M Legrée

V Charbonnier

S Al Bacha

K Asano

K Sakaki

I Aubert

F Mauvy

J Sabatier

J.- L Bobet

Hydrogen generation performances and electrochemical properties of Mg alloys with 14H Long Period Stacking Ordered structure

Keywords: Magnesium, LPSO, hydrolysis, Corrosion, Electrochemical Impedance Spectroscopy, hydrogen production

Hydrolysis of Magnesium with water is an excellent way to produce H2 at high rate, especially when alloyed with nobler elements that accelerate its corrosion. In this work, the hydrogen generation properties of four Magnesium based alloys with Long Period Stacking Ordered (14H-LPSO) structures (Mg91Cu4Y5; Mg91Ni4Y5; Mg91Ni4Gd5 and Mg91Ni4Sm5) are studied through standard electrochemical methods. The hydrolysis experiment on coarse powder are coherent with the electrochemical measurements performed on polished surface. The rare earth (Y, Gd or Sm) and transition metal (Ni or Cu) involved in the LPSO are found to have a significant influence on the corrosion and hydrolysis properties of alloys with 14H-LPSO structures.

Introduction

With twelve protons, magnesium (Mg) is one of the lightest metals (1.738 g/cm 3 ) and among the most available on our planet (7 th most abundant element on earth's crust) [START_REF] Fleischer | Recent estimates of the abundances of the elements in this earth's crust[END_REF]. In addition, it has good electrical and thermal conductivities, interesting mechanical properties and is biocompatible (Mg 2+ is the 4 th most abundant cation in human body) [START_REF] Mordike | Magnesium Properties -applications -potential[END_REF][START_REF] Monteiro | The Influence of Alloy Element on Magnesium for Electronic Devices Applications -A Review[END_REF]. These characteristics make magnesium a very interesting material for many applications such as lightweight structure materials, transport, electronics, energy, fracturing ball technologies or biodegradable implants [START_REF] Mordike | Magnesium Properties -applications -potential[END_REF][START_REF] Monteiro | The Influence of Alloy Element on Magnesium for Electronic Devices Applications -A Review[END_REF][4][START_REF] Wang | Enhanced mechanical properties and degradation rate of Mg-Ni-Y alloy by introducing LPSO phase for degradable fracturing ball applications[END_REF][START_REF] Esmaily | Fundamentals and advances in magnesium alloy corrosion[END_REF][START_REF] Zong | Corrosion Behaviors of Long-Period Stacking Ordered Structure in Mg Alloys Used in Biomaterials: A Review[END_REF]. Widening Mg alloys usability has motivated extensive scientific research on their oxidation and corrosion properties [START_REF] Esmaily | Fundamentals and advances in magnesium alloy corrosion[END_REF][START_REF] Samaniego | On the evidence for univalent Mg[END_REF][9][START_REF] Birbilis | On the corrosion of binary magnesiumrare earth alloys[END_REF][START_REF] Bland | Effect of Crystallographic Orientation on the Corrosion of Magnesium: Comparison of Film Forming and Bare Crystal Facets using Electrochemical Impedance and Raman Spectroscopy[END_REF][START_REF] Gusieva | Corrosion of magnesium alloys: The role of alloying[END_REF][START_REF] Banjade | Fundamental Investigation of Magnesium Corrosion Using Experiments and Simulation[END_REF][START_REF] Atrens | Advances in Mg corrosion and research suggestions[END_REF][START_REF] Li | Anodic hydrogen evolution on Mg[END_REF][START_REF] Frankel | Evolution of hydrogen at dissolving magnesium surfaces[END_REF][START_REF] Fajardo | Effect of impurities on the enhanced catalytic activity for hydrogen evolution in high purity magnesium[END_REF][START_REF] Huang | The anodically polarized Mg surface products and accelerated hydrogen evolution[END_REF][START_REF] Delgado | A Measuring Approach to Assess the Corrosion Rate of Magnesium Alloys Using Electrochemical Impedance Spectroscopy[END_REF][START_REF] Samaniego | Hydrogen evolution during anodic polarization of Mg alloyed with Li, Ca, or Fe[END_REF][START_REF] King | Accurate Electrochemical Measurement of Magnesium Corrosion Rates; a Combined Impedance , Mass-Loss and Hydrogen Collection Study[END_REF][START_REF] Leleu | Corrosion rate determination of rare-earth Mg alloys in a Na2SO4 solution by electrochemical measurements and inductive coupled plasma-optical emission spectroscopy[END_REF]. There are several key features that make magnesium and its alloys a peculiar topic in the field of corrosion science. First, magnesium has a very low corrosion potential ( -1.7 V/SCE (Saturated Calomel Electrode) in aqueous solutions) and is highly sensitive to corrosion, particularly to galvanic corrosion [START_REF] Asmar | The Mg-Rich Phase NdNiMg15: Structural and Magnetic Properties[END_REF]. It is frequently used as sacrificial anode in the industry. Furthermore, enhanced hydrogen evolution occurs under anodic polarization [START_REF] Samaniego | On the evidence for univalent Mg[END_REF][START_REF] Li | Anodic hydrogen evolution on Mg[END_REF][START_REF] Thomas | Corrosion mechanism and hydrogen evolution on Mg[END_REF][START_REF] Esmaily | Anomalous hydrogen evolution on AZ31, AZ61 and AZ91 magnesium alloys in unbuffered sodium chloride solution[END_REF]] and a discrepancy is thus observed between corrosion rates measured from electrochemical methods and mass loss and H2 collection [START_REF] Pardo | Electrochemical estimation of the corrosion rate of magnesium/aluminium alloys[END_REF]. Due to its strong ability to corrode, Mg has attracted a lot of interest for di-hydrogen (H2) production via hydrolysis reaction [START_REF] Uesugi | Production of hydrogen storage material MgH2 and its application[END_REF][START_REF] Kantürk Figen | Hydrogen generation from waste Mg based material in various saline solutions (NiCl2, CoCl2, CuCl2, FeCl3, MnCl2)[END_REF][START_REF] Grosjean | Hydrogen production from highly corroding Mg-based materials elaborated by ball milling[END_REF][START_REF] Sabatier | A New Device for Hydrogen Production on Demand with Application to Electric Assist Bike: Description, Production Characteristics and Basic Control[END_REF][START_REF] Oh | Fabrication of Mg-Ni-Sn alloys for fast hydrogen generation in seawater[END_REF][START_REF] Bacha | Hydrogen generation via hydrolysis of ball milled WE43 magnesium waste[END_REF][START_REF] Hou | Comparative investigation on feasible hydrolysis H2 production behavior of commercial Mg-M (M = Ni, Ce, and La) binary alloys modified by high-energy ball milling-Feasible modification strategy for Mg-based hydrogen producing alloys[END_REF] Among Mg alloys, Long Period Stacking Ordered (LPSO) is the name given to a family of structures found in some Mg-Transition Metal (TM)-Rare Earth (RE) systems [START_REF] Saal | Thermodynamic stability of Mg-based ternary long-period stacking ordered structures[END_REF][START_REF] Kishida | Crystal structures of highly-ordered longperiod stacking-ordered phases with 18R, 14H and 10H-type stacking sequences in the Mg-Zn-Y system[END_REF][START_REF] Mi | New polytypes of long-period stacking ordered structures in Mg-Co-Y alloys[END_REF][START_REF] Kim | Compositional evolution of long-period stacking ordered structures in magnesium studied by atom probe tomography[END_REF]. Because of the presence of TM in the structure, LPSO shows higher corrosion potential than pure Mg which leads to galvanic coupling between the two phases (i.e. Mg and LPSO) [START_REF] Li | Effect of volume fraction of LPSO phases on corrosion and mechanical properties of Mg-Zn-Y alloys[END_REF][START_REF] Liu | Role of the LPSO structure in the improvement of corrosion resistance of Mg-Gd-Zn-Zr alloys[END_REF]. In the literature, the Mg-Y-Zn system (in which Inoue et al. [START_REF] Inoue | Novel hexagonal structure and ultrahigh strength of magnesium solid solution in the Mg-Zn-Y system[END_REF] discovered Mg-LPSO structure for the first time) is by far the most studied one [START_REF] Kim | Compositional evolution of long-period stacking ordered structures in magnesium studied by atom probe tomography[END_REF][START_REF] Inoue | Novel hexagonal structure and ultrahigh strength of magnesium solid solution in the Mg-Zn-Y system[END_REF][START_REF] Abe | Long-period ordered structure in a high-strength nanocrystalline Mg-1 at% Zn-2 at% Y alloy studied by atomic-resolution Z-contrast STEM[END_REF][START_REF] Kawamura | Formation and mechanical properties of Mg97Zn1RE 2 alloys with long-period stacking ordered structure[END_REF][START_REF] Yamasaki | Highly ordered 10H-type long-period stacking order phase in a Mg-Zn-Y ternary alloy[END_REF][START_REF] Cheng | Effect of the morphology of long-period stacking ordered phase on mechanical properties and corrosion behavior of cast Mg-Zn-Y-Ti alloy[END_REF]. Mg-Gd-Zn, Mg-Gd-Y-Zn and Mg-Y-Ni systems were studied for biodegradable implants and fracturing ball applications [START_REF] Liu | Role of the LPSO structure in the improvement of corrosion resistance of Mg-Gd-Zn-Zr alloys[END_REF][START_REF] Zong | Effects of Li on Microstructures, Mechanical, and Biocorrosion Properties of Biodegradable Mg94-xZn2Y4Lix Alloys with Long Period Stacking Ordered Phase[END_REF][START_REF] Han | The Anodic Role of Ni-Containing LPSO Phases During the Microgalvanic Corrosion of Mg 98 Gd 1.5 Ni 0.5 Alloy[END_REF]. In both cases, corrosion resistance is not desired too important because the alloy is required to dissolve after use, this is why researchers focus on controlling the degradation rate of the material. Beyond these few examples, reported works on corrosion of LPSO containing other elements than zinc (Zn) and yttrium (Y) remain very scarce.

Additionally, recent study led in our group suggested that LPSO can be very efficient materials for H2 generation via hydrolysis [START_REF] Legrée | Hydrogen generation by hydrolysis reaction using magnesium alloys with long period stacking ordered structure[END_REF].

It is common practice to lead hydrolysis experiment with Mg powder or end of life alloys scraps with a roughened surface for optimized hydrogen generation performances (i.e. activated by ball milling) [START_REF] Grosjean | Hydrogen production from highly corroding Mg-based materials elaborated by ball milling[END_REF][START_REF] Bacha | Effect of ball milling strategy (milling device for scaling-up) on the hydrolysis performance of Mg alloy waste[END_REF][START_REF] Tayeh | Production of hydrogen from magnesium hydrides hydrolysis[END_REF]. In such conditions, the reacting surface is not well defined and it evolves importantly throughout the reaction. The present work aims at comparing the behavior of Mg-LPSO alloys under hydrolysis conditions (i.e. as gross powders) with electrochemical measurements led on bulk surfaces. To achieve this goal, four samples were prepared by induction melting of pure elements and annealing, namely: Mg91Cu4Y5, Mg91Ni4Y5, Mg91Ni4Gd5 and Mg91Ni4Sm5. Samples were characterized physically and chemically with X-Ray Diffraction (XRD), Scanning Electrochemical Microscopy (SEM) and Energy Dispersive X-ray Spectroscopy (EDXS). Hydrolysis reactions with powders are presented and further interpreted in the light of standard electrochemical characterization on bulk materials surfaces. The techniques involved are chronopotentiometry (Open Circuit Potential (OCP) recording versus time), Electrochemical Impedance Spectroscopy (EIS) and potentiodynamic polarization (data is analyzed using Tafel method).

Materials and methods

The compounds involved in the present study were synthesized using chunks of magnesium (99.9%) from Kojundo Chemical Laboratory Co., Ltd., copper (Cu) (99.99%) and nickel (Ni) (99.9%) from Furuuchi Chemical Co., and all Rare Earths (RE) (Yttrium (Y), Gadolinium (Gd) and Samarium (Sm)), 99.9% from Nippon Yttrium Co., Ltd. The pure elements were melted together in an induction furnace under helium atmosphere. Due to its high vapor pressure, an excess of 7 wt.% of Mg was systematically added. The obtained ingots were subsequently annealed for 10 days under argon atmosphere at 500°C in a tantalum foil. Synthesized alloys were characterized by XRD using a Philips PANalytical X'Pert (PW1820) diffractometer with Cu Kα1 radiation (λ=1.5405 Å). SEM observations were made using a TESCAN VEGA3 SB and SERON TECHNOLOGY microscope equipped with Secondary Electron and Back Scattered Electron detectors (hereafter referred to as SE and BSE modes respectively). Furthermore, the elemental composition of the surface was estimated using EDX spectrometer.

All electrochemical measurements were performed in simulated seawater solution (i.e. 0.6 mol/L NaCl) at room temperature with a three electrodes set-up (Working Electrode: bulk samples were abrased up to 4000 grit SiC with ethanol as lubricant prior to each electrochemical test; Counter Electrode: Titanium net; Reference Electrode: Saturated Calomel Electrode (SCE) -Hg/Hg2Cl2). EIS and polarization measurements were made after 40 min OCP stabilization (except in the case where immersion time is varied as a parameter). OCP and potentiodynamic polarization measurements were done with VersaSTAT 4 potentiostat and controlled via VersaStudio software. Anodic and cathodic polarization curves were recorded separately with a scan rate of 0.5 mV/s as follows: (i) a first sweep from +50 mV to -250 mV vs OCP for the cathodic polarization measurement and (ii) from -50 mV to +250 mV vs OCP for the anodic polarization measurement. EIS diagrams were recorded from 100 kHz to 50 mHz with 10 mV amplitude versus OCP on a Biologic SP-150 potentiostat because of its good stability in AC mode. The data was collected on EC-Lab software and fitted using the commercial Zview software while Kramers-Kronig relations and Distribution of Relaxation Times (DRT) calculations were done with in-house CANELEIS software using Tikhonov regularization method with weighing parameter λ=10 -2 [START_REF] Zeimetz | Using the Distribution of Relaxation Times for analyzing the Kramers-Kronig Relations[END_REF]. In order to help emphasize electrochemical features from these measurements, different immersion times prior to measurements and different rotation speeds of the working electrode were tested. Rotating Disc Electrode (RDE) measurements were carried out with an Origatrod electrode connected to an Origaflex potentiostat.

The reaction of magnesium with water (hydrolysis process) considered in this study is:

Mg + 2H2O = Mg(OH)2 + H2 (1)
Kinetic study of hydrolysis reactions was carried out in a closed 100 mL three-neck round bottom flask connected to a burette that dips in a beaker of water. The experimental setup was previously described in ref [START_REF] Pighin | Nanostructured Mg for hydrogen production by hydrolysis obtained by MgH2 milling and dehydriding[END_REF]. Water from the beaker is initially sucked up into the burette. Then, the di-hydrogen (H2) produced during the experiment induces water displacement back to the beaker and a balance connected to a computer records the mass variation, allowing to follow in real time the hydrogen generation. Bulk samples were filed and about 30 mg of material were taken to react with 20 mL of simulated seawater solution (i.e. 0.6 mol/L NaCl). The solution was rapidly stirred with a magnetic bar and during the whole experiment, the temperature was kept at a constant value of 25°C.

After measurement, 1 mL of 2.4 mol/L HCl was added to complete the hydrolysis process and thereby generate the maximum H2 amount the sample could release. The yield of reaction was calculated from this amount of H2.

Results and discussion

Material characterization

Four LPSO compounds have been selected and characterized by X-ray diffraction (XRD):

Mg91Cu4Y5 ; Mg91Ni4Y5 ; Mg91Ni4Gd5 ; Mg91Ni4Sm5. The associated patterns are reported on figure 1 and show that the four alloys synthesized formed LPSO with a 14H structure. LPSO structures can be described as alternating layers of Mg and Mg-TM-RE layers containing L12-type atomic clusters (TM6RE8). In the case of 14H studied here, Mg layers are four atomic planes wide. The "H" stands for "Hexagonal" and the "14" gives the number of atomic planes in a unit cell (counted along c direction).

Detailed information on LPSO structures can be found in dedicated works [START_REF] Kishida | Crystal structures of highly-ordered longperiod stacking-ordered phases with 18R, 14H and 10H-type stacking sequences in the Mg-Zn-Y system[END_REF][START_REF] Yokobayashi | Enrichment of Gd and Al atoms in the quadruple close packed planes and their in-plane long-range ordering in the long period stacking-ordered phase in the Mg-Al-Gd system[END_REF][START_REF] Kishida | The crystal structure of the LPSO phase of the 14 H-type in the Mg-Al-Gd alloy system[END_REF][START_REF] Jiang | Phase equilibria of the long-period stacking ordered phase in the Mg-Ni-Y system[END_REF][START_REF] Charbonnier | Hydrogenation Properties of Mg 83.3 Cu 7.2 Y 9.5 with Long Period Stacking Ordered Structure and Formation of Polymorphic γ-MgH 2[END_REF][START_REF] Inoue | Correlative atom probe tomography and scanning transmission electron microscopy reveal growth sequence of LPSO phase in Mg alloy containing Al and Gd[END_REF]. The microstructures of the elaborated samples have been characterized using SEM micrographs and EDXS analysis. Main and secondary phases identification, surface proportions and deduced experimental compositions are summarized in Table 1. The discrepancy observed between measured and theoretical LPSO compositions has already been reported and may be explained by Mg insertion or substitution in the TM6RE8 cluster [START_REF] Legrée | Hydrogen generation by hydrolysis reaction using magnesium alloys with long period stacking ordered structure[END_REF][START_REF] Charbonnier | Hydrogenation Properties of Mg 83.3 Cu 7.2 Y 9.5 with Long Period Stacking Ordered Structure and Formation of Polymorphic γ-MgH 2[END_REF][START_REF] Kishida | A formation criterion for Order-Disorder (OD) phases of the Long-Period Stacking Order (LPSO)-type in Mg-Al-RE (Rare Earth) Ternary Systems[END_REF]. It can be deduced from BSE-SEM images of figure 2 that all studied alloys present both LPSO (bright zones) and Mg (dark zones) phases with Mg surface proportions ranging from 15% to 27%. These values were estimated using contrast thresholding with Image J software (with an accuracy of about 2%). In addition, Figure 2 shows that the size and the morphology of Mg inclusions are quite large and well defined in alloys containing Y as RE, while alloys with Sm and Gd present a very intricate Mg phase. According to the EDXS analysis, the white particles visible on the BSE-SEM micrographs correspond to RE2O3 and Mg2TM. The surface proportion for these secondary phases has been estimated to be inferior to 4% by image analysis (Table 1). EDXS measurements revealed variations in the LPSO composition of each alloy. It results in very different TM/RE ratios in the 14H-LPSO phases with different elements, that range from 0.6 for Mg91Ni4Y5 to 1.1 for Mg91Ni4Sm5 (the 14H LPSO theoretical value being 0.75). It should be noted here that EDXS allows only for semi-quantitative characterization. The values discussed here are thus used to compare the different alloys but should not be taken as exact composition values.

Di-hydrogen generation

All four alloys were filed into coarse powders to generate di-hydrogen gas (H2) by water hydrolysis. The associated kinetic curves are displayed in figure 3 along with the performances of pure Mg and Mg-Ni-C powder. Mg-Ni-C mixture is considered as reference material whose preparation has been optimized in previous works [START_REF] Grosjean | Hydrogen production from highly corroding Mg-based materials elaborated by ball milling[END_REF][START_REF] Awad | Effect of carbons (G and CFs), TM (Ni, Fe and Al) and oxides (Nb 2 O 5 and V 2 O 5 ) on hydrogen generation from ball milled Mg-based hydrolysis reaction for fuel cell[END_REF][START_REF] Alasmar | Hydrogen generation from Nd-Ni-Mg system by hydrolysis reaction[END_REF]. It consists in a ball milling of 3 hours at 250 rpm of 95%wt.

Mg powder (99.98%), 5%wt. Ni powder (99.98%) and 5%wt. graphite. Even without prior surface activation, Mg91Ni4Sm5 and Mg91Ni4Gd5 gross powders present hydrogen generation performances (i.e.

Electrochemical measurements

Open Circuit Potential

Open Circuit Potentials (OCP) were recorded versus time in 0.6 mol/L NaCl solution and are presented in figure 4. OCP of pure Mg is also reported as a matter of comparison. It is interesting to note that Mg91Ni4Y5, Mg91Ni4Gd5 and Mg91Ni4Sm5 alloys have almost identical stabilized OCP values (close to -1.35V/ECS). The nature of the RE involved in the LPSO phase does not seem to influence much the steady state potential of the alloy. This result can be analyzed considering that, in NaCl solutions, RE elements have relatively close corrosion potentials neighboring that of magnesium [START_REF] Chen | Technical note: Experimental survey of corrosion potentials for rare earth metals Ce, Er, Gd, La, and Nd as a function of pH and chloride concentration[END_REF][START_REF] Del | the Role of Intermetallic Phases in the Corrosion of Magnesium-Rare Earth Alloys[END_REF].

OCP recording versus time shows that thermodynamic equilibrium of Cu containing alloy is closer to that of pure Mg as compared to Ni containing alloys (figure 4). Mg in Mg91Cu4Y5 may thus be less likely to corrode than in Mg91Ni4RE5 alloys with nobler nature. This statement is consistent with hydrolysis experiments in which reaction kinetic was clearly lower for Mg91Cu4Y5. The hydrolysis kinetics of the three Mg91Ni4RE5 alloys were slightly different and their stabilized OCP about even. It can be deduced here that the transition metal strongly influences the OCP voltage in the Mg91TM4RE5 alloys while the RE does not. Hydrolysis experiments for Mg91Ni4RE5 alloys may thus be understood from kinetics perspective explored in the next sections. After 2400s of immersion, the surfaces of the samples are in major part covered of black corrosion product. This phenomenon have been assigned to magnesium hydroxide precipitation :

MgOH2 formation has been confirmed by XRD in previous work [START_REF] Legrée | Hydrogen generation by hydrolysis reaction using magnesium alloys with long period stacking ordered structure[END_REF]. Electrochemical measurements (EIS and polarization) have been performed when the OCP are stabilized, i.e. after 2400s of immersion.

Electrochemical Impedance Spectroscopy

The EIS diagrams were recorded between 0.1 Hz and 100 000 Hz with 10 mV of amplitude. In order to verify the stationarity of the system, Kramers-Kronig relations were used [START_REF] Zeimetz | Using the Distribution of Relaxation Times for analyzing the Kramers-Kronig Relations[END_REF][START_REF] Dierickx | How the distribution of relaxation times enhances complex equivalent circuit models for fuel cells[END_REF]. As shown in figures 5A and 5B, the results have good correlation both for real and imaginary parts, which indicates that the data is consistent and causal in the range 1Hz -100 000Hz. According to the literature, in the low frequency range ( i.e. freq → 0 Hz), the real part of the impedance can be assigned to the electrode polarization resistance, which is inversely proportional to the corrosion rate of the alloy (Jcorr=B/Rp) [START_REF] King | Accurate Electrochemical Measurement of Magnesium Corrosion Rates; a Combined Impedance , Mass-Loss and Hydrogen Collection Study[END_REF]. In the present study, alloys with LPSO phase are corroding so rapidly that a steady state (e.g.

𝐶𝑒𝑓𝑓 = (R * Q) 1/α R (2)
with Ceff the effective capacity, R the corresponding resistance parallel to the CPE, Q the CPE capacity and α the exponent that reflects the impedance distribution. Figure 8 shows the equivalent circuit used as well as Nyquist plot of Mg91Ni4Gd5 and HF and MF contributions simulated independently. According to the preliminary analysis of Mg impedance diagram and the literature on magnesium corrosion [START_REF] Bland | Effect of Crystallographic Orientation on the Corrosion of Magnesium: Comparison of Film Forming and Bare Crystal Facets using Electrochemical Impedance and Raman Spectroscopy[END_REF][START_REF] King | Accurate Electrochemical Measurement of Magnesium Corrosion Rates; a Combined Impedance , Mass-Loss and Hydrogen Collection Study[END_REF][START_REF] Curioni | Correlation between electrochemical impedance measurements and corrosion rate of magnesium investigated by real-time hydrogen measurement and optical imaging[END_REF], an equivalent circuit with two parallel CPE was used to model EIS data. Initially designed for the corrosion of a metal covered with organic coating, the chosen model appeared to be relevant and widely used for corrosion of bare metals [START_REF] King | Accurate Electrochemical Measurement of Magnesium Corrosion Rates; a Combined Impedance , Mass-Loss and Hydrogen Collection Study[END_REF][START_REF] Curioni | Correlation between electrochemical impedance measurements and corrosion rate of magnesium investigated by real-time hydrogen measurement and optical imaging[END_REF][START_REF] Córdoba-Torres | On the intrinsic coupling between constant-phase element parameters alpha and Q in electrochemical impedance spectroscopy[END_REF][START_REF] Ivanova | Electrochemical Impedance Spectroscopy (EIS): A Review Study of Basic Aspects of the Corrosion Mechanism Applied to Steels[END_REF]. The serial resistance Rs is associated to the electrolyte resistance and is measured when frequency tends toward infinite. The double-layer capacitance Cdl accounts for charge separation at the electrolyte-material interface. This capacitive behavior shortcuts other phenomena at HF. In order to avoid over-fitting, the different possible interfaces (oxide/electrolyte, hydroxide/electrolyte …)

were not distinguished; their overall contribution is described by a unique CPE. Since the porous oxide/hydroxide film is not perfectly insulating the surface, a film resistance Rf accounts for the current passing through this layer. The adsorption capacitance Cads is the pseudo-capacitive phenomena linked to adsorption of species on the metal surface and occurs in the MF range. The charge transfer resistance Rct at the metal/electrolyte interface is associated in parallel to Cads.

The fitted parameters obtained for all alloys and pure Mg using the equivalent circuit of figure 6 are reported in Table 2. Typical oxide layer capacity is in the order of few tens of µF/cm² [START_REF] Baril | An Impedance Investigation of the Mechanism of Pure Magnesium Corrosion in Sodium Sulfate Solutions[END_REF][START_REF] Jorcin | CPE analysis by local electrochemical impedance spectroscopy[END_REF][START_REF] Fontana | Corrosion Engineering[END_REF][START_REF] Perez | ELECTROCHEMISTRY AND CORROSION SCIENCE[END_REF],

such values are in good agreement with the Cdl measured in the present study (Table 2). The particularly high value of 172 µF/cm² obtained for Mg91Ni4Sm5 indicates that the effective reacting surface of this material has significantly increased during the initial immersion time of 2400s, which is most likely due to crevices and porous hydroxide layer formation. In contrast, the low double layer capacity Cdl of pure Mg (11 µF/cm²) corresponds to an oxide coating capacity [START_REF] Baril | An Impedance Investigation of the Mechanism of Pure Magnesium Corrosion in Sodium Sulfate Solutions[END_REF][START_REF] Jorcin | CPE analysis by local electrochemical impedance spectroscopy[END_REF], meaning that less corrosion occurred on this material surface during the immersion. Indeed, after this immersion period all LPSO are mostly covered of black corrosion products while pure Mg still shows silvery parts which is the color of air formed MgO. This oxide layer covering pure Mg material also explains its higher value of Rf compared to those of LPSO that are covered by Mg(OH)2 porous layer. The total resistance measured by EIS for LPSO compounds are all very close (Rs + Rf + Rct ≈ 11 ohm.cm²) which makes comparisons thorny. However, it can be noticed that Mg91Ni4Sm5 is the only alloy with such low Rct and high Cads (Table 2), which indicates a strong interaction affinity between metallic surface and ions from the solution. As corrosion is a faradaic process, there is a direct link between the quantity n(H2) (mol/cm²) of H2 produced and the corrosion rate of the alloy :

Potentiodynamic polarization

𝑛(𝐻2) = 𝐽𝑐𝑜𝑟𝑟 * 𝑡 𝐹 * 𝑧 (3)
with Jcorr the corrosion current (A/cm²), t the time (s), F the Farraday constant (96485 C/mol) and z the valance of the species (2). Determining Jcorr for a material is thus a direct characterization of its ability to produce hydrogen by hydrolysis.

Linear sweep voltametry experiment allowed the construction ot Tafel plots (Log|J| = f(E))

shown on figure 9. Tafel extrapolation method (figure 9A) was used to determine the corrosion currents densities Jcorr listed in table 3. Current measured under anodic polarization is characteristic of the dissolution kinetic of the metal (i.e. M oxidizes into M n+ + ne -). In the case of Mg, interpretation of this measurement is not that trivial because of the anomalous Hydrogen Evolution (HE) observed on anodically polarized surfaces (also known as Negative Difference Effect) [START_REF] Esmaily | Fundamentals and advances in magnesium alloy corrosion[END_REF][START_REF] Atrens | Advances in Mg corrosion and research suggestions[END_REF][START_REF] Li | Anodic hydrogen evolution on Mg[END_REF][START_REF] Esmaily | Anomalous hydrogen evolution on AZ31, AZ61 and AZ91 magnesium alloys in unbuffered sodium chloride solution[END_REF]. Even though not predicted by conventional corrosion theory, Mg undergoes an increasing HE with increasing potential toward positive values. For this reason, anodic current is underestimated since part of electrons recombine to form H2 and do not flow in external circuit. Even after IR drop correction, the alloys tend to exhibit higher current densities under anodic polarization than under cathodic. This may be partly due to a non neglectable part of the surface being covered by H2 bubbles during cathodic polarization.

Anomalous HE and partial surface coverage by H2 bubbles may explain the discrepancies reported for corrosion current values measured by electrochemical means and mass loss and hydrogen collection methods. [START_REF] Esmaily | Fundamentals and advances in magnesium alloy corrosion[END_REF][START_REF] Pardo | Electrochemical estimation of the corrosion rate of magnesium/aluminium alloys[END_REF]. Despite these points, corrosion current densities reported in Table 3 are in good agreement with hydrolysis experiments since alloys with higher corrosion currents also have higher hydrolysis kinetics. Indeed, while all above-mentionned bias may lead to underestimation of the corrosion current density values determined by Tafel method, the comparisons between alloys is possible since they all undergo these phenomena.

Corrosion current density of pure Mg by this method has already been reported in the same conditions [START_REF] Alasmar | Hydrogen generation from Mg-NdNiMg 15 composites by hydrolysis reaction[END_REF] and is three times inferior to Mg91Cu4Y5, the LPSO with lowest corrosion rate (table 3). The different corrosion rates (and hydrolysis kinetics) observed between the three Mg91Ni4RE5 alloys may be explained by materials characterizations. It has indeed been demonstrated that phase repartition has a major role on corrosion rates [START_REF] Alasmar | Hydrogen generation from Mg-NdNiMg 15 composites by hydrolysis reaction[END_REF], the galvanic corrosion being favored in alloys with more intricate phases. Mg91Ni4Y5 has a microstructure with larger Mg phases domains (figure 2B) compared to Mg91Ni4Gd5 and Mg91Ni4Sm5 (figures 2C and 2D), its corrosion rate and hydrolysis kinetic are also smaller. Even though visible on figures 2C and 2D, microstructural differences between Mg91Ni4Gd5 and Mg91Ni4Sm5 are more difficult to analyze. The actual chemical composition of the LPSO may explain the slight corrosion kinetics between these two alloys. EDXS analyses of LPSO given in table 1 show that Sm-14H (LPSO in Mg91Ni4Sm5) contains a higher Ni proportion compared to Gd-14H (LPSO in Mg91Ni4Gd5). This LPSO nobility difference may lead to both stronger micro galvanic coupling with Mg and higher inner LPSO corrosion (stronger coupling between Mg and Mg-RE-TM layers within the LPSO).

Electrochemical measurements using Rotating Disk Electrode (RDE)

Using Rotating Disc Electrode (RDE) allows controlling the convection effect in the electrolyte and leads to electrochemical measurements on a rotating surface that helps the diffusion of species in the electrolyte near the reacting surface. If the corrosion mechanism is limited by mass transport, the inverse of the measured current should evolve as the inverse of the squared root of the electrode rotation speed [START_REF] Amira | Synthesis, characterization, and electrochemical behavior of a new Nd1.9Sr0.1Ni0.9Co0.1O4 ± Δ material as electrocatalyst for the oxygen reduction reaction[END_REF][START_REF] Amira | Carbon-based lanthanum nickelate material La2-x-yNdxPryNiO4+Δ (x = 0, 0.3, and 0.5; y = 0 and 0.2) as a bifunctional electrocatalyst for oxygen reduction in alkaline media[END_REF]. However, according to Tafel plots reported on figure 10 for Mg91Ni4Y5 at 25°C in 0.6 mol/L NaCl solution it seems that the rotation speed has no effect on the polarization curves recorded on Mg91Ni4Y5 electrode material. Indeed, anodic polarization domain (E> -1.2V/ECS) of all rotation speed (0 to 1500 rpm) superimposes. On the cathodic side, the differences of current density values are assigned to hydrogen bubbles covering (randomly) the surface from electrolyte. This is also the reason why these measurements are noisy. The same experiment has been done on Mg91Ni4Gd5 with rotation speed in the range 2000 to 5000 rpm and similar results were obtained. Electrochemical characterization using RDE confirms that mass transport (i.e. diffusion) is not the limiting step in the corrosion process of LPSO and cannot explain the difference in corrosion current densities between the compounds.

Effect of immersion time

Figure 11A shows the OCP evolution for Mg91Cu4Y5 with corrosion currents estimated by Tafel method at different immersion times. The associated current density recorded at t=500 ; 2400 and 4000s have been reported on the figure 11A. In addition, EIS diagrams have also been measured at the same selected immersion times (see figure 11B). It can be noticed that the corrosion current density values tend to increase with immersion time for two reasons :

(i)

The longer the sample is immersed the less passivated regions remain and the more regions start to corrode (surface is "activated"). Electrochemical measurements analytically enlightened the different hydrolysis properties observed on powders. Potentiodynamic polarization (i.e. Jcorr determination) emphasized the kinetic differences recorded between the alloys during hydrolysis experiment. OCP measurement showed that thermodynamic stability of the systems is more dependent on the presence of transition metal than that of rare earth. The electrochemical impedance spectroscopy did not show major corrosion mechanism differences between Mg and LPSO (i.e. same shape of Nyquist diagram) but an important difference of impedance magnitude, which showed the very small corrosion resistance of LPSO alloys. When analyzing corrosion on bulk material, corrosion rates increases and impedance diminish because the reacting surface increases. However, in hydrolysis experiment on powders, the kinetic is maximal in the initial stage and decreases as available reactant is consumed. Despite these different testing conditions, the results obtained with electrochemical tests are coherent with hydrolysis experiment.

Conclusion

Four alloys with high 14H-LPSO content (Mg91Cu4Y5 ; Mg91Ni4Y5 ; Mg91Ni4Gd5 ; Mg91Ni4Sm5) were synthesized from different elements (Mg + Ni or Cu + Y, Gd or Sm) in order to compare their electrochemical properties with their hydrogen generation performances as powders. The main results of this study are summarized as follows:

-Good correlation was observed between hydrolysis experiments and electrochemical measurements, highlighting the relevance of surface characterization on bulk materials for understanding hydrogen generation by hydrolysis with powder.

-14H LPSO structures have high corrosion current densities and low impedance values compared to pure Mg, reflecting their strong ability to corrode in seawater. These results explain the good hydrogen generation kinetics obtained from powders of these materials.

-Changing the nature of the rare earth (Y, Gd or Sm) and of the transition element (Ni or Cu) in the alloy resulted in both microstructural and LPSO chemical composition differences.

-These differences in turn influenced the corrosion rates determined by Tafel method as the alloys with most intricate Mg-LPSO phases and LPSO with higher Ni content happened to corrode faster.

-The double layer capacity Cdl, measured by impedance spectroscopy after a given immersion time informed, in an indirect manner, on the alloy ability to corrode by informing on the actively reacting surface.

-14H-LPSO corrosion rate is not controlled by diffusion processes.

Depicting further the fundamental corrosion mechanisms would be greatly helped with lower scale characterization of LPSO.

Figure 2 :

 2 Figure 2: BSE-SEM micrographs of : A-Mg91Cu4Y5, B-Mg91Ni4Y5, C-Mg91Ni4Gd5 and D-Mg91Ni4Sm5. The clear zones (majoritarian) are LPSO phases, the dark zones are Mg phase and the small bright areas are Mg2TM or Y2O3.

Figure 1 :

 1 Figure 1 : X-ray diffraction patterns on powders filed from bulk alloys.

  kinetics and yields) comparable with an optimized mixture. Hydrolysis experiment confirms the strong potential of LPSO materials as efficient hydrogen production materials. Mg91Cu4Y5 has a much lower H2 generation kinetic compared to alloys containing Ni as TM. For this alloy, two stages with different gas generation speeds can be distinguished on figure3. The transition between the two kinetic stages that occurs when about 30%at. of the alloy is consumed could be linked to the prior reaction of Mg phase ( which would be consistent with SEM estimation). The alloys with Ni as TM also present different kinetics depending on the RE involved.The hydrolysis reactions of the different alloys showed very different features depending on the TM and RE involved in the LPSO phase. Hydrogen generation properties of the alloys will be further investigated in the next sections with electrochemical characterization of bulk material's surface.

Figure 3 :

 3 Figure 3 : Hydrolysis at 25°C in 0.6 mol/L NaCl solution with approximately 5% accuracy.

Figure 4 :

 4 Figure 4 : Open Circuit Potentials (OCP) versus time at 25°C in 0.6 mol/L NaCl solution with approximately 1% accuracy.

  stable OCP, same surface condition …) cannot be held for the long time required by low frequencies measurements. Thus, Jcorr could not be determined by EIS. The Nyquist diagram of Mg of figure6Aand the corresponding Bode plot of figure 6B both show two well-separated relaxation frequencies, in the high frequency (HF) range and in the medium frequency range (MF). In contrast to pure Mg, LPSO bode plots (figure6B) are flattened at HF. The DRT calculation method allowed to evidence the relaxation times from EIS data[START_REF] Zeimetz | Using the Distribution of Relaxation Times for analyzing the Kramers-Kronig Relations[END_REF][START_REF] Ivers-Tiffée | Evaluation of electrochemical impedance spectra by the distribution of relaxation times[END_REF]. The DRT spectra of figure 7 confirm the presence of two relaxation times in the case of Mg (at ca. 40 Hz and 1000 Hz) while for LPSO a multiplicity of peaks with lower intensity appears (illustrated with Mg91Ni4Y5 on figure7). As LPSO containing alloys corrode more than pure Mg, their impedance magnitude is smaller (figure 6A compares pure Mg and Mg91Cu4Y5 Nyquist plots) and different time constants may appear because of the irregularity of the surface, roughened by corrosion. Surface irregularities lead to impedance distribution and result in a strong convolution of the HF and MF peaks that cannot be distinguished in the case of alloys with LPSO (figure5B). The multiple DRT peaks of LPSO should not be attributed to different mechanisms but to spreading of relaxation time due to irregular surface. The EIS data of alloys with LPSO is thus fitted with the same model as pure Mg, with only two contributions (HF and MF). Constant Phase Elements (CPE) have been used to account for the distributed impedance responses by introduction of α exponent (0 < α < 1). Capacity values discussed below are effective capacities calculated as follows:

Figure 5 :

 5 Figure 5: Impedance spectra and Kramers-Kronig (KK) tests for A-Real and B-Imaginary parts.

Figure 6 :

 6 Figure 6 : A-Nyquist plot of Mg and Mg91Cu4Y5 and B-Phase bode plots for Mg and LPSO alloys.

Figure 7 :

 7 Figure 7 : DRT plots of Mg (top) and Mg91Ni4Y5 (bottom) calculated with Tikhonov regularization method (λ=10 -2 ).

Figure 8 :

 8 Figure 8 : Nyquist plot of Mg91Ni4Gd5 and equivalent circuit used for fitting EIS data.

Figure 9 :

 9 Figure 9 : A-Tafel extrapolation method (Mg91Ni4Gd5) and B-Polarization curves at 25°C in 0.6 mol/L NaCl solution.

Figure 10 :

 10 Figure 10 : Polarization curves of Mg91Ni4Y5 at 25°C in 0.6 mol/L NaCl solution for different rotation speeds.

  (ii)During the OCP immersion where corrosion propagates, the actual surface of the sample increases with forming pits and crevices. The surface by which recorded current is normalised is the geometrical surface measured prior to the test and is necessarily smaller than the real surface.

Figure 11 :

 11 Figure 11 : Effect of immersion time in 0.6 mol/L NaCl solution on Mg91Cu4Y5 : A-OCP and corrosion currents and B-Nyquist plot of EIS data.

Figure 12 :

 12 Figure 12 : Evolution of Mg91Cu4Y5 surface with immersion time : A-500s (BSE), B-5000s (BSE) and C-5000s (SE).

Table 1 :

 1 Studied alloys and their corresponding characterization features.

Table 2 :

 2 Fitting results (with approximately 5% accuracy) for EIS measurement performed in 0.6 mol/L NaCl after

Table 3 :

 3 Values obtained from OCP and polarization measurements at 25°C in 0.6 mol/L NaCl solution, average on at least 3 measures and standard deviation in brackets.

Table 4 :

 4 Fitting results (with approximately 5% accuracy) for EIS measurement performed in 0.6 mol/L NaCl at different immersion times for Mg91Cu4Y5.

Acknowledgment

This work was financially supported by Direction Générale de l'Armement and Région Nouvelle Aquitaine through a grant to Manuel Legrée.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.


Travis West

Pitch Fingering Systems and the Search for Perfection

Keywords: pitch fingering systems, optimization, brute force search, DMI design CCS Concepts, Applied computing → Arts and Humanities → Sound and music computing

In the search for better designs, one tool is to specify the design problem such that globally optimal solutions can be found. I present a design process using this approach, its strengths and limitations, and its results in the form of four pitch fingering systems that are ergonomic, simple, and symmetric. In hindsight, I emphasize the subjectivity of the design process, despite its reliance on objective quantitative assessment.

you may reasonably constrain the design space to those fingering systems used on saxophones. Fair enough. For other individuals, different criteria will take priority.

For instance, (according to Marcelo Wanderley 1 [START_REF] Wanderley | Interaction Musicien-Instrument: Application Au Contrôle Gestuel de La Synthèse Sonore[END_REF]), Gerard Beauregard developed the fingering system of the Bleauregard instrument with "[the goal to] propose a fingering that could be more easily learned by beginners." Martin Marier developed the fingering system for the Sponge in search of an idiomatic approach for the instrument [START_REF] Marier | Designing Mappings for the Sponge: Towards Spongistic Music[END_REF]. Performers with physical disabilities have their own unique set of constraints; Larsen and colleagues provide an overview of designs for this population in their 2016 article [START_REF] Larsen | The Prospects of Musical Instruments For People with Physical Disabilities[END_REF].

Obviously there is no ultimate pitch fingering system that is perfect for all people. But for any given set of goals, it may be a useful exercise to quantitatively describe the important evaluation criteria. This may be seen as an early alternative to empirical evaluation, providing one form of evidence of the merits of a design. In this discussion, I will consider some strengths and weakness of this approach; final judgement is left to the reader.

In the rest of this paper, I recount the search for a perfect pitch fingering system.

Starting with a review of related work, especially the work of David Hartvigsen that lays the foundation for my approach, I describe my design principles and evaluation methodology, provide a narrative account of the design process, and summarize the final design decisions in the clarity of hindsight. Discussion and conclusions consider the merits and limitations of the design process and possible directions for future work.

Related Work

Musical control of pitch is an important topic that can be approached in numerous ways. Goudard and colleagues previously presented a good overview of approaches and concerns when performing monodic pitch [START_REF] Goudard | On The Playing of Monodic Pitch in Digital Music Instruments[END_REF]. However, the main features of the present discussion are not prevalent in the literature on digital musical instruments: enumeration of all possible designs, quantitative modelling of the design problem, and optimization. Although enumeration has been approached in the context of isomorphic keyboards by Steven Maupin and colleagues [START_REF] Maupin | Isomorphic Tessellations for Musical Keyboards[END_REF], its use for other sorts of pitch fingering systems has not to my knowledge previously been attempted, likely due to the seemingly intractable quantity of possibilities in these contexts.

More similar work, employing quantitative models and optimization to propose designs, is found in the context of text-entry keyboards rather than musical ones. Since the ubiquitous QWERTY keyboard standard was not designed with ergonomics, typing speed, or error minimization in mind, alternative keyboard arrangements have been proposed frequently since at least as early as the 1930s, with the work of August Dvorak [START_REF] Dvorak | Typewriting Behavior: Psychology Applied to Teaching and Learning Typewriting[END_REF]. More recently, various heuristic models of advantageous keyboard design have been proposed, and optimization algorithms ranging from simulated annealing [START_REF] Light | Typewriter Keyboards via Simulated Annealing[END_REF] to ant colony optimization [START_REF] Wagner | Ergonomic Modelling and Optimization of the Keyboard Arrangement with an Ant Colony Algorithm[END_REF] have been applied to find approximate solutions. The approach to design adopted in these works bears strong similarity to the present work; the designers are focused on ergonomic efficiency, speed, and error reduction, and employ a quantitative cost function to try to find an optimal solution to a heuristic mathematical model of the design problem. The main difference of these works is that the design space of text-entry keyboards is significantly larger than the design space explored here, making exact search impractical.

David Hartvigsen's Optimal Instruments

Most similar to the current discussion is the work of David Hartvigsen [10][11], whose approach to the design of pitch fingering systems provided the initial motivation for me to take this line of inquiry.

Hartvigsen considers the pitch fingering systems used to play one discrete pitch at a time, where each pitch is selected by pressing a certain combination of buttons, as on a woodwind instrument. His stated goal is to develop an instrument that is optimally easy to play. Fingerings can be represented by a sequence in binary, where a 1 corresponds to a button being pressed and a 0 to released (or equivalently vice versa). Hartvigsen mainly uses 4 buttons, so fingerings look like 0010, 1101, 1001, and so on, for example. Hartvigsen quantifies the difficulty of moving from one fingering to another by the hamming distance between the two fingerings. For fingerings and , in a system with buttons, the hamming distance is given by the following equation:

Where refers to the boolean value of the button with index in the fingering . So for example, from 1001 to 1101 is a hamming distance of 1 (only the second button changed), or from 1101 to 0010 is a hamming distance of 4 (all four buttons changed)-.

Given some sequence of notes , the cost of playing the sequence with an instrument , where gives the fingering of the note , can be calculated by

f 1 f 2 B (1) dist(f , f ) = 1 2 ||f (b) - b ∑ B 1 f (b)|| 2 f(b) b f {s , s , ..., s } 0 1 q M M (s) s
accumulating the cost of each note transition in terms of the hamming distance:

For a collection of sequences, the total cost is the sum of the cost to play each sequence:

Thus the difficulty associated with performing a certain pitch fingering system is defined relative to the collection of pitch sequences that are used to evaluate it.

Hartvigsen's approach can be thought of as having two main components: a function that evaluates the difficulty of transitioning from one fingering to another (hamming distance), and the selection or curation of a set of pitch sequences that embody the kind of fingering transitions that a performer is likely to encounter (e.g. common scales [START_REF] Hartvigsen | Optimal Electronic Musical Instruments[END_REF]). The total difficulty of performing with the system is a kind of product between fingering-transition cost and fingering-transition likelihood.

Evaluation Procedure

My own evaluation procedure is roughly analogous to David Hartvigsen's. A set of initial constraints are adopted that reduce the size of the search space and improve the simplicity of the resulting pitch fingering systems. A cost function is then used to operationalize the notion of performance easiness/difficulty in terms of the likelihood of performing a movement, and the difficulty of performing that movement. Unlike

Hartvigsen, I develop a more compact model of movement likelihood that allows systems to be evaluated in constant time with respect to the size of the pitch sequence dataset used, and I use empirical measurements of movement difficulty in order to incorporate physiological concerns in the model. Furthermore, due to a different set of initial constraints, the design space investigated here is small enough that a brute force search is readily accessible by using parallel computing techniques.

Initial Constraints

Hartvigsen considers systems in which four buttons are used to select pitch class and a separate set of buttons is used to determine octave. The number of possible pitch fingering systems satisfying this constraint is very large (described in appendix B below), and Hartvigsen writes [START_REF] Hartvigsen | Fingering Systems for Electronic Musical Instruments[END_REF] "enumerating the mappings to find the best for a [START_REF] Wanderley | Interaction Musicien-Instrument: Application Au Contrôle Gestuel de La Synthèse Sonore[END_REF] cost (s , s , ..., s ) =

M 0 1 q dist(M(s ), M(s )) i=0 ∑ q-1 i i+1
(3)

T otalCost (C) = M cost (S) S∈C ∑ M
particular collection of note sequences is not possible." For my purposes, I wanted to examine a design problem in which a globally optimal solution could be determined without complex proof, such as by an exhaustive search where every possible system is enumerated.

I therefore chose to examine systems with 12 buttons, where each button is assigned a pitch class 2 . Pitches are selected by pressing the assigned button, and if multiple buttons are pressed then multiple pitches are played. A separate mechanism is used to determine octave, described below. Although it is possible to play multiple pitches on such a keyboard, in this process I have only optimized for playing one note at a time, as in a melody.

Additional Benefits

As well as decreasing the size of the search space, using 4 or 12 buttons to select pitch class with a separate control over octave provides an additional benefit: in these systems, the number of different fingerings required to play a given sequence in a different key or octave is minimized. This should be advantageous when learning to play these systems.

A sequence of pitches can be identified irrespective of octave and transposition by converting it to a sequence of interval classes. On 4 or 12 button systems as described, there are exactly 12 ways to perform any sequence of interval-classes (one for each transposition). This is a meaningful reduction compared to most instruments, where, as well as differing by transposition, fingering may also differ depending on octave or other factors such as alternate fingerings.

Selecting pitch class separately from octave provides a marked reduction in possible "things" that can be played, from strong octave-equivalence and lack of alternate fingerings. In principle this should make a pitch fingering system easier to learn, since there is simply less to learn.

Prototype Interface

Having constrained the design space to instruments with 12 buttons as described, I wished to firmly ground the design exploration in a practical context. For this purpose, I developed the simplest possible prototype interface I could imagine. The interface consists of a Planck ortholinear text-input keyboard 3 with an orientation sensor attached to the bottom. 12 buttons are chosen to control the interface-six per hand, in four columns and three rows, highlighted in the schematic diagram below. This layout of buttons influenced several later design decisions, so it is important to note. Note also that throughout the paper, images are represented as if looking in a mirror to facilitate the reader "playing along" with their imagination, as if they were holding the prototype themselves.

Q W E R T " ' Y U I O P A S D F G H J K L : ; Z X C V B N M < , > . ? /
Image 1 Schematic of the prototype as seen in a mirror when holding the device to play.

The keys used for performing are highlighted. The orientation sensor is seen attached to the bottom of the device.

The octave is controlled based on the inclination of the keyboard as illustrated in the following diagram and in the demonstrations in later sections and appendix A.

Image 2 Image of the prototype device showing how the device is held when performing. The instrument is stabilized by pressure from the palms of the hands pressing in on the sides of the device.

Fingering systems in the remainder of the paper are notated based on the layout of the buttons as seen by the player when looking in a mirror:

Movement Likelihood

One half of the cost function used to evaluate potential pitch fingering systems is based on the likelihood that a movement from one button to another will be required 4 .

Hartvigsen models this probability distribution by directly accumulating his movement Image 3 Illustration of the octave control mechanism. Octave is controlled by tilting the device up and down.

difficulty metric over a set of sequences of pitches. This means that the time to compute the overall cost for each system grows in proportion with the size of the pitch sequence dataset.

I adopt a simplified model of movement likelihood based on the strong octaveequivalence of the pitch-class-oriented systems considered, and the quality of 12 tone equal temperament tuning that sequences transposed by an interval are in some sense "the same". Rather than directly storing the pitch sequences of interest, we can instead collect sequences of interval classes derived from the pitch sequence dataset.

By simply counting the relative frequency of each interval class, we end up with a compact model of how likely you are to play any given interval class.

Assuming two pitches and are labelled with their MIDI note number, the interval class between them can be calculated as follows:

interval_class = mod(b, 12) -mod(a, 12) if interval < 0: interval_class = interval + 12 elif interval > 6: interval_class = 12 -interval
The interval class corresponds to the smallest interval between two pitch classes, where the pitch class of a pitch is given by . For example a minor third (3 semitones, e.g. A to the next C) is in the same interval class as a major sixth ( 9semitones, e.g. C to the next A) or a minor tenth (15 semitones, e.g. A to after the next C). There are 6 interval classes in 12 tone equal temperament tuning.

The overall likelihood of each interval class can be accumulated as follows:

interval_likelihood = [0,0,0,0,0,0,0] N = len(pitch_sequence) for n in range(1, N): a = pitch_sequence[n-1] b = pitch_sequence[n] interval_likelihood[interval_class(b -a)] += 1 / (N-1)
The final model consist of just six numbers, regardless of the size of the original dataset. If the probabilities are stored in a list, as above, then the probability of an interval class with semitones is given by . So for instance, the probability of repeating a note is , the probability of playing a semitone is , a whole tone is (since a whole tone is equivalent to two semitones) and so on. probability [START_REF] Wanderley | Interaction Musicien-Instrument: Application Au Contrôle Gestuel de La Synthèse Sonore[END_REF] Since I am interested in performing jazz melodies, I calculated the relative frequency of interval classes found in the Weimar Jazz Database using the MeloSpy tool, both from the Jazzomat project [START_REF] Hartvigsen | Optimal Electronic Musical Instruments[END_REF]. The resultant interval class probabilities are thus:

interval_likelihood = [0.06, 0.27, 0.28, 0.18, 0.10, 0.09, 0.02]

Movement Cost

The other aspect of ease/difficulty is the operationalization of the cost of transitioning from one fingering to another. Here we wish to model the full complexity of the physiological system of the hand, wrist, arm, and so on, used to play the system.

Rather than attempt to model this intricate system, I propose to simply measure the difficulty of each fingering transition empirically. This is feasible thanks to our initial constraints on the design; because there are only twelve buttons, each associated with one pitch class, there are only 66 possible transitions from one fingering to another (

).

One approach to this measurement is to perform each of the 66 possible trills/tremolos between two fingerings for a certain amount of time (e.g. 10 seconds) and count the number of transitions. Based on the assumption that easier transitions will be easier to trill, a higher number of transitions should be measured in the same period of time.

Dividing the number of transitions by the measurement period gives the rate of transitions. The inverse of this rate provides the period of each transition; easier fingering transitions should have smaller periods.

This measurement procedure provides a very specific snapshot where the final measures incorporate all of the influences on the difficulty of performing the system, ranging from the physical layout and design of the keyboard to the particular physiology of the subject or subjects performing the instrument in the measurement trials. Every individual player could in principle provide their own measurements, such that the final pitch fingering system is optimal to their unique physical capabilities.

Obviously if I wished to develop a system that was optimal in a general way, numerous measurements would be required from a reasonable number of participants in a controlled experiment. However, since my goal is to develop a pitch fingering system primarily for myself, I only measured my own movement difficulty, when performing with the prototype interface described above. I accept that this limits the generality of the design results, but argue that it does not limit the merit of the design strategy.

12choose2 = 66

System Cost

Given the movement likelihood and movement cost as described, the overall cost associated with a pitch fingering system can be calculated as follows. First, a list of button transitions is initialized:

button_transitions = [(0,1), (0,2), (0,3), ..., (0,11), (1,2), (1,3), ..., (1,11), (2,3), ..., [START_REF] Wanderley | Interaction Musicien-Instrument: Application Au Contrôle Gestuel de La Synthèse Sonore[END_REF][START_REF] Hartvigsen | Fingering Systems for Electronic Musical Instruments[END_REF], ... [START_REF] Hartvigsen | Fingering Systems for Electronic Musical Instruments[END_REF][START_REF] Hartvigsen | Fingering Systems for Electronic Musical Instruments[END_REF]]

Each element of this list is a pair of numbers referring to buttons. Thus, for example, the transition labelled "(1,3)" represents the transition from the button labelled 1 to the button labelled 3. The button labels used here are shown in the section above related to the prototype interface, but any 0-based integral indexing system can be used as long as it is consistent. The list of movement costs may be initialized so that the cost of a given transition is located at the same address as the corresponding transition in the button transition list. In this way, the cost of the th transition is given by .

A pitch fingering system is represented by a permutation of the numbers 0 to 11. If the permutation is loaded into a list 5 , then the pitch associated with the button labelled in this system can be accessed by looking up the element with address in the list, . In other words, the order of pitches in the permutation allows pitches to be matched to buttons, since the buttons have a fixed order.

A series of transformations then leads from the button transition list to a list of interval classes, and then to the cost of the system. Using Python syntax, the procedure is approximately as follows: not evaluating the systems, but simply generating them. The systems are represented as permutations, which are normally generated one after another serially. Typically, you start with a trivial permutation ( ), and then apply some procedure to generate the next permutation, and then recursively apply the same procedure to generate all permutations. This approach is incompatible with parallel computation, so a somewhat more unusual method is adopted.

Permutations are generated from an integer representing the order of a particular permutation in the list of all permutations ordered lexicographically. This transformation is accomplished by first converting the integer to factoradic or factorial number system representation. The factoradic representation of the integer is then treated as a Lehmer code in order to transform it into a permutation. More details of this algorithm can be found in Keith Schwarz implementation in his Archive of Interesting Code [START_REF] Hartvigsen | Fingering Systems for Electronic Musical Instruments[END_REF].

This approach is easy to parallelize. Each parallel processor only needs to be told what index to start on and how many permutations to generate, and can independently generate its permutations, evaluate them, and write the results (including the system cost and the permutation representing the system) to shared memory where it can later be collected.

My full implementation using OpenCL can be found here. Running on a consumer gaming laptop, using both the CPU and GPU, this implementation is able to generate and evaluate all possible pitch fingering systems satisfying the constraints of my design space in less than 10 seconds. Running on only a consumer laptop CPU, the whole search takes less than 3 minutes.

The Search for Perfection

In this section I recount how the plan outlined above, developed relatively early in the design process, was inevitably forced to change when put into action. At some point early in the process, before the difficulty measurement procedure was implemented, I recorded a pitch fingering system that I found particularly promising; I may have developed it ad-hoc, or found it using the evaluation procedure described with ad-hoc cost numbers, before measurements were taken. After the difficulty measurement procedure and full cost function described above were implemented, compared to this early recorded system, I found the results disappointing. The systems found seemed random, disorganized, and were unpleasant to explore. An example follows, with its inversion. The inversion is used in the video demonstration, which shows a chromatic [0, 1, 2, [START_REF] Marier | Designing Mappings for the Sponge: Towards Spongistic Music[END_REF][START_REF] Larsen | The Prospects of Musical Instruments For People with Physical Disabilities[END_REF][START_REF] Goudard | On The Playing of Monodic Pitch in Digital Music Instruments[END_REF][START_REF] Maupin | Isomorphic Tessellations for Musical Keyboards[END_REF][START_REF] Dvorak | Typewriting Behavior: Psychology Applied to Teaching and Learning Typewriting[END_REF][START_REF] Light | Typewriter Keyboards via Simulated Annealing[END_REF][START_REF] Wagner | Ergonomic Modelling and Optimization of the Keyboard Arrangement with an Ant Colony Algorithm[END_REF][START_REF] Hartvigsen | Optimal Electronic Musical Instruments[END_REF][START_REF] Hartvigsen | Fingering Systems for Electronic Musical Instruments[END_REF] scale played on the prototype and as seen in the Pure Data synthesizer made for the prototype. I thought perhaps I could derive more smooth and symmetric solutions by smoothing out the movement cost data. I had noticed while performing trills that I was able to perform more quickly with my right hand than my left, and that certain finger movements were easier than others: alternating between hands was easier than alternating between fingers which was in turn easier than moving one finger back and forth between a pair of buttons in two different columns. I tried averaging the movement cost data across hands, then averaging the cost data within types of finger movements, but neither of these changes produced the kinds of results I was looking for.

Curious how my preferred early result compared to the optimal-but-unsatisfying solutions I had found since then, I ran it through the evaluation routine: it was rated almost as well as the "more optimal" solutions, worse by less than one percent.

0:00

Video 1 Video demonstration of one of the systems found before additional constraints were placed on the search space, used to play a chromatic scale. The slider in the Pure Data UI gives an indication of whether the octave control signal is above or below the most recently activated pitch. https://youtube.com/shorts/XVVUm4aOcr Y?feature=share

Considering this, and the low accuracy of my movement difficulty measurement protocol, I became concerned that very good potential solutions were being masked by measurement noise and an overzealous optimization algorithm. To combat this, I truncated the resolution of the cost evaluation to be more in line with resolution of my measurement protocol; this was achieved by multiplying the final cost by 100 and rounding to the nearest integer, effectively discarding decimal place values a thousandth and less. Behold, my favorite pitch fingering system from the beginning now emerged as one of the optimal solutions, along with several hundred other solutions.

Since the quantity of results was now too large, I considered how to quantitatively describe the properties of the pitch fingering system that I was attracted to, to reduce the solution set to systems with these desirable properties. The main difference I noted was that my preferred solution was symmetric. If a sequence of finger movements were performed with either hand, either hand would produce the same pitch-class sequence transposed by a minor third relative to each other. I added a step to the evaluation procedure that would severely penalize pitch fingering systems not possessing this symmetry property, so that they would be filtered out of the solution set.

Now my preferred fingering system was one of just a few optimal solutions. Considering all of the results at this step, I continued to significantly prefer the same solution. It differed from most other results in that buttons in the same column were always separated by a semitone from their neighboring buttons in the column, giving a sense of smoothness to this solution. I added this as a final constraint, reducing the number of results to two canonical pitch fingering systems, plus their inversions. The fourth system is used in the video demonstration that follows. The reader is encouraged to explore the other optimal solutions under each set of constraints using my implementation of the described procedure available on github (DocSunset/pitch-fingering-brute).

An Approximate Search

One of the apparent advantages of a design process driven by a quantitative evaluation function is the semblance of objective mathematical rigor. Claims can be made about a solution being globally optimal or "the best possible" (to quote myself from earlier in the paper), with the implication that these solutions were determined by an impartial and unerring computational system and cannot be rationally called into question. This appearance of rigor and rationality is deeply threatened by the honest messiness of the design process. It is clear in the above account that past a certain point in the process, I knew exactly what kind of solution I wanted to arrive at and deliberately modified the evaluation procedure to favor that result. When this became clear to me, I began to doubt the foundations of my design process. Is my science a sham?! No. Horror is not necessary. I realized that, although this kind of result picking would be gruesomely inappropriate in many contexts, in this and most design processes it is perfectly normal and necessary. After all, the goal of this design is to develop a pitch fingering system that is as close to perfect (for me personally) as possible. As such, the cost function is not the final arbiter: I am. The cost function (which I developed based on my own subjective goals and interests) is one indicator of value, but my own judgement as a designer is the final test. The fact that the results of early searches 1: the number of solutions can be reduced by keeping only one canonical form of solutions that are equivalent under transposition and inversion 2: optimal results are found by evaluating possible solutions using the method described above based on movement likelihood and movement difficulty, with final scores truncated to the hundredths place 3: examine only 12-button systems where each button is assigned a single pitch class, and octave is determined by a separate mechanism (in this case the inclination of the keyboard)

4: require movements played with one hand to perform the same interval-class sequence when played with the other hand 5: require buttons within the same column to be separated by a semitone from their neighbors were dissatisfying doesn't indicate that I am irrational, but that the evaluation procedure was not modelling important properties of the design problem.

Since the final results of my design process are almost entirely determined by the final set of constraints, it is reasonable to question whether the use of a cost function and search procedure is useful or necessary. I argue that the use of a cost function motivated several key design choices, such as the use of only 12 buttons, and the final selection of systems; with both the "smooth" and "symmetric" constraints imposed, there are four distinct canonical solutions (plus their inversions). The cost function favors two of them, leading to the selection of the final two systems (plus their two inversion). The cost function also retains its usefulness as a comparative metric; the final chosen systems score more favorably than 99.996% of possible systems, including 99.996% of symmetric systems. These solutions, along with 972 other distinct optimal solutions, are given the lowest possible score of 203 by the cost metric employed here.

For comparison, the average score is 218, and the worst score possible is 235; the scores are distributed with a roughly Gaussian bell-curve throughout the whole design space.

Adding additional constraints is a natural way to model important properties, in this case symmetry and smoothness. Another designer with different goals might choose other constraints. This is where the broad design strategy employed here begins to really shine. This approach allows designers to unambiguously describe the design space so that meaningful design choices can be more readily understood, communicated, and built upon. Iterative community-based development is facilitated by such a clear common language.

It is crucial to recognize that the overall design process is inherently subjective.

Although each iteration of the process may involve an impartial exhaustive search, the higher order process in which these global searches are situated is an approximate search. One thing this illuminates is that it is useful and necessary to question and modify the evaluation criteria. Another key insight is that at some point the designer must manually stop the search and accept a design that is "good enough", or the approximate search may never halt in its quest to find the "best possible".

Future Work

Although it initially appeared impractical to search 4-button-combination based pitch fingering systems by brute force, the performance of the parallel search implementation presented here suggests that it may be approachable. This distinct design space could be explored using the same methods presented here. Another distinct design space worth exploring might be an alternative button arrangement. The reader may yet imagine other design spaces.

The cost function presented here emphasizes melodic playing, one note at a time. It assumes that empirical measurement of trills can be generalized to longer sequences of notes. This assumption could be investigated further by examining empirical measurements of longer sequences of button transitions. Alternative methods could also be developed for quantitatively evaluating the easiness/difficulty of performing chord progressions, based on the same premise of likelihood multiplied by difficulty.

Such alternative cost functions could be used to re-evaluate the systems presented here, or to search for other interesting systems.

Finally, of most pressing interest, the practical merit of the presented pitch fingering systems could be deeply investigated by learning to play music using them, and developing a regular musical practice with them.

Conclusion

I present four new pitch fingering systems for an instrument with 12 buttons arranged in three rows and two columns, where each hand plays 6 buttons. The systems are well suited to performing melodies, they are highly ergonomic and symmetric, and very simple to learn. By using a separate control (such as the inclination of the keyboard) to determine the octave, there are exactly 12 ways to play any interval-class sequence-sometimes fewer if also considering the left/right-hand symmetry of the systems.

The design process makes use of an objective quantitative cost function based on movement likelihood derived from a jazz music dataset and movement cost determined by empirical measurement. The use of quantitative evaluation motivated important design choices, leading to results that are more ergonomic and symmetric than the vast majority of possible solutions.

Although the use of brute force search to find "globally optimal" solutions to the design problem implies impartial objectivity, the overall design process remains inherently subjective, and therefore approximate. During the process, additional constraints were added to improve the subjective qualities of the optimal solutions found. Ultimately, the four resulting pitch fingering systems are determined almost entirely by the design constraints; the subjective quality of these results thus depend mainly on whether the evaluator agrees with the design constraints.

The flexibility and un-ambiguity of the overall design approach is especially compelling. Future work may consider other design spaces, alternative cost functions, and novel constraints. Future work may also investigate the merits of the presented systems in musical practice.

example of how the chosen design constraints make the system potentially a little easier to learn.

Appendix B: Number of Possibilities

In either case, 4 buttons (as employed by Hartvigsen) or 12 (as explored here), if the button combinations or individual buttons that determine a pitch class are placed in a fixed order (the actual order is inconsequential), a pitch fingering system can be represented by a permutation of the 12 pitch classes. There are possible permutations, and this corresponds to the number of possible pitch fingering systems when using 12 buttons in the way described above.

For 4 button systems, an additional step is required: with 4 buttons, there are different button combinations, from which 12 must be selected to assign pitch classes to. This adds an additional choice with different possibilities, increasing the total number of possible systems to [START_REF] Hartvigsen | Fingering Systems for Electronic Musical Instruments[END_REF].

Footnotes Citations

  pitch_transitions = [ (permutation[n], permutation[m]) for n, m in button_transitions ] interval_classes = [ interval_class(b -a) for a, b in pitch_transitions ] costs = [ movement_likelihood[ic] * movement_cost[n] for n, ic in enumerate(interval_classes) ] system_cost = sum(costs)Parallel Search ProcedureIn order for an exhaustive brute force search of the whole design space to be feasible, parallel computing methods must be employed. The challenging part of this is actually

  . because I could not find a copy of Beauregard's own thesis: Beauregard, G. T.(1991). Rethinking the Design of Wind Controllers. Dartmouth College. ↩ 2. 12 tone equal temperament is implied. Other tuning systems could be approached similarly, although brute force search may not be feasible if the number of pitch classes is much larger. ↩ 3. "What is a Planck Keyboard?" https://youtu.be/bEPg8kk84gw ↩ 4. Since we are concerned with melody playing, movements from two or more buttons to two or more buttons, as for instance when moving from one chord to another, are not considered. ↩ 5. or any data structure providing a zero-based indexing operation ↩ 1. Cook, Perry. 2001. "Principles for Designing Computer Music Controllers." In Proceedings of the CHI'01 Workshop on New Interfaces for Musical Expression (NIME-01), 3-6. ↩

	12! = 479, 001, 600
	2 = 4	16
	16choose12 = 1280
	16choose12 * 12! = 871, 782, 912, 000

1

Acknowledgements

Thanks to David Hartvigsen for his work on the subject that provided my initial excitement for this topic. Thanks to my brother for getting into system administration and providing the initial motivation to try a brute force search on this problem. Thanks to my spouse for continued encouragement and for showing me at least one better solution when I thought none was possible.

Summary

The final design choices are summarized in the table below with some information on the size of the design space as different constraints are applied.

Ethics Statement

The prototype constructed in this project used only materials available on hand, minimizing electronic waste. No non-environmental ethical concerns are noted.

Appendix A: Major Scales

All 12 major scales are demonstrated in the following video. The current tonic is noted in the bottom left corner.

Notice that due to the symmetry between hands and within columns, the major scales can be grouped based on how they are played into three groups. Scales within the same group are played with the same sequence of columnar patterns. For example, C, Eb, Gb, and A major scales all start with the bottom and top button in one column, the middle and top button in the next column, only the middle button in the next column, and then the bottom and top button in the final column. This provides an early positive 0:00 Video 3 All major scales played on one of the final pitch fingering systems. https://youtu.be/nXexwuUR1xo