A Ruellan

A Guinault

C Sollogoub

G Chollet

A Ait-Mada

V Ducruet

S Domenek

Industrial vegetable oil by-products increase the ductility of polylactide

Keywords: biodegradable polymers, poly(lactic acid), ductility, vegetable oil by-products, crazing

The use of industrial by-products of the vegetable oil industry as ductility increasing additives of polylactide (PLA) was investigated. Vegetable oil deodorization condensates were melt-blended by twin-screw extrusion up to a maximum inclusion quantity of 20 wt% without preliminary purification. Sample films were obtained by single screw cast extrusion. Compounded PLA films featured largely improved ductility in tensile testing with an elongation at break up to 180%. The glass transition temperature remained higher than room temperature. The native mixture of molecules, which composed the deodorization condensates, had superior performance compared to a synthetic mixture of main compounds. The investigation of the correlation between composition of the additives and the ductility of the PLA blends by Principal Component Analysis showed synergy in property improvement between fatty acids having a melting point below and beyond the room temperature. Furthermore, a compatibilizing effect of molecules present in the native mixture was evidenced. Oil deodorization condensates, which are a price competitive by-product of the vegetable oil industry, are therefore a very promising biobased and biodegradable additive for improving the ductility of PLA.

Introduction

Biodegradable and/or biobased polymers are widely studied to replace petroleum-based materials with the aim of contributing to higher sustainability of plastics. However, in many cases these new products still do not meet economical or functional requirements to be competitive in commodity applications. Polylactide (PLA) is one of the polymers which can potentially enter high volume markets, mostly in the food packaging [START_REF] Auras | An overview of polylactides as packaging materials[END_REF][START_REF] Auras | Evaluation of oriented poly(lactide) polymers vs. existing PET and oriented PS for fresh food service containers[END_REF][START_REF] Petersen | Potential of biobased materials for food packaging[END_REF] or textile [START_REF] Solarski | Designing polylactide/clay nanocomposites for textile applications: Effect of processing conditions, spinning, and characterization[END_REF] sector. Polylactide is a biobased and compostable polymer [START_REF] Zhang | Sun X.: Mechanical and thermal properties of poly(lactic acid)/starch blends with dioctyl maleate[END_REF][START_REF] Södergård | Properties of lactic acid based polymers and their correlation with composition[END_REF][7] issued from polymerization of lactic acid produced by fermentation of starch, potatoes or beets [7,[START_REF] Kaplan | Biopolymers from renewable resources[END_REF]. Although it has been known before [START_REF] Lowe | Preparation of high molecular weight polyhydroxyacetic ester[END_REF], production cost reduction in the 1990s and 2000s thanks to breakthroughs in the polymerization technology [START_REF] Gruber | Continuous process for the manufacture of lactide and lactide polymers[END_REF][START_REF] Gruber | Continuous process for manufacture of lactide polymers with purification by distillation[END_REF], made it economically competitive with petroleum-based materials. PLA offers many interests such as a glass transition higher than room temperature, ease of processing, high transparency, printability, glossy aspect [START_REF] Auras | An overview of polylactides as packaging materials[END_REF][START_REF] Avérous | Monomers, polymers and composites from renewable resources[END_REF][START_REF] Siracusa | Biodegradable polymers for food packaging: A review[END_REF]. However, its mechanical properties still remain an obstacle for many applications. In fact, PLA features high tensile strength and modulus but also high brittleness [START_REF] Liu | Research progress in toughening modification of poly(lactic acid)[END_REF]. In order to enhance the PLA ductility, melt-blending with rubbery materials as poly(ether)urethane [START_REF] Li | Toughening of polylactide by melt blending with a biodegradable poly(ether)urethane elastomer[END_REF]16], polyamide elastomer [START_REF] Zhang | Surprising shape-memory effect of polylactide resulted from toughening by polyamide elastomer[END_REF], acrylonitrile-butadiene-styrene copolymer, or various impact modifiers [START_REF] Byrne | The effect of masterbatch addition on the mechanical, thermal, optical and surface properties of poly(lactic acid)[END_REF][START_REF] Murariu | Polylactide compositions. Part 1: Effect of filler content and size on mechanical properties of PLA/calcium sulfate composites[END_REF][START_REF] Murariu | Polylactide (PLA) and highly filled PLA -Calcium sulfate composites with improved impact properties[END_REF][START_REF] Hughes | Polymeric blend comprising polylactic acid[END_REF][START_REF] Brulé | Composite based on polylactic acid and polyamide, having improved impact resistance, its manufacturing process and use[END_REF][START_REF] Markarian | Biopolymers present new market opportunities for additives in packaging[END_REF][START_REF] Jin | Blending of poly(L-lactic acid) with poly(cis-1,4-isoprene)[END_REF][START_REF] Li | Improvement in toughness of poly (L-lactide) (PLLA) through reactive blending with acrylonitrile-butadiene-styrene copolymer (ABS): Morphology and properties[END_REF][START_REF] Mccarthy | Polylactic acid based blends[END_REF] is effective. Furthermore, plasticizers such as citrate esters [START_REF] Baiardo | Thermal and mechanical properties of plasticized poly(L-lactic acid)[END_REF][START_REF] Ljungberg | Film extrusion and film weldability of poly(lactic acid) plasticized with triacetine and tributyl citrate[END_REF][START_REF] Murariu | Polylactide (PLA) designed with desired end-use properties: 1. PLA compositions with low molecular weight ester-like plasticizers and related performances[END_REF][START_REF] Lemmouchi | Plasticization of poly(lactide) with blends of tributyl citrate and low molecular weight poly(D,L-lactide)-b-poly(ethylene glycol) copolymers[END_REF][START_REF] Wang | Morphology, mechanical properties, and durability of poly (lactic acid) plasticized with di(isononyl) cyclohexane-1,2-dicarboxylate[END_REF][START_REF] Sierra | Relationship between properties, citrate content and postproduction time for a plasticized polylactic acid[END_REF][START_REF] Courgneau | Analysis of the structure-properties relationships of different multiphase systems based on plasticized poly(lactic acid)[END_REF][START_REF] Labrecque | Citrate esters as plasticizers for poly (lactic acid)[END_REF][START_REF] Ljungberg | Film extrusion and film weldability of poly(lactic acid) plasticized with triacetine and tributyl citrate[END_REF][START_REF] Yu | Fabrication and characterization of poly(lactic acid)/acetyl tributyl citrate/carbon black as conductive polymer composites[END_REF][START_REF] Ruellan | Solubility factors as screening tools of biodegradable toughening agents of polylactide[END_REF], polyethylene glycols [START_REF] Baiardo | Thermal and mechanical properties of plasticized poly(L-lactic acid)[END_REF][START_REF] Courgneau | Analysis of the structure-properties relationships of different multiphase systems based on plasticized poly(lactic acid)[END_REF][START_REF] Kulinski | Plasticization of poly(L-lactide) with poly(propylene glycol)[END_REF][START_REF] Kulinski | Crystallization, structure and properties of plasticized poly(L-lactide)[END_REF][START_REF] Hu | Crystallization and phase separation in blends of high stereoregular poly(lactide) with poly(ethylene glycol)[END_REF][START_REF] Jacobsen | Plasticizing polylactide -The effect of different plasticizers on the mechanical properties[END_REF][START_REF] Martin | Poly(lactic acid): Plasticization and properties of biodegradable multiphase systems[END_REF][START_REF] Pillin | Thermo-mechanical characterization of plasticized PLA: Is the miscibility the only significant factor?[END_REF][START_REF] Sheth | Biodegradable polymer blends of poly(lactic acid) and poly(ethylene glycol)[END_REF] or manifold other molecules [START_REF] Murariu | Polylactide (PLA) designed with desired end-use properties: 1. PLA compositions with low molecular weight ester-like plasticizers and related performances[END_REF][START_REF] Pillin | Thermo-mechanical characterization of plasticized PLA: Is the miscibility the only significant factor?[END_REF][START_REF] Martino | Thermal and mechanical characterization of plasticized poly(L-lactide-co-D,L-lactide) films for food packaging[END_REF][START_REF] Martino | Processing and characterization of poly(lactic acid) films plasticized with commercial adipates[END_REF][START_REF] Ljungberg | Thermomechanical film properties and aging of blends of poly(lactic acid) and malonate oligomers[END_REF][START_REF] Ren | Dynamic mechanical and thermal properties of plasticized poly(lactic acid)[END_REF] can be used. Unfortunately, impact modifiers are often not biodegradable nor biobased and plasticizers are generally derived from fossil resources, degrading one important environmental advantage of employing PLA. Therefore biobased and biodegradable additives have been and are investigated for toughening PLA. Chemically modified vegetable oils have received important research interest, because ester or epoxy groups can be degraded by microorganisms [START_REF] Rahman | The plasticizer market: An assessment of traditional plasticizers and research trends to meet new challenges[END_REF], maintaining the biodegradability of blends with PLA. Polymerized soybean oil derivatives prepared by crosslinking double bonds of alkyl chains afforded ductility increase after meltblending with PLA and using a compatibilizer [START_REF] Robertson | Toughening of polylactide with polymerized soybean oil[END_REF]. Improvements were also obtained with conjugated soybean oil which was reactively compatibilized with PLA by unsaturated triglycerides [START_REF] Gramlich | Reactive compatibilization of poly(l-lactide) and conjugated soybean oil[END_REF]. Epoxidized soybean oil [START_REF] Ali | Thermal, mechanical and rheological properties of poly (lactic acid)/epoxidized soybean oil blends[END_REF][START_REF] Xu | Mechanical and rheological properties of epoxidized soybean oil plasticized poly(lactic acid)[END_REF][START_REF] Xiong | Preparation and characterization of poly(lactic acid)/starch composites toughened with epoxidized soybean oil[END_REF][START_REF] Al-Mulla | New biopolymer nanocomposites based on epoxidized soybean oil plasticized poly(lactic acid)/fatty nitrogen compounds modified clay: Preparation and characterization[END_REF] and epoxidized palm oil [START_REF] Silverajah | Mechanical, thermal and morphological properties of poly(lactic acid)/epoxidized palm olein blend[END_REF][START_REF] Silverajah | A comparative study on the mechanical, thermal and morphological characterization of poly(lactic acid)/epoxidized palm oil blend[END_REF][START_REF] Al-Mulla | Properties of epoxidized palm oil plasticized polytlactic acid[END_REF] also showed some positive effects on PLA toughness. Another approach consists in using rigid particles, which usually increased stiffness but is ineffective, or even detrimental for the ductility. Nonetheless, NatureWorks [START_REF]Polylactic acid containing fillers and fibres[END_REF] reported the use of EMforce ® , a mineral additive, able to change the polymer failure mode from brittle to ductile. In fact, the efficient crack initiation in glassy PLA, which was provided by the well-dispersed additive with good interfacial adhesion to the matrix, enhanced the PLA elongation at break. Expertise from plasticizing polyvinylchloride teaches that mixtures of different plasticizers are efficient for increasing toughness, because they make use of different mechanisms available for ductility improvement [START_REF] Rahman | The plasticizer market: An assessment of traditional plasticizers and research trends to meet new challenges[END_REF]. For example, a mixture of tributyl citrate (TBC) and a block copolymer PLA-g-polyethyleneglycol yielded a material with high elongation at break and satisfying stress at yield [START_REF] Lemmouchi | Plasticization of poly(lactide) with blends of tributyl citrate and low molecular weight poly(D,L-lactide)-b-poly(ethylene glycol) copolymers[END_REF]. Al-Mulla et al. [START_REF] Al-Mulla | New biopolymer nanocomposites based on epoxidized soybean oil plasticized poly(lactic acid)/fatty nitrogen compounds modified clay: Preparation and characterization[END_REF] successfully used modified nanoclays in combination with epoxidized soybean oil to increase stiffness and ductility. The combination of PLA with rubber and compatibilizers has also been investigated by several authors, showing important gains in ductility, with an elongation at break of up to 160% for high rubber concentrations [START_REF] Bitinis | Poly(lactic acid)/ natural rubber/cellulose nanocrystal bionanocomposites. Part II: Properties evaluation[END_REF] or bicontinous phases [START_REF] Yuan | Crosslinked bicontinuous biobased PLA/NR blends via dynamic vulcanization using different curing systems[END_REF], and up to 200% when using compatibilizers [START_REF] Thepthawat | Improving the properties of polylactic acid by blending with low molecular weight polylactic acid-g-natural rubber[END_REF][START_REF] Ayutthaya | Thermal and mechanical properties of poly(lactic acid)/natural rubber blend using epoxidized natural rubber and poly (methyl methacrylate) as co-compatibilizers[END_REF]. However, these solutions often require the inclusion of several additives, chemically modified and costly components that increase the price of the final formulation, which is inappropriate for a large volume production. Finding a low cost biobased and biodegradable toughening agent would therefore help PLA to conquer new markets. Vegetable oils contain a number of potentially interesting molecules for PLA toughening and the refinery of vegetable oils gives rise to several by-products. Among those, oil deodorization condensates represent a high-volume by-product, which is chemically close to the vegetable oil. They are obtained by distillation in the aim of purifying the raw oil from odorous compounds, which are negative for the sensorial properties of vegetable oils. Deodorization condensates contain free fatty acids, glycerides and unsaponifiable molecules. The objective of this study was to use vegetable oils and the byproduct deodorization condensate as produced for toughening PLA. The individual molecules which make up oils and deodorization condensates were tested on the mechanical properties and several vegetable oils were screened.

Experimental 2.1. Materials

PLA 4060D was supplied by NatureWorks (U.S.A.) and consists of 89±1% L-lactic acid and 11±1% Dlactic acid units, making it unable to crystallize under common conditions [START_REF] Auras | An overview of polylactides as packaging materials[END_REF][START_REF] Hughes | Improved flexibility of thermally stable poly-lactic acid (PLA)[END_REF]. The glass transition temperature (T g ) obtained from differential scanning calorimetry (DSC) measurements was 56°C. Average molecular weights obtained from size exclusion chromatography (SEC) measurements were M w = 236 800 g•mol -1 , M n = 103 700 g•mol -1 and dispersity M w /M n = 2,28. Hydrogenated palm oil (HPO), Hydrogenated Copra Oil (HCO), Rapeseed Oil Deodorization Condensate (RODC), Soybean Oil Deodorization Condensate (SODC), Olive Oil Deodorization Condensate (OODC) and Palm Oil Deodorization Condensate (PODC) were supplied by ITERG (Bordeaux, France). Palmitic acid (purity >98%) i.e. hexadecanoic acid (C16:0), oleic acid (purity >96%) i.e. 9-octadecenoic acid (C18:1), squalene (purity >98%) i.e. (6E,10E,14E,18E)-2,6,10,15,19,23-hexamethyltetracosa-2,6,10,14,18,22-hexaene, and alpha-toco-pherol (purity >96% ) i.e. (2R)-2,5,7,8-tetramethyl-2-[(4R,8R)-(4,8,12-trimethyltridecyl)]-6-chromanol, were supplied by Sigma-Aldrich (France).

Film fabrication

Prior to melt-blending, PLA pellets were dried at 60°C for 24 h under dried air using a SOMOS 60L. Relative humidity of dried pellets was controlled to be lower than 350 ppm using an Aboni FMX Hydrotracer (France). Melt-mixing of PLA with or without additives was carried out using a twin screw extruder (Thermo Haake Ptw 16-40D, France), having a screw diameter of 16 mm and a length to diameter ratio (L/D) 40:1. A temperature of 180°C was used to process by twin-screw extrusion the PLA pellets as reference material, while the temperature profile of the 7 heating zones decreased from 180 to about 150/130°C along the extrusion flow, depending on the nature and the amount of additive. Screw speed was 300 rpm. The exact temperature profile for each formulation is given in Table 1. To properly control the feed rate of additives, which are solid at room temperature, a home-made apparatus consisting in a heated syringe was used. Obtained pellets were stored into hermetic sealed metalized bags to avoid PLA rehydration. Films of about 0.8 mm thickness containing palmitic acid, oleic acid, hydrogenated palm oil, hydrogenated copra oil, squalene, !-tocopherol and some combinations of these products were obtained using a single screw extruder (Scamex Rheoscam, France), mounted with a screw of 20 mm diameter and a length to diameter ratio (L/D) 12:1 and a flat die of 40 mm width and 1 mm thickness. Films were stretched and cooled with chill rolls. Due to the short length of the screw, an increasing temperature profile from 180 to 195°C over the 3 heating zones was used for PLA, while it was decreasing from 180 In the case of the deodorization condensates of the vegetable oils (RODC, SODC, OODC, PODC), the available quantity was insufficient for carrying out a single screw extrusion in the aim to obtain films.

In that case PLA sheets of 1 mm thickness were thermo-moulded by compression at 220 bar (Laboratory Press Gibitre Instruments 20 tons, Italy). For this, pellets were pre-melted at 180°C without pressure during 180 seconds then heating plates were closed with progressive increase in pressure during 120 seconds to eliminate air bubbles.

Chemical and physical chemical characterization of vegetable oils and deodorization condensates

Glyceride composition of fats was determined according to the IUPAC 6.002 and EN 14105 standards using a Shimadzu GC-2010 Plus gas chromatograph (France) equipped with a Zebron ZB 5 HT Inferno (15 m, 0.25 mm, 0.1 "m) column and a flame ionization detector set at 380°C. The vector gas was H 2 at a flow rate of 1.17 mL•min -1 . Both the injector and the oven temperature were set at 60°C for 3 min, then raised to 370°C at 10°C•min -1 and held at 370°C for 12 min. Direct on-column injection was performed. Fatty acid composition was determined according to the ISO 12966-2 standard using a Shimadzu GC-2010 Plus gas chromatograph (France) equipped with a BPX70 (50 m, 0.22 mm, 0.25 "m) column, and a flame ionization detector set at 250°C. The vector gas was H 2 at a flow rate of 0.32 mL•min -1 . The oven temperature was set at 60 C for 2 min, raised to 170°C at 20°C•min -1 , held at 170°C for 25 min, raised to 230°C at 4°C•min -1 and held at 230°C for 10 min. The injector temperature was set at 250°C and a split ratio of 200 was used. Acid value was determined according to the ISO 660 standard using a mixture of ethanol 95% and diethylic ether as solvent, potassium hydroxide 0.5 mol•L -1 in ethanol 95% as titrant and alkali blue 6B as indicator. Saponification value was determined according to the ISO 3657 standard. Samples were saponified with potassium hydroxide 0.5 mol•L -1 in ethanol 95% boiled under reflux during 2 h. Hydrochloric acid 0.5 mol•L -1 was used as titrant and alkali blue 6B as indicator. Water content was measured using a Mettler Toledo HB43 S Halogen Moisture Analyzer set (France) at 103°C. Melting point of free fatty acids was determined from literature [START_REF] Bradley | Jean-Claude Bradley double plus good (highly curated and validated) melting point dataset[END_REF]. Estimation of mono, di-and triglycerides average chemical structure of oil deodorization condensates was based on the averages unsaturation number and the alkyl chains length of the corresponding free fatty acids composition profile (Table 3). Average glycerides melting point was estimated calculating a weighted average of the specific melting points of the contained molecules taken from references [START_REF] Bradley | Jean-Claude Bradley double plus good (highly curated and validated) melting point dataset[END_REF][START_REF] Vereeckeen | Fat structuring with partial acylglycerols: Effect on solid fat profiles[END_REF][START_REF] Sato | Kinetic aspects in polymorphic crystallization and transformation of fats mixed and dispersed systems. in 'Physical properties of fats, oils, and emulsifiers[END_REF]. Physical properties of molecules are given in Table 4.

Calculation of solubility parameters

Molar volumes and molar attraction constants of polylactide and additives were determined according to the van Krevelen and Hoftyzer atomic group contribution method [START_REF] Van Krevelen | Cohesive properties and solubility[END_REF]. Average molar volumes and average molar attraction constants of mono, di and triglycerides of oil deodorization condensates were estimated based on average chemical structures previously determined. Hansen Solubility parameters were calculated using Equations ( 1), ( 2) and (3) [START_REF] Van Krevelen | Chemical structure and properties of coal. XXVIII. Coal constitution and solvent extraction[END_REF][START_REF] Van Krevelen | Practical evaluation of the [)]-M relationship. III. Estimation of the exponent A[END_REF]. Used molar constant values [START_REF] Van Krevelen | Cohesive properties and solubility[END_REF] are presented in Table 5:

(1) (2)
(3

)
where ! d is the dispersion component of the solubility parameter in J 1/2 •cm -3/2 , ! p the polar component of the solubility parameter in J 1/2 •cm -3/2 , ! h the hydrogen bonding component of the solubility parameter in J 1/2 •cm -3/2 , F d the dispersion contribution of the molar attraction constant in (J 1/2 •cm -3/2 )•mol -1 , F p the polar contribution of the molar attraction constant in (J 1/2 •cm -3/2 )•mol -1 , E h the hydrogen bonding energy contribution of the molar attraction constant in J•mol -1 and V the molar volume contribution of the chemical group involved in cm 3 •mol -1 .

The solubility of the molecules in PLA (Table 4) was assessed using the HSP Relative Energy Difference (RED) from Equations ( 4) and ( 5):

(5) RED 5

Distance Radius where ! d , ! p , and ! h are the components of the solubility parameter of the molecules and the PLA obtained from Equations ( 1), ( 2) and ( 3), where the radius value is the maximal distance obtained from [START_REF] Abbott | Chemical compatibility of poly (lactic acid): A practical framework using hansen solubility parameters. in 'Poly(lactic acid): Synthesis, structures, properties, processing, and applications[END_REF] beyond which the molecules are not miscible anymore with the polymer. Therefore, the closer the RED value to zero, the better the compatibility. A RED value higher than 1 means a theoretical nonmiscibility of the additive with PLA.

d h 5 " gE hi gV i d p 5 " gF Pi 2 gV i d d 5 g F dt g V i d d 5 g F dt g V i d p 5 " gF Pi 2 gV i d h 5 " gE hi gV i RED 5 Distance Radius

Physical-chemical characterization PLA blends

Differential Scanning Calorimetry (DSC) analyses were performed using a Mettler Toledo (France) DSC1 STARe System under nitrogen atmosphere (50 mL•min -1 ) in 40 "L standard Aluminum pans (Mettler). Calibration of the device was done using Indium and Zinc standards. Calorimetric scans of additives were performed from -80 to 220°C at a heating rate of 2°C•min -1 . Experiments were carried out in duplicate. Calorimetric scans of blended PLA samples were done at a heating/cooling rate of 10°C•min -1 . The first heating scan for sample rejuvenation was performed from 25 to 75°C with an isotherm at 75°C during 2 min. Then, the cooling scan was from 75 to -20°C and the second heating scan from -20 to 100°C. The glass transition temperature (T g ) was taken at the midpoint of the second heating scan. Experiments were done in triplicate. Tensile properties were investigated at 23°C, relative humidity (RH) 50±10 % and cross-head speed of 25 mm•min -1 , using an universal tensile machine (Instron model 4301, France) equipped with a load cell 1000±1 N and without extensiometer. The dog bone shaped samples (ISO 527-2, type 5A) were directly cut from the materials. Prior to tensile testing, samples were conditioned at 23°C and 50±10% RH for at least 72 h. Each mechanical characteristic value is an average of 8 measurements. Uniaxial deformation under an optical microscope was done in cutting rectangular samples of 20 mm length, 4 mm width, and 0.8 mm thickness from blank PLA, PLA + 10 wt% C16:0, PLA + 10 wt% C18:1 and PLA + 10 wt% PODC films. They were stretched at 5 mm•min -1 using a homemade tensile machine placed under an Olympus Japan optical microscope mounted with WHK 10$/20 L ocular lens and MD Plan 10 0.25 objective lens. Observations were performed in optical transmission light mode using a Sony CCD-IRIS Model DXC-107P camera (France). The morphology of the dispersed phase of PLA + 10 wt% C16:0, PLA + 10 wt% C18:1 and PLA + 10 wt% PODC materials before and after being stretched was observed with Scanning Electron Microscopy (SEM) (Hitachi 4800 II, France). Observations were directly conducted on the longitudinal surface of the dog bone shaped samples without any previous preparation.

Principal component analysis (PCA)

PCA was carried out considering the contents of each free fatty acid, mono, di and triglyceride types, hydrocarbons, sterols and water, the acid and saponification values and the measured elongation at break [%], Young modulus and T g values of the PLA formulations tested. PCA was done with XLstat software.

Results and discussion

Effects of free fatty acids, glycerides,

"-tocopherol and squalene on PLA ductility Individual components present in vegetable oils, which could feature toughening ability for PLA, as free fatty acids, glycerides, and the unsaponifiable

[(J 1/2 •cm -3/2 )•mol -1 ] F p 2 [(J•cm -3 )•mol -1 ] E h [J•mol -1 ]

Molar volume

[cm components as !-tocopherol and squalene were tested. In order to achieve the most efficient formulation, the solubility of a molecule in the polymer matrix is one determinant for its plasticizing power [START_REF] Ruellan | Solubility factors as screening tools of biodegradable toughening agents of polylactide[END_REF]. Therefore the Hansen Solubility Parameters (HSP) of each compound was calculated. HSP (Table 4) shows low solubility of the different compounds in the PLA matrix. All the obtained values are close to one, which marks the solubility limit. The highest solubility (although still modest) was displayed by small fatty acids and !-tocopherol. The mechanical and thermal properties of the corresponding compounds are given in Table 6. The raw data of typical stress/strain curves of blends are presented in Figure 1. The DSC thermograms of neat additives in Figure 2 and of blends in Figure 3 for further information.

Literature studies on the use of vegetable oil as toughening agent of PLA showed that generally chemical modification is required to obtain positive effects [50, 52-54, 56-58, 72]. Not surprisingly due to their low solubility in PLA, the impact of the tested fatty acids (C16:0, C18:1), vegetable oils (HPO and HCO) and unsaponifiable compounds on T g was negligible. However, despite the low solubility of the compounds, PLA ductility was increased in some cases. In particular, HCO, which contains mainly triglycerides, but with short chain length, improved the PLA ductility and lowered the apparent Young modulus, stress and elongation at yield. HPO, which mostly contains triglycerides and some diglycerides with higher chain length did not bring significant ductility enhancement nor T g decrease. HSP of di-and triglycerides from HPO and triglycerides from HCO were close, while the average molar volume and molecular weight of triglycerides from HCO are smaller than the ones from HPO (Table 2). Testing of single fatty acids with longer chain length (C16:0, C18:1) showed that they were not able to increase PLA ductility. This is coherent with existing literature. For example, Jacobsen and Fritz [START_REF] Jacobsen | Plasticizing polylactide -The effect of different plasticizers on the mechanical properties[END_REF] studied PLA blends with fatty acid esters. No significant ductility improvement was observed. The blending with the unsaponifiable compound !-tocopherol showed small gains in elongation at break, which extends already existing knowledge. It was already shown that the use of !-tocopherol as a natural antioxidant in PLA at small quantities (<4 wt%) induced a slightly T g decrease [START_REF] Manzanarez-López | Release of !-tocopherol from poly(lactic acid) films, and its effect on the oxidative stability of soybean oil[END_REF] but no increase in elongation at break [START_REF] Hwang | Poly(L-lactic acid) with added !-tocopherol and resveratrol: Optical, physical, thermal and mechanical properties[END_REF]. Here, no elongation at break improvement upon was found using greater !-tocopherol amounts, despite solubility was assessed to be higher than other molecules (such as HPO and HCO, Table 4) and a T g decrease was observed (Table 3). Squalene, which is a natural tri-terpene found in vegetable oils, showed some toughening abilities (Table 6) despite its low solubility in PLA (Table 4). In fact, the creation of a dispersed phase promoting crazing was most probably responsible for this effect [START_REF] Ruellan | Solubility factors as screening tools of biodegradable toughening agents of polylactide[END_REF]. The most interesting result was obtained upon mixing different compounds. A mix of C18:1 and C16:0 fatty acids increased the PLA ductility much more than using them separately. The Young modulus and the T g remained higher than using only C18:1 likely because of its lower content involved, while the stress and elongation at yield were lessened as much as adding only C18:1. Mixture made of [95 wt% (50 wt% C16:0 + 50 wt% C18:1) + 5 wt% HPO] at a total content of 10 wt% in PLA, further enhanced the ductility. There was apparently a synergistic effect between compounds, which can be possibly exploited.

Properties of PLA blends with vegetable oil deodorization condensates

Deodorization oil condensates are mixtures of different molecules contained in vegetable oils, the composition of which depends on the botanic source and oil refinery process. Four kinds of oil deodorization condensates (RODC, SODC, OODC and PODC) were blended with PLA, each at four concentrations (5, 10, 15 and 20 wt%). As shown in Table 3, content in free fatty acids, mono di or triglycerides, sterols (mainly alpha-tocopherol) and hydrocarbons (mainly squalene) was highly variable, as well as unsaturation degree of fatty acids. From the composition, an average HSP was calculated and is given in Table 4. As observed in Table 4, the longer and/or more unsaturated the alkyl chain is, the lower the solubility in PLA. However, except for very short fats alkyl chains and organic compounds as !-tocopherol or squalene, HSP of involved molecules appear to be rather similar. Mechanical and thermal properties of PLA/ oil deodorization condensates blends are given in Table 6 and typical curves of raw data are shown in the Figure 1. Significant increases in elongation at break were obtained, especially with the PODC. Addition of too much additive, i.e. about 20 wt%, induced a stagnation or decrease in the elongation at break. All the deodorization condensates led to a lowering in the stress and elongation at yield. Interestingly, the apparent Young modulus of materials remained high. The stress/strain curves (Figure 1) showed that the yielding peak shrunk and flattened with the increase in additive content, as if the yield critical stress would tend to meet the plateau value where the stress remains constant with the strain.

There is thus an important gain of using native mixtures of fatty acids present in deodorisation condensates for PLA ductility increase.

Study of the deformation mechanisms involved in PLA/oil deodorization condensates blends

In all samples the T g remained higher than the measurement temperature, the main deformation mechanism was thus crazing of the glassy polymer [START_REF] Ruellan | Solubility factors as screening tools of biodegradable toughening agents of polylactide[END_REF]. The yield process in glassy amorphous polymers can be described as a stress induced glass-transition. In fact, because PLA Poisson's ratio is less than 0.5 (" PLA = 0.36) [START_REF] Jamshidian | Poly-lactic acid: Production, applications, nanocomposites, and release studies[END_REF], the volume of PLA increases when subjected to tensile stress. Correspondingly, samples experienced important stress whitening, which appeared simultaneously with yielding. Stress whitening is caused by the formation of sizeable microvoids in the polymer matrix due to cavitation and crazes [START_REF] Lampman | Fracture and fractography. in 'Characterization and failure analysis of plastics[END_REF]. Crazes can also be initiated inside non-miscible inclusions in the polymer matrix or on the interface between inclusion and polymer in the case of low compatibility. The addition of oil deodorization condensates initiated crazes at stress levels substantially below those of the brittle failure of neat PLA, which propagated perpendicularly to the stretching direction. Morphological analysis of the form and size of inclusions of PODC, C16:0 and C18:1 was done to help interpretation. Figure 4 shows SEM micrographs of [PLA/C16:0], [PLA/ C18:1] and [PLA/PODC] 90 wt%/ 10 wt% blends.

[PLA/C16:0] blends exhibit rods of about 5 to 10 "m length and 1 to 2 "m width, corresponding to fatty acid crystals. [PLA/C18:1] showed small spher-ical domains of fatty acids of about 0.5 "m diameters and some aggregates of about 3 to 5 "m diameters. The [PLA/PODC] blend micrograph showed both rods and aggregates of droplets, where the droplets appear to be distributed all around the crystals. Figure 5 shows optical micrographs in transmission mode of stretched samples under the microscope taken at different percentages of elongation. Many small cracks started appear in neat PLA when approaching the yield peak, which is common for glassy amorphous polymers. Addition of C16:0, i.e. rod like crystals, increased both the occurrence and the length of cracks. Cracks quantity upon stretching caused the material to become opaque. Failure of the blend revealed PLA fibrils. C18:1, i.e. small spherical liquid domains, induced fewer but much larger cracks as if their extensibility was eased. PODC, i.e. mostly a combination of C16:0 and C18:1 and some minor constituents, showed a high number of cracks but with smaller width as if using only C18:1. The mixture of palmitic acid crystals and liquid inclusions of oleic acid had thus a synergistic effect allowing for efficient craze initiation by the crystals and by cavitation inside the liquid inclusions. The superior performance of the PODC blends seems thus to be linked to the chemical composition in fatty acids and the presence of the minor constituents.

Role of the chemical composition of deodorization condensates in PLA ductility improvement

A Principal Components Analysis (PCA) was carried out including composition data and physicochemical characteristics of the deodorization condensates and of the PLA samples. Figure 6 shows the principal components projection plot of F1 and F2 of the data set. Only 50% of the total variance in these PLA samples is extracted according to F1 and F2 axes. In fact, scores of formulations containing 7. The positive part of F1 axis appears to be mainly governed by the elongation at break and as opposed to its negative part with the T g . The F2 axis mostly separates fats as a function of the unsaturation and length of their alkyl chains. Looking at the elongation at break improvement, unsaturation of alkyl chains appears to be required, but the lower the ratio, the larger the enhancement was. Medium alkyl chain length, i.e. from lauric (C12) to stearic (C18) acid, also tends to be preferable than the long Therefore, efficiency of fats in toughening PLA appeared to be linked to medium melting point properties. A mixture of crystalline and liquid fatty acids at room temperature seemed favorable. This could explain the ability of HCO triglycerides to increase the PLA ductility while HPO did not, although their HSP were similar, as their ability to depress the T g of PLA. In fact, due to short alkyl chains, HCO triglycerides are waxy at room temperature. The DSC thermogram (Figure 2) shows a broad melting peak going from 8 to 42°C of HCO while it is beyond 55°C for HPO. Literature shows that combining liquid and solid additives can sometimes be effective for PLA [START_REF] Notta-Cuvier | Tailoring polylactide (PLA) properties for automotive applications: Effect of addition of designed additives on main mechanical properties[END_REF]. Acid and saponification values are related to the elongation at break improvement (Figure 7). Therefore, the higher the esterified fats and/or the free fatty acid contents, the better the efficiency of the additive. In opposite, unidentified compounds, i.e. mostly unsaponifiable compounds, are not favorable to the ductility increase. Therefore, the superiority in toughening abilities of the PODC compared to the alike mixture made of [95 wt% (50 wt% C16:0 + 50 wt% C18:1) + 5 wt% HPO] can be attributed to the complex and favorable mixture of fatty acids having different chain length.

Conclusions

By-products of the vegetable oils industry, namely deodorization condensates, were investigated as biobased additives for ductility improvement of PLA. The deodorization condensates improved substantially the elongation at break of PLA up to 180%. The glass transition was merely decreased which brought ductility to the still glassy polymer. The deformation mechanism was efficient craze initiation delaying failure. Most importantly, it was shown that the industrial by-product of the palm oil refinery had superior properties compared to mixtures of fatty acids and vegetable oils because it contained a mixture of fatty acids with melting points below and beyond room temperature. This makes deodorization condensates efficient and low price additives for PLA able to be used in commodity applications such as food packaging.

Table 4 .

 4 Physical properties and solubility parameters of fatty acids, mono-, di-, and triglycerides contained in the hydrogenated vegetable oils and oil deodorization condensates a from [65], b from [

Figure 1 .

 1 Figure 1. Stress/strain curves of PLA blends with free fatty acids, hydrogenated vegetable oils (HPO and HCO), unsaponifiable compounds (squalene, !-tocopherol), and vegetable oil deodorization condensates (palm oil PODC, olive oil OODC, soybean oil SODC and rapeseed oil RODC); (-) neat PLA; (== ==) PLA + 10 wt% C16:0; (= =) PLA + 10 wt% C18:1; (== •) PLA+ 10 wt% [50 wt% C16:0 + 50 wt% C18:1]; (== • •) PLA + 10 wt% [95 wt% (50 wt% C16:0 + 50 wt% C18:1) + 5 wt% HPO]; (--) PLA + 5 wt% additive; (---) PLA + 10 wt% additive; (• •) PLA + 15 wt% additive; (-• •) PLA + 20 wt% additive

Figure 2 .

 2 Figure 2. Thermograms (first heating scan) of free fatty acids, hydrogenated vegetable oils (HPO and HCO), unsaponifiable compounds (squalene, !-tocopherol), and vegetable oil deodorization condensates (palm oil PODC, olive oil OODC, soybean oil SODC and rapeseed oil RODC); (--) C16:0; (--) C18:1; ($) HCO; (%) HPO; (&) !-tocopherol; (') squalene; (+) RODC; (() SODC; (!) OODC; (!) PODC

Figure 3 .

 3 Figure 3. Thermograms (second heating scan) of PLA blends with free fatty acids, hydrogenated vegetable oils (HPO and HCO), unsaponifiable compounds (squalene, !-tocopherol), and vegetable oil deodorization condensates (palm oil PODC, olive oil OODC, soybean oil SODC and rapeseed oil RODC); (-) neat PLA; (--) PLA + 10 wt% C16:0; (--) PLA + 10 wt% C18:1; (---) PLA + 10 wt% [50 wt% C16:0 + 50 wt% C18:1]; (--) PLA + 10 wt% [95 wt% (50 wt% C16:0 + 50 wt% C18:1) + 5 wt% HPO]; ($) PLA + 10 wt% HCO; (%) PLA + 10 wt% HPO; (&) PLA + 10 wt% !-tocopherol; (') PLA + 10 wt% squalene; (+) PLA + 10 wt% RODC; (() PLA + 10 wt% SODC; (!) PLA + 10 wt% OODC; (!) PLA + 10 wt% PODC

Figure 4 .

 4 Figure 4. SEM micrographs of (a) C16:0, (b) C18:1 and (c) PODC blends with PLA at 10 wt%

Figure 5 .Figure 6 .

 56 Figure 5. Optical micrographs of uniaxially stretched PLA and PLA blends with free fatty acids (C16:0, C18:1) and palm oil deodorization condensate (PODC) at different strain

Table 1 .

 1 Processing conditions of melt-mixing twin screw extrusion

		Formulation	Temperature profile [°C]
	Neat PLA	175/180/190/190/190/190/180
	PLA + 10 wt% C16:0	175/180/180/170/170/170/170
	PLA + 10 wt% C18:1	175/180/180/170/170/170/170
	PLA + 10 wt% [50 wt% C16:0 + 50 wt% C18:1]	175/180/180/170/170/170/170
	PLA + 10 wt% [95 wt% (50 wt% C16:0 + 50 wt% C18:1) + 5 wt% HPO]	175/180/180/170/170/170/170
		5 wt% HCO	175/180/180/180/180/170/170
	PLA +	10 wt% HCO	175/180/180/180/180/170/170
		15 wt% HCO	175/180/180/180/170/170/160
		5 wt% HPO	175/180/180/180/180/170/170
	PLA +	10 wt% HCO	175/180/180/180/180/170/170
		15 wt% HCO	175/180/180/180/180/170/170
		5 wt% !-tocopherol	175/180/180/180/180/180/180
	PLA +	10 wt% !-tocopherol	175/180/180/180/180/180/170
		15 wt% !-tocopherol	175/180/180/180/180/170/170
		5 wt% squalene	175/180/180/180/180/180/180
	PLA +	10 wt% squalene	175/180/180/180/180/170/170
		15 wt% squalene	175/180/180/180/180/170/170
		5 wt% RODC	175/180/180/180/180/175/170
	PLA +	10 wt% RODC 15 wt% RODC	175/180/180/180/180/170/160 175/180/180/180/180/170/160
		20 wt% RODC	175/180/180/180/170/160/150
		5 wt% SODC	175/180/180/180/180/175/170
	PLA +	10 wt% SODC 15 wt% SODC	175/180/180/180/180/170/160 175/180/180/180/180/170/160
		20 wt% SODC	175/180/180/180/170/160/150
		5 wt% OODC	175/180/180/170/170/170/170
	PLA +	10 wt% OODC 15 wt% OODC	175/180/180/170/170/170/170 175/180/180/170/170/160/160
		20 wt% OODC	175/180/180/170/160/150/150
		5 wt% PODC	175/180/180/180/180/175/170
	PLA +	10 wt% PODC 15 wt% PODC	175/180/180/180/170/160/155 175/180/180/180/170/160/150
		20 wt% PODC	175/180/180/170/160/150/135

Table 2 .

 2 Processing conditions of cast films extrusion

		Formulation	Temperature profile [°C]	Die temperature [°C]
	Neat PLA	180/190/195	185
	PLA + 10 wt% C16:0	180/180/175	170
	PLA + 10 wt% C18:1	180/180/175	170
	PLA + 10 wt% [50 wt% C16:0 + 50 wt% C18:1]	180/180/175	170
	PLA + 10 wt% [95 wt% (50 wt% C16:0 + 50 wt% C18:1) + 5 wt% HPO]	180/180/175	170
		5 wt% HCO	180/180/180	180
	PLA +	10 wt% HCO	180/170/170	170
		15 wt% HCO	180/170/170	170
		5 wt% HPO	180/180/180	170
	PLA +	10 wt% HPO	180/170/170	170
		15 wt% HPO	180/170/170	170
		5 wt% !-tocopherol	180/180/180	180
	PLA +	10 wt% !-tocopherol	180/180/180	180
		15 wt% !-tocopherol	180/175/170	170
		5 wt% squalene	180/180/180	180
	PLA +	10 wt% squalene	180/180/175	170
		15 wt% squalene	180/180/170

Table 3 .

 3 Composition of hydrogenated vegetable oils and vegetable oil deodorization condensates Distance 5 "41d d molec 2 d d PLA 2 2 1 1d p molec 2 d p PLA 2 2 1 1d h molec 2 d h PLA 2 2 Distance 5 "41d d molec 2 d d PLA 2 2 1 1d p molec 2 d p PLA 2 2 1 1d h molec 2 d h PLA 2 2

	Type			HPO	HCO	RODC	SODC	OODC	PODC
					Free fatty acids		0.0	0.0	39.0	43.1	39.2	95.4
	Glyceride	composition	[%]	Monoglycerides Diglycerides Triglycerides Sterols (!-tocopherol) Hydrocarbons (Squalene)	0.2 10.6 88.9 0.0 0.0	0.0 0.0 100 0.0 0.0	11.2 2.7 12.2 25.7 0.0	2.9 9.0 16.8 9.8 13.8	2.0 8.4 33.5 1.6 13.0	1.7 2.2 0.7 0.0 0.0
					Unidentified		0.3	0.0	9.2	4.6	2.3	0.0
	Acid value [mg KOH•g -1 ]		0.1	0.1	65.3	68.4	47.7	201.0
	Saponification value [mg KOH•g -1 ]		198.6	255.2	125.1	157.9	162.2	205.7
	Water content [%]		0.11	0.10	0.55	0.34	0.37	0.16
					Caproic acid	C6:0	0.0	0.5	0.0	0.0	0.0	0.0
					Caprylic acid	C8:0	0.0	6.8	0.0	0.0	0.0	0.0
					Capric acid	C10:0	0.0	5.7	0.0	0.0	0.1	0.0
					Lauric acid	C12:0	0.5	47.5	0.0	0.8	0.0	0.4
	Fatty acid composition	[%]	Myristic acid Palmitic acid Palmitoleic acid Stearic acid Oleic acid Linoleic acid Linolenic acid Arachidic acid	C14:0 C16:0 C16:1 C18:0 C18:1 C18:2 C18:3 C20:0	1.2 43.6 0.0 53.8 0.0 0.0 0.0 0.5	18.1 9.4 0.0 10.8 1.0 0.0 0.0 0.1	0.0 7.4 0.0 3.4 27.3 42.4 1.5 0.5	0.4 12.3 0.0 4.1 21.7 49.7 6.4 0.3	0.0 11.3 0.0 2.5 69.5 10.9 0.6 0.4	1.3 49.8 0.2 4.1 35.2 7.8 0.3 0.3
					Eicosenoic acid	C20:1	0.0	0.0	0.3	0.2	0.4	0.1
					Behenic acid	C22:0	0.1	0.0	1.0	0.5	0.0	0.0
					Lignoceric acid	C24:0	0.1	0.0	0.5	0.2	0.1	0.0
					Unidentified		0.5	0.1	0.2	15.7	13.4	4.2
	Average alkyl chain carbon quantity		17.1	13.0	17.9	17.7	17.8	16.9
	Average alkyl chain unsaturation quantity	0.0	0.0	1.4	1.5	1.0	0.5

Table 5 .

 5 Used group contribution molar constants

	Group	F d

  3

Table 6 .

 6 Mechanical and thermal properties of formulations

		Formulation	Elongation at break [%]	app. Young modulus a [MPa]	Elongation at yield [%]	Stress at yield [MPa]	T g [°C]
	PLA (twin screw pellets)	6±1	1760±90	4.2±0.4	64±5	56.3±0.2
	PLA + 10 wt% C16:0	4±1	1720±60	3.2±0.1	50±3	49.3±0.1
	PLA + 10 wt% C18:1	16±5	1440±50	2.2±0.1	28±2	45.2±0.2
	PLA + 10 wt% [50 wt% C16:0 + 50 wt% C18:1]	45±12	1590±80	2.2±0.1	31±2	40.7±0.3
	PLA + 10 wt% [95 wt% (50 wt% C16:0 + 50 wt% C18:1) + 5 wt% HPO]	74±15	1550±120	2.2±0.2	25±4	39.8±0.4
		5 wt% HCO	9±3	1680±70	3.6±0.2	51±1	54.1±0.2
	PLA +	10 wt% HCO	9±6	1540±80	3.6±0.2	42±2	54.1±0.1
		15 wt% HCO	11±5	1460±70	3.8±0.2	38±2	54.8±0.3
		5 wt% HPO	55±20	1610±50	2.7±0.1	38±1	52.6±0.2
	PLA +	10 wt% HPO	85±15	1420±70	2.6±0.2	26±2	54.4±0.4
		15 wt% HPO	91±14	1300±80	2.4±0.1	26±1	53.9±0.3
		5 wt% !-tocopherol	6±2	1150±90	3.8±0.2	41±4	51.3±0.2
	PLA +	10 wt% !-tocopherol	5±2	1070±60	4.0±0.4	31±3	49.8±0.2
		15 wt% !-tocopherol	8±3	990±70	3.9±0.3	29±4	47.1±0.2
		5 wt% squalene	38±7	1180±90	3.3±0.4	36±7	51.4±0.2
	PLA +	10 wt% squalene	51±9	990±60	3.1±0.9	28±3	49.5±0.2
		15 wt% squalene	29±7	890±80	2.9±0.7	19±2	49.4±0.1
		5 wt% RODC	22±3	1530±80	3.2±0.2	40±3	52.6±0.2
	PLA +	10 wt% RODC 15 wt% RODC	55±7 65±15	1490±140 1380±155	2.4±0.2 2.3±0.3	28±4 23±4	49.1±0.1 48.8±0.1
		20 wt% RODC	73±20	1270±150	2.2±0.3	20±4	48.5±0.3
		5 wt% SODC	29±8	1550±110	3.3±0.3	41±7	50.6±0.1
	PLA +	10 wt% SODC 15 wt% SODC	73±18 80±13	1510±90 1340±80	2.3±0.2 2.2±0.3	25±5 23±3	49.4±0.3 48.7±0.1
		20 wt% SODC	52±15	1220±100	2.1±0.4	16±4	47.5±0.1
		5 wt% OODC	49±6	1430±90	3.1±0.3	34±4	49.5±0.2
	PLA +	10 wt% OODC 15 wt% OODC	85±17 88±16	1490±90 1360±70	2.6±0.2 2.1±0.2	31±2 20±3	49.0±0.1 48.5±0.1
		20 wt% OODC	67±16	1370±110	2.2±0.2	21±3	48.7±0.1
		5 wt% PODC	51±17	1680±110	2.5±0.2	36±2	45.3±0.1
	PLA +	10 wt% PODC 15 wt% PODC	132±18 179±15	1460±40 1180±130	2.1±0.2 2.3±0.3	24±1 18±2	39.2±0.2 35.1±0.2
		20 wt% PODC	84±31	1130±60	2.0±0.4	14±3	33.8±0.2

a apparent Young modulus for comparison reasons obtained without extensiometer

Acknowledgements

This work was supported by Brodart Packaging and the BIP-ADEME French project CREABioM. The authors greatly thank the laboratory Institut des Corps Gras (ITERG -Bordeaux, France) for its technical support.


Mattia G Bergomi

Riccardo D Jadanza

Alessandro Portaluri

A mathematical model for voice leading and its complexity

We give a formal de nition of the musical concept of voice leading in mathematical terms, based on the idea of partial permutations of certain ordered multisets. Then we associate a partial permutation matrix with a voice leading in a unique way and write an algorithm to easily transform any musical composition into a sequence of such matrices; we then generalise it in order to include in the model also rhythmic independence and rests. From that we extract a vector whose components return information about the movements of the voices in the piece and hence about the complexity of the voice leading. We provide some examples by analysing three compositions, also visualising complexity as a point cloud for each piece. Finally, we interpret the sequence of complexity vectors associated with each composition, thus considering the position of each observation with respect to time. The Dynamic Time Warping allows us to compute the distance between two pieces and to show that our approach distinguishes well the examples that we took into account, exhibiting a strong indication that the notion of complexity we propose is a good tool to identify and classify musical pieces.

Introduction

In Music, the study of harmony and voice leading concerns the dynamical relation among chords and melodic lines. Chords can be seen as vertical entities supporting a main melodic line, whereas compositional styles can be characterised by the interplay of the various voice lines in horizontal motion. Hence, voice leading can be interpreted as the writing of several melodies interacting together in two ways: voices moving simultaneously a ect the listener as chords, whilst the independence in terms of both rhythm and pitch creates a horizontal ow, perceived as a superposition of di erent themes.

The concurrent motion of two voices is classically referred to as contrapuntal motion and is traditionally divided into four classes, illustrated in Figure 3 on page 7. In contrary motion the voices move in opposite directions; this gives them contrast and independence [START_REF] Aldwell | Harmony and voice leading[END_REF]Chapter 6], creating an interesting "soundscape" for the listener, as the Canadian composer Raymond M. Schafer calls it. Parallel motion occurs when the interval between the voices is kept constant along the movement; when applied to thirds, sixths and tenths it can be considered among the most powerful voice leading techniques. However, in some cases it impedes the growth in independence of the voices: this is one reason why it is generally avoided when not even forbidden for unisons, octaves and fths. In oblique motion actually only one voice is moving whilst the other remains at the same pitch; this one and similar motion, where the two voices proceed in the same direction, convey surely less independence than contrary motion, but undoubtably more than the parallel one.

For practical purposes, voices can be grouped in ranges: from the highest to the lowest, we have:

Soprano : from C 4 to G 6 , Alto : from G 3 to C 5 , Tenor : from C 3 to G 4 , Bass : from E 2 to C 4 .
Di erent compositional styles are characterised by di erent types of motion: the ve species of counterpoint represented in Figure 1 arise from various combinations of rhythmic choices and of the amplitudes of the intervals between two consecutive notes. A particular movement of voices is the (voice) crossing, that occurs when two voices exchange their relative positions -for instance when the Soprano moves below the Alto. This kind of dynamics is generally not desirable because it conveys a sense of discomfort, albeit it is considered less problematic when it involves inner voices (Alto and Tenor) for a few chords.

Given a sequence of chords, an important question is how to transform them into a superposition of voices according to a certain contrapuntal style (see Figure 2). Our aim is to formalise the concept of voice leading in mathematical terms and to build a computationally e cient model for dealing with voice leading based on sparse matrices that encode the motions of the involved voices. This representation allows us to de ne a notion of complexity of the voice leading and to classify di erent contrapuntal styles, by representing an entire composition as a static point cloud or as a multi-dimensional time series. This second interpretation o ers also the possibility to compare two pieces by making use of the dynamic time warping, a tool that measures the similarity between them in terms of complexity.

Voice leadings, multisets and partial permutations

In general, it is possible to describe a melody as a nite sequence of ordered pairs (p i , p i+1 ) i∈I , where I is a nite set of indices. In order to model the voice leading in a mathematical way it is necessary to introduce rst the concept of multiset, a generalisation of the idea of set. (This approach was already considered by D. Tymoczko in [START_REF] Tymoczko | The geometry of musical chords[END_REF].) Roughly speaking, we can think of it as of a list where an object can appear more than once, whilst the elements of a set are necessarily unique. More formally, a multiset M is a couple (X, µ) composed of an underlying set X and a map µ : X → N, called the multiplicity of M , such that for every x ∈ X the value µ(x) is the number of times that x appears in M . We de ne the cardinality |M | of M to be the sum of the multiplicities of each element of its underlying set X. Observe, however, that a multiset is in fact completely de ned by its multiplicity function: it su ces to set M := dom(µ), µ .

If we interpret a set of n singing voices (or parts played by n instruments, or both) as a multiset of pitches of cardinality n, then a voice leading can be mathematically described as follows.

De nition 2.1. Let M := (X M , µ M ) and L := (X L , µ L ) be two multisets of pitches with same cardinality n and arrange their elements into n-tuples (x 1 , . . . , x n ) and (y 1 , . . . , y n ) respectively. 1 A voice leading of n voices between M and L, denoted by (x 1 , . . . , x n ) → (y 1 , . . . , y n ), is the multiset Z := (x 1 , y 1 ), . . . , (x n , y n ) , whose underlying set is X Z := X M × X L and whose multiplicity function µ Z is de ned accordingly, by counting the occurrences of each ordered pair.

Remark 2.2. Observe that the de nition just given is not linked to the particular type of object (pitches): it is possible to describe voice leadings also between pitch classes, for instance. Note that it is also possible to describe a voice leading as a bijective map from the multiset M to the multiset L, i. e. as a partial permutation of the union multiset

M ∪ L := (X M ∪ X L , µ M ∪L ), where µ M ∪L := max{µ M χ M , µ L χ L }
and χ M and χ L are the characteristic functions of X M and X L , respectively. 2 We recall that a partial permutation of a nite multiset S is a bijection between two subsets of S. In general, if S has cardinality n then this map can be represented as an n-tuple of symbols, some of which are elements of S and some others are indicated by a special symbol -we use -to be interpreted as a "hole" or an "empty character". However, since we are not dealing with subsets of a xed multiset, we shall use the cycle notation to avoid ambiguity and confusion.

Remark 2.3. In order to be able to do computations with partial permutations, it is fundamental to x an ordering among the elements of the union multiset M ∪ L.

We henceforth give M ∪ L the natural ordering of real numbers, being its elements pitches. Indeed, in classical music with equal temperament, one de nes the pitch p of a note as a function of the fundamental frequency ν (measured in Hertz) associated with the sound; more precisely, as the map p : (0, +∞) → R given by p(ν) := 69 + 12 log 2 ν 440 .

This can be done also in the case where the elements of the union multiset are pitch classes: the ordering is induced by the ordering of their representatives belonging to a same octave. However, in this paper we shall not follow this practice and shall instead restrict to pitches only.

Example 2.4. The voice leading

(G 2 , G 3 , B 3 , D 4 , F 4 ) → (C 3 , G 3 , C 4 , C 4 , E 4 ) (2.1)
is described by the partial permutation of the ordered union multiset

(G 2 , C 3 , G 3 , B 3 , C 4 , C 4 , D 4 , E 4 , F 4 )
de ned by

G 2 C 3 G 3 B 3 C 4 C 4 D 4 E 4 F 4 C 3 G 3 C 4 C 4 E 4 . (2.2)
Thus, a voice leading between two multisets of n voices can be seen as a partial permutation of a multiset whose cardinality is less than or equal to 2n. The next step is to associate a representation matrix with the partial permutation. Let V be an n-dimensional vector space over a eld F and let E := {e 1 , . . . , e n } be a basis for V . The symmetric group S n acts on E by permuting its elements: the corresponding map S n × E → E assigns (σ, e i ) → e σ(i) for every i ∈ {1, . . . , n}. We consider the well-known linear representation ρ : S n → GL(n, F) of the group S n given by

ρ(1 i) :=        0 1 1 . . . 1 1 0 1 . . . 1        ,
where the 1's in the rst row and in the rst column occupy the positions 1, i and i, 1 respectively. The map ρ sends each 2-cycle of the form (1 i) to the corresponding permutation matrix that swaps the rst element of the basis E for the i-th one. Note that each row and each column of a permutation matrix contains exactly one 1 and all its other entries are 0. Following this idea and [2, De nition 3.2.5, p. 165], we say that a matrix P ∈ Mat(m, R) is a partial permutation matrix if for any row and any column there is at most one non-zero element (equal to 1). When dealing with a voice leading M → L, the dimension m of the matrix P is equal to the cardinality of the multiset M ∪ L.

Remark 2.5. In general, the partial permutation matrix associated with a given voice leading is not unique. This is due to the fact that we are dealing with multisets: if M → L is a voice leading it is possible that some components of L have the same value, i. e. that di erent voices are playing or singing the same note.

For this reason we introduce the following convention.

Convention 2.6. Let M := (x 1 , . . . , x n ) → L := (y 1 , . . . , y n ) be a voice leading and suppose that more than one voice is associated with a same note of L. To this end, let

(x i 1 , . . . , x i k ) be the pitches of M (with i 1 < • • • < i k ) that are mapped to the pitches (y j 1 , . . . , y j k ) of L, with y j 1 = • • • = y j k and j 1 < • • • < j k .
In order to uniquely associate a partial permutation matrix P := (a ij ) with the above voice leading, we assign the value 1 to the corresponding entries of P by following the order of the indices, that is by setting

a i 1 j 1 = 1, . . . , a i k j k = 1.
Thus, we shall henceforth speak of the partial permutation matrix associated with a given voice leading.

Example 2.7. The partial permutation matrix associated with the cycle representation (2.2) of voice leading (2.1) is

             
0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0

              .
Therefore, if M → L is a voice leading, if both M and L are thought of as ordered tuples and if P is its partial permutation matrix, we have that P M = L; in addition, the "reversed" voice leading L → M is obviously described by the transpose P T of P : P T L = M . This representation has the advantage of providing objects that are much handier than a multiset of couples, speaking in computational terms. Algorithm 1 presents the pseudocode for the computation of the partial permutation matrix of a voice leading.

Algorithm 1 Computing the partial permutation matrix

Input:

M → L Source (M ) and target (L) multisets describing the voice leading Output: P Partial permutation matrix associated with the voice leading Evaluate multiplicities of all x ∈ M and all y ∈ L; Generate the ordered multiset U := M ∪ L; Initialise P ∈ Mat |U | , R by setting P (i, j) = 0 for all i, j;

1: for i, j ∈ {1, . . . , |U |} do 2: if U (i) → U (j)

Relative motion of the voices and complexity of a voice leading

We have seen in the previous section how the partial permutation matrix associated with a voice leading contains the information about the passage from one note to the next one for each voice. Here we are going to illustrate that, in fact, the tool that we have built also encodes the direction of motion of the di erent voices, including the crossings. On the one hand, in Music one distinguishes between three main behaviours (cf. Figure 3; we omit parallel motion because that is not involved in our analysis):

• Similar motion, when the voices move in the same direction;

• Contrary motion, when the voices move in opposite directions;

• Oblique motion, when only one voice is moving.

On the other hand, with reference to a partial permutation matrix (a ij ), it is possible to describe the motion of a voice by noting three conditions, which are immediate consequences of the ordering of the union multiset:

1) If there exists an element a ij = 1 for i < j then the i-th voice is moving "upwards";

2) If there exists an element a ij = 1 for i > j then the i-th voice is moving "downwards";

3) If there exists an element a ii = 1 then the i-th voice is constant.

The connection between the two worlds is the following:

• If either Condition 1) or Condition 2) is veri ed by two distinct elements then we have similar motion;

• If both Condition 1) and Condition 2) hold for two distinct elements then we are facing contrary motion;

• The case of oblique motion involves Conditions 1) and 3) or Conditions 2) and 3), for at least two distinct elements.

As we mentioned in Section 1, voice crossing is a particular case of these motions where the voices swap their relative positions. This phenomenon can be described in terms of multisets as follows.

De nition 3.1. Let (x 1 , . . . , x n ) → (y 1 , . . . , y n ) be a voice leading (n ∈ N). If there exist two pairs (x i , y i ) and (x j , y j ) such that x i < x j and y i > y j or such that x i > x j and y i < y j then we say that a (voice) crossing occurs between voice i and voice j.

The partial permutation matrix retrieves even this information, as the following proposition shows. Proposition 3.2. Consider a voice leading of n voices and let P := (a ij ) be its associated partial permutation matrix. Choose indices i, j, k, l ∈ {1, . . . , n} such that a ij = 1 and a kl = 1. Then there is a crossing between these two voices if and only if one of the following conditions hold: i) i < k and j > l;

ii) i > k and j < l.

Furthermore, the total number of voices that cross the one represented by a ij is equal to the number of 1's in the submatrices (a rs ) and (a tu ) of P determined by the following restrictions on the indices: r > i, s < j and t < i, u > j.

Proof. In a partial permutation matrix the row index of a non-zero entry denotes the initial position of a certain voice in the ordered union multiset, whereas the column index of the same entry represents its nal position after the transition. It is then straightforward from De nition 3.1 that for a voice crossing to exist either condition i) or condition ii) must be veri ed. Every entry a kl satisfying one of those conditions refers to a voice that crosses the one represented by a ij , hence the number of crossings for a ij equals the amount of 1's in positions (r, s) such that r > i and s < j, summed to the number of 1's in positions (t, u) such that t < i and u > j.

Remark 3.3. The fact that the number of crossings with a given voice equals the number of 1's in the submatrices determined by the entry corresponding to that voice (as explained in the previous proposition) holds true only because we assumed Convention 2.6. Indeed, if we did not make such an assumption, the submatrices could contain positive entries referring to voices ending in the same note but that do not produce crossings.

From what we have shown thus far it emerges that it is possible to characterise a voice leading by counting the voices that are moving upwards, those that are moving downwards, those that remain constant and the number of crossings. We summarise these features in a 4-dimensional complexity vector c de ned by c := #upward voices, #downward voices, #constant voices, #crossings , (3.1) so that we are now able to classify and distinguish voice leadings by simply looking at these four aspects.

Example 3.4. Similar motion. The voice leading

(C 1 , E 1 , G 1 ) → (D 1 , F 1 , A 1 ) is represented by        
0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0

       
and its complexity vector is (3, 0, 0, 0). Oblique motion. The voice leading

(G 2 , G 2 , C 3 ) → (C 3 , C 3 , C 3 ) is associated with       0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1      
and its complexity vector is (2, 0, 1, 0). Voice crossing. The voice leading

(C 1 , E 1 , G 1 ) → (G 1 , C 1 , E 1 ) is represented by   0 0 1 1 0 0 0 1 0  
and its complexity vector is (1, 2, 0, 2).

By virtue of these tools it is straightforward to analyse an entire piece of music: it is enough to divide it into pairs of notes for each voice and apply the procedure described above for each passage. The concatenation of all the consecutive passages results then in a sequence of partial permutation matrices, whence one can extract a sequence of complexity vectors. This last piece of information can be visualised as a set of points in a 4-dimensional space -or rather as one or more of its 3-dimensional projections (see Subsection 3.1). In fact, if one wants to represent the complexity of the whole composition as a point cloud, one should take into account that di erent matrices can produce the same complexity vector; therefore we have a multiset of points in R 4 (with non-negative integer components).

Complexity analysis of two Chartres Fragments

We are going to analyse two pieces that are parts of the Chartres Fragments, an ensemble of compositions dating back to the Middle Ages: Angelus Domini and Dicant nunc Judei; both of them are counterpoints of the rst species and involve only two voices. The musical interest in these compositions consists in the introduction of a certain degree of independence between the voices and the use of a parsimonious voice leading, i. e. an attempt to make the passage from a melodic state to the next as smooth as possible. Note how the independence of the voices is re ected by the presence of contrary motions and crossings, which can then be interpreted as a rough measure of this feature. For a complete treatise on polyphony and a historical overview we refer the reader to [START_REF] Taruskin | Music in the Nineteenth Century: The Oxford History of Western Music[END_REF].

In what follows, we represent the multiplicity of each complexity vector c as a circle of centre c ∈ R 4 and radius equal to the normalised multiplicity µ(c)/n of c, where µ(c) is the number of occurrences of c in the analysed piece and n is the total number of notes played or sung by each voice in the whole piece.

Angelus Domini. The fragment under examination is depicted in Figure 4; here is the list of its rst four voice leadings, as they are generated by the pseudocode described in Algorithm 1:

Voice Leading: ['F4', 'C4'] ['G4', 'D4'] [2, 0, 0, 0] -similar motion up Voice Leading: ['G4', 'D4'] ['A4', 'E4'] [2, 0, 0, 0] -similar motion up Voice Leading: ['A4', 'E4'] ['G4', 'F4'] [1, 1, 0, 0] -contrary motion Voice Leading: ['G4', 'F4'] ['F4', 'G4'] [1, 1, 0, 1] -contrary motion -1 crossing Table 1 on page 12 contains the the complexity vectors and their occurrences in the piece; the point cloud associated with this multiset is represented in Figure 5. Observe how the projection that neglects the component of c corresponding to the number of constant voices (Figure 5b) gives an immediate insight on the relevance of voice crossing in the piece.

Dicant nunc Judei. The rst part of the output of Algorithm 1 produces the following analysis: The complexity vectors arising in the whole piece and their multiplicities are again collected in Table 1; see Figure 7 instead for a visualisation of the point cloud describing the piece. Note how the voice crossing is more massive than in the point cloud describing Angelus Domini. In addition, the point (0, 0, 0) in Figure 7b corresponds to the point (0, 0, 2, 0) ∈ R 4 , that represents trivial voice leadings where both parts do not vary.

Voice Leading: ['F4', 'C4'] ['G4', 'E4']

Rhythmic independence and rests

The examples analysed in Subsection 3.1 are counterpoints of the rst specieswhich is the simplest case, in that the voices follow a note-against-note ow. It is however possible to study more complex scenarios by introducing rhythmic Table 1: Complexity vectors of the analysed fragments and their occurrences.

Angelus Domini

c µ(c) (0, 1, 1, 0) 2 (0, 1, 1, 1) 2 (0, 2, 0, 0) 4 (1
, 0, 1, 1) 2 (1, 1, 0, 0) 6 (1, 1, 0, 1) 4 (2, 0, 0, 0) 4

Dicant nunc Judei c µ(c) (0, 0, 2, 0) 1 (0, 2, 0, 0) 7 (1, 0, 1, 0) 5 (1
, 0, 1, 1) 1 (1, 1, 0, 0) 9 (1, 1, 0, 1) 15 (2, 0, 0, 0) 4

independence between voices and rests in the melody, in any case reducing nonsimultaneous voices to the simplest case.

If the voices play at di erent rhythms or follow rhythmically irregular themes, we consider the minimal rhythmic unit u appearing in the phrase and homogenise the composition based on that unit: if a note has duration ku, with k ∈ N, we represent it as k repeated notes of duration u (see Figure 8 for an example). This transformation of the original counterpoint introduces only oblique motions and does not alter the number of the other three kinds of motion.

In musical terms, if a voice is silent it is neither moving nor being constant and it cannot cross other voices. Therefore, in order to include rests in our model it is necessary to slightly modify Algorithm 1 by introducing a new symbol (p) in the dictionary of pitches; we also choose to indicate a rest in the matrices associated with a voice leading by the entry -1. We adopt the following convention concerning the ordered union multiset. greater than any other pitch symbol. Remark 4.3. Note that when introducing the -1's in the matrix associated with a voice leading we are no longer dealing with partial permutation matrices. However, to study voice leadings with rhythmic independence of the voices as before (thus ignoring rests) it is enough to consider the minor of the matrix obtained by deleting all rows and columns containing -1 (which is obviously again a partial permutation matrix).

We extend the complexity vector de ned previously in Formula (3.1) by adding a fth component that counts the number of voices that are silent at least once in the voice leading, i. e. it counts the number of negative (-1) entries of the associated matrix. Furthermore, we slightly modify also the notion of normalised multiplicity of a complexity vector c, needed for the representation of the complexity of a piece in the form of a point cloud, now dividing the number µ(c) of occurrences of c in the piece by the total number of notes per voice after the homogenisation. Retrograde Canon c µ(c) c µ(c) (0, 0, 1, 0, 1) 2 (1, 0, 0, 0, 1) 2 (0, 0, 2, 0, 0) 8 (1, 0, 1, 0, 0) 43 (0, 1, 0, 0, 1) 2 (1, 0, 1, 1, 0) 1 (0, 1, 1, 0, 0) 43

(1, 1, 0, 0, 0) 14 (0, 1, 1, 1, 0) 1 (1, 1, 0, 1, 0) 3 (0, 2, 0, 0, 0) 11 (2, 0, 0, 0, 0) 11 beginning of which is reproduced in Figure 9. We homogenise the rhythm by expressing each note in eighths and we apply Algorithm 1. Here is the output of the rst four meaningful voice leadings:

Concatenation of voice leadings and time series

The paradigmatic point cloud associated with a voice leading gives a useful 3-dimensional representation of the piece; however, this analysis is just structural, as it does not take into account the way in which voice leadings have been concatenated by the composer. It is possible to introduce this temporal dimension by looking at the sequence of complexity vectors from a di erent viewpoint. The concatenation of observations in time can be seen as a time series, that is a sequence of data concerning observations ordered according to time. In our case each piece of music can be described as a 5-dimensional time series, whose observations are the complexity vectors associated with each voice leading. More speci cally, we use the so-called dynamic time warping (DTW), a method for comparing time-dependent sequences of di erent lengths: it returns a measure of similarity between two given sequences by "warping" them non-linearly (see Figure 11 for an intuitive representation). We invite the reader to consult [START_REF] Senin | Dynamic time warping algorithm review[END_REF] for a detailed review of DTW algorithms.

Dynamic time warping analysis

Let F be a set, called the feature space, and take two nite sequences X := (x 1 , . . . , x n ) and Y := (y 1 , . . . , y m ) of elements of F, called features (here n and m are natural numbers). In order to compare them, we need to introduce a notion of distance between features, that is a map C : F × F → R, also called a cost function, that meets at least the following requirements: i. C(x, y) 0 for all x, y ∈ F; ii. C(x, y) = 0 if and only if x = y;

iii. C(x, y) = C(y, x) for all x, y ∈ F. Now, if we apply C to the features X and Y , we can arrange the values in an n×m real matrix C := C(x i , y j ) , where i ranges in {1, . . . , n} and j in {1, . . . , m}.

A (n, m)-warping path in C is a nite sequence γ := (γ 1 , . . . , γ l ) ∈ R l , with l ∈ N, such that: ).

An optimal warping path on X and Y is a warping path realising the minimum total cost (see Figure 12). We are now ready to de ne the DTW distance between X and Y :

DT W (X, Y ) := min { C γ (X, Y ) | γ is a (n, m)-warping path } .

Remark 5.1. Note that the minimum always exists because the set is nite.

We computed the DTW distance between each pair of the three examples that we analysed in Subections 3.1 and 4.1, choosing as cost function the Euclidean distance in R 5 . We embedded the 4-dimensional complexity vectors in R 5 by adding a fth component and setting it to 0. The results of the comparison are shown in Table 3. Although we analysed only three compositions, it is possible to observe how the DTW distance segregates the two pieces belonging to the Chartres fragments.

Conclusion

Our analysis showed that our de nition of complexity in terms of the relative movements of the voices and especially of crossing is suitable for characterising a musical piece. Point-cloud representation yields a "photograph" of complexity, a sort of ngerprint that lets clearly emerge what are the main features of the examined composition, noticeable even at rst glance. Dynamic time warping provides then further support to this evidence by directly measuring the distance between the complexities of two pieces, considering each complexity vector as an observation of the piece in time, and giving a quantitative description of the

Figure 1 :Figure 2 :

 12 Figure1: Five di erent degrees of independence between voices, from the tied one-to-one in the rst species of counterpoint to the complete independence of the fth species.

  then

Figure 3 :

 3 Figure 3: Motion classes for two voices. Similar: same direction but di erent intervals; parallel: same direction and same intervals; oblique: only one voice is moving; contrary: opposite directions.

Figure 4 :

 4 Figure 4: Angelus Domini

  (a) Projection neglecting the crossing component of the complexity vectors. (b) Projection neglecting the constant voices component of the complexity vectors.

Figure 5 :

 5 Figure 5: Three-dimensional projections of the complexity cloud of the paradigmatic voice leading Angelus Domini. The radius of each circle represents the normalised multiplicity of the corresponding complexity vector.

Figure 6 :

 6 Figure 6: Dicant nunc Judei, Chartres fragment.

[ 2 ,

 2 0, 0, 0] -similar motion up Voice Leading: ['G4', 'E4'] ['F4', 'D4'] [0, 2, 0, 0] -similar motion down Voice Leading: ['F4', 'D4'] ['E4', 'C4'] [0, 2, 0, 0] -similar motion down Voice Leading: ['E4', 'C4'] ['D4', 'D4'] [1, 1, 0, 1] -contrary motion -1 crossing

  (a) Projection neglecting the crossing component of the complexity vectors. (b) Projection neglecting the constant voices component of the complexity vectors.

Figure 7 :

 7 Figure 7: Three-dimensional projections of the complexity cloud of the paradigmatic voice leading Dicant nunc Judei. The radius of each circle represents the normalised multiplicity of the corresponding complexity vector.

Convention 4 . 1 .

 41 We choose rests to be the last elements in the ordered union multiset associated with a voice leading. In other words, we declare p to be strictly Counterpoint of the fth species.

  oe oe oe oe oe oe oe oe oe oe oe oe oe oe oe oe oe oe oe oe oe oe oe oe oe oe oe oe oe (b) Reduction to the rst species.

Figure 8 :

 8 Figure 8: Reduction of rhythmically independent voices to a counterpoint of the rst species.

Example 4 . 2 .

 42 The voice leading (p, D 4 , D 5 ) → (D 4 , C 3 , C 3 ) corresponds to the matrix

4. 1 Figure 9 :

 19 Figure 9: The Retrograde Canon (bars 1-4), a palindromic canon belonging to the Musikalisches Opfer by J. S. Bach.

  Voice Leading: ['D4', 'D4'] ['D4', 'F4'] c = [1, 0, 1, 0, 0] -oblique motion Voice Leading: ['D4', 'F4'] ['F4', 'A4'] c = [2, 0, 0, 0, 0] -similar motion up Voice Leading: ['F4', 'A4'] ['F4', 'D5'] c = [1, 0, 1, 0, 0] -oblique motion Voice Leading: ['F4', 'D5'] ['A4', 'C#5'] c = [1, 1, 0, 0, 0] -contrary motion Table 2 collects the complexity vectors and their multiplicities; they are displayed in the form of point clouds in Figure 10.

  (a) Projection on the rst three components of the complexity vector. (b) Projection on the upward, downward and crossing components of c. (c) Projection on the upward, downward and rest components of c.

Figure 10 :

 10 Figure 10: Three-dimensional projection of the 5-dimensional point cloud representing the complexity of the Retrograde Canon. The radius of each circle represents the normalised multiplicity of each complexity vector.

Figure 11 :

 11 Figure 11: Comparing two time series with DTW.

1 .

 1 γ k := (γ x k , γ y k ) ∈ {1, . . . , n} × {1, . . . , m} for all k ∈ {1, . . . , l}; 2. γ 1 := (1, 1) and γ l := (n, m); for all k ∈ {1, . . . , l -1}; 4. γ k+1 -γ k ∈ (1, 0), (0, 1), (1, 1) for all k ∈ {1, . . . , l -1}. The total cost of a (n, m)-warping path γ over the features X and Y is de ned as C γ (X, Y ) := l k=1 C(x γ x k , y γ y k

Figure 12 :

 12 Figure 12: Optimal warping path on Angelus Domini and Dicant nunc Judei.

Table 2 :

 2 Complexity vectors of the Retrograde Canon and their occurrences.

Table 3 :

 3 DTW distance matrix for the three time series of complexity vectors.

		Angelus Dicant Canon
	Angelus	0.00	0.62	1.34
	Dicant	0.62	0.00	1.16
	Canon	1.34	1.16	0.00

These are in fact the images of two bijective maps ψM : {1, . . . , n} → M and ψL : {1, . . . , n} → L.

For a multiset S we assume that µS(x) = 0 if x / ∈ XS. With this understanding, the function µM∪L is de ned on the whole of XM ∪ XL.

dissimilarity of the time series describing the pieces.


Henri Gouin
email: henri.gouin@univ-amu.fr

Archimedes' famous-theorem 1 Archimedes' famous-theorem

In his treatise addressed to Dositheus of Pelusium [START_REF] Calinger | A Contextual History of Mathematics[END_REF]), Archimedes of Syracuse obtained the result of which he was the most proud: a sphere has two-thirds the volume of its circumscribing cylinder (see Fig. 1). At his request a sculpted sphere and cylinder were placed on his tomb near Syracuse [START_REF] Rorres | Tomb of Archimedes Illustrations[END_REF]). Usually, it is admitted that to find this formula, Archimedes used a half polygon inscribed in a semicircle; then he performed rotations of these two figures to obtain a set of trunks in a sphere. This set of trunks allowed him to determine the volume. In our opinion, Archimedes was so clever that he found the proof with shorter demonstration. Archimedes did not need to know π to prove the result and the Pythagorean theorem was probably the key to the proof.

Keywords: Archimedes, science history MSC numbers: 01A20, 03-03 -------------------------------- Proof See Fig. 2a:
The area generated by the rotation of segment AC around -→ Oz at level ℓ is proportional to the square of radius r (the proportionality coefficient is π).

The area generated by the rotation of segment AB around -→ Oz at level ℓ is proportional to the radius square of a radius which is, on the basis on the Pythagorean theorem, √ r 2 -ℓ 2 . The area generated by the rotation of segment BC around

-→ Oz at level ℓ is proportional to r 2 -(r 2 -ℓ 2 ) = ℓ 2 .
See Fig. 2b: Volume V 1 of the domain included between the cylinder and the sphere is therefore twice the volume of a cone whose the base is that of the cylinder and the height is the radius of the sphere. (Archimedes knew the volume of the cone which is the extension of the volume of a pyramid with n faces, (3 ≤ n ∈ N )).

Consequence:

Two times the volume of the cone is V 1 = (2/3) r S where S is the surface of the cone basis.

The volume of the cylinder is V 2 = 2r S.

The volume of the sphere is

V 3 = V 2 -V 1 Figure 1.
A sphere has 2/3 the volume of its circumscribing cylinder. The radius of the sphere is r.

Figure 2 .

 2 Figure 2. Sphere, cylinder and cone.

† dedicato ai miei amici italiani


Morgan Le Chénéchal
email: morgan.lechenechal@b-com.com

Thierry Duval
email: thierry.duval@telecom-bretagne.eu

Valérie Gouranton
email: valerie.gouranton@irisa.fr

Jérome Royan
email: jerome.royan@b-com.com

Bruno Arnaldi
email: bruno.arnaldi@irisa.fr

The Stretchable Arms for Collaborative Remote Guiding

Keywords: Descriptors (according to ACM CCS): Information Interfaces and Presentation (e, g, HCI) [H, 5, 2]: User Interfaces-Computer Graphics [I, 3, 6]: Methodology and Techniques-

Figure 1: The stretchable guiding arms, controlled by the remote expert's location, from the agent's viewpoint. On the left, the arms are extended and red because the guide is ahead of the agent. On the right

Introduction

Collaborative Virtual Environments (CVE) allow multiple users to share a common virtual world to achieve collaborative tasks. For a long time, the literature leveraged the issue to get the user to reach a specific viewpoint in order to show him some artifact. Indeed, in the real world, natural awareness features such as gaze direction [SWM * 08] allows people to communicate without the need to explain explicitly but these are rarely available in Mixed Reality (MR) applications. Thus, different techniques must be proposed to handle this useful feature: to guide someone to reach a specific viewpoint.

We propose a novel interaction technique based on a MR system in which a remote expert using a Virtual Reality (VR) application guides an agent using an Augmented Reality (AR) application. The agent must perform a physical task that is guided by virtual arms collocated with his shoulders, controlled by the remote expert. Before performing the actual arm guidance, the remote expert must reach the correct location in the virtual world while showing it to the agent in his augmented real world. To perform this initial location guidance, we propose to extend the GoGo technique initially developed for single user selection and manipulation tasks in VR [START_REF] Poupyrev | The go-go interaction technique: non-linear mapping for direct manipulation in VR[END_REF]. We added the arms representation as in [FBHH99] and removed the non-linear scaling, instead controlling it directly with the expert's interactions.

Section 2 introduces proposed techniques in the literature that have achieved a similar goal and their limitations. Section 3 presents our stretchable guiding arms technique as well as its context of use, and Section 4 discusses the results of a pilot user study. Finally, Section 5 concludes this paper.

Related Work

Making a user reach a specific viewpoint is not only used in CVE. Indeed, many single user MR applications need to provide such a feature. Recently, Sukan et al. proposed the parafrustum approach to guide a user in an AR application to reach a set of acceptable viewpoints [SEO * 14]. It presents an advanced technique based on a 3D augmentation that shows the user the viewpoint he must reach. Compared to classical approaches such as the use of head-up display (HUD), this technique greatly eases the reaching process because of a fully collocated guidance in the real interaction space. However, Sukan et al. highlight the complexity of defining the shape of the 3D augmentation, and thus it seems complicated to extend it for collaborative purposes with an online definition of the shape by one user to another.

In CVE, we distinguish two approaches: asynchronous and synchronous guidance. First, asynchronous guidance is often used when a user defines some interesting viewpoint to another user, who can switch from one predefined viewpoint to another. In [START_REF] Duval | Collaborative exploration of 3d scientific data[END_REF], these interesting viewpoints are represented with virtual cameras all around a set of scientific data visualizations. Second, synchronous guidance is the category to which our technique belongs. It is utilized by a user who interactively defines a viewpoint for another user to reach. Many metaphors have been proposed [NDF13]: directional arrows, compass, lit path. However, these techniques work well in an asymmetric setup, i.e. when the guide has a global view of the scene and while the visitor is immersed in it, but are hardly extendable for a symmetric, fully collocated setup. Indeed, even if their implementations can be fully automated, i.e. the guide does not need to define the parameters of the guidance cues himself because the system is able to compute them automatically, they suffer from strict limitations: the loss of naturalness in the guidance process for the guided user because of the interpretation of specific visual cues and the eye strain that can be generated by the proximity of visual cues according to the guided viewpoint.

The Stretchable Guiding Arms Technique

The stretchable guiding arms technique has been developed to overcome the limitations found in the techniques from the literature in a specific context: the expert and the agent share a fully collocated setup. Moreover, the final goal of the system is to guide the agent in performing a physical task with the use of virtual arms controlled by the remote expert. In this setting, the first step consists of viewpoints collocation. Then, the expert must show which object to move, and thus he often needs to move the agent's viewpoint forward in order to grasp the virtual cloned object. The existing technique based on directional arrows can achieve this purpose, as illustrated in Figure 2, but first users who tested our system complained about eye strain because of the proximity of the visual cues, including the remote expert's frustum. Moreover, it increases the cognitive load on the agent when having to follow arms guidance and location guidance at the same time using different visual cues. To avoid this, we take advantage of the already existing virtual arms initially present for gesture-based guiding purposes only. They can also be used for location reaching guidance in a non-intrusive way, contrary to other methods. Figure 3 Here, it is added to understand the expert's interaction. From the expert's viewpoint, his virtual arms do not scale and also stay collocated with his shoulders.

the expert chooses to switch to stretch mode, his forward motion makes the virtual arms seen from the agent's viewpoint start to scale up to keep the virtual hands always collocated with the expert's ones, while maintaining the virtual shoulders fixed according to the agent's shoulders. From the expert's viewpoint, his virtual arms do not scale up, and also stay collocated with his shoulders.

The main advantage of this technique is the naturalness of the guidance. Indeed, the agent can directly perceive the forward motion of the remote expert, thanks to the scaling up of his virtual arms. Then, to follow the expert, the agent simply moves his viewpoint in order to retrieve the initial length of his virtual arms. In addition to the scaling cue, a coloring code smoothly changes the color of the independent virtual arms from green to red, according to their scale. They return to green when the scale is close to its initial value, which means that the agent has reached the expert's viewpoint (cf. Figure 1). If the agent overtakes the expert's location, the virtual arms keep their initial length and color, but their bending indicates to the agent that he must move backward to retrieve a correct pose.

In parallel, the expert can use a virtual camera with a different viewpoint of his VE in order to get visual feedback on the agent following. This camera is displayed in a corner of the expert's viewport and is enabled when the stretchable guiding arms technique is activated. The remote expert can switch between different available viewpoints: a side or front view based on his head location, or the viewpoint of the agent slightly behind him when performing a forward motion guidance. Moreover, the color change for virtual arms from green to red is used to inform the remote expert explicitly about his distance with the guided agent.

Implementation

Currently available see-through AR glasses do not provide a wide enough field of view. Thus, we use a see-through HMD equipped with a pair of stereo cameras. The prototype has been developed with Unity3D TM to be used with an Oculus Rift TM DK2. On the expert side, the virtual arms are controlled with a Razer TM Hydra that provides two handles with 6DOF tracking and joysticks for navigation.

Pilot User Study

We compared reaching performances and subjective feelings of (C1) our technique (cf Figure 1) with (C2) the frustum with 3D directional arrows based approach (cf. Figure 2). In order to evaluate the agent's performance in the location reaching process, we try to remove bias generated by the expert's interactions. Thus, we recorded the interactions of a real expert in a pre-process phase in order to replay it for the experiment, removing possible expert's interaction bias due to changing behavior. We postulate that the expert's interactions are not really affected by the agent's ones. Indeed, the expert must reach a location to grab a virtual object, and thus must move forward to achieve the goal assuming that the agent is following.

Measurements and Hypotheses

We collected three objective measures: completion time (hit target around 0.15m), mean distance between the replayed expert and the subject, and final precision (triggered by the subject) in location and orientation. Moreover, subjects fill in a final subjective questionnaire (7 point Likert-scale) to express their impressions of the naturalness of the system, their perceived cognitive load and eye-strain, amongst other considerations. We stated the following hypotheses:

• H1: Completion time, mean distance and precision are equivalent in both conditions C1 and C2 • H2: C1 is more comfortable to use than C2

Protocol

We design our experiment in a counter-balanced way using a set of 20 pre-defined targets (10 per condition) in a range from 1 to 3 meters from the home location and a horizontal angle range from -30 to 30 degrees. Their altitude is defined around a mean value of 1.5m with a range of ±0.3m. Each participant passes both conditions twice alternatively, with a different set of 2x5 virtual targets per condition. These targets are not displayed for the replay process because the subjects must reach them following only the recorded guide. We simulate the initial viewpoints collocation, illustrated in step 1 of the process described in Figure 3, with a re-targeting of the replayed data according to the subject's height and exact home location.

Results

Results have been collected from 10 subjects aged from 20 to 30 (mean = 24.4, sd = 4.03). A few iterations are not part Figure 4: Boxplots of the delay between the target hitting by the guide and the subject, their mean distance until the hit, and their final position and orientation differences.

of the analysis because of tracking issues. Figure 4 illustrates the collected quantitative results. We found a significant difference in the target hitting delay between the guide and the subject (F(1, 194) = 20.47, p < .01). This delay is shorter with (C1) the guiding arms (M = -0.03, sd = 1.28) than with (C2) the frustum (M = 0.75, sd = 1.61). In the same way, the mean distance between the guide and the subject until the hit is significantly smaller

Discussion

H1 is not proved. Actually, we found that the guiding arms (C1) are better for closely following the remote guide, while the frustum-based guiding (C2) is more accurate for final precision. This degradation of preciseness using C1 can be explained by the complexity of human body's DOFs that generate variations between the relative pose of the agent's head and the expert's hands. Thus, the coupling of C1 with an alternative guiding technique for final positioning should be studied to improve preciseness without decreasing naturalness of the guidance. Concerning the qualitative results (cf. Figure 5), we validated H2 because of better visual comfort using our approach (C1). Moreover, subjects prefer our approach (C1) in terms of originality and guide presence. For other criterion (efficiency, naturalness and cognitive load), results are similar for both conditions C1 and C2.

Conclusion

We presented an extension of the GoGo interaction technique for collaborative purposes. We use the same idea of virtual arms scaling up, but not in a selection and manipulation context. Rather, we use it to guide a remote user to reach a location defined by the location of another one. Solutions in the literature require the use of specific visual cues that decrease the naturalness as well as the immersion of the application. Our stretchable guiding arms technique takes advantage of already existing virtual arms that are used to guide a physical task using remote arm gestures. We believe our technique is more appropriate in this context, and the results of the pilot user study show that following performances are even better using our approach as well as enhancing comfort of use. Future work will focus on improving final precision of our guiding technique.

Figure 2 :

 2 Figure 2: A simple guiding technique based on directional arrows and viewing frustum of the remote expert.

  illustrates the principle of the proposed stretchable guiding arms technique. When c The Author(s)

Figure 3 :

 3 Figure 3: Principle of the stretchable guiding arms technique. Illustration from the agent side in blue. The red circle represents the expert's head but is not actually displayed.Here, it is added to understand the expert's interaction. From the expert's viewpoint, his virtual arms do not scale and also stay collocated with his shoulders.

  (F(1, 194) = 20.39, p < .01) using (C1) the guiding arms (M = 0.21, sd = 0.008) than (C2) the frustum (M = 0.26, sd = 0.007). In terms of precision, the frustum-based guiding (C2) is significantly more accurate than the guiding arms (C1) for both location (F(1, 194) = 11.34, p < .01 ; (C1): M = 0.19, sd = 0.006; (C2): M = 0.15, sd = 0.01) and orientation (F(1, 194) = 55.37, p < .01; (C1): M = 31.10, sd = 122.55; (C2): M = 16.54, sd = 252.88).

  Figure 5: Synthesis of subjects' qualitative appreciations.

c The Author(s)