diameter to that of its crosssection, and by the Reynolds number Re based on the cross-section diameter. Experimental investigations of Roshko [START_REF] Roshko | On the development of turbulent wakes from vortex streets[END_REF] for Re up to 500 or Inoue et al. [START_REF] Inoue | An experimental study on a wake behind a torus using the UVP monitor[END_REF] at Re = 1500 with tori and Bearman and Takamoto [START_REF] Bearman | Vortex shedding behind rings and discs[END_REF] with rings of trapezoidal cross-section for Re between 10 4 and 7 × 10 4 have evidenced a distinction between high and low aspect ratios, that has been conrmed by the numerical simulations of Sheard et al. [START_REF] Sheard | From spheres to circular cylinders: the stability and ow structures of blu ring wakes[END_REF][START_REF] Sheard | From spheres to circular cylinders: non-axisymmetric transitions in the ow past rings[END_REF] or Lazeroms et al. [START_REF] Lazeroms | Low reynolds number ow around tori of dierent slenderness Γ[END_REF]. The limit is found at an aspect ratio around 4 or 5. For high aspect ratios, the previously cited studies and also the experimental works of Leweke and Provansal [START_REF] Leweke | The ow behind rings: blu body wakes without end eects[END_REF] reported counter-rotating annular vortices parallel to the torus alternately shed from inside and outside of the torus. This axisymmetric periodic wake is analoguous to the well known von Kármán vortex street behind a straight circular cylinder but appears at slightly higher critical values of Re and has lower frequencies by a few percent, both quantities asymptotically tending to the values for the cylinder as the aspect ratio is increased. Lower frequencies are due to the self-induced velocity of the vortex rings which results in a slow-down of the vortex convection in the torus wake [START_REF] Leweke | The ow behind rings: blu body wakes without end eects[END_REF]. Shedding of pair of helical vortices, analoguous to the oblique vortices shed in the wake of a cylinder, has also been observed but their appearance required a "violent perturbation" of the axisymmetric wake [START_REF] Leweke | The ow behind rings: blu body wakes without end eects[END_REF]. Unsteady regimes of low aspect ratio tori wake appear similar to what is observed for a sphere or a disk. They present a planar symmetry and consist of hairpin vortices periodically and alternately shed from opposite sides of the outer diameter of the torus [START_REF] Roshko | On the development of turbulent wakes from vortex streets[END_REF][START_REF] Sheard | From spheres to circular cylinders: non-axisymmetric transitions in the ow past rings[END_REF][START_REF] Wang | An ecient immersed boundary-lattice boltzmann ux solver for simulation of 3d incompressible ows with complex geometry[END_REF]. For all aspect ratios, unsteady regimes described above are preceded by steady attached ow up to Re of order unity then steady separated ows up to Reynolds numbers of order 100, both critical Re values depending on the aspect ratio [START_REF] Sheard | From spheres to circular cylinders: the stability and ow structures of blu ring wakes[END_REF][START_REF] Sheard | From spheres to circular cylinders: non-axisymmetric transitions in the ow past rings[END_REF][START_REF] Lazeroms | Low reynolds number ow around tori of dierent slenderness Γ[END_REF][START_REF] Yu | Steady ow past a torus with aspect ratio less than 5[END_REF]. The second regime is characterized by recirculation zones in the wake with a special feature at aspect ratios around 2 where a detached recirculation bubble forms on the symmetry axis [START_REF] Sheard | From spheres to circular cylinders: the stability and ow structures of blu ring wakes[END_REF][START_REF] Wang | An ecient immersed boundary-lattice boltzmann ux solver for simulation of 3d incompressible ows with complex geometry[END_REF][START_REF] Yu | Steady ow past a torus with aspect ratio less than 5[END_REF]. The creeping ow, at very low Re, has been the subject of various analytical works as [START_REF] Pell | On stokes ow about a torus[END_REF][START_REF] Goren | Asymmetric creeping motion of an open torus[END_REF]. Fall of tori, and more generally of rings, in a uid has received little attention. Amarakoon et al. [START_REF] Amarakoon | Drag measurements for axisymmetric motion of a torus at low reynolds number[END_REF] and Roger and Hussey [START_REF] Roger | Stokes drag on a at annular ring[END_REF] dropped tori and holed disks, respectively, in very viscous liquids to deduce the drag they experience in the Stokes limit. For this low Re range, never reached during the present study, the rings underwent a steady vertical translation. The same motion is reported in the numerical work of Yu et al. [START_REF] Yu | A three-dimensional hybrid meshfree-cartesian scheme for uid-body interaction with complex geometry[END_REF] for a torus of aspect ratio of 2 falling at Re = 25 or 50, and a detached axisymmetric recirculation bubble forms in the wake. This study also showed that when the torus is released with an angle of 30 • it returns to the horizontal (0 • ) after a few damped oscillations. Monson [START_REF] Monson | The eect of transverse curvature on the drag and vortex shedding of elogated blu bodies at low reynolds number[END_REF] conducted fall experiments using various tori and uids with the aim to investigate the dierent regimes of the wake ow but not the torus motions. By means of ow visualizations he reported vortex patterns similar to those observed behind xed tori described above. That is for aspect ratios higher than about 4 or 5, pair of vortex rings, and more rarely pair of helical vortices, were seen, while for lower aspect ratios, the torus wake behaves like those of spheres or disks with shedding of vortex loops. More recently, Vincent et al. [START_REF] Vincent | Holes stabilize freely falling coins[END_REF] and Bi et al. [START_REF] Bi | Experimental study on the vortex structure and path instability of free falling annular disks[END_REF] experimentally studied the eect of a central hole on the fall of disks. Vincent et al. [START_REF] Vincent | Holes stabilize freely falling coins[END_REF] showed how the phase diagram is modied by the hole and Bi et al. [START_REF] Bi | Experimental study on the vortex structure and path instability of free falling annular disks[END_REF] reported a new descent mode, that appears for holed disks of low moment of inertia around a diameter, where the ring fall vertically with periodic velocity uctuations as annular vortices are shed in its wake. To the author knowledge, the present study is the rst to investigate the dynamics of falling tori. Experimental results on tori horizontally released in water at rest are presented and, in particular, the eects of the dimensionless parameters that control the system are considered. The experimental setup is presented in section 2. The results are provided in section 3 and discussed in the last section 4.

Experimental details and parameters

Experiments were performed in a tank with transparent walls, of 0.4 × 0.4 m 2 horizontal cross-section and lled with a height of water of 1 m (see Fig. 1). An open torus is geometrically dened by it cross-sectional diameter d and mean diameter D > d, and is hereafter referred to as d × D where the two diameters are expressed in millimeters.

The release mechanism of the tori is shown in Fig. hole for the moveable rod. The support plate was held partly immerged at the water surface to avoid free surface eects above the release site. The torus was maintained horizontally at the end of the rods, placed inside the torus hole, through a tension exerted by the moveable rod. It was released applying a small displacement of this latter towards the torus center.

The torus fall was recorded on a white background in grayscale images with a camera operating at a frame rate up to 50 s -1 . And since we are interested here in the fully developed fall regimes, the eld of view was of about 0.7 m high from 0.2 m below the release site to 0.1 m above the tank bottom. Two perpendicular views (x, z) and (y, z) were simultaneously captured using a vertical mirror carefully positioned at 45 • as shown in Fig. 1. Consecutive frames of the video recordings were then processed to deduce the coordinates of the torus center x(t), y(t) and z(t) as function of time. The processing consisted in inverting and applying a threshold to the images to obtain a binarized images in which the two torus projections appear white on a black background. Then ellipses were tted to these projections. With this method, the coordinates of the torus center, whose projections correspond to the ellipse centers, were determined to within 1 pixel that is to ±0.2 mm. The calibrations for the conversions from pixel to length unit were determined with images of vertical and horizontal scales xed inside the tank facing the two projection planes. The estimated resolution for the vertical velocities is of about 10 -2 m.s -1 . The tori were released without initial velocity and initial angle (i.e. with their axis of revolution aligned with the vertical axis z) in water at rest. To be more precise, the initial angle was measured a posteriori on the video recordings and was found to be less than 5 degrees, but it has been veried (see section 3.2) that such a small initial angle did not aect the nal state of the torus descent. Moreover, the successive experiments were conducted leaving tori at the bottom of the tank to avoid introducing further perturbations and with a time delay of at least ten minutes allowing the water to return to a quiescent state. After a series of experiments, the tori were recovered and the tank was left at rest for many hours. The heigth of fall is supposed to be high enough to reach a fully developed fall regime. This is supported by the results reported in section 3 showing that the tori have reached a limit vertical velocity (see section 3.1) and that the amplitudes of periodic utters are constant (see Fig. 6) suggesting saturated states. Morevover the tank cross-section is assumed to be large enough for wall eect to be negligible that is the measured limit velocity in the tank is the same as the the velocity in innite medium. This point is critical because most of the quantitative results presented in the following are related to this velocity. The wall effect has been the object of numerous studies dating back to Newton in The Principia, most of them have dealt with the fall of spheres as e.g. Fidleris and Whitmore [START_REF] Fidleris | Experimental determination of the wall eect for spheres falling axially in cylindrical vessels[END_REF], Chhabra et al. [START_REF] Chhabra | Some further observations on the hindered settling velocity of spheres in the inertial ow regime[END_REF] or Arsenijevi¢ [START_REF] Lj | Wall eects on the velocities of a single sphere settling in a stagnant and counter-current uid and rising in a co-current uid[END_REF]. The wall eect depends on and decreases with increasing Reynolds and decreasing blockage ratio, which is the ratio of the projected area of the body to the tank crosssection. Fidleris and Whitmore [START_REF] Fidleris | Experimental determination of the wall eect for spheres falling axially in cylindrical vessels[END_REF] found that the wall eect for blockage ratios of 25 × 10 -4 and 10 -2 is less than 1% as soon as Re exceeds 5 and 30 respectively. Here the blockage ratio is between 9 × 10 -4 and 2.2 × 10 -2 with much higher Re values exceeding 130 and 1900 respectively (section 3.1). For this reason small wall eect can be anticipated, at least smaller than the data dispersion obtained as the experiments were repeated which is between 3 and 8%. The height of fall is supposed to be high enough to reach a fully developed fall regime and, because the Reynolds number of this regime is always more than 100 , the tank cross-section is assumed to be large enough for connement and With these conditions, the fall of a torus is dened by six physical parameters, namely the mean D and cross-sectional d diameters of the torus, and its material density ρ, the density ρ w < ρ and kinematic viscosity ν w of water, and the gravitational acceleration g. The problem is then governed by three dimensionless parameters, we used here the torus aspect ratio D/d, the density ratio ρ/ρ w and the Galileo number

Ga = (ρ/ρ w -1)gd 3 /ν w (1)
that measures the ratio of net gravitational to viscous forces.

During the present tests, the water temperature was of 16 • C so that its density was ρ w = 1000 kg.m -3 and its kinematic viscosity ν w = 1.1 × 10 -6 m 2 .s -1 . Temporal variation of the water temperature during the experiments was lower than 1 • C and its spatial variation from the top to the bottom lower than 0.5 • C. Thirty-two tori of dierent sizes were tested, 17 were made of nitrile rubber (O-ring) of density ρ = 1250 kg.m -3 and 15 of stainless steel of density ρ = 7720 kg.m -3 . The homogeneity of the torus material was not tested but tori with visible defects were eliminated and for most of the tori, experiments were repeated with two or more samples with similar results. Each torus was released at least twenty times.

Inuences of the three dimensionless parameters were investigated by considering tori of cross-sectional diameter 2 < d < 10 mm, mean diameter 14 < D < 110 mm. The parameter values were in the ranges 3.5 D/d 16 and 120 < Ga < 1430 for the rubber tori ρ/ρ w = 1.25, and in the ranges 6 D/d 16 and 1200 < Ga < 7400 for the steel tori ρ/ρ w = 7.72.

Results

Descent velocity and drag coecient

When a torus is released with no velocity in quiescent water, it rst undergoes a phase of acceleration then reaches a limit mean vertical velocity U L hereafter referred to as descent velocity. The measured descent velocities, derived from the temporal evolution of the vertical position of the torus center z(t), are found to range from 0.07 to 0.23 m.s -1 for the rubber tori ρ/ρ w = 1.25 and from 0.59 to 1.05 m.s -1 for the steel tori ρ/ρ w = 7.72. They are displayed in Fig. 3 under the non dimensional form of a Reynolds number based on the cross-sectional diameter d, Re = U L d/ν w , as function of the Galileo number Ga. This Reynolds number is useful for comparisons with xed body studies for which it is a control parameter. It varies between 130 and 2080 for ρ/ρ w = 1.25 and between 1610 and 9520 for ρ/ρ w = 7.72. The gure shows the monotonic increase of the descent Reynolds number Re with the Galileo number Ga and the continuity of the two curves of equal aspect ratio D/d = 6 and 11 as ρ/ρ w is changed suggests that the parameter ρ/ρ w alone has no eect on Re. Moreover, the gure, and particularly the The descent velocity results from the balance between net gravitational forces (ρ -ρ w )gd 2 Dπ 2 /4 and drag forces ρ w C D dDU 2 L π/2 in such a way that the drag coecient C D is deduced from the measured descent velocities with

C D = π 2 ρ ρ w -1 dg U 2 L . (2)
The experimental drag coecients C D are plotted in Fig. 4 as a function of the Reynolds number Re together with the drag coecient for a xed circular cylinder (i.e. a torus of innite aspect ratio) in an uniform ow.

The empirical expression derived in [START_REF] Sucker | Fluiddynamik bei quer angeströmten zylindern[END_REF] is used for the drag coecient of circular cylinders. Drag coecient C D for the falling tori is found to evolve between 0.7 and 1.4 for 10 2 < Re < 10 4 , rst decreasing down to Re ≈ 2000 then increasing, following the same tendency as the drag coecient of a circular cylinder. But C D values of falling tori appear smaller than those of the cylinder and approach them as the aspect ratio D/d is increased as already noted by Sheard et al. [START_REF] Sheard | Computations of the drag coecients for low-reynolds-number ow past rings[END_REF] for xed tori. Moreover, results of this latter numerical

Descent modes

When tori of dierents sizes or densities are horizontally released in water, a variety of descent modes is observed. The observation of the dierent modes for the 32 tori tested here are reported in the three-dimensional phase diagram (Ga, D/d, ρ/ρ w ) in Fig. 5. Two main categories of motion are distinguishable, namely vertical translation and periodic utter, utter can be of zigzagging or spiraling type. The dierent descent modes are illustrated in Fig. 6 with stroboscopic visualizations of the torus fall along two perpendicular views and paths of the torus center. Periodic behaviors are characterized by their frequency f expressed under the dimensionless form of a Strouhal number St = f d/U L (Fig. 7). Frequential analysis was only performed for the rubber tori because the maximum frame rate of 50 fps and transit times in the view eld of less than or order 1 s for the steel tori give a maximum frequency and a frequency resolution not compatible with the characteristic frequencies of the steel torus fall. Indeed, the frequencies estimated by extrapolation of the results obtained with the rubber tori, and presented in Fig. 7, would be of a few hertz for zigzagging steel tori and a few tens hertz for steel tori in vertical translation.

For both tested values of the density ratio, vertical translation, corresponding to the closed symbols in the phase diagram of Fig. 5, is observed for the largest aspect ratio values. In this mode the torus descends without any rotation, its rotational symmetry axis remains vertical (Fig. 6(a)), and its center does not show signicant horizontal displacement (Fig. 6(d)). However, the vertical velocity of the torus center presents low amplitude periodic oscillations whose nondimensionalized frequencies St are displayed, as function of Re, with closed circles in Fig. 7, they are found to increase from 0.18 to 0.22 with Re. Dimensionless frequency of the parallel vortices shed in the wake of a xed circular cylinder is also reported on the graph (thick line in Fig. 7) using the analytical expression introduced in [START_REF] Williamson | A series in 1/ √ Re to represent the Strouhal-Reynolds number relationship of the cylinder wake[END_REF]. Frequencies of the vertical velocity uctuations of the falling tori appear lower than the vortex shedding frequency behind a cylinder and they approach this value as the aspect ratio is increased. During vertical translation the tori are constantly normal to the relative ow exactly as a torus normally xed in an incoming ow, making relevant a comparison between both congurations. And the frequency of the annular vortices shed behind xed tori has been found to follow the same trend as the frequency of the velocity oscillations of the translating tori, namely it is lower by a few percent to the frequency of the parallel vortex shed behind a cylinder and approaches it as the aspect ratio is increased [START_REF] Roshko | On the development of turbulent wakes from vortex streets[END_REF][START_REF] Sheard | From spheres to circular cylinders: the stability and ow structures of blu ring wakes[END_REF][START_REF] Leweke | The ow behind rings: blu body wakes without end eects[END_REF]. The comparison suggests that the velocity oscillations observed during the vertical translation mode are driven by the shedding of the vortex rings in the wake of the tori. A behavior that as been conrmed for thin holed disks for aspect ratios higher than about 9, where the aspect ratio is dened here as the ratio of the dierence between the outer radius and the inner radius to the mean diameter, by means of simultaneous torus and ow visualizations [START_REF] Bi | Experimental study on the vortex structure and path instability of free falling annular disks[END_REF]. It should be pointed out that no velocity uctuations were detected for the two heaviest rubber tori, this could be due to their high translational inertia that should result in too small oscillation amplitudes for the velocity resolution.

Small aspect ratio tori fall following utter motions (open symbols in Fig. 5). A uttering torus undergoes periodic oscillations while descending and its center follows an elliptical helical path. Except for the 4 × 14 rubber torus (thick open circle in Fig. 5) which is the tested torus of the lowest aspect ratio, helical paths of the center of the uttering tori have a high ellipticity, as seen on the projection into the horizontal plane (x, y) in Fig. 6(e). Therefore this motion appears quasiplanar and is referred to as zigzagging utter in contrast to the spiraling utter of the 4 × 14 rubber torus for which the helical path of the center is almost circular (Fig. 6(f)). Note that the orientation of the vertical plane of oscillation for the zigzagging utter has been seen to be arbitrary as the experiments were repeated. Dimensionless characteristic frequencies of both these types of utter are reported in Fig. 7 where it can be seen that zigzagging and spiraling utters dier also by their characteristic frequency, St ≈ 0.024 for the former and of about 0.043 for the latter. The inset in Fig. 7 shows the frequency of zigzagging tori normalized using the outer diameter (D + d) and the velocity scale

((1 -ρ w /ρ)g(D + d)) 1/2 , namely St * = f (D + d)/ ((1 -ρ w /ρ)g(D + d)) 1/2
, rather than d and U L used to dene St. This dimensionless frequency St * has a value of about 0.1 almost independent of the Reynolds number. This is similar to the experimental results reported by Fernandes et al. [START_REF] Fernandes | Oscillatory motion and wake instability of freely rising axisymmetric bodies[END_REF] for zigzagging rising disks where (D + d) must be replaced by the disk diameter in the denition of St * . In Fig. 6(d-f) the origin (x = 0, y = 0) coincides with the horizontal position of the torus center at the top of the eld of view i.e. at z ≈ 0.2 m and can dier from its horizontal position at the release site z = 0. Actually, during experiments a small drift and/or damped oscillations are sometimes observed during the rst stage of the fall that are suspected to be due to a small initial angle between the vertical and the rotational symmetry axis of the torus which is controlled to a few degrees in the present study. Some experiments are performed for investigating the eect of such an initial angle showing that the nal descent mode is not aected as long as the initial angle is not "too large". A similar behavior has been observed in experimental studies of falling thin disks [START_REF] Willmarth | Steady and unsteady motions and wakes of freely falling disks[END_REF][START_REF] Lee | Experimental investigation of free falling thin disks. part ii: Onset of three dimensional motion from zigzag to spiral[END_REF], thick disks [START_REF] Brosse | The motion of an axisymmetric body falling in a tube at moderate reynolds number[END_REF] and also in numerical simulations on a falling torus released with an initial angle of 30 • [START_REF] Yu | A three-dimensional hybrid meshfree-cartesian scheme for uid-body interaction with complex geometry[END_REF]. Eect of an initial angle is illustrated in Fig. 8(a) where the eld of view is extended up to the release site. The stroboscopic visualization of the fall of a 4 × 34 rubber torus released with an angle of about 35 • exhibits horizontal drift and damped oscillations before reaching the vertical translation descent mode. However, for still larger initial angle, as in Fig. 8(b) for the 4 × 24 rubber torus with an initial angle of about 70 • , the torus falls vertically.

Concluding discussion

A phase diagram for the fall of tori in a viscous uid has been experimentally established (Fig. 5). In the explored region of the control parameters, the diagram reveals a distinction, as previously reported for the wake ow of tori held in a uniform ow, in the behavior of tori of low and high aspect ratios with a limit which depends on the Galileo number Ga. The limit is found at 8.5 < D/d < 11 for Ga ≈ 1870, 6 < D/d < 8.5 for Ga ≈ 360 and lower than D/d = 6 at Ga ≈ 230. For D/d above this limit the tori fall vertically straight with periodic velocity oscillations due to the shedding of annular vortices, analoguous to the straight vortices behind a cylinder. In contrast, low aspect ratio tori utter periodically during their descent describing either a quasi-planar zigzagging motion or a spiraling three-dimensional motion. Most of the studies on falling bodies use a dimensionless moment of inertia and the descent Reynolds number as parameters, although this latter is not a true control parameter since the descent velocity is not known a priori. To compare, the phase diagram is presented in Fig. 9 as function of ρ/ρ w , Re = U L d/ν w and the dimensionless moment of inertia I * = I/(ρ w (D + d) 5 ) where I is the moment of inertia of a torus around a mean diameter I = (ρπ 2 d 2 D/32)(D 2 + 5d 2 /4). Here I * lies between 10 -3 and 3 × 10 -2 , it ranges from 5 × 10 -4 to 3 × 10 -3 and from 8 × 10 -3 to 2 × 10 -1 in the holed disks experiments of Bi et al. [START_REF] Bi | Experimental study on the vortex structure and path instability of free falling annular disks[END_REF] and Vincent et al. [START_REF] Vincent | Holes stabilize freely falling coins[END_REF], respectively; in these studies the Reynolds number is twice Re because it is based on the length scale dened by the outer minus inner diameter.

Although the torus dier of the holed thin disk by a nite thickness d and by rounded corners, the comparison of the phase diagram of Fig. 9 with those in [START_REF] Vincent | Holes stabilize freely falling coins[END_REF][START_REF] Bi | Experimental study on the vortex structure and path instability of free falling annular disks[END_REF] and even with the one for the whole disk in [START_REF] Zhong | Experimental study of freely falling thin disks: Transition from planar zigzag to spiral[END_REF] reveals similarities. First, comparison shows that the periodic vertical translation is never observed for falling disks, coherently with the fact that disks does not present unsteady axisymmetric wake, and is only seen for holed disks of small moment of inertia I * 10 -3 [START_REF] Bi | Experimental study on the vortex structure and path instability of free falling annular disks[END_REF]. This is in line with our own observations where it maintains up to much higher Reynolds number of order 10 5 . Secondly, zigzagging and spiraling utter modes experienced by tori have also been reported for both whole [START_REF] Zhong | Experimental study of freely falling thin disks: Transition from planar zigzag to spiral[END_REF][START_REF] Chrust | Numerical simulation of the dynamics of freely falling discs[END_REF] and holed disks [START_REF] Vincent | Holes stabilize freely falling coins[END_REF][START_REF] Bi | Experimental study on the vortex structure and path instability of free falling annular disks[END_REF]. Further investigations are required to complete the phase diagram and to check the existence of others modes as those observed for falling disks with or without central hole as steady vertical translation, tumbling and chaotic descent. The experiments of Amarakoon et al. [START_REF] Amarakoon | Drag measurements for axisymmetric motion of a torus at low reynolds number[END_REF] have shown that tori descending at very low Reynolds numbers, for which the ow does not separate, fall vertically and steadily. This descent mode could be also expected for tori falling in the Re range where the wake consists in steady recirculation regions which has been found to extend up to Re ≈ 90 at D/d = 3.5 and up to Re ≈ 50 at D/d = 20 for xed tori [START_REF] Sheard | From spheres to circular cylinders: the stability and ow structures of blu ring wakes[END_REF] well below the values measured during the present work for corresponding aspect ratios. The investigation of the region of the low Reynolds number or equivalently, according to Fig. 3, of the low Galileo number, required to work either with tori of smaller crosssectional diameter d and/or made of a lighter material or using more dense and/or more viscous uid. Finally, the works of Vincent et al. [START_REF] Vincent | Holes stabilize freely falling coins[END_REF] and Bi et al. [START_REF] Bi | Experimental study on the vortex structure and path instability of free falling annular disks[END_REF] suggest that chaotic and tumbling descents could be obtained for tori of larger moment of inertia I * than considered in the present study.

2 .Figure 1 :

 21 Figure 1: Schematic of the experimental setup with the coordinate system and torus dimensions.

Figure 2 :

 2 Figure 2: Release mechanism.

Figure 3 :

 3 Figure 3: Reynolds number Re as function of Galileo number Ga, circles are for ρ/ρw = 1.25 and triangles for ρ/ρw = 7.72, open and closed symbols refer to utter and translational fall, respectively (see next section). Insets: close-up around Ga = 360 and 1867.

Figure 4 :

 4 Figure 4: Drag coecient CD as function of the Reynolds number Re, symbols as in Fig. 3.

Figure 5 :

 5 Figure 5: Phase diagram: triangles are for ρ/ρw = 1.25, circles for ρ/ρw = 7.72, closed symbols refer to translational fall, thin open symbols to zigzagging utter and thick open circle to spiraling utter.

Figure 6 :

 6 Figure 6: Descent modes of tori in water: Two perpendicular stroboscopic visualisations (top) with a time interval of 1/10 s and corresponding paths of the torus center (bottom, where the scale in the horizontal x-and y-direction is stretched with respect to the vertical z-direction) for the vertical translation of the 4×44 rubber torus(a,d), the zigzagging 6×36 (b,e) and spiraling 4 × 14 (c,f) rubber tori.

Figure 7 :

 7 Figure 7: Characteristic dimensionless frequencies of the torus motion as function of the descent Reynolds number, symbols as in Fig. 5.

Figure 8 :

 8 Figure 8: Eect of an initial angle: Stroboscopic visualisations of the 4 × 34 rubber torus released with an angle of about 35 • (time interval of 1/30 s) (a) and of the 4 × 24 with an angle of about 70 • (time interval of 1/15 s) (b).

Figure 9 :

 9 Figure 9: Phase diagram in the (Re, I * , ρ/ρw) space, symbols as in Fig. 5.


Bruno Gazengel
email: bruno.gazengel@univ-lemans.fr

Jean Pierre Dalmont

Jean Francois Petiot

Link between objective and subjective characterizations of Bb Clarinet reeds

Keywords: Musical acoustics, clarinet, reed, subjective tests, mechanical characterization 00-01, 99-00

This paper deals with the subjective and mechanical characterizations of single cane reeds.

Subjective tests performed by a single expert gave an estimation of the subjective descriptors "ease of playing", "brightness", "roundness", "projection" and "global quality". The mechanical characterization involved measurements of static compliance, dynamic compliance, resonance frequency and Q factor of the first vibration mode. Two sets of clarinet reeds were studied. The first set contained 50 reeds identical in cut, brand and strength. The second set contained 150 reeds with 3 different cuts but identical in brand, strength and static compliance. Results show that the perceptual space is almost bi-dimensional for the first set of reeds and mono-dimensional for the second set. For the first set, the two dimensions were largely created by the descriptors ease of playing (highly correlated to brightness) and roundness. For the second set (with a constant static compliance), the main dimension was created by all the descriptors, that are correlated. The study of correlation coefficients between subjective descriptors and mechanical parameters shows that the static and dynamic compliance are correlated with the descriptor ease of playing for set 1. For set 2, the correlations are too low to make a link between the perceptual space and the objective parameters.

Introduction

The musical quality of woodwind instruments such as the clarinet or the saxophone depends strongly on the reed quality. Today, musicians use the strength of the reed (indicated on the box by a number or letter), along with the type of cut, to choose a reed that suits their mouthpiece and playing. However, the mechanical properties of single cane reeds with the same strength and same cut may vary from one reed to another, which lead to differences in the playability and in the sound perceived by the player and listener.

The following questions were raised :

• is it possible to distinguish perceptively reeds with the same strength, assuming that the strength is measured accurately ?

• is it possible to explain perceived differences using physical parameters other than the strength ? Currently, experimental characterization of reed mechanical properties by reed makers is performed by measuring the mechanical stiffness, the reed being submitted to a static force at a particular location from the tip. This measurement enables an estimation of the reed strength which is indicated on the box for the clarinet or saxophone player. This method is considered necessary to sort reeds of different strengths and to indicate to the musician whether or not a reed can be played with a particular mouthpiece. However, this approach cannot explain the great differences perceived by musicians between reeds with the same strength and the same cut [START_REF] Petiot | Modeling of the subjective quality of saxophone reeds[END_REF].

The aim of the present work is to try to make the link between the physical properties of a reed and subjective descriptors.

The objective characterization of a reed's physical properties has been studied using different approaches (visualization of cane cells, mechanical measurement of vibration response, optical holography to identify the vibrational modes of the reed). Kolesik [START_REF] Kolesik | Anatomical characteristics affecting the musical performance of clarinet reeds made fromarundo donaxl.(gramineae)[END_REF] studied the anatomical characteristics of cane using confocal laser scanning microscopy. Mukhopadhyay et al. [START_REF] Mukhopadhyay | Saxophone reed inspection employing planar electromagnetic sensors[END_REF] proposed to characterize the quality of saxophone reeds using planar electromagnetic sensors. Obataya et al. [START_REF] Obataya | Acoustic properties of a reed (arundo donax l.) used for the vibrating plate of a clarinet[END_REF] studied the effect of relative humidity on the dynamic Young's modulus of the reed using a free-free beam flexural vibration method and by measuring small plates made with cane. Pinard et al. [START_REF] Pinard | Musical quality assessment of clarinet reeds using optical holography[END_REF] observed the vibrational modes of 24 clarinet reeds in both dry and wet conditions using holographic interferometry. Picart et al. [START_REF] Picart | Tracking high amplitude auto-oscillations with digital fresnel holograms[END_REF] observed the modes of a clarinet reed using holography and measured the displacement of the reed tip during auto-oscillations created in an artificial mouth. Taillard et al. [START_REF] Taillard | Statistical Estimation of Mechanical Parameters of Clarinet Reeds Using Experimental and Numerical Approaches[END_REF] studied the mechanical behaviour of reeds using holography. These experiments have enabled a better understanding of the reed material parameters and the effect of these parameters on the reed vibration, but they do not make the link between mechanical parameters and the subjective evaluation by the musician.

The subjective characterization of reeds has been the subject of fewer studies. Recently, Petiot et al [START_REF] Petiot | Modeling of the subjective quality of saxophone reeds[END_REF] conducted an experiment with 20 tenor saxophone reeds of identical brand and cut played by 10 saxophone players. The musicians were asked to evaluate the ease of playing, the brightness and the general quality of the reeds. In parallel, different signals (mouth pressure, acoustic pressure at the saxophone bell) were recorded during saxophone playing and different playing parameters were estimated. Results show that there is a noticeable agreement between the musicians for the descriptors "ease of playing" and "brightness", whereas the "global quality" descriptor leads to important inter-individual differences. The study of correlations between the subjective descriptors and the objective parameters reveals that the ease of playing is correlated with the threshold pressure and that the subjective brightness is correlated with the spectral centroid of the acoustic pressure. Predictive models of the "ease of playing" and "brightness" using the objective parameters in playing conditions are proposed and show interesting performances. However, there are still no objective parameters related to the reed mechanics that are clearly representative of the subjective dimensions described by players.

The aim of the present work is to explain by physical measurements why particular reeds with the same strength (which corresponds to common playing conditions) produce different sounds (more or less bright) and provide different feelings for the player in terms of their playability (more or less easy to play). In this study, each reed is measured dry and the hygrometry, which can lead to important physical changes as shown by Taillard [START_REF] Taillard | Statistical Estimation of Mechanical Parameters of Clarinet Reeds Using Experimental and Numerical Approaches[END_REF], is not controlled. However, we assume that differences in reed quality are greater than the differences due to hygrometric variations. Finally, it is assumed that large differences in reed quality remain constant over long time periods.

In order to measure reed parameters, two systems are used. First, to measure the parameters under static conditions, a static bench was developed by the reed manufacturer.

Meanwhile, to measure the parameters under dynamic conditions, a vibroacoustic bench has been developed. This system is designed so that it could be used in a factory where many measurements must be done easily. It is based on the measurement of the vibroacoustic response of reeds free to vibrate without any mouthpiece nor artificial lip.

On the other hand, subjective tests have been conducted with a single expert musician in order to assess the reeds according to subjective descriptors. The study of correlations between objective parameters and subjective descriptors enables a performance assessment of the vibroacoustic and static measurement benches for predicting the subjective descriptors. Section 2 of the paper reviews the physical parameters which could potentially explain how a musician perceives a reed when playing (both in terms of playability and sound produced). Section 3 presents two mechanical systems used for characterizing reeds as well as the subjective tests that were conducted using two different reed sets. In the last section, the subjective descriptors are analyzed and the correlations between these descriptors and the mechanical parameters are discussed.

Background

The aim of this section is to describe the role of the reed in the functioning of the instrument and to define which mechanical parameters should be considered in order to predict the subjective parameters.

Physical Quantities

While playing, a clarinet player receives different information as a result of his/her interaction with the instrument and the environment (room) as shown in figure 1. In this situation, the player gets feedback from :

-the mouth pressure -the lip pressure -the radiated sound (direct) and reverberant field due to the room acoustics -the vibrations of the mouthpiece inducing bone conduction and the bore vibrations (sensed by the hands)

In the context of a simplified mouthpiece model, the main physical parameters which govern the functioning of the instrument itself are (see figure 2 Assuming a lumped element model for the reed, the usual reed parameters are the effective mass M , effective damping R, and effective stiffness K as shown for example in [START_REF] Chatziioannou | Estimation of clarinet reed parameters by inverse modelling[END_REF]. Moreover, the reed volume velocity U r , defined by U r = S r dy dt , where S r is the reed effective area, can be taken into account.

Effects of reed on playing parameters

The aim of this section is to recall the link between the reed and mouthpiece parameters (mass, damping, stiffness, reed tip opening, effective vibrating area) and the physical parameters (called playing parameters) which govern the clarinet player's perception (figure 3). The physical parameters which a priori govern the perception of the player are :

-the fundamental frequency of the perceived sound f 0 , related to the frequency of the oscillation signal (pressure in the mouthpiece)

-the threshold pressure P th [START_REF] Backus | Small-vibration theory of the clarinet[END_REF], which is the minimum pressure in the mouth that the musician needs to supply in order to play the instrument, which is probably related to the ease of playing perception [START_REF] Petiot | Modeling of the subjective quality of saxophone reeds[END_REF] -the lip pressure P L used by the musician to change the reed opening H and effective area S r -the properties of the perceived sound (related to the radiated pressure and to the room conditions), which can be described by different objective parameters as explained in [START_REF] Krimphoff | Caractérisation du timbre des sons complexes. ii. analyses acoustiques et quantification psychophysique[END_REF][START_REF] Auvray | Effect of changing the vocal tract shape on the sound production of the recorder : An experimental and theoretical study[END_REF]. The radiated pressure is directly related to the acoustic pressure in the mouthpiece and to the radiation properties of the instrument. According to Barthet [START_REF] Barthet | From clarinet control to timbre perception[END_REF], for a clarinet player the important parameters are the attack of the tones (Attack Time AT ), the spectral richness (Spectral Centroid SC, third Tristimulus coefficient T S), and the irregularity of the spectrum (Odd/Even Ratio OER),

-The properties of the sound perceived by bone conduction, a priori related to the acceleration a m of the mouthpiece, due to the beating of the reed on the lay as observed in [START_REF] Picart | Tracking high amplitude auto-oscillations with digital fresnel holograms[END_REF] The reed stiffness K is a major parameter that affects the playing frequency [START_REF] Dalmont | Some aspects of tuning and clean intonation in reed instruments[END_REF][START_REF] Nederveen | Acoustical aspects of woodwind instruments[END_REF][START_REF] Coyle | Predicting playing frequencies for clarinets : A comparison between numerical simulations and simplified analytical formulas[END_REF] and the threshold pressure [START_REF] Dalmont | Nonlinear characteristics of single-reed instruments : Quasistatic volume flow and reed opening measurements[END_REF][START_REF] Dalmont | An analytical prediction of the oscillation and extinction thresholds of a clarinet[END_REF][START_REF] Dalmont | Oscillation and extinction thresholds of the clarinet : Comparison of analytical results and experiments[END_REF][START_REF] Ferrand | Seuils d'oscillation de la clarinette : validité de la représentation excitateur-résonateur[END_REF]. The reed mass and damping may also affect to a lesser extent the threshold pressure [START_REF] Fritz | Some aspects of the harmonic balance method applied to the clarinet[END_REF]. Finally, the timbre of the clarinet depends on the reed resonance frequency [START_REF] Thompson | The effect of the reed resonance on woodwind tone production[END_REF][START_REF] Benade | The clarinet spectrum : Theory and experiment[END_REF][START_REF] Barthet | On the relative influence of even and odd harmonics in clarinet timbre[END_REF] and on the extent to which the reed beats against the mouthpiece lay (observed numerically [START_REF] Avanzini | Modelling the mechanical response of the reed-mouthpiecelip system of a clarinet. part i. a one-dimensional distributed model[END_REF] and experimentally [START_REF] Picart | Study of the mechanical behaviour of a clarinet reed under forced and auto-oscillations with digital fresnel holography[END_REF]).

These results show that the reed properties affect the playing in different ways. However, the relationship between reed mechanical parameters and subjective descriptors is not well established yet. The following sections aim at investigating the relationship between objective parameters describing the reed mechanics and subjective descriptors.

Materials and methods

This section presents the two methods used for measuring the reed mechanical parameters as well as the subjective tests performed to derive subjective descriptors.

Basic assumption of the study

The results of previous studies ( §2) show that the linear reed parameters (stiffness, mass, damping) while playing, plus the beating of the reed against the mouthpiece lay, affect the playing parameters (threshold pressure, timbre, vibrations).

As we wish to explain subjective descriptors along with reed mechanical parameters measured during the manufacturing process, we assume in this study that the linear parameters of the reed alone (not in a playing situation) can affect the playing parameters.

Reed static characterization

In this section, we present the principle of the static measurement bench which enabled us to measure the static compliance StC of the reed used in this paper. The static compliance is used to deduce the strength of the reed (range 1 to 5) defined by the manufacturer on the reed box and used by the musician.

The reed static compliance is measured using a flexural bench as follows. The reed is maintained on a plate with a screw pressing the stock (the geometry of the reed is defined in Figure 4). The vamp of the reed is bent using a spring with a known stiffness. Therefore, a known displacement of the spring imposes a known force on the reed. At the same time, the displacement of the reed tip is measured at a distance x from the reed tip (with x ∈ [START_REF] Kolesik | Anatomical characteristics affecting the musical performance of clarinet reeds made fromarundo donaxl.(gramineae)[END_REF][START_REF] Pinard | Musical quality assessment of clarinet reeds using optical holography[END_REF] mm depending on the reed cut).

This measurement enabled us to deduce the compliance (inverse of the stiffness), defined as the ratio between the measured displacement and force at the reed tip. The compliance range is from 8 (very hard reeds, strength = 5 or strength = "H") to 24 (very soft reeds, strength = 1 or strength = "S"). As the static bench is not calibrated, this compliance is given in arbitrary units.

Reed dynamic characterization

The vibroacoustic (VA) measuring bench enables a characterization of single cane reed mechanical properties. An important feature is that this bench could be used for industrial applications. For this reason, this bench does not use any artificial lip nor any mouthpiece as presented in previous work [START_REF] Gazengel | Objective and subjective characterization of saxophone reeds[END_REF].

Principle of the VA bench

The vibroacoustic bench is shown in figure 5. It excites the single cane reed with an acoustic field and measures the reed tip displacement at different points.

The measuring bench is composed of the following parts :

-A plate which controls the vibrating length of the reed and the dimensions of the gap around the reed boundary. In this experiment, the vibrating length is 30 mm and the width of the gap is 1 mm (see figure 5) -A clamping system made of a linear support equipped with a spring that applies a force on the reed. The position of the linear support can be modified along the x axis.

-An acoustic system composed of a loudspeaker (tweeter) placed in the back cavity (volume ∼ 4cm 3 ) and excited with a sweep signal.

-A displacement sensor Philtec RC25 [START_REF] Philtec | Displacement sensor model rc25[END_REF] measuring the reed displacement. This sensor can be displaced in the longitudinal (x) and transversal (z) directions in order to measure different points. It can also be moved in the vertical (y) direction in order to use the linear region of the sensor sensitivity, the optical sensor being non linear.

-A 1/4 inch microphone placed in the back cavity near the reed in order to measure the excitation pressure. The dimensions of the back cavity enables to assume constant pressure in the cavity up to 6kHz.

The equivalent reed parameters are estimated by measuring the reed response y i /p, where y i is the displacement of a point i of the reed and p is the acoustic excitation pressure in the cavity, assumed to be homogeneous. Three reed responses H i m (m for "measured") are measured for three points placed on the reed tip (left, center, right) as shown in figure 6.

According to the displacement sensor dimensions and to the time needed for measuring the different reeds (up to 150 reeds for one experiment), it was assumed that three points enabled to study the transverse vibration behaviour of the reeds. However, the analysis of the transverse modes did not give any relevant results so that the three transfer functions obtained at the three points are used for estimating the first mode properties only.

An example of the measurement results is given in figure 7 for a vibrating length of 30 mm. It shows that the first vibration mode (first flexural mode) is around 2000 Hz and that it is predominant (60 dB greater than the amplitudes of the higher modes). At resonance, the order of magnitude of the reed tip displacement is 1 µm/P a. Knowing that the pressure level in the cavity is around 120 dB SPL, the amplitude of the reed tip is around 20 µm at the resonance frequency. This shows that the measurement is performed using a smaller displacement of the reed tip than that observed in playing conditions (0.1 to 0.5 mm typically).

The second mode is a torsional mode, the third mode is the second flexural mode and mode 4 is a combination of flexural and torsional effects. These results are similar to those obtained by Taillard [START_REF] Taillard | Statistical Estimation of Mechanical Parameters of Clarinet Reeds Using Experimental and Numerical Approaches[END_REF], the main difference being the number of modes observed.

Estimation of reed parameters : principle

Given that the reed displacement y i is measured at three different points (i = 1, 2, 3) and assuming a linear behaviour of the reed (no mouthpiece, no lip), the reed displacement can be written as

M Ÿ + B Ẏ + KY = P, (1)
where

Y =        y 1 y 2 y 3        is the displacement vector, P =        p 1 p 2 p 3       
is the pressure vector, p i being the excitation pressure at point i, and where M , B, K are respectively the mass, damping and stiffness matrices.

Assuming the excitation pressure is identical at the three points i, the excitation pressure can be written P = E.p, where E is the unit vector and p is the acoustic pressure measured by the reference microphone. Using this assumption, the theoretical reed response can be written

H th =        y 1 p y 2 p y 3 p       
. Finally, the observation of the measured FRF suggests that the modes are uncoupled (figure 7) so that the reed Frequency Response Function vector can be written as (subscript th for theoretical)

H th = N n=1 C n 1 + j ω ωnQn -( ω ωn ) 2 , ( 2
)
where N is the number of modes observed and C n is the compliance vector for mode n written as

C n =        c n1 c n2 c n3        .
Parameters C n , ω n and Q n are estimated by adjusting the theoretical FRF H th to the measured transfer function H m using a least mean square (LMS) estimator described by Piranda [START_REF] Piranda | Analyse modale expérimentale[END_REF]. Figure 8 shows an example of measured and estimated responses.

As the first vibration mode of the reed is predominant (see figure 7), only the parameters of the first mode (C 1 , ω 1 and Q 1 ) are used. In the next part of the paper, the parameters used for studying the correlation with subjective indicators are : -the dynamic compliance DyC defined as DyC = , or average compliance estimated at the three measurement points.

-the first resonance frequency F r = f 1 = ω 1 2π . -the quality factor of the first resonance Q = Q 1 .

In the next part, all measurements are done with a reed vibrating length equal to 30 mm. Modal parameters for higher modes were also considered for measuring the correlation between mechanical parameters and subjective descriptors but did not give any relevant results.

Accuracy of the bench

The accuracy of the bench was studied in order to estimate the uncertainty in the reed parameters for the first mode. This value is an indicator of the ability of the vibroacoustics bench to classify reeds in different families. 50 reeds having the same strength and same cut were measured by two operators for two consecutive days (operator 1 during day 1, operator 2 during day 2). For each reed, the relative differences between the 2 measurements (day1, day2) was computed. Then the standard deviation of these relative differences (in %) led to the estimation of the measurement error between the two measurements. For the first mode compliance DyC, the standard deviation is 2.5%. Assuming that the measurement results follow a normal distribution, the relative error for DyC is estimated to be ±5% with a 95% confidence.

The estimation of uncertainties for the resonance frequency and the quality factor show that the resonance frequency is well estimated, the 95% confidence interval being ±1% for the resonance frequency and ±2% for the quality factor.

Comparison between static and dynamic measurements

In order to compare the dynamic compliance DyC and the static compliance StC, 50 reeds with variable static compliance between 8 (strength = 5) and 24 (strength = 1) were measured. The result is shown in figure 9. In this measurement, the dynamic compliance value DyC is within the interval 15 -45 nm/P a and shows a high correlation between the two measurements (R 0.99). The relationship between the two measurements does not seem to be linear for high static compliance (greater than 20).

Strength = 5 Strength = 1

Subjective tests

The aim of the subjective tests was to obtain a reliable quantitative assessment of the perception of different reeds by a musician.

Reed samples

Two sets of Bb clarinet reeds were used for this study. The first set (referred to as "unselected reeds") contained 50 reeds of the same strength but different static compliance (variation ±10%). The second set (referred to as "selected reeds") contained 150 reeds of the same strength and approximately identical static compliance (variation ±1%).

Finally, when compared with the full range of commercially available reed strengths (from 1 to 5), it can be seen that the compliance variations of these two groups is small or very small (see Figure 10). and set 2 (selected reeds).

Procedure

One expert musician participated in the subjective tests. This expert is a skilled clarinet player, working regularly with the reed manufacturer as a professional tester. With only one expert, we are of course aware that it is not possible to generalize the results of the experiment nor to make general conclusions concerning the perception of reeds by musicians. So, the work focuses on an interpretation of the link between the subjective evaluation of a given expert (professional tester) and the mechanical parameters of the reeds. Based on previous studies [START_REF] Gazengel | Objective and subjective characterization of saxophone reeds[END_REF][START_REF] Petiot | Modeling of the subjective quality of saxophone reeds[END_REF], we propose different subjective descriptors to assess the reeds depending on the reed sets. These descriptors are detailed later in §4.1 and §4.2.

After the reed was mounted on the mouthpiece by an external operator, the musician was asked to play freely to assess all the descriptors (free playing conditions) without knowing the reed type. The evaluation phase used pencil+paper to assess the reeds on graphical quantitative scales. After playing the reed, the expert was invited to put a mark on the rating scale of each subjective descriptor, corresponding to his perceived sensation of the intensity of the descriptor. The mark was next transformed into a quantitative rating by measuring the distance in cm between the left end of the scale and the location of the mark.

The reeds presentation order was randomized. 3 repetitions of each reed were proposed, in 3 independent blocks, for a total of 150 evaluations for set 1 and 450 for set 2.

The assessment of this huge number of evaluations has been possible given the particular abilities of the musician, who is a professional tester. If necessary, the expert was allowed to make slight adjustments of the reed on the mouthpiece, but of course without watching the type of reed used (hidden by the ligature). The consistency of the mounting of the reed was unfortunately not checked, it was only dependent on the ability of the operator. This source of variation is assumed to be weak.

Data analysis

Each descriptor was analyzed using a one-way variance analysis. Reeds were treated as a fixed effect. Differences were considered significant at p < 0.05.

The subjective evaluations of the expert were analyzed with classical methods in sensory analysis [START_REF] Naes | Statistics for Sensory and Consumer Science[END_REF]. The evaluations are quantitative ratings of each product (the reed) according to different descriptors, during 3 repetitions. The objective of the analysis is to understand and describe the differences between the products (the reeds). The first way to analyse the data is to consider each descriptor one by one (univariate analysis). In this case, the analyst wants to know if the differences between the average values of the reeds (across the repetitions) are important enough, given the repetition error. The classical method to perform this analysis is ANOVA (Analysis of variance). Analysis of variance (ANOVA) is a general method to study the influence of one or several factors (with different levels) on a quantitative response [START_REF] Lea | Analysis of Variance for Sensory Data[END_REF]. The method belongs to the general frame of the linear model, that proposes to fit a linear model from a set of observations. In our case, the factor under study is the product (the reed, with a number of levels equal to the number of reeds) and the response is the ratings given by the expert for a given descriptor. Assumptions are made according to the fitting error of the model (normal distribution) that allow the definition of statistical tests to decide whether or not the differences between the average ratings are important enough. The classical statistics in ANOVA is the Fisher variable F (or F-ratio) that makes the ratio between the between-reeds variability to the within-reed variability. A statistical test (F-test) on the Fisher variable is carried out, to determine the false rejection probability (p-value or p) of the hypothesis H0. In our experiment, H0 is stated as : All the reeds obtain on average the same ratings (in other words : there is no effect of the reeds on the ratings). For a given significance threshold (5%, or 1%), critical values F 5% (or F 1% ) can be computed with the Fisher-Snedecor distribution law, considering H0 true. The next stage of the test is to compare the observed value of F (Fobs with the actual ratings) to the critical values F 5% . If Fobs is larger than F 5% , H0 can be rejected with a false rejection probability lower than 5%. Another way to present the results of the test is to give the pvalue (p) that corresponds to the probability with which Fobs= F p% . If p is low, H0 can be rejected with a very weak risk to take a false decision. The effect of the reeds on the ratings is then considered as significant. ANOVA and the F-test is used in the next section for each subjective descriptor, to detect if the differences in the ratings of the expert are important enough to be significant. The second way to analyse and understand the ratings is to study the differences between the reeds, by considering all the descriptors together (multivariate analysis). The most applied approach for getting an overview of multivariate data is to use Principal Component Analysis (PCA) on the reeds-by-descriptors matrix, after having averaged out replicates. The products are considered as individuals, and the descriptors as variables. PCA provides a transformation to convert variables into a set of uncorrelated principal components. The transformation is defined in such a way that the first principal component accounts for as much of the variability in the data as possible (% of inertia), the succeeding components accounting for the remaining variance in a decreasing way. In this way, a graphical representation of the data can be made in a low dimensional space (2 or 3 dimensions) without much loss of information. This representation is interesting to reveal the underlying structure of the data, to explain the main differences between products and to visualize the correlations between variables. Interpretation of the principal components can be provided by an examination of the correlations between the descriptors and the principal components, and with the projection on the factorial plane of additional variables. In the next section, PCA is used to describe and interpret the subjective ratings for the two sets of reeds. A principal component analysis (PCA) was performed on the correlation matrix of the expert sample means. Objective parameters were plotted on the factorial space as additional variables.

Results

Set 1 : unselected 50 reeds

A set of 50 reeds for Bb clarinet of identical brand, cut and strength (strength = 3.5) was taken from 5 commercial boxes of 10 reeds each. The name given by the makers to the cut of reed that was used in this set is Bb 1 . The static compliance StC was measured using the bench described in §3.2. The static compliance range ( max(StC)-min StC mean(StC)

) in this set was 20% and the dynamic compliance range was 32%. The variation range defined by 2σ, where σ is the standard deviation, was 2σ = 12% for the static compliance and 2σ = 15% for dynamic compliance.

Subjective tests

For this set , 3 descriptors were considered :

-Ease of playing. Representing an assessment of the ease of producing a sound, rated on a 5-level structured scale : very difficult -difficult -average -easy -very easy (the higher the rating, the easier the reed) -Brightness. Representing an assessment of the overall brightness of the sounds produced by this reed, rated on a 5-level structured scale : very dark -dark -neutralbright -very bright (the higher the rating, the brighter the reed).

-Roundness. Representing an assessment of the overall roundness of the sounds produced by this reed, rated on a 3-level structured scale : nasal -neutral -round (the higher the rating, the rounder the reed).

The results of the subjective tests on ease of playing, brightness and roundness are presented respectively in figures 11, and 12 and 13. The figures plot the average ratings distribution for the three descriptors. They represent the histogram of the average ratings of the 50 reeds, by considering 9 intervals (bins) for the ease of playing and brightness, and 7 intervals for roundness. To facilitate the interpretation of the histogram, the levels used in the structured scales are mapped onto the corresponding intervals of the histogram.

For ease of playing, the expert focused strongly on the level 'easy', for which the estimation error (proportional to the standard deviation of the three evaluations) was much lower than for other levels. For brightness, the expert focused less strongly on the level 'Neutral' but the estimation error for level 'bright' was smaller. For roundness, most of the reeds were considered as 'sneutral'.

Global significant differences between reeds were studied by ANOVA. The F-ratios of the one-way ANOVA are presented for each descriptor in table 1.

Descriptor ease of playing brightness roundness Reed F-ratio 4.57 (***) 2.75 (***) 2.18 (***) * p < 0.05 -** p < 0.01 -*** p < 0.001

Table 1: Results of the one-way ANOVA for each descriptor (F-ratio) for the 50 unselected reeds. All the descriptors showed a significant product effect with large F-ratios (p < 0.001), the effect being the most important for ease of playing and the least important for roundness.

Given that the F-ratio represents the between-reeds variability to the within-reed variability, the larger the F-ratio, the larger the variability due to the reeds in comparison to the variability due to the repetitions (repetition error of the expert). The assessments of the expert are repeatable enough to show a significant reed effect, particularly for ease of playing.

A principal component analysis was conducted on the mean ratings to show the correlations between the subjective descriptors and to explain the main differences between the reeds. The first two components account for more than 87% of variance. This high percentage signifies that the reeds locate mainly in a bi-dimensional space and that interpretations of the differences between reeds can be made in this space, the loss of information being weak (13%). To help the interpretation of the factorial plane, a VARIMAX rotation in the factorial plane is provided. The principle of VARIMAX is to rotate the two axis of the plane of the first two principal components in order to associate (as much as possible) each descriptor to at most one axis. After VARIMAX, the data are not represented according to the two principal components any more, but according to a new system of coordinates (D1, D2) that facilitates the interpretation. Figure 13 presents the map of the variables according to D1 and D2 (47% of variance is accounted by D1, and 40% by D2). Given the orientation of the descriptors in the plane, the first dimension is largely created by the two descriptors ease of playing and brightness which are correlated. The second dimension is largely created by the descriptor roundness.

Correlation Objective parameters /subjective descriptors

To provide an interpretation of the subjective descriptors, a one to one linear correlation between each subjective descriptor and each objective parameter is proposed. The linear coefficient of correlation is given in table 2 with the p-value of the significance test.

The most important correlations are between the ease of playing and the static and dynamic compliance (R = 0.74 ; R = 0.75).

A causality is suspected between compliance (static or dynamic) and ease of playing.

Indeed, the physical model of the reed predicts a monotonous relationship between the compliance of the reed and the threshold pressure, which may govern the descriptor ease of playing [START_REF] Gazengel | Objective and subjective characterization of saxophone reeds[END_REF]. This result confirms the representativeness of the physical model.

Figure 15 shows the ease of playing versus the dynamic compliance. For all the reeds, it shows a monotonous relation between the two variables but the difficult reeds (low DyC)

show a large dispersion which might be due to the high uncertainty in the expert's rating. The correlation between static and dynamic measurements shows a high correlation (R = 0.99) for a large range of strength (1 to 5) as shown in §3.4 and on figure 9. In practice, such a range can not be played by one musician who is using only a single strength value (3 or 3.5 usually). For the set of 50 'unselected reeds', the correlation between static and dynamic measurement is lower (R = 0.80). The fact that the range for static compliance is ± 10%

and that the uncertainty in dynamic compliance is ± 5% might explain the relatively lower correlation.

Finally, as dynamic compliance and static compliance are correlated (R 1 for strength 1 -5 and R 0.8 for strength 3.5), the dynamic compliance doesn't give additional information compared to the static compliance to predict the descriptor ease of playing.

For brightness, the correlation with the compliance (static, dynamic) is lower but still significant. Soft reeds are easy to play and bright, which can be explained by the fact the reed can beat more easily on the mouthpiece, leading to higher harmonics in the radiated sound.

For roundness, correlation coefficients with compliance (static, dynamic) are low but still significant, at the 5% level. The negative sign of the coefficient makes sense, a round sound not being bright, may then be produced by a stiff reed. This result must be confirmed by further experiments.

These conclusions are also supported by observing the projection of the four objective parameters (static compliance StC, dynamic compliance DyC, resonance frequency and quality factor of the first mode F r and Q) onto the factorial map as additional variables (figure 14).

Set 2 : 150 selected reeds

In this section, we investigate whether the expert is able to differentiate reeds with very similar static compliance. For this, reeds were selected such that the static compliance, measured using the bench described in §3.2, remains in an interval ±1 % around the mean value (StC = [10.1 -10.3]). However, the dynamic compliance range remains larger than the static compliance range (DyC = [18 -22]) with a standard deviation of 0.8).

This set was made up of 150 reeds for Bb clarinet with same strength (strength = 3.5) and brand. Reeds were selected with three different cuts, named '56', 'V 12'and 'Bb 2 ' by the makers, with 50 reeds of each cut used.

Subjective tests

Four descriptors were defined, in collaboration with the expert who prefered 'projection'

to 'roundness' :

-Ease of playing, similar to set 1 -Brightness, similar to set 1 -Projection. This dimension was proposed by the expert and corresponds to the feeling that the sound propagates far away from the instrument, rated on a 5 level structured scale : very weak -weak -neutral -strong -very strong (the higher the rating, the stronger the projection).

-Global quality. This dimension is related to the overall quality of the reed, linked to the experience and the preference of the musician. It was rated on a 5-level structured scale : very bad -bad -average -good -very good.

-In addition to these descriptors, the expert had to indicate which type of reed he thought he was playing (V 12, 56, or Bb 2 ). This information was introduced to assess the ability of the expert to recognize the reed type. Given the abilities of the expert (he is the in-house musician for the control of playing qualities of reeds in a company),

no training phase was necessary to present the reeds types that he knows perfectly.

The objective was to assess the perceptual differences between this set of very similar reeds from a compliance point of view. The F-ratios of the one-way ANOVA are presented for each descriptor and each reed type in table 3.

V 12

Bb 2

Ease of playing F-ratio 3. Except for the descriptor brightness for reeds '56', all the effects were significant. The most important effects were for the 'Bb 2 ' reed type, and for the descriptor ease of playing, which obtained the most important F-ratios. The assessments of the expert are repeatable enough to show a discriminant assessment of the reeds, since they have very similar strength.

In general, the most important effects were obtained for Bb 2 reeds, particularily for the descriptor projection (proposed by the expert). The F-ratio was smaller for reeds 56 and V 12, the assessment being less reliable.

The recognition rate of the expert was very high (100% for the family 56, 96% for V 12 and 99% for Bb 2 ), a sign of a very good ability to identify the reeds in a blindfold test.

The principal component analysis was conducted on the Bb 2 reed data to show the correlation between the subjective descriptors. Two factors were sufficient to take account for 95% of variance. So the reeds locate mainly in a bi-dimensional space and the descriptors are correlated. Figure 16 represents the map of the variables according to the two principal components F1 and F2 (F1 : 81.19% of variance, F2 : 14.55%). The first factor is largely created by the three descriptors ease of playing, projection and global quality which are correlated.

The second factor is more difficult to interpret, given that it is created by almost all the descriptors (largely brightness) and given that the percentage of variance on this factor is low (14.55%). As a conclusion, the perceptual space of the reeds is highly dominated by one dimension (that can be interpreted as the general quality of the reeds). The consequence of the selection of reeds according to the static compliance (set 2) is that the perceptual differences between the reeds become mono-dimensional. The proposed descriptors and the ratings of the expert are unable to provide a second dimension that could explain the differences between the reeds. Similar results were obtained for the reeds 56 and V 12.

The factorial map (Figure 16) shows the descriptors that describe the dimensions of the perceptive space.

Correlation Objective/subjective

Table 4 shows the correlation coefficient between objective parameters obtained with the VA bench and the four subjective descriptors in the case of Bb 2 reeds. It shows a significant but low correlation between ease of playing and compliance DyC, which conforms to the previous experiment and suggests that DyC and StC are not strictly equivalent.

Finally, the analysis of the correlation coefficient did not give an additional explanation about the perception of the reed behaviour. This can be explained by the fact that the reeds were intentionally selected with a constant static compliance (which is highly correlated with the dynamic compliance). Similar results were obtained for 56 and V 12 reeds.

DyC

Conclusion

In this work, two sets of reeds were characterised subjectively by an expert and measured using a static bench and a vibroacoustics bench.

Results obtained with a set of 50 reeds (strength 3.5) show that the subjective tests showed significant differences between the reeds the descriptors ease of playing, brightness and roundness, the effect being the highest for ease of playing and the lowest for roundness.

A principal component analysis shows that the perceptual space is almost bi-dimensional.

The first dimension was largely created by the two descriptors ease of playing and brightness (which are correlated). The second factor is largely created by the descriptor roundness.

Correlation between subjective indicators and objective parameters show that ease of playing is correlated with the static compliance and dynamic compliance of mode 1. Altough the reed strength is constant, this experiment showed that the reed compliance (static or dynamic) can be used for selecting reeds according to a scale going from soft and bright reeds to hard and dark reeds.

Results obtained with a set of 150 selected reeds made from three families (56, V 12, Bb 2 )

with the same static compliance (range ±1%) showed that the expert can recognise the reed family with a recognition rate higher than 96%. The subjective tests showed significant effects of the reeds for all descriptors except for brightness and reeds 56. The expert perceived a larger variability for the Bb 2 reed type, and for the descriptors ease of playing and projection, that obtained the most important F-ratios in the ANOVA. However, the ease of playing descriptor still dominates despite the reed having the same static compliance. When the reeds are very similar in term of static compliance (set2), the Principal Component Analysis showed that the perceptual differences between the reeds rated by the expert is almost mono-dimensional (general reed quality). Due to the very small variability of static compliance, the correlation coefficients between subjective descriptors and mechanical parameters, including higher mode properties (compliance, quality factor, frequency, symmetry), did not explain the perceptual space in this case.

Finally, this work showed that the mechanical parameters (static compliance, vibration characteristics) of the reed alone are strongly correlated with the ease of playing and brightness for reeds having the same strength. This property is already used by manufacturers to offer musicians the possibility of selecting reeds more accurately using the measurement of the static compliance. However the experiment with reeds of equal static compliance showed that the ease of playing is not fully explained by reed compliance. This suggests that the interaction between the reed and the mouthpiece lay during playing can not be ignored, particularily because the reed displacement amplitude is much higher while playing than during the measurement process with mechanical benches. So, the important conclusion of the work is that it is necessary to make measurements at high displacement corresponding to playing conditions using the measurement of important physical quantities such as mouthpiece pressure, mouth pressure and reed tip displacement.

  ) : -p : acoustic pressure in the mouthpiece -P m : mouth pressure -y : reed tip displacement -U i : volume flow velocity entering the mouthpiece -U r : volume flow velocity induced by the reed movement -U : volume flow velocity entering the bore -H : opening of the reed at rest with the lip pressure

Figure 1 :Figure 2 :

 12 Figure 1: Information received by the clarinet player while playing.

Figure 3 :

 3 Figure 3: Systemic representation of the clarinet playing (input -output).

Figure 4 :

 4 Figure 4: View of the different parts of the reed (from [26]).

Figure 5 :

 5 Figure 5: Description of the vibroacoustic bench (VA bench) (front and top view).

Figure 6 :

 6 Figure 6: View of the three positions used for measuring the reed displacements y 1 , y 2 and y 3 .

Figure 7 :

 7 Figure 7: Example of reed response measurement H i m . The reference for the amplitude level is 1 µm/P a.

Figure 8 :

 8 Figure 8: Example of transfer function identification using the method described by Piranda [29] (o : measurement H m , -: theoretical FRF H th ). The amplitude reference level is 1 µm/P a.

3 k=1 c 1k 3

 3

Figure 9 :

 9 Figure 9: Comparison between the dynamic compliance DyC and the static compliance StC for a set of 50 reeds.

Figure 10 :

 10 Figure 10: Range of compliance variation for commercial reeds (strength 1 to 5), set 1 (unselected reeds)

Figure 11 :

 11 Figure 11: Distribution of ease of playing mean values for set 1 (50 unselected reeds).

Figure 12 :

 12 Figure 12: Distribution of brightness mean values for set 1 (50 unselected reeds).

Figure 13 :

 13 Figure 13: Distribution of roundness mean values for set 1 (50 unselected reeds)

Figure 14 :

 14 Figure 14: PCA of the subjective assessments (map of the variables, with VARIMAX rotation)

Figure 15 :

 15 Figure 15: ease of playing for set 1 (50 unselected reeds) as a function of dynamic compliance. Plot of mean value and uncertainty as error bar.

Figure 16 :

 16 Figure 16: PCA of the subjective descriptors -reeds type Bb 2 (map of the variables).

  07 -0,38 (**) * p < 0.05 -** p < 0.01 -*** p < 0.001

Table 2 :

 2 linear coefficient of correlation between the objective parameters and the subjective descriptors.

Table 3 :

 3 Results of the one-way ANOVA for each descriptor and each reed type (F-ratio).

Table 4 :

 4 Linear coefficient of correlation between the objective parameters and the subjective descriptors for reeds Bb 2 .

Acknowledgements

The authors would like to thank Marilyn Twell for proofreading and correcting the English and the reviewers for helpful comments. Thanks to Alberto Muñoz for drawing figure 1.