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Multistability of saxophone oscillation regimes and itsinfluence on sound production

Tom Colinot, Christophe Vergez, Philippe Guillemain, Jean-Baptiste Doc

To cite this version:Tom Colinot, Christophe Vergez, Philippe Guillemain, Jean-Baptiste Doc. Multistability of saxophoneoscillation regimes and its influence on sound production. Acta Acustica, Les Ulis, France : LesEditions de Physique, 2021, 5, pp.33. �10.1051/aacus/2021026�. �hal-03319784�

Multistability of saxophone oscillation regimes and its influence on

sound production

Tom Colinota, Christophe Vergeza, Philippe Guillemaina, Jean-Baptiste Docba Aix Marseille Univ, CNRS, Centrale Marseille, LMA, Marseille, France

b Laboratoire de Mecanique des Structures et des Systemes couples, Conservatoire Nationaldes Arts et Metiers, Paris, France

Summary1

The lowest fingerings of the saxophone can lead to2

several different regimes, depending on the musician’s3

control and the characteristics of the instrument. This4

is explored in this paper through a physical model5

of saxophone. The Harmonic Balance Method shows6

that for many combinations of musician control pa-7

rameters, several regimes are stable. Time-domain8

synthesis is used to show how different regimes can9

be selected through initial conditions and the initial10

evolution (rising time) of the blowing pressure, which11

is explained by studying the attraction basin of each12

stable regime. These considerations are then applied13

to study how the produced regimes are affected by14

properties of the resonator. The inharmonicity be-15

tween the first two resonances is varied in order to find16

the value leading to the best suppression of unwanted17

overblowing. Overlooking multistability in this de-18

scription can lead to biased conclusions. Results for19

all the lowest fingerings show that a slightly positive20

inharmonicity, close to that measured on a saxophone,21

leads to first register oscillations for the greatest range22

of control parameters. A perfect harmonicity (inte-23

ger ratio between the first two resonances) decreases24

first register production, which adds nuance to one25

of Benade’s guidelines for understanding sound pro-26

duction . Thus, this study provides some a posteriori27

insight into empirical design choices relative to the28

saxophone.29

1 Introduction30

A classic endeavor in musical acoustics consists in31

the systematic study of sound production features32

of a musical instrument. Early studies use an arti-33

ficial mouth to replace the musician (on the clarinet34

[45, 1, 60, 20] or the bassoon [28]) in order to bet-35

ter describe and understand the physical phenomena36

at play during sound production. Later on, artificial37

mouths have been robotized to provide a complete38

mapping of the instrument’s behavior, aiming at un-39

derstanding how the instrument must be acted on to40

produce different sounds [34, 42] or describing the41

influence of an acoustical parameter of the resonator 42

on sound production [23]. The objective of this last 43

study is shared by other works using a rather differ- 44

ent approach to systematic description of the instru- 45

ment’s behavior: using a physical model. Based on 46

oscillation thresholds for instance[27], some conclu- 47

sions can be drawn as to the acoustical characteristics 48

facilitating the production of sound. Numerical res- 49

olution of the model’s equations also constitute a re- 50

peatable way to map the produced sound to the char- 51

acteristics of the instruments, which has direct appli- 52

cations in instrument making [58, 25]. However, from 53

a mathematical perspective, as nonlinear dynamical 54

systems, wind instruments models often admit multi- 55

ple solutions for a given set of parameters. The ques- 56

tion of the stability of each of these solutions holds 57

great importance when aiming to describe or predict 58

the playability of an instrument based on its physical 59

model. But some important questions remain unan- 60

swered, even for ideal cases where the stability or in- 61

stability of each regime would be known. For instance, 62

which regime is produced if two regimes are stable for 63

the same control parameters combination? 64

In the case of such coexistence of stable solutions, 65

denominated multistability hereafter, the convergence 66

towards one or the other solution depends on the ini- 67

tial conditions. Indeed, each solution is associated 68

with a region of attraction or attraction basin, defined 69

as the region of the phase space where all initial con- 70

ditions converge towards this solution [52, 50]. For in- 71

stance, attraction basins are studied in walking mod- 72

els [51, 43], where the ‘walking’ (periodic) regime al- 73

most always coexists with a stable equilibrium, corre- 74

sponding to falling. In this case, describing attraction 75

basins informs control strategies in robotics [61, 62]. 76

Attraction basins are also studied for classic dynami- 77

cal oscillators, such as Chua’s circuit [44], with exper- 78

imental explorations of the attraction basins [48] as 79

well as numerical investigations [55]. As strongly non- 80

linear self-oscillating systems capable of multiple os- 81

cillating regimes, wind instrument models are among 82

the systems for which studying attraction basins can 83

shed light on their rich behavior and help understand 84

control strategies used by musicians. However, to our 85

Colinot et al., p. 2

knowledge, no study on the attraction basins of mu-86

sical instruments has been produced, although sev-87

eral studies explore their multistability. Experimen-88

tal work on the clarinet [37] and a numerical study of89

several idealized woodwind resonators [56] illustrate90

in particular the hysteresis between regimes, which91

is a consequence of multistability. On the flute, con-92

tinuation and synthesis have been used to investigate93

the hysteresis between regimes, notably depending on94

inharmonicity [57].95

Describing the attraction basins and comparing96

their sizes is expected to give information on which97

regime is most likely produced, assuming some prob-98

abilistic repartition of the initial conditions in the99

phase space [10]. However, an exhaustive description100

is almost impossible for a complete model of instru-101

ment, where the phase space is of very large dimen-102

sion. In such cases, attraction basins may be partially103

explored, based on a reduction of the phase space to104

one or two dimensions. For instance, the infinite-105

dimensional phase space of a delayed system can be106

partially described along two dimensions [59]. In the107

case of musical instruments, a reduction of the phase108

space is proposed in this paper, based on knowledge109

of typical musical scenarios. Throughout this work,110

the case of a model of saxophone is considered, and111

two scenarios are studied: transition from another es-112

tablished limit-cycle (scenario number 1), and first113

attack transient of a note, where the blowing pres-114

sure parameter goes from 0 to a certain final value115

(scenario number 2).116

Section 2 presents the physical saxophone model117

and the two numerical methods used to solve its118

equations: the Harmonic Balance Method and time-119

domain synthesis. Next, multistability is introduced120

by computing the bifurcation diagram with the har-121

monic balance method and continuation (Asymptotic122

Numerical Method) and exhibiting hysteresis cycles123

using time-domain synthesis in Section 3 (control sce-124

nario number 1). Then, in Section 4, a simple test-125

case of scenario number two is presented to study126

sound production, where the blowing pressure in-127

creases from 0 to its final value over different du-128

rations. We show how this duration can influence129

the final regime in multistability regions, and explain130

these results by presenting the attraction basin of each131

regime. Section 5 demonstrates how the awareness of132

multistability can lead to a better description of the133

behavior of the model. Depending on the inharmonic-134

ity of the resonator, the size of the control parame-135

ter regions where each regime appears in synthesis136

is described, taking into account multistability. This137

provides an interpretation to the inharmonicity value138

measured on the saxophone by showing that it corre-139

sponds to an optimum in periodic regime production.140

2 Numerical simulation frame- 141

work 142

2.1 Saxophone model 143

The saxophone model used in this study is comprised 144

of three main elements: a one degree-of-freedom os- 145

cillator representing the reed, a regularized nonlin- 146

ear characteristic giving the flow through the reed 147

channel, and a modal description of the measured 148

impedance of the resonator. Similar models solved 149

by time-domain synthesis (Section 2.2) are used in 150

conjunction with analytical techniques to study the 151

playing frequency [19] and spectrum [49] of clarinets, 152

as well as their radiated power with a comparison to 153

measurements [33]. The Harmonic Balance Method 154

(Section 2.3) can also be applied to this model to 155

study its dynamic behavior, for instance to quantify 156

the effect of neglecting reed contact [18]. 157

Dimensionless [28, 35] acoustical Kirchhoff vari-ables (p, u) are used in this work:

p =p

pM, u = Zc

u

pM, (1)

where the hat notation indicates the variable in phys-ical unit, pM is the static pressure necessary to closethe reed channel completely and Zc is the characteris-tic input impedance of the resonator for plane waves.Similarly, the reed displacement from equilibrium isgiven in dimensionless form

x =x

H(2)

where H is the distance between the reed and themouthpiece lay at rest. With this formalism, the reedchannel is closed when x ≤ −1. In this work, the onlytime-varying control parameter [60] is the dimension-less blowing pressure γ:

γ(t) =pm(t)

pM, (3)

where pm is the physical value of the pressure in the 158

mouth of the musician. We leave all other control pa- 159

rameters constant in order to limit the dimensionality 160

of the study. The values and names of the parameters 161

are summarized in Table 1 and detailed below through 162

the model description. Their values are drawn from 163

[46] for the reed parameters qr and ωr, from [13] for 164

the order of magnitude of the contact stiffness Kc. 165

2.1.1 The reed model 166

Following [13], the reed is modeled by a single de-gree of freedom oscillator including a nonlinear con-tact force accounting for the mouthpiece lay

x

ω2r

+qrωrx+ x = p− γ + Fc(x+ 1), (4)

Colinot et al., p. 3

Name Symbol Value

Blowing pressure γ VariableReed opening at rest ζ 0.6

Reed damping qr 1Reed eigenfrequency ωr 4224 rad.s−1

Mouthpiece lay stiffness Kc 100

Contact regularization η 10−3

Number of modes Nm 8Number of harmonics H 20

Table 1: Parameters of the numerical model: musi-cian control parameters γ and ζ, reed parameters qrand ωr, contact parameter Kc and parameters inher-ent to the numerical implementation η, Nm and H.

where the two parameters of the reed are its angulareigenfrequency ωr and its damping coefficient qr, andthe contact force is a function of the dimensionlessreed opening x+ 1 and is taken from [8],

Fc(x+ 1) = Kcmin(x+ 1, 0)2, (5)

where Kc = 100. Since x+ 1 is the distance betweenthe reed and the mouthpiece lay, Fc can be interpretedas a quadratic stiffness activated whenever the reedtouches the lay. The ramp function min(x + 1, 0) isregularized using a parameter η = 10−3 to avoid non-differentiability at x = −1 (reed closure)

min(x+ 1, 0) 'x+ 1−

√(x+ 1)2 + η

2. (6)

The regularization controlled by parameter η is nec-167

essary for the system to fit the quadratic formalism168

required by the implementation of the harmonic bal-169

ance method and asymptotic numerical continuation170

in the MANLAB software, which produces the bifur-171

cation diagrams of the present article. Although the172

parameter η is not necessary for the time-domain syn-173

thesis method to function, it is kept for comparison174

purposes.175

2.1.2 The reed channel176

The flow at the input of the resonator is deduced fromBernoulli’s law [2, 36] applied to the reed channel andturbulent mixing into the mouthpiece

u = ζmax(x+ 1, 0)sign(γ − p)√|γ − p| (7)

where ζ is the dimensionless control parameter ac-counting for reed opening at rest

ζ = ZcwH

√2

ρ, (8)

w being the effective width of the reed channel and ρthe density of the medium. We choose to ignore theflow due to the speed of the reed [19, 21] in the present

model, as it only has a small effect on the playingfrequency, which is not discussed here. The absolutevalue and ramp function in Eq. (7) are regularizedwith the same parameter η as in Eq. (6)

|γ − p| '√

(γ − p)2 + η, (9)

max(x+ 1, 0) 'x+ 1 +

√(x+ 1)2 + η

2. (10)

2.1.3 The resonator 177

The input impedance is used to represent the res-onator’s acoustical response. The dimensionless in-put impedance Z(ω) of a Buffet-Crampon Senzo altosaxophone is measured with the CTTM impedancesensor [22]. The saxophone is measured withoutits mouthpiece, placing the reference plane of theimpedance measurement at the input cross-section ofthe crook. A cylindrical tube is added by TransferMatrix Method [12] in post-processing to representthe mouthpiece, before using it in synthesis. Thelength of the cylinder is 60 mm and the radius is thesame as the input radius of the crook, 6 mm. Thetotal volume of the added cylinder approximately fitsthat of the missing cone apex, as per a classical aca-demic approximation [38]. In order to use this inputimpedance for the two numerical synthesis methodspresented above, it is decomposed into modes [19] sothat

Z(ω) =

Nm∑n=1

Cnjω − sn

+C∗n

jω − s∗n, (11)

where Cn and sn are the estimated complex modalresidues and poles [54] andNm is the number of modesretained in the simulation. In this paper Nm = 8modes are used. This translates into the time do-main by describing the pressure as a sum of complexmodal components pn, whose evolution depends onthe modal coefficients, such that

pn(t)− snpn(t) = Cnu(t), ∀n ∈ [1, Nm], (12)

p(t) = 2

Nm∑n=1

Re(pn(t)). (13)

The flow u in (12) is given by (7). Figure 1 displays 178

the measured impedance and the associated modal 179

reconstruction according to Eq. (11) for the D] fin- 180

gering used throughout the rest of this article. The 181

corresponding modal coefficients Cn and poles sn are 182

summarized in Table 2. Note that the choice of 183

a modal formalism over a direct resolution of partial 184

differential equations in the resonator is made here be- 185

cause of its lower computational cost and number of 186

variables, which facilitates large-scale numerical stud- 187

ies such as those presented in Section 5. Addition- 188

ally, the modal formalism involves a limited number 189

of parameters, which are directly tied to the acoustics 190

Colinot et al., p. 4

Residues Cn (unitless) Poles sn (rad.s−1)176.1 −17.59 + j1195470.5 −35.50 + j2483649.4 −65.30 + j3727328.7 −269.34 + j4405541.5 −70.32 + j5153224.9 −166.0 + j6177382.2 −94.49 + j6749409.9 −116.5 + j7987

Table 2: Modal parameters for the fingering D] of thealto saxophone.

of the resonator, instead of the complete geometri-191

cal description that a wave propagation model would192

require.193

Figure 1: Measured input impedance (solid) andmodal reconstruction (dashed) for the fingering D]of the alto saxophone.

2.2 Time-domain synthesis194

Equations (4), (7) and (12) are discretized using195

finite-difference approximations for the time-domain196

derivatives, following a discretization scheme first ap-197

plied to simple waveguides [29] and then adapted to198

a modal formalism [19]. The reader can find a de-199

tailed description of this discretization scheme in a200

recent document [16]. The sampling rate used in the201

simulation is Fs = 176400 Hz, four times higher than202

the standard audio sampling rate. Such a high sam-203

pling rate is required, given the chosen finite differ-204

ence scheme, to give precise results that match those205

obtained with the harmonic balance method.206

As an illustrative result, Figure 2 shows an exam-207

ple of the synthesized pressure signal and its spectro-208

gram. Note that the signal shown is a portion of the209

signal used in Figure 4, with control scenario num- 210

ber one. It corresponds to the first occurrence of 211

the oscillations at a blowing pressure value γ ' 0.45. 212

At this point, the system jumps from equilibrium to 213

the first register and passes through fleeting second 214

register and quasi periodic regimes. The spectrogram 215

(Figure 2 (b)) shows the second register to be the oc- 216

tave (double the fundamental frequency) of the first 217

register. The quasi-periodic portion of the signal dis- 218

plays amplitude variations, seen in the envelope of the 219

signal (Figure 2 (a)) and on the odd harmonic com- 220

ponents of the spectrogram. Quasi-periodic regimes 221

are well-known on saxophone-like instrument models, 222

documented for instance in [20, 24, 23]. 223

(a)

-60

-50

-40

-30

-20

-10

0

(b)

Figure 2: Time-domain synthesized pressure signal.(a) temporal envelope (black) and blowing pressureparameter γ (red). (b) Normalized spectrogram(dB) with regime names indicated (unstable ones be-tween parentheses) : equilibrium (Eq.), second reg-ister, quasi-periodic and first register. The signal isextracted from the same blowing pressure ramp as inFigure 4, between γ = 0.45 and γ = 0.51 (at first oc-currence of oscillation).

Colinot et al., p. 5

2.3 Harmonic balance and numerical224

continuation225

The harmonic balance method is an analysis methodparticularly adapted to the study of musical instru-ment models [26], since it focuses on periodic solu-tions, which correspond to the produced notes. As-suming periodicity of the solution allows expandingall variables in Fourier series [39, 47] up to order H,such that the i-th variable Xi expands to

Xi(t) 'H∑

h=−H

Xi,h exp(jhω0t), (14)

where there are 2H + 1 complex Fourier coefficientsXi,h per variable, and ω0 is the fundamental angularfrequency of the signal. In this work, the number ofharmonics retained is H = 20. Applying the methodto a differential system transforms it into an algebraicsystem of which the unknowns are the Fourier coeffi-cients of the variables and the solution’s fundamentalfrequency, of the form

R({Xi,h}, ω0) = 0. (15)

A numerical continuation method such as the Asymp-226

totic Numerical Method can then be applied to the re-227

sulting algebraic system [15, 31] to find how the solu-228

tion changes for other constant values of a chosen con-229

trol parameter, for instance the blowing pressure pa-230

rameter γ. More precisely, knowing a solution to the231

system for a given value of γ, the continuation method232

finds a solution for a slightly higher or lower values233

of γ, and therefore progressively maps out the solu-234

tions of the system for a given range of blowing pres-235

sures. In this work, simulations were carried out us-236

ing the MANLAB software (http://manlab.lma.cnrs-237

mrs.fr/). This yields the value of the Fourier coef-238

ficients of the oscillating solutions along several val-239

ues of a control parameter. The Fourier coefficients240

can then be used to reconstruct the time-domain solu-241

tions. This evolution can be summarized by a bifurca-242

tion diagram, which represents the variation of some243

descriptor, for instance the amplitude, of the solu-244

tions of the system with respect to the chosen control245

parameter. In addition, the stability of the solutions246

is determined using Floquet theory (for more details247

refer to [41, 5, 32]).248

3 Multistability249

This section presents the blowing pressure ranges250

where the model can produce each regime by study-251

ing their stability with the harmonic balance method.252

This result is summarized in the bifurcation dia-253

gram on which multistability zones appear as inter-254

vals where several regimes are stable. Signals are255

also synthesized with time-domain synthesis to ex- 256

hibit how multistability leads to hysteresis. The cor- 257

respondance between the two methods (the harmonic 258

balance method and time-domain synthesis) is also 259

checked. 260

3.1 Overlapping stability zones on the 261

bifurcation diagram 262

The bifurcation diagram is computed for the (writ-ten) low D] fingering of an alto saxophone. Thewritten D] produces the heard note F]3 at 185 Hz asits first register regime. This intermediate fingeringof the first register is chosen as test case because itexhibits both first and second register regimes, but nostable third register regime and few double two-stepphenomena [17]. Figure 3 shows the L2-norm of thepressure signal

||p||2 =1

T

√∫ T

0

p(t)2dt, (16)

where T is the period of the signal, and identifies 263

which regime each branch corresponds to. The blow- 264

ing pressure parameter γ spans the interval between 265

0 and 2. This bifurcation diagram contains branches 266

corresponding to the so-called equilibrium, where no 267

sound is produced, for the lowest and highest γ val- 268

ues. The equilibrium at low γ corresponds to the mu- 269

sician not blowing hard enough into the instrument 270

to obtain a sound, while at high γ equilibrium means 271

that the reed channel remains closed due to the large 272

pressure difference between mouth and mouthpiece. 273

For intermediate γ values, the first and second regis- 274

ter both appear. The first register is the fundamental 275

pitch obtained with a given fingering, and the sec- 276

ond register, sometimes referred to as overblowing, is 277

pitched one octave higher than the first register. Even 278

though the saxophone has an octave key facilitating 279

the production of the second register, musicians know 280

how to produce second register regimes without ac- 281

tivating it. Note that contrary to the clarinet, the 282

register key of the saxophone controls the two regis- 283

ter holes, opening one or the other depending on all 284

the pressed keys on the instrument. It is therefore 285

not surprising that both regimes appear on the same 286

fingering. In the present case, the γ interval between 287

0 and 2 contains all the studied limit cycles of the 288

model, and at its bounds, only the equilibrium solu- 289

tion exists and is stable. The diagram in Figure 3 290

displays several zones of coexistence between stable 291

regimes (i.e. multistability). These regions of coex- 292

istence are often bounded by bifurcation points, that 293

mark qualitative changes in the oscillating regimes. In 294

the present work, the system encounters several types 295

of bifurcations, which we define succinctly in terms of 296

the regime changes they correspond to. More formal 297

definitions and characterizations of these bifurcations, 298

Colinot et al., p. 6

notably in terms of critical values of the eigenvalues299

of the jacobian matrix of the system, can be found300

in [40, 7]. The Hopf bifurcation marks the emergence301

of an oscillating solution from equilibrium. The fold302

bifurcation corresponds to a stable and unstable solu-303

tion branch colliding and disappearing, which can be304

better seen on the bifurcation diagrams as limit points305

in the solution branches. Neimark-Sacker bifurcations306

correspond to a periodic regime becoming unstable307

and being replaced by a quasi-periodic regime. A de-308

generate case of the Neimark-Sacker bifurcations is309

the period-doubling bifurcation, where a periodic so-310

lution of halved frequency emerges from an oscillating311

solution. On saxophone models, period doubling bi-312

furcations transform a second register regime into a313

first register. As is discussed in the next paragraph314

for the saxophone, Hopf and fold bifurcations often315

delimit coexistence between the equilibrium and an316

oscillating regime, while Neimark-Sacker and period-317

doubling bifurcations mark the limits of multistability318

regions between two oscillating regimes.319

Figure 3: Bifurcation diagram obtained with har-monic balance method and numerical continuation:L2-norm of the acoustical pressure depending on theblowing pressure parameter γ for the low written D]fingering of an alto saxophone. Thick lines: stablesolution, thin lines: unstable solutions. Black: equi-librium, green: 1st register regimes, red: 2nd registerregimes. Multistability zones are shaded: light yel-low where 1st and 2nd register coexist, darker grayfor equilibrium and 1st register. Blue circles specifythe location of bifurcations. Vertical black lines corre-spond to those in Figure 6 (c), and (from left to right)to phase diagrams 7 (a), (b) and (c).

Starting from low blowing pressure values, the first320

coexistence zone appears between the first register321

and the equilibrium. It is delimited by the fold bi-322

furcation F1 of the first register around γ = 0.3 and 323

the inverse Hopf bifurcation H1 at γ = 0.4 where 324

the equilibrium becomes unstable. The second coexis- 325

tence zone is between first and second register, in the 326

interval where the second register is stable between 327

the Neimark-Sacker bifurcation NS1 and the period- 328

doubling bifurcation PD1 (respectively at γ = 0.66 329

and γ = 0.79). The Neimark-Sacker bifurcation NS1 330

mark the destabilization of the second register and 331

the emergence of a quasi-periodic regime (not repre- 332

sented here), sometimes called multiphonics by musi- 333

cians. The next coexistence zone occurs in the in- 334

terval between the two period-doubling bifurcations 335

PD1 and PD2 on the second register branch, where a 336

stable double two-step solution [17] emerges. This co- 337

existence zone is not shaded on the figure, as it could 338

represent less of a musical issue, since double two-step 339

regimes have roughly the same frequency as standard 340

first register regimes. The fourth coexistence zone is 341

more complicated: it starts between first and second 342

register at the period-doubling bifurcation PD2, and 343

then the equilibrium also becomes stable at the Hopf 344

bifurcation H4. The limit of the last coexistence zone 345

is made by the two fold bifurcations F2 and F3 where 346

the first and second register solutions cease to exist. 347

The diagram in Figure 3 shows that coexistence 348

zones between stable regimes span most of the range 349

in γ where oscillating solutions exist, including ar- 350

guably crucial γ values like the lowest for which an 351

oscillating regime exists. Multistability is not an iso- 352

lated phenomenon, but rather corresponds to the gen- 353

eral situation, at least for this fingering. 354

3.2 Time-domain synthesis with blow- 355

ing pressure ramps (control sce- 356

nario number one) 357

Once multistability zones are identified, time-domain 358

synthesis can be used to exhibit their role when play- 359

ing the instrument. One of the main phenomena mul- 360

tistability entails is hysteresis: for several values of the 361

blowing pressure, a different regime is produced de- 362

pending on whether the blowing pressure is increasing 363

or decreasing. Various multistable regimes are exhib- 364

ited using this method in [56] on woodwind models. 365

Figure 4 shows the hysteresis cycles obtained by using 366

ramps of γ: in control scenario number one, the pa- 367

rameter γ is progressively increased from 0 to 2 and 368

then decreased back to 0. Each one of the increasing 369

and decreasing phases of the synthesis has a duration 370

of 60 s. This duration was chosen after several trials, 371

sufficiently long to let stable regimes establish while 372

keeping a γ slope steep enough to limit dynamical 373

bifurcation delays [6]. 374

Figure 4 shows that the synthesized signal starts 375

from γ = 0 at equilibrium, its L2-norm being zero. 376

Then, at Hopf bifurcation H1, the equilibrium be- 377

comes unstable, which causes the system to start os- 378

Colinot et al., p. 7

cillating. At this point, the synthesis goes through a379

transient represented in Figure 2 (between t = 14.2 s380

and t = 14.8 s), shortly passing by unstable second381

register and quasi-periodic regimes before reaching382

the first register. Once the first register is estab-383

lished, the branch is followed all the way to extinc-384

tion (around γ = 1.75 on Figure 4), because the first385

register does not become unstable until the fold bifur-386

cation F3. At this point, the system returns to equi-387

librium until the highest value of the ramp, γ = 2.388

The blowing pressure γ then starts descreasing, and389

the system stays at equilibrium until Hopf bifurca-390

tion H3 is reached, for γ ' 1.05. There, the sys-391

tem jumps to the stable second register regime. The392

second register branch is followed until the period-393

doubling bifurcation PD2, where the system briefly394

follows the double two-step branch. The double two-395

step branch appears on the L2-norm very close to the396

second register branch, and looks like a small bump.397

The two branches cross again at period-doubling bi-398

furcation point PD1. Then, something rather surpris-399

ing occurs: the L2-norm of the time-synthesized signal400

seems to follow the second register branch of the bifur-401

cation diagram further than Neimarck-Sacker bifur-402

cation NS1, although the second register branch has403

become unstable. This is because the quasi-periodic404

regime emerging at NS1 is actually a stable attractor,405

and the associated L2 norm happens to be close to406

that of the second register. Branches of quasi-periodic407

regimes have not been computed so as not to clutter408

Figure 3, but note that it is possible with harmonic409

balance method and Manlab [31]. When the quasi-410

periodic regime becomes unstable (around γ = 0.55)411

the system jumps back onto the first register branch,412

which is followed until fold bifurcation F1, beyond413

which the stable equilibrium is the unique solution of414

the model. The path described precedently is highly415

hysteretic: the sequence of regimes produced for in-416

creasing and decreasing γ are very different. Actually,417

the two paths only coincide in three regions: the low-418

est and highest γ intervals, for which only the equilib-419

rium is stable, and a very small region around γ = 0.5420

where only the first register is stable.421

The hysteresis phenomenon observed here in time-422

domain synthesis can be interpreted as the first step423

in attraction basin description: once a certain stable424

regime is reached, it is followed until extinction or425

loss of stability, even when other regimes are simul-426

taneously stable. This confirms that a stable peri-427

odic regime is part of its own attraction basin. Re-428

constructing stable parts of the bifurcation diagram429

using time-domain synthesis and comparing them to430

those obtained using the harmonic balance method431

also provides validation for the numerical discretiza-432

tion scheme in this context. Here, it shows that time-433

domain synthesized signals are not perturbed by nu-434

merical artifact due to time discretization and can435

be trusted to describe properties of the model. Note436

F1

F3

Figure 4: L2-norm of the time-domain synthesis sig-nal (dark blue line) for control scenario number one,a ramp of the blowing pressure parameter γ from 0to 2 and down, overlaid with the bifurcation diagramof Figure 3 (thick stable solutions and thin unstablesolutions, of the first register in green and second reg-ister in red). Solid dark blue line indicates the firstpart of the signal (increasing γ) and dashed dark blueline the second (decreasing γ).

that even subtle details of the behavior such as the 437

tiny branch of double two-step solution between pe- 438

riod doubling PD1 and PD2 are found by the time- 439

domain synthesis. 440

This exploration of the blowing pressure space us- 441

ing a long ramp is very useful to exhibit the hysteresis 442

phenomenon, as well as test the coherence between 443

the two synthesis methods. However, this kind of 444

sound is extremely artificial and far from anything a 445

musician would use in everyday practice (provided it 446

is even possible for a musician to produce it). There- 447

fore, we frame the conditions of the rest of the study 448

so that they can be interpreted in terms of selection 449

of one regime over another. 450

4 Regime emergence in a mul- 451

tistable context 452

It is very likely that musicians learn to select between 453

coexisting stable regimes, adjusting their control so 454

that the established regime in a multistability region 455

is the one they desire. This idea provides the layout 456

for control scenario number two: we study the effect 457

of a parametrized transient control or initial condi- 458

tions on the established steady-state regime that fol- 459

lows when the control is constant. To provide an- 460

other way to interpret the selection process between 461

Colinot et al., p. 8

multistable regime, a third control scenario is used,462

where only initial conditions are varied and all con-463

trol parameters are constant.464

4.1 Control scenario number two: in-465

creasing blowing pressure466

One way to study the attraction basins more thor-oughly is to run many simulations with initial con-ditions spanning the whole phase space. However,since the considered model has a 2Nm+2 dimensionalphase space, a complete exploration is not possible.Moreover, many of the possible initial conditions areunlikely to be created by the musician. More inter-esting is the exploration of the regions of the phasespace that are crossed by the system when a givencontrol pattern is varied. Here, we focus on a mono-tonic increase of the blowing pressure γ at the attack:without using the tongue, the player starts blow-ing progressively harder into the instrument. Sucha scenario was proposed in [53]. Note that instru-mented mouthpiece measurements performed on thesaxophone, such as those presented in [30], often showa different profile including a pressure overshoot be-fore the apparition of the oscillations. However, inthe present study, we omit that overshoot so that thecontrol scenario is entirely defined by a single param-eter. In control scenario number two, the blowingpressure starts from 0 and rises up until stabilizing ata certain value γf , during a certain time determinedby the parameter τg. The temporal variation of γ isgiven by the

γ(t) =γf2

(1 + tanh

(t− 5τgτg

)), (17)

which is differentiable infinitely many times. Figure467

5 displays four examples of such transients.468

Other envelopes (sigmoid, sine branch) were tested,469

only causing small quantitative changes to the results.470

Figures 6 shows which established regimes appear in471

time-domain synthesis depending on τg, for final val-472

ues γf belonging to the multistability zones described473

in Figure 3. Each dot on the figure represents the474

type of established regime after five seconds of time-475

domain synthesis. This synthesis duration was chosen476

to be sufficiently long so that the transient is com-477

pleted and the established regimes can be observed.478

Regime types are estimated using an energy-based479

criterion for equilibrium (if the energy of the pres-480

sure signals in the last ten periods is less than that of481

the first ten, the regime is classified as equilibrium)482

and a fundamental frequency estimator to determine483

the first and second register. To detect quasi-periodic484

regimes, an attribute that can be observed in Figure485

2 is used: the fact that the amplitudes of the harmon-486

ics of a quasi-periodic signal vary temporally. For the487

classification, if the mean variance of the harmonic488

Figure 5: Four examples of blowing pressure varia-tions in control scenario number two, as described by

Eq. (17), for γf = 0.4 and 0.6 and τg = τ[1]g = 10 ms

and τ[2]g = 20 ms.

amplitudes is more than a certain threshold (here set489

to 10−6), then the regime is considered quasi-periodic.490

Figure 6 (a) focuses on the first multistability re- 491

gion (highlighted in gray in Figure 3), near the first 492

Hopf bifurcation H1. The two stable regimes in this 493

region are the equilibrium ( ) and the first regis- 494

ter ( ). For final values γf between 0.38 and 0.4, 495

the system can converge to both regime depending on 496

the characteristic rising time τγ . It is interesting to 497

note that equilibrium is reached for the longest rising 498

times, i.e. the slowest γ variation, whereas the oscil- 499

lating regime is reached for the shortest rising times. 500

This is understandable as a quick γ increase tends to 501

drive the system away from equilibrium, and there- 502

fore possibly out of its attraction basin. Note that 503

some of the oscillating regimes near the limit, for the 504

longest attack times, are classified as quasi-periodic. 505

This is due to the transient being extremely long in 506

this particular region: the steady-state regime is not 507

yet established at the end of the synthesized signals. 508

This classification particularity, which could be seen 509

as an error, is not corrected because, from a musi- 510

cian’s perspective, a regime still varying five seconds 511

after the start of the attack will arguably not be con- 512

sidered periodic. 513

The second zone of multistability is explored in Fig- 514

ure 6 (b). The first and second register are separated 515

by some stable quasi-periodic regimes ( ). This is 516

the same quasi-periodic regime that appears in time- 517

domain synthesis in Figure 2, which overlays the un- 518

stable portion of the second register branch. There is 519

a particular range of characteristic time τg that seems 520

to produce the first register for a larger range of γf . 521

Colinot et al., p. 9

(a)

8 (a) 8 (b) 8 (d) 8 (e)

(b)

8 (c) 7 (a) 7 (b) 7 (c) 8 (f)

(c)

Figure 6: Classification of the steady-state regimesproduced depending on the blowing pressure transientparameters: final value γf and characteristic timeτg. Multistability zones (deduced from Figure 3): (a)equilibrium and first register (b) first register

and second register (c) all three regime types.Blue triangles indicate quasi-periodic regimes. Thehorizontal line on graph (b) shows τ = 1/f1. The ver-tical lines highlight the γ values of phase diagrams inFigure 7 and 8.

The corresponding values of τg are close to the period 522

of the first register, represented by the horizontal line 523

in 6 (b).524

In the last multistability (0.9 ≤ γ ≤ 1.25), three525

regimes may be stable for the same parameter val-526

ues, as shown on Figure 3. However, Figure 6 re-527

veals that there is no γf region where all three are528

produced. This can be explained by analyzing the529

attraction basins (see Section 4.2 Figure 8).530

4.2 Control scenario number three: 531

varying initial conditions 532

The results concerning the influence of the blowing 533

pressure parameter can be better understood by ex- 534

amining the region of the phase space leading to each 535

regime. A point in the phase space represents the 536

current state of the system, meaning the value of the 537

state variables and their derivatives. Since the sys- 538

tem is deterministic, a given point in the phase space 539

will always lead to the same stable established regime. 540

Therefore regions of the phase space can be associated 541

with each regime. These regions are called attraction 542

basins. The attraction basins are only rigorously 543

defined in a context where all control parameters are 544

constant. However, to interpret the results obtained 545

with a control parameter transient such as control sce- 546

nario number two, it is useful to observe the layout of 547

attraction basins obtained for the final blowing pres- 548

sure value γf . Specifically, at the beginning of control 549

scenario number two, the blowing pressure is subject 550

to fast variations, making a direct approach based on 551

attraction basins ill-defined. However, after the tran- 552

sient, the blowing pressure stabilizes around its final 553

value γf . To elucidate the behavior of the system 554

at this moment, subsection 4.2 performs a systemat- 555

ical analysis of its convergence with constant control 556

parameters, in which case the attraction basins rep- 557

resentations are relevant. Therefore, a third control 558

scenario is devised, where the control parameters are 559

kept strictly constant and only the initial values of 560

certain state variables of the system are modified. 561

Because the phase space is of dimension 2Nm + 2(all modal components and their derivatives, plus reedposition and speed), it is necessary to choose a pro-jection to represent the attraction basins. After sometrials, a projection of the phase space on the two firstmodal components (see Eq. (12)) and the derivativeof the second one, (p1, p2, p2), was chosen as a three-dimensional projection. These variables were chosennot only because of their physical or mathematicalmeaning, as they relate respectively to the first andsecond register, but also because they allow for theclearest visual separation of the limit cycles and at-traction basins that could be obtained by the authors.To estimate the attraction basins, time-domain syn-thesis is launched with initial conditions spanning theprojected phase space and constant control parame-ters (control scenario number three) . A total of 256initial conditions are scattered in a Latin hypercubesampling into a rectangular parallelepiped such that

pI1 ∈ [−0.2, 0.2], pI2 ∈ [−2, 2], p2I ∈ [−707, 707].

(18)

These bounds should be understood with respect to 562

Colinot et al., p. 10

the amplitude of the limit cycle along each dimension 563

(that can be seen in Figures 7 and 8). They were cho- 564

sen so that whenever a regime is stable, it is obtained 565

in synthesis at least once. All the other modal pres-566

sure components and their derivatives, as well as the567

reed speed x, are initially zero. So that there is no568

discontinuity when starting the synthesis, the initial569

values of the variables p, then x and u are computed570

accordingly through equations (13), (4) and (7). All571

control parameters, including the blowing pressure γ,572

are kept constant during these simulations: recall573

that the attraction basin analysis is only strictly valid574

for constant values of the control parameters. There-575

fore, the deductions that are made using the projected576

attraction basins (control scenario number three) on577

the results of control scenario number two are subject578

to caution.579

Figure 7 shows these initial conditions in the plane580

(p2, p2), associated with the regimes they lead to, for581

three values of the blowing pressure parameter γ. All582

the phase space points the system passes through dur-583

ing the transient are also part of the attraction basin,584

so they are represented in the figure as well. The last585

period of synthesized signals is plotted as the limit586

cycles. The three values of γ are chosen near the587

last Hopf bifurcation (H3 and H4 in Figure 3), where588

the equilibrium becomes stable again. Graph 7 (a) is589

computed at γ = 1, before the Hopf bifurcations, so590

only register one and two are stable. Therefore, none591

of the initial conditions lead to equilibrium (no black592

dots on the figure). Each attraction basin is located593

around the corresponding limit cycle. The attraction594

basins seem to overlap, but it is merely an effect of595

the projection of the phase space. Graph 7 (b) corre-596

sponds to γ = 1.1, right above the Hopf bifurcations597

H3 and H4. One can see that some initial conditions598

located near the origin now lead to equilibrium. This599

plot can seem surprising when compared to Figure600

6 (c), which only shows equilibrium and register one601

for this value of γf using the control scenario number602

two, although the attraction basin of the second regis-603

ter (red points in Figure 7 (b)) seems larger than that604

of the other regimes. This is due to two effects. First,605

the projection spreads the second register but shrinks606

the first register. Secondly, control scenario number607

two starts from the origin of the phase space: it is608

ensured by setting all variables and their derivatives609

at 0 at the start of the synthesis. Therefore, attrac-610

tion basins surrounding the origin, such as that of the611

first register, are more likely to be entered . Here612

we see that considering only the size of the attraction613

basins and ignoring their position provides only little614

information. This data needs to be interpreted in615

terms of musician action, notably pertaining to the616

ability of the musician to actually reach the region of617

the phase space. Here, the fact that the attraction618

basin is vast is effectively countered by the fact that619

it is far from the origin, making it hard to reach using620

the studied control scenario number two. Graph 7 (c)621

represents the same results for γ = 1.2, where Figure622

6 (c) showed more occurences of the equilibrium than623

for γ = 1.1. This is explained by the attraction basin 624

of the equilibrium being larger – there are many more 625

black dots on Figure 7 (c) than on (b). Notice that 626

the attraction basin of the first register expands (also 627

many more green dots than on Figure 7 (b)), while 628

the attraction basin of the second register (red dots) 629

shrinks. This process (not represented here) contin- 630

ues until the second register ceases to be stable at fold 631

bifurcation F3 (see Figure 3). 632

Figure 8 shows the attraction basins and limit cy- 633

cles, in a three-dimensional projection of the phase 634

space (p1, p2, p2), at particular values of γ highlighted 635

in Figure 6. Graphs 8 (a), (b), (d) and (e) should be 636

read as further information on the regime map 6 (b), 637

at the beginning of the multistability zone between 638

first and second registers. 639

Graph 8 (a) corresponds to γ = 0.6, and confirms 640

that the first register is the only stable regime: it 641

was the only one to appear in the regime map 6 (b) 642

(for γf = 0.6). Then, a quasi-periodic attractor ap- 643

pears in Graph 8 (b), for γ = 0.63. Although the 644

associated attraction basin seems smaller than that 645

of the first register, it seems to almost surround the 646

origin of the phase diagram. It was observed in syn- 647

thesis that the transient of control scenario number 648

two does not send the system very far from the ori- 649

gin in the phase space – compared, for instance, with 650

the size of the limit cycles of register one and two. 651

This is linked to it necessarily starting from the ori- 652

gin of the phase space. The fact that control scenario 653

number two tends to lead the system to phase space 654

points near the origin explains why regime map 6 (b) 655

displays more quasi-periodic regimes than first regis- 656

ter. A similar interpretation can be formulated with 657

regards to graphs 8 (d) and (e), for γ = 0.645 and 658

γ = 0.72 respectively. For these values of γ on Figure 659

6 (b), there is more second register than first register. 660

On graphs 8 (d) and (e), it can be seen that although 661

the size of the first register’s attraction basin seems 662

comparable to that of the second register, the latter 663

clearly holds a central position around the origin of 664

the phase space. When the second register attraction 665

basin grows in Graph 8 (e), this translates to the dis- 666

appearance of the first register from the regime map 667

6 (b). Note that Graph 8 (e) confirms that the first 668

register is still stable, as announced by the harmonic 669

balance method in Figure 3. Graph 8 (c) γ = 0.9 il- 670

lustrates a slightly different explanation of a similar 671

case of only second register appearing in the regime 672

map 6 (c): in this case, the attraction basin of the first 673

register is just too small, it only makes up for a few 674

points in Figure 8 (c). Graph 8 (f) (γ = 1.25) is com- 675

parable to (d) in that many regimes are stable, but 676

only the one with the most central attraction basin 677

appears in the corresponding regime map on Figure 6 678

Colinot et al., p. 11

(a)

(b)

(c)

Figure 7: Projection of the attraction basins obtainedwith control scenario number three . Large dots areinitial conditions, small dots are points the synthe-sis goes through. The dots’ colors indicate the finalregime they lead to (black: equilibrium, green: firstregister, red: second register). Blue lines representthe limit cycles. The small one on the inside is thefirst register and the one on the outside is the secondregister. The blowing pressure γ is (a) 1, (b) 1.1, (c)1.2 (highlighted in figures 6 and 3).

(c). In this case this regime is the equilibrium, whose 679

attraction basin in Graph 8 (f) surrounds the origin, 680

although it appears smaller than the others. To com- 681

plete the study, the full evolution sequence of the at-682

traction basins obtained with control scenario number683

three can be found as an animation in multimedia file684

Supp1.mp4. The authors suggest frequently pausing685

the animation to observe precisely how the attraction 686

basins develop along multistability zones. 687

5 Effect of the resonator’s in- 688

harmonicity on regime pro- 689

duction 690

The concept of multistability and the attraction 691

basins are presented and explored here because they 692

seem to be a very important part of the observed be- 693

havior of a saxophone model. The present section 694

offers a succinct description of the behavior of the 695

model, applied to an instrument design problematic. 696

It also illustrates how ignoring multistability can af- 697

fect the description of the behavior of a model. 698

Before using the analysis of a woodwind physical 699

model to develop new instruments, it can be very in- 700

formative to apply it to existing instruments, in the 701

idea of a reverse engineering procedure. If the analysis 702

method can explain a posteriori some design choices 703

made on instruments with satisfying sound produc- 704

tion characteristics, then it might help guide further 705

innovative design choices in the right direction. In the 706

present case, the produced regimes are studied for the 707

seven lowest first register fingerings, and one acous- 708

tical parameter is varied artificially: the inharmonic- 709

ity between first and second resonance. The original 710

data corresponds to measured impedance for the cor- 711

responding fingerings of a Buffet-Crampon Senzo pro- 712

fessional alto saxophone. According to the so-called 713

Bouasse-Benade prescription [9, 4, 3], near perfect in- 714

harmonicity between the resonances is cited as a con- 715

dition for good playability of the instrument. This 716

prescription is also discussed in recent studies [27, 11]. 717

On the saxophone, experimental studies using an ar- 718

tificial mouth have shown that varying inharmonic- 719

ity greatly affects regime production [23, 20]. In this 720

work we define inharmonicity as the ratio between the 721

second and first resonance f2/f1, or =(s2)/=(s1) in 722

terms of the parameters of Eq. (11). On a saxo- 723

phone this ratio is close to two. Many definitions 724

of the inharmonicity can be devised, possibly taking 725

into account more resonances. The present definition 726

of the harmonicity has the advantage of being very 727

short, and easy to modify in the modal formalism by 728

adjusting only one modal frequency. 729

Specifically for the purpose of the following study, 730

optimal regime production conditions are defined 731

crudely in terms of how often each regime appears 732

in synthesis. This definition differs from that em- 733

ployed in [27]. On the lowest fingerings, in this 734

study, we simply consider that optimizing regime pro- 735

duction means maximizing the appearance of the first 736

register while minimizing that of the second register 737

Colinot et al., p. 12

(a) γ = 0.6 (b) γ = 0.63 (c) γ = 0.9

(d) γ = 0.645 (e) γ = 0.72 (f) γ = 1.25

Figure 8: Attraction basins (dots) and limit cycles (lines) in a 3D projection of the phase space obtained withcontrol scenario number three for different blowing pressures γ. Black: equilibrium, green: first register, red:second register, blue: quasi-periodic.

and quasi-periodic regimes. Indeed, one of the chal- 738

lenges many beginner saxophone players face on the 739

lowest fingerings is controlling the instrument so that 740

the first register can be produced, and not another741

regime. Quasi-periodic regimes are largely consid-742

ered undesirable in common musical practice, how-743

ever they are a common issue on the lowest fingerings744

of the saxophone.745

5.1 Regime production regions746

Expanding on the idea in Figure 6, one can study747

the produced regime across the two-dimensional pa-748

rameter space (γf , ζ), while still varying the charac-749

teristic time τg of control scenario number two. Fig-750

ure 9 shows the classification of obtained regimes for751

several combinations (γf , ζ) and several characteris-752

tic times. Note that for a musician using their lower753

lip to control the instrument, it is difficult to control754

the reed opening parameter ζ without also varying755

the properties of the reed ωr and qr. Leaving the756

reed parameters constant in this study amounts to a757

simplification of the musician control. For readability758

reasons, the resolution of the cartography presented759

here is rather coarse: only eight values of γf and ζ and760

three characteristic times τg, for a total of 192 synthe-761

sized signals. The range in parameter ζ is inspired762

by the range measured in [23] for artificial mouth ex-763

periments, using the method proposed in [21]. This764

method relies on measuring the flow rate as a function765

of the mouth pressure, and estimating ζ based on the 766

maximum flow using the nonlinear characteristic of 767

Eq. (7). As a case study, two maps computed with 768

different inharmonicity values are presented on Figure 769

9 (a) and (b), so that they can be compared. In the 770

modal formalism, the inharmonicity is changed very 771

simply by modifying the value of the second modal 772

frequency. Note that changing the length of the 773

saxophone mouthpiece constitutes another method to 774

vary the inharmonicity. However, it modifies all the 775

modal frequencies simultaneously, which makes some 776

interpretations less robust. Therefore, only the results 777

obtained by varying the second modal frequency are 778

presented here. Similar results can be obtained by 779

varying the length of the mouthpiece. Two typical 780

values of inharmonicity are chosen: one that could be 781

called null, f2/f1 = 2; and the value measured on the 782

saxophone which is slightly higher, f2/f1 = 2.065. 783

Focusing on Figure 9 (a), several features can be de- 784

scribed, and recognized from the situations explored 785

in Section 4 with a fixed ζ. Coexistence regions can be 786

noticed on most of the map, with a given (γf , ζ) cou- 787

ple leading to different regimes depending on the char- 788

Colinot et al., p. 13

acteristic time. This further demonstrates that mul- 789

tistability is a very common phenomenon across the 790

control parameter space in woodwind models. A par- 791

ticular case of coexistence occurs on all regime maps, 792

where long attack times lead to the system remaining793

at equilibrium, while fast attacks can trigger oscilla-794

tions. This is the same phenomenon as in Figure 6795

(a). These phenomena are located, as for Figure 6 (a),796

at the boundaries between equilibrium and oscillation797

regimes, which correspond to the Hopf bifurcations of798

the model (where the equilibrium becomes unstable).799

On the regime maps, points containing both equilib-800

rium regimes and oscillating regimes are seen on the801

vertical threshold near γ = 0.4, the horizontal thresh-802

old near ζ = 0.2 and the extinction threshold around803

γ = 1.1.804

Coexistence situations similar to Figure 6 (b) can805

also be seen on Figure 9 (a), for example at ζ = 1.2806

and γf ' 0.8, where the short and long characteristic807

times lead to the second register, while the medium808

time leads to a first register. This possibly indi-809

cates that the second register attraction basin almost810

surrounds the origin, as seen on Figure 8, in all the811

multistability zones between first and second register812

for that fingering.813

Figure 9 (b) (f2/f1 = 2) displays a lot more second814

register ( ) than Figure 9 (a) (f2/f1 = 2.065). Con-815

trary to what could be expected, a null inharmonic-816

ity, where the second resonance frequency is twice the817

first, does not lead to more first register production.818

Since an exact integer ratio between resonances does819

not facilitate the production of the first register, one820

can ask if the model shows a particular value of inhar-821

monicity which favors the production of first register.822

5.2 Rate of produced regimes: influ-823

ence of the rise time on global824

regime production825

To study the question of the inharmonicity favoringfirst register production, regime maps are computedfor all the fingerings of the saxophone that shouldproduce first register regimes, meaning those wherethe register holes are closed. For each fingering, thesecond modal frequency f2 is varied from 1.96f1 to2.15f1, by steps of 0.01f1. A regime map containingNp = 192 points, as for Figure 9, is then computed foreach value of inharmonicity. The produced regimesare counted for the whole map, and a rate is computedfor each of them with respect to the total number ofoscillating regime as

Ri =Np,iNp

, (19)

where regime i can either be first register, second reg-826

ister or quasi-periodic regimes, Np,i is the number of827

points corresponding to regime i in the regime map828

(a)τg = 0.1 ms, R1

τg = 3.2 ms, R2

τg = 100 ms, Eq.γf = 1.16, ζ = 1.2

(b)

Figure 9: Classification of the regimes produced( equilibrium, first register, second register,

quasi-periodic) depending on control parameters:γf and ζ. Each rectangle corresponds to a couple(γf , ζ) and the points inside indicate the regime foreach characteristic time τg (bottom 0.1 ms, middle3 ms and top 100 ms). One rectangle on graph (b)is annotated as an example. Graphs correspond totwo inharmonicities for low written D] fingering (a)f2 = 2.065f1, the value measured on a real saxophoneand (b) f2 = 2f1.

and Np,o is the total number of points correspond-829

ing to any oscillating regime (i.e. all regimes but830

equilibrium). Note that this description ignores non-831

oscillating regimes.832

Figure 10 depicts the rate of each oscillating regime 833

produced depending on inharmonicity for the lowest 834

fingering of the first register, the written B[ which 835

produces the heard note C3 at 131 Hz. On this fig- 836

ure the regimes are counted separately for each char- 837

acteristic time τg before being summed for the whole 838

Colinot et al., p. 14

map to produce an averaged rate. On the averaged 839

rate, optimal points are highlighted by triangles, cor- 840

responding to the maximum of first register and min- 841

imum of second register produced, respectively. Both 842

appear for values slightly above f2/f1 = 2. The inhar-843

monicity values maximizing first register production844

do not correspond to exactly harmonic resonances,845

but a second resonance slightly higher than the oc-846

tave of the first. The proportion of quasi-periodic847

regimes is also displayed on the figure. Note that848

inharmonicity values around two lead to less quasi-849

periodic regimes, which is corroborated by existing850

results [24, 23]. On Figure 10, it can be seen that the851

region of low rate of production of the second register852

for f2/f1 between 2.05 and 2.09 almost coincides with853

a region of high rate of production of quasi-periodic854

regimes, for f2/f1 between 2.06 and 2.1. In this case,855

minimizing second register production seems to favor856

the production of quasi-periodic regimes.857

Figure 10 shows that ignoring multistability, by858

using only one value of characteristic time, could859

have lead to conclusions similar to those drawn with860

production rates averaged over several characteristic861

times. Indeed, all the characteristic times yield qual-862

itatively similar production rates. However, this is863

not always the case, and Figure 11 shows two exam-864

ples where neglecting multistability and studying only865

one characteristic time can lead to biased conclusions.866

Figure 10: Rate of produced regimes (Eq. (19)) forwritten low B[ fingering. Green: first register, red:second register, blue: quasi-periodic. Linestyles in-dicate the characteristic time. Dotted: τg = 0.1 ms,dash-dot: τg = 3.2 ms, dashed: τg = 100 ms, solid:averaged rate. An upward triangle marks the maxi-mum first register averaged rate, a downward trianglemarks the minimum second register rate.

For the D] fingering (Figure 11 (a)), one can see867

that depending on the chosen increase duration τg the868

production ratio varies greatly (from 20% to 60%). If869

any quantitative interpretation is to be expected from870

these results, it can be changed dramatically depend-871

ing on the chosen attack time. Notice that the lowest872

rate of first register corresponds to the longest attack873

time τg = 100 ms. Figure 9 shows that the first regis-874

ter regimes are produced on the edges of the zone of875

oscillation, in a multistability region between the first 876

register and the equilibrium. The longest attack times 877

in this region tend not to lead to a first register, but 878

instead to an equilibrium due to its attraction basin 879

surrounding the origin (see for instance Figure 7 (c) or 880

multimedia file Supp1.mp4). Figure 11 (b) shows the 881

results for the written D fingering. This case exhibits 882

an outlier: the shortest attack time yields a optimal 883

inharmonicity value of 2.08, whereas the others point 884

to 2.04. In this case, considering several attack times 885

is a way to smooth out outliers due to a particular 886

value of the attack time. 887

Note that Graph 11 (b) can also be subject to an 888

interesting interpretation in terms of musician control 889

strategies: the fact that certain attack time values 890

seem to markedly decrease the rate of production of a 891

certain regime could be used by the musician to avoid 892

producing it. 893

5.3 Inharmonicity of the saxophone894

In this section, the optimal inharmonicity in terms of895

regime production is studied for the seven lowest fin-896

gerings of the instrument. Higher fingerings are not897

represented because they add no relevant information:898

first register regime production rates are close to 100%899

for all the studied inharmonicity values. This corre-900

sponds to the saxophonists’ experience that the high901

notes of the instrument’s first register are often eas-902

ier to produce than the low notes, and to the fact903

that the first impedance peak is much higher than904

the others on the high fingerings [14]. The optimums905

are compared with the inharmonicity value measured906

on the saxophone on which the model is based. Fig-907

ure 12 summarizes the production ratios for all the908

fingerings. The optimal inharmonicity seems to vary909

across the fingerings. It is always greater than two:910

null inharmonicity does not favor first register pro-911

duction on the low fingerings of the saxophone. The912

two optimums are close to the measured inharmonic-913

ity. Additionally, the trend is respected, with optimal914

and measured harmonicities increasing for higher fin-915

gerings. Note that the optimum for the E fingering916

is very far from the measured inharmonicity, but the917

production ratios are almost constant. Overall, the918

simulation shows that the most first register and least919

second register is produced by the model for values of920

inharmonicity near those measured on the saxophone.921

This result sheds some light on the empirical choice of922

the acoustical properties on the saxophone. Indeed, if923

the inharmonicity was far from the values observed on924

saxophones, the model predicts more second register925

would be produced. This effect is arguably undesir-926

able. However, this choice comes with a compromise,927

Colinot et al., p. 15

(a)

(b)

Figure 11: Rate of first register regime (Eq. (19)) pro-duced for (a) written low D] fingering and (b) writtenlow D fingering. Linestyles indicate the characteristictime. Dotted: τg = 0.1 ms, dash-dot: τg = 3.2 ms,dashed: τg = 100 ms, solid: averaged rate. An up-ward triangle marks the maximums of first registerrates.

as it also favors quasi-periodic regime production (see928

Figure 10), which are a known issue in low saxophone929

fingerings.930

6 Conclusion931

In the studied saxophone model, stable regimes coex-932

ist throughout large regions in the musician control933

parameter space. Thus, even an exhaustive descrip-934

tion of the stability or instability of each oscillating935

regime in the control parameter space is only an in-936

complete answer, as it doesn’t suffice to predict which937

regime emerges in these multistability zones. In par-938

ticular, in the event that the attraction basin of a sta-939

ble regime proves too small or unattainable by the mu-940

sician, this regime should be considered unreachable,941

almost like an unstable regime. This work demon-942

Figure 12: Rate of produced regimes for the lowest fin-gerings of the alto saxophone (written pitch). Green:first register, red: second register. An upward trian-gle marks the maximum first register averaged rate,a downward triangle marks the minimum second reg-ister rate. Vertical lines mark the measured inhar-monicity values.

strates that saxophone regime stability is a nuanced943

topic.944

An adapted combination of two radically different945

numerical methods provides a more complete descrip-946

tion of the model’s behavior: the stability study per-947

formed by harmonic balance method is completed by948

time-domain synthesis, which quantifies multistabil- 949

ity by outlining the attraction basin of each regime. 950

A varying control scenario explores which regime is 951

produced in multistability zones, with the advantage 952

that it can be tied to plausible musician actions. How- 953

ever, a varying control scenario only provides a very 954

partial view of the attraction basins, and its results 955

deserve to be explicited by representing the attraction 956

basins in the phase space using different initial condi- 957

tions. Dedicated experimental work, out of the scope 958

of this paper, could help design more realistic control 959

scenarios. 960

Accounting for multistability, the study of synthe- 961

sized regimes may explain an acoustical choice made 962

by instruments makers: the inharmonicity of the sax- 963

ophone. An integer ratio between the first and second 964

resonance frequencies does not favor the production 965

of first register. Note that this result adds nuance 966

one of Benade’s guidelines stating that an oscillation 967

Colinot et al., p. 16

is favored if the impedance is large at its fundamental 968

and its harmonic frequencies [4]. This work shows 969

that competition between registers also comes into 970

play, depending on more than solely the impedance 971

magnitude at the playing frequency and its harmon- 972

ics. Instead, an integer ratio between the first and 973

second resonance frequencies tends to favor the pro-974

duction of second register, which is arguably unde-975

sirable for a first register fingering. Carefully tuned976

inharmonic resonances, where the second frequency977

is higher than twice the first, can lead to more first978

register production. The optimal inharmonicity value979

found on the model is close to harmonicities measured980

on saxophone resonators. This result provides an a981

posteriori interpretation of the acoustical characteris-982

tics of the saxophone, as chosen empirically by instru-983

ment makers, as the acoustical characteristic leading984

to easier production of the first register. Such results985

are among the first steps towards applying numerical986

simulations as predictive tools to estimate playability987

in instrument design.988

Acknowledgments989

The authors would like to thank Erik Petersen for cor-990

recting the English. This work has been carried out in991

the framework of the Labex MEC (ANR-10-LABEX-992

0092) and of the A*MIDEX project (ANR-11-IDEX-993

0001-02), funded by the Investissements d’Avenir994

French Government program managed by the French995

National Research Agency (ANR). This study has996

been supported by the French ANR LabCom LIAMFI997

(ANR-16-LCV2-007-01).998

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List of Figures 1279

1 Measured input impedance (solid) and 1280

modal reconstruction (dashed) for the 1281

fingering D] of the alto saxophone. . 4 1282

2 Time-domain synthesized pressure sig- 1283

nal. (a) temporal envelope (black) and 1284

blowing pressure parameter γ (red). 1285

(b) Normalized spectrogram (dB) with 1286

regime names indicated (unstable ones 1287

between parentheses) : equilibrium 1288

(Eq.), second register, quasi-periodic 1289

and first register. The signal is ex- 1290

tracted from the same blowing pressure 1291

ramp as in Figure 4, between γ = 0.45 1292

and γ = 0.51 (at first occurrence of os- 1293

cillation). . . . . . . . . . . . . . . . . 4 1294

3 Bifurcation diagram obtained with 1295

harmonic balance method and numeri- 1296

cal continuation: L2-norm of the acous- 1297

tical pressure depending on the blow- 1298

ing pressure parameter γ for the low 1299

written D] fingering of an alto sax- 1300

ophone. Thick lines: stable solu- 1301

tion, thin lines: unstable solutions. 1302

Black: equilibrium, green: 1st regis- 1303

ter regimes, red: 2nd register regimes. 1304

Multistability zones are shaded: light 1305

yellow where 1st and 2nd register co- 1306

exist, darker gray for equilibrium and 1307

1st register. Blue circles specify the lo- 1308

cation of bifurcations. Vertical black 1309

lines correspond to those in Figure 6 1310

(c), and (from left to right) to phase 1311

diagrams 7 (a), (b) and (c). . . . . . . 6 1312

4 L2-norm of the time-domain synthesis 1313

signal (dark blue line) for control sce- 1314

nario number one, a ramp of the blow- 1315

ing pressure parameter γ from 0 to 2 1316

and down, overlaid with the bifurca- 1317

tion diagram of Figure 3 (thick stable 1318

solutions and thin unstable solutions, 1319

of the first register in green and second 1320

register in red). Solid dark blue line in- 1321

dicates the first part of the signal (in- 1322

creasing γ) and dashed dark blue line 1323

the second (decreasing γ). . . . . . . 7 1324

5 Four examples of blowing pressure vari- 1325

ations in control scenario number two, 1326

as described by Eq. (17), for γf = 0.4 1327

and 0.6 and τg = τ[1]g = 10 ms and 1328

τ[2]g = 20 ms. . . . . . . . . . . . . . . 8 1329

Colinot et al., p. 20

6 Classification of the steady-state1330

regimes produced with control sce-1331

nario number two, depending on the1332

blowing pressure transient parameters:1333

final value γf and characteristic time1334

τg. Multistability zones (deduced from1335

Figure 3): (a) equilibrium and first1336

register (b) first register and second1337

register (c) all three regime types.1338

Blue triangles indicate quasi-periodic1339

regimes. The horizontal line on graph1340

(b) shows τ = 1/f1. The vertical1341

lines highlight the γ values of phase1342

diagrams in Figure 7 and 8. . . . . . . 91343

7 Projection of the attraction basins ob-1344

tained with control scenario number1345

three . Large dots are initial condi-1346

tions, small dots are points the syn-1347

thesis goes through. The dots’ colors1348

indicate the final regime they lead to1349

(black: equilibrium, green: first regis-1350

ter, red: second register). Blue lines1351

represent the limit cycles. The small1352

one on the inside is the first register1353

and the one on the outside is the sec-1354

ond register. The blowing pressure γ1355

is (a) 1, (b) 1.1, (c) 1.2 (highlighted in1356

figures 6 and 3). . . . . . . . . . . . . 111357

8 Attraction basins (dots) and limit cy-1358

cles (lines) in a 3D projection of the1359

phase space obtained with control sce-1360

nario number three for different blow-1361

ing pressures γ. Black: equilibrium,1362

green: first register, red: second reg-1363

ister, blue: quasi-periodic. . . . . . . 121364

9 Classification of the regimes produced1365

(equilibrium, first register, second reg-1366

ister, quasi-periodic) depending on1367

control parameters: γf and ζ. Each1368

rectangle corresponds to a couple1369

(γf , ζ) and the points inside indicate1370

the regime for each characteristic time1371

τg (bottom 0.1 ms, middle 3 ms and top1372

100 ms). One rectangle on graph (b) is1373

annotated as an example. Graphs cor-1374

respond to two inharmonicities for low1375

written D] fingering (a) f2 = 2.065f1,1376

the value measured on a real saxo-1377

phone. (b) f2 = 2f1. . . . . . . . . . . 131378

10 Rate of produced regimes (Eq. (19)) 1379

for written low B[ fingering. Green: 1380

first register, red: second register, blue: 1381

quasi-periodic. Linestyles indicate the 1382

characteristic time. Dotted: τg = 1383

0.1 ms, dash-dot: τg = 3.2 ms, dashed: 1384

τg = 100 ms, solid: averaged rate. An 1385

upward triangle marks the maximum 1386

first register averaged rate, a down- 1387

ward triangle marks the minimum sec- 1388

ond register rate. . . . . . . . . . . . 14 1389

11 Rate of first register regime (Eq. (19)) 1390

produced for (a) written low D] fin- 1391

gering and (b) written low D finger- 1392

ing. Linestyles indicate the character- 1393

istic time. Dotted: τg = 0.1 ms, dash- 1394

dot: τg = 3.2 ms, dashed: τg = 100 ms, 1395

solid: averaged rate. An upward trian- 1396

gle marks the maximums of first regis- 1397

ter rates. . . . . . . . . . . . . . . . . 14 1398

12 Rate of produced regimes for the lowest 1399

fingerings of the alto saxophone (writ- 1400

ten pitch). Green: first register, red: 1401

second register. An upward triangle 1402

marks the maximum first register aver- 1403

aged rate, a downward triangle marks 1404

the minimum second register rate. Ver- 1405

tical lines mark the measured inhar- 1406

monicity values. . . . . . . . . . . . . 15 1407

List of Tables 1408

1 Parameters of the numerical model: 1409

musician control parameters γ and ζ, 1410

reed parameters qr and ωr, contact pa- 1411

rameter Kc and parameters inherent to 1412

the numerical implementation η, Nm 1413

and H. . . . . . . . . . . . . . . . . . . 3 1414

2 Modal parameters for the fingering D] 1415

of the alto saxophone. . . . . . . . . . 4 1416