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Regime change thresholds in flute-like instruments:influence of the mouth pressure dynamics

Soizic Terrien, Remi Blandin, Christophe Vergez, Benoît Fabre

To cite this version:Soizic Terrien, Remi Blandin, Christophe Vergez, Benoît Fabre. Regime change thresholds in flute-likeinstruments: influence of the mouth pressure dynamics. Acta Acustica united with Acustica, HirzelVerlag, 2015, pp.300-316. �10.3813/AAA.918828�. �hal-00964988�

Regime change thresholds in flute-like instruments: influence of the

mouth pressure dynamics

Soizic Terrien1) , Remi Blandin1),2) , Christophe Vergez1) , Benoıt Fabre3)

1) LMA, CNRS, UPR 7051, Aix-Marseille Univ, Centrale Marseille,

F-13402 Marseille Cedex20, France.2) currently at: Gipsa-lab, CNRS, UMR 5216, Grenoble INP, Universite Joseph Fourier,

Universit Stendhal, Grenoble Campus, 11 rue des Mathematiques,

BP 46, 38402 Saint Martin d’Heres Cedex, France3) LAM, Sorbonne Universites, UPMC Univ Paris 06, CNRS, UMR 7190

Institut Jean Le Rond d’Alembert, 11 rue de Lourmel F-75015 Paris, France.

Abstract

Since they correspond to a jump from a given noteto another one, the mouth pressure thresholdsleading to regime changes are particularly impor-tant quantities in flute-like instruments. In thispaper, a comparison of such thresholds betweenan artificial mouth, an experienced flutist and anon player is provided. It highlights the abilityof the experienced player to considerabily shiftregime change thresholds, and thus to enlarge itscontrol in terms of nuances and spectrum. Basedon recent works on other wind instruments and onthe theory of dynamic bifurcations, the hypothe-sis is tested experimentally and numerically thatthe dynamics of the blowing pressure influencesregime change thresholds. The results highlightthe strong influence of this parameter on thresh-olds, suggesting its wide use by experienced musi-cians. Starting from these observations and froman analysis of a physical model of flute-like instru-ments, involving numerical continuation methodsand Floquet stability analysis, a phenomenolog-ical modelling of regime change is proposed andvalidated. It allows to predict the regime changethresholds in the dynamic case, in which timevariations of the blowing pressure are taken intoaccount.

1 Introduction and problem

statement

In flute playing, the phenomenon of regimechange is particularly important, both becauseit corresponds to a jump from a given note to

another one (in most cases an octave higher orlower) and because it is related to the blowingpressure, directly controlled by the musician. Asan example, a regime change from the first reg-ister to the second register (periodic oscillationregimes synchronized on the first and the secondresonance mode of the instrument, respectively),occurs when the musician blows harder enoughin the instrument, and is characterized by a fre-quency leap approximately an octave higher (seefor example [1]).

It is well known that register change is accom-panied by hysteresis (see for example [1, 2, 3]):the mouth pressure at which the jump betweentwo registers occurs (the so-called regime changethreshold) is larger for rising pressures than for di-minishing pressures. For musicians, the hysteresisallows a greater freedom in terms of musical per-formance. Indeed, it allows them both to playforte on the first register and piano on the sec-ond register, leading to a wider control in termsof nuance and timbre. Numerous studies havefocused on both the prediction and the experi-mental detection of such thresholds [1, 2]. Otherstudies have focused on the influence of differentparameters on regime change thresholds, such asthe geometrical dimensions of channel, chamfersand excitation window of recorders or organ fluepipes [4, 5, 6], the importance of nonlinear losses[2], or the convection velocity of perturbations onthe jet [2]. However, it seems that few studieshave focused, in terms of regime change thresh-olds, on other control parameters (i.e. relatedto the musician) than the slowly varying blowingpressure.

1

Since it has important musical consequences,one can wonder if flute players develop strategiesto change the values of regime change thresh-olds and to maximize the hysteresis. To testthis hypothesis, increasing and decreasing profilesof blowing pressure (crescendo and decrescendo)were performed on the same alto recorder and fora given fingering (corresponding to the note F4),by an experienced flutist, a non player, and anartifical mouth [7]. Both experienced musicianand non musician have been instructed to stayas long as possible on the first register and onthe second register for crescendo and decrescendorespectively. The different experimental setupswill be described in section 2. The representa-tion of the fundamental frequency of the soundwith respect to the blowing pressure, displayedin figure 1, highlights that the musician obtainedan increasing threshold 213 % higher and a de-creasing threshold 214 % higher than the artificialmouth, whereas the differences between the nonmusician and the artificial mouth are of 9 % forthe increasing threshold and 32 % for the decreas-ing threshold. As highlighted in figure 2, similarcomparisons on other fingerings (G4, A4, B

b4 and

B4) show that thresholds reached by the musicianare at least 95 % higher and up to 240 % higherthan thresholds observed on the artificial mouth.On the other hand, thresholds obtained by thenon musician are at most 13.3 % lower and 29 %higher than thresholds of the artificial mouth.

Figure 3 presents the comparison between theexperienced flutist, the non musician and the arti-ficial mouth in terms of hysteresis. For the threecases, the difference between the thresholds ob-tained performing an increasing and a decreasingblowing pressure ramp are represented for the fivefingerings studied. One can observe that the mu-sician reaches hysteresis between 169 % and 380% wider than the artifical mouth for the F4, G4,A4 and Bb

4 fingerings, and up to 515 % wider thanthe artificial mouth for the B4 fingering. The hys-teresis observed for the non musician are between27 % and 233 % wider than the hysteresis ob-tained with the artificial mouth. One can notethat the maximum relative difference of 233 % isobtained for the B4 fingering. For all the otherfingerings, the relative differences with the artifi-cial mouth remain between 27% and 65%. In allcases, the hysteresis obtained by the experiencedflutist are at least 84 % wider than that observedfor the non musician.

As a first conclusion, one can consider that thebehaviour of a given instrument played by the ar-tificial mouth and by a non musician is not sig-nificantly different in terms of increasing regimechange thresholds. In terms of hysteresis, if theresults are not significantly different for the F4,A4 and Bb

4 fingerings, more important differencesare observed for both the G4 and B4 fingerings.However, the values measured for the experiencedflutist remain significantly higher, both in termsof thresholds and hysteresis, than that obtainedfor the non player and the artificial mouth. Anexperienced flutist is able to significantly and sys-tematically modify these thresholds, and thus toenlarge the hysteresis, which presents an obviousmusical interest.

100 200 300 400 500 600 700 800 900

300

350

400

450

500

550

600

650

700

750

Blowing pressure (Pa)

freq

uenc

y (H

z)

Experienced flutistNon musicianArtificial mouth

Figure 1: Oscillation frequency with respect tothe blowing pressure, for the F4 fingering ofan alto Zen recorder, played by an experiencedflutist, a non musician and an artificial mouth.Oscillations around 350 Hz and 740 Hz corre-spond to the first and second register, respec-tively.

Which parameters does the musician use tocontrol the regime change thresholds?If the influence of the blowing pressure has been

widely studied under hypothesis of quasi-staticvariations [1, 2, 3, 4, 5, 6, 8, 9] (called hereafterthe static case), and if studies have focused onthe measurement of various control parameters[10, 11, 12] to the authors’ knowledge, no studyhas ever focused on the influence of the blowingpressure dynamics on the behaviour of flute-likeinstruments. Moreover, recent works have shownthe strong influence of this parameter on oscilla-tion thresholds of reed instruments [13, 14], and

2

F4 G4 A4 Bb4 B4

200

300

400

500

600

700

800

900

1000

1100In

crea

sing

pre

ssur

e th

resh

old

(Pa)

(jum

p be

twee

n th

e fir

st a

nd th

e se

cond

reg

iste

rs)

artificial mouthnon musicianexperienced flutist

Figure 2: Increasing pressure thresholds corre-sponding to the jump from the first to the secondregister of an alto recorder played by an expe-rienced flutist, a non musician and an artificialmouth, for five fingerings.

F4 G4 A4 Bb4 B40

50

100

150

200

250

300

350

400

450

hyst

eres

is o

n th

e re

gim

e ch

ange

thre

shol

dbe

twee

n th

e tw

o fir

st r

egis

ters

(P

a)

artificial mouthnon musicianexperienced flutist

Figure 3: Hysteresis on the jump between the twofirst registers of an alto recorder played by an ex-perienced flutist, a non musician and an artificialmouth, for five fingerings.

thus suggest that it could be a control parameterfor musicians. In the same way, as recent studies[3, 9] have highlighted that the phenomenon ofregister change in flute-like instruments is relatedto a bifurcation of the system, corresponding toa loss of stability of a periodic solution branch, itsuggests to consider the results of the theory ofdynamic bifurcations [15]. This theory takes intoaccount time evolution of the bifurcation param-eters.

This paper focuses on the influence of the

dynamics of linearly increasing and decreasingramps of the blowing pressure on the regimechange thresholds between the two first registersof flute-like instruments. In section 2, the state-of-the-art physical model for flutes is briefly pre-sented, as well as the instrument used for experi-ments, and the numerical and experimental toolsinvolved in this study. Experimental and numeri-cal results are presented in section 3, highlightingthe strong influence of the slope of a linear rampof the blowing pressure on the thresholds. Finally,a phenomenological modelling of regime change isproposed and validated in section 4, which couldlead to a prediction of regime change thresholdsand associated hysteresis.

2 Experimental and numerical

tools

In this section, experimental and numerical toolsused throughout the article are introduced.

2.1 Measurements on musicians

For the present study, an alto Bressan Zen-Onrecorder adapted for different measurements andwhose geometry is described in [16] has beenplayed by the professional recorder player Ma-rine Sablonniere. As illustrated in figure 4, twoholes were drilled to allow a measurement ofboth the mouth pressure Pm, through a capillarytube connected to a pressure sensor HoneywellASCX01DN, and the acoustic pressure in the res-onator (under the labium), through a differentialpressure sensor endevco 8507C-2.

2.2 Pressure controlled artificial

mouth

Such experiments with musicians do not allowa systematic and repeatable exploration of theinstrument behaviour. To play the instrumentwithout any instrumentalist, a pressure controlledartificial mouth is used [7]. This setup allowsto control precisely the blowing pressure, and tofreeze different parameters (such as the configu-ration of the vocal tract or the distance betweenthe holes and the fingers) which continuously varywhen a musician is playing. As described infigure 5, a servo valve connected to compressedair controls the flow injected in the instrumentthrough a cavity representing the mouth of the

3

musician. Every 40 µs, the desired pressure (thetarget) is compared to the pressure measured inthe mouth through a differential pressure sensorendevco 8507C-1. The electric current sent tothe servo valve, directly controlling its openingand thus the flow injected in the mouth, is thenadjusted using a Proportional Integral Derivativecontroller scheme. It is implemented on a DSPcard dSpace 1006 [7].

Figure 4: Experimental setup with the adaptedrecorder, allowing to measure both the pressure inthe mouth of the flutist and the acoustic pressureunder the labium.

Figure 5: Schematic representation of the princi-ple of the artificial mouth. The opening of theservo valve, controlling the flow injected in themouth, is adapted every 40 µs in order to mini-mize the difference between the measured and thedesired values of the pressure in the mouth.

2.3 Physical model of the instrument

In parallel of experiments, the behaviour of thestate-of-the-art model for flute-like instruments isstudied through time-domain simulations and nu-merical continuation, and qualitatively comparedbelow to experimental results, giving rise to a bet-ter understanding of the different phenomena in-volved.

As for other wind instruments, the mechanismof sound production in flute-like instruments canbe described as a coupling between a nonlinearexciter and a linear, passive resonator, the laterbeing constituted by the air column contained inthe pipe [17, 18]. However, they differ from otherwind instruments in the nature of their exciter:whereas it involves the vibration of a solid ele-ment for reed and brass instruments (a cane reedor the musician’s lips), it is constituted here bythe nonlinear interaction between an air jet anda sharp edge called labium (see for example [19]),as illustrated in figure 6.

More precisely, the auto-oscillation process ismodeled as follows: when the musician blows intothe instrument, a jet with velocity Uj and semi-half width b is created at the channel exit. As thejet is naturally unstable, any perturbation is am-plified while being convected along the jet, fromthe channel exit to the labium. The convectionvelocity cv of these perturbations on the jet is re-lated to the jet velocity itself through: cv ≈ 0.4Uj

[20, 21, 22]. The duration of convection intro-duces a delay τ in the system, related both tothe distance W between the channel exit and thelabium (see figure 6) and to the convection ve-locity cv through: τ = W

cv. Due to its insta-

bility, the jet oscillates around the labium witha deflection amplitude η(t), leading to an alter-nate flow injection inside and outside the instru-ment. These two flow sources Qin and Qout inphase opposition (separated by a small distanceδd, whose value is evaluated by Verge in [23]) actas a dipolar pressure source difference ∆psrc(t) onthe resonator [1, 23, 24], represented through itsadmittance Y . The acoustic velocity vac(t) of thewaves created in the resonator disrupts back theair jet at the channel exit. As described above,this perturbation is convected and amplified alongthe jet, toward the labium. The instability is am-plified through this feedback loop, leading to self-sustained oscillations. This mechanism of soundproduction can be represented by a feedback loop

4

system, represented in figure 7.

Figure 6: Schematic representation of the jet be-haviour, based on Fabre in [19]. (a) Perturbationof the jet at the channel exit by the acoustic fieldpresent in the resonator. (b) Convection and am-plification of the perturbation, due to the unsta-ble nature of the jet. (c) Jet-labium interaction:oscillation of the jet around the labium, whichsustains the acoustic field.

According to various studies describing the dif-ferent physical phenomena involved ([5, 20, 21,22, 25] for the jet, [1, 23, 24] for the aeroacousticsource), the state-of-the-art model for flute-likeinstruments [19] is described through system 1,in which each equation is related to a given ele-ment of the feedback loop system of figure 7:

Figure 7: Basic modeling of sound productionmechanism in flute-like instruments, as a systemwith a feedback loop [26, 19].

η(t) =h

UjeαiW vac(t− τ)

∆p(t) =∆psrc(t) + ∆plos(t)

=ρδdbUj

W

d

dt

[tanh

(η(t)− y0

b

)]

− ρ

2

(vac(t)

αvc

)2

sgn(vac(t))

Vac(ω) = Y (ω) · P (ω)

=

[a0

b0jω + c0+

p−1∑

k=1

akjω

ω2k − ω2 + jω ωk

Qk

]· P (ω)

(1)

In these equations, αi is an empirical coefficientcharacterizing the amplification of the jet pertur-bations [20, 25], ρ is the air density, and y0 theoffset between the labium position and the jet cen-terline (see figure 6). Vac and P are respectivelythe frequency-domain expressions of the acousticvelocity at the pipe inlet vac(t) and the pressuresource δp(t).In the second equation, the additional term

∆plos = −ρ2

(vac(t)αvc

)2sgn(vac(t)) models nonlin-

ear losses due to vortex shedding at the labium[27]. αvc is a vena contracta factor (estimated at0.6 in the case of a sharp edge), and sgn representsthe sign function.The admittance Y (ω) is represented in the

frequency-domain as a sum of resonance modes,including a mode at zero frequency (the so-calleduniform mode [26]). The coefficients ak, ωk andQk are respectively the modal amplitude, the res-onance pulsation and the quality factor of the kth

5

resonance mode, ω is the pulsation, and a0, b0and c0 are the coefficients of the uniform mode.For the different fingerings of the recorder usedfor experiments, these coefficients are estimatedthrough a fit of the admittance. These admit-tances are estimated through the measure of thegeometrical dimensions of the bore of the recorderand the use of the software WIAT [28]. Thelength corrections related to the excitation win-dow of the recorder (see figure 6) are subsequentlytaken into account using the analytical formulasdetailed in chapter 7 of [26].

2.3.1 Numerical resolution methods

Time-domain simulations of this model are car-ried out through a classical Runge-Kutta methodof order 3, implemented in Simulink [29]. A highsampling frequency fs = 23 × 44100 Hz is used.This value is chosen both because the solutionis not significantly different for higher samplingfrequencies, and because it allows an easy resam-pling at a frequency suitable for audio productionsystems.

In parallel, equilibrium and periodic steady-state solutions of the model are computed us-ing orthogonal collocation (see for example [30])and numerical continuation [31]. Starting from agiven equilibrium or periodic solution, continua-tion methods, which rely on the implicit functiontheorem [32], compute the neighbouring solution,i.e the solution for a slightly different value ofthe parameter of interest (the so-called contin-uation parameter), using a prediction-correctionmethod. This iterative process is schematicallyrepresented in figure 8. It thus aims at followingthe corresponding branch (that is to say ”fam-ily”) of solutions when the continuation param-eter varies. For more details on these methodsand their adaptation to the state-of-the-art flutemodel, the reader is referred to [33, 34] and [9].The stability properties of the different parts ofthe branches are subsequently determined usingthe Floquet theory (see for example [35]).

For a given dynamical system, the compu-tation of both the different branches of equi-librium and periodic solutions and their stabil-ity, here achieved with the software DDE-Biftool[36, 37, 34], leads to bifurcation diagrams. Suchdiagrams ideally represent all the branches ofequilibrium and periodic solutions as a functionof the continuation parameter, and provide ac-

Figure 8: Schematic representation of theprinciple of numerical continuation through aprediction-correction algorithm [31, 36]. Startingfrom a known part of the branch, the neighbour-ing solution (for a slightly different value of thecontinuation parameter λ) is predicted and cor-rected. By succesive iterations, it leads to thecomputation of the complete solution branch ofequilibrium or periodic solutions. x represents acharacteristic of the solution, such as its frequencyor its amplitude.

cess to specific information that are not possibleto access experimentally or in time-domain sim-ulations: unstable parts of the branches, coexis-tence of different solutions, and bifurcations aris-ing along the different branches. Thereby, a bi-furcation diagram provides a more global knowl-edge of the system dynamics and an easier in-terpretation of different phenomena observed ex-perimentally and in time-domain simulations, asillustrated for example in [38, 39, 9]. This willbe illustrated by figure 11 provided in section 3,which represents such a diagram of the state-of-the-art model of flute-like instruments, in termsof oscillation frequency of the periodic solutionswith respect to the blowing pressure.

3 Linear ramps of the blowing

pressure: experimental and

numerical results

3.1 Influence of the slope of blowing

pressure ramps on thresholds

As highlighted in section 1, important differencesarise, in terms of regime change thresholds andhysteresis, between experienced flutist and arti-ficial mouth or non musician, which remain un-explained. Recent works [13, 14] have demon-

6

strated the strong influence of the dynamics ofcontrol parameters on the oscillation threshold ofreed instruments. Particularly, it has highlighted,in such instruments, the phenomenon of bifurca-tion delay, corresponding to a shift of the oscil-lation threshold caused by the dynamics of thecontrol parameter [15]. Although we focus hereon transitions between the two first registers (i.e.between two different oscillation regimes), and al-though flute-like instruments are mathematicallyquite different dynamical systems from reed in-struments, these former studies suggest that thetemporal profile Pm(t) of the pressure dynamicscould considerabily influence the regime changethresholds. We focus in this section on the com-parison of regime change thresholds between thestatic case and the dynamic case, the latter cor-responding a time varying blowing pressure.

To test this hypothesis, linearly increasing anddecreasing blowing pressure ramps Pm(t) = Pini+a · t, with different slopes a, have been runboth through time-domain simulations and ex-periments with the artificial mouth (the reader isreferred to appendix A for a table of notations).Figure 9 represents, for the F4 fingering, the dy-namic pressure thresholds Pdyn corresponding tothe jump between the two first registers, with re-spect to the slope a. The positive and negativevalues of a correspond to increasing and decreas-ing ramps of Pm(t) respectively. For each valueof a, the experimental threshold is a mean valuecalculated for three realisations. In this paper,the value of Pdyn is determined through a fun-damental frequency detection using the softwareYin [40]: Pdyn is defined as the value of Pm atwhich a jump of the fundamental frequency is ob-served. The temporal resolution of the detectionis 0.0016 s for experimental signals and 0.0007s for simulation signals, which corresponds to aresolution of 0.8 and 0.36 Pa (respectively), inthe case of a slope a = 500 Pa/s of the blowingpressure. Despite the dramatic simplifications ofthe model, these first results higlight that the realinstrument and the model present similar qualita-tive behaviours. Surprisingly enough, the experi-mental and numerical behaviours are also quanti-tatively similar, with typical relative differencesbetween 3 % and 28 % on the thresholds ob-served for rising pressure (called increasing pres-sure threshold Pdyn 1→2). For the decreasing pres-sure threshold Pdyn 2→1, observed for diminishingpressure, the difference is more important, with a

−500 0 500100

150

200

250

300

350

400

a (Pa/s)

Pdy

n (P

a)

artificial mouthtime−domain simulation

Figure 9: Dynamic regime change threshold be-tween the two first registers of the F4 fingering,with respect to the slope a of linear ramps: arti-ficial mouth and time-domain simulation.

typical relative deviation of about 50 %. More-over, the strong influence of a on both Pdyn 1→2

and Pdyn 2→1 is clearly pointed out: with the ar-tificial mouth, a = 400 Pa/s leads to a valueof Pdyn 1→2 45% higher than a = 10 Pa/s, andto a value of Pdyn 2→1 16% lower. Similarly, fortime-domain simulations, a = 400 Pa/s leads to avalue of Pdyn 1→2 15.5% higher and to a value ofPdyn 2→1 18% lower than a = 10 Pa/s. Increasinga thus enlarges the hysteresis; indeed Pdyn 1→2

and Pdyn 2→1 are respectively increased and de-creased. This can be compared (at least quali-tatively) with phenomena observed on an experi-enced flutist, presented in section 1.

Figure 10 represents, as previously, the meanvalue of the regime change thresholds Pdyn 1→2

and Pdyn 2→1 obtained for three experiments,with respect to the slope a, for the other fingeringsalready studied in section 1. It higlights that thebehaviour observed in figure 9 for the F4 fingeringlooks similar for other fingerings of the recorder.Indeed, depending on the fingering, the increaseof a from 20 Pa/s to 400 Pa/s leads to an in-crease of Pdyn 1→2 between 13 % and 43 % and toa decrease of Pdyn 2→1 from 3 % to 15 %. Again,these results can be qualitatively compared withthe results presented in section 1 for an experi-enced musician.

7

−400 −300 −200 −100 0 100 200 300 400 50050

100

150

200

250

300

350

400

450

500

550

600

a (Pa/s)

Pdy

n (P

a)

F4 fingering

G4 fingering

A4 fingering

B4 fingering

Bb4 fingering

Figure 10: Transition between the two first reg-isters of an alto recorder played by an artificialmouth, for five different fingerings: representationof the dynamic regime change thresholds with re-spect to the slope a of linear ramps of the blowingpressure.

3.2 Influence of the slope of blowing

pressure ramps on oscillation fre-

quency and amplitude

As observed for the oscillation threshold inclarinet-like instruments [13], we show in this sec-tion that a modification of the regime changethreshold does not imply a strong modificationof the characteristic curves, observed in the staticcase, linking the oscillation amplitude and the os-cillation frequency to the blowing pressure. Fornumerical results, this feature can be easily il-lustrated through a comparison between the re-sults of time-domain simulations and the bifur-cation diagrams obtained through numerical con-tinuation. This is done in figure 11, in terms offrequency with respect to the blowing pressurePm, for modal coefficients corresponding to theG4 fingering. In this figure, the two periodic so-lution branches correspond to the first and thesecond registers, and solid and dashed lines rep-resent stable and unstable parts of the branches,respectively. As the computation of such a bi-furcation diagram relies on the static bifurcationtheory, the point where the first register becomesunstable, at Pm = 311.5Pa, corresponds to thestatic threshold Pstat 1→2 from first to second reg-ister. It thus corresponds to the threshold thatwould be observed by choosing successive con-stant values of the blowing pressure, and letting

the system reach a steady-state solution (here,the first or the second register). In the sameway, the point at which the change of stabilityof the second register is observed corresponds tothe static threshold from second to first registerPstat 2→1 = 259Pa. Figure 11 shows that for highvalues of a, the system follows the unstable partof the branch corresponding to the first register:the maximum relative difference between the fre-quency predicted by the bifurcation diagram andthe results of time-domain simulations is 9 cents.In the dynamic case, the system thus remains onthe periodic solution branch corresponding to the”starting” regime (the first register in figure 11),after it became unstable.

Providing, for the A fingering, the oscillationamplitude as a function of Pm for different val-ues of a, figure 12 highlights that the same prop-erty is observed experimentally. In both cases, thevalue of a considerabily affects the register changethresholds. However, far enough from the jumpbetween the two registers, the oscillation ampli-tude only depends on the value of Pm, and doesnot appear significantly affected by the value ofa.

In figure 12, the comparison between the twoslowest ramps (20 Pa/s and 100 Pa/s) and thetwo others is particularly interesting. Indeed, forthe two slowest ramps, an additional oscillationregime, corresponding to a quasiperiodic sound(called multiphonic sound in a musical context)[6, 41, 42, 3, 43], is observed for blowing pressurebetween 300 Pa and 400 Pa for a = 20 Pa/s, andbetween 340 and 400 Pa for a = 100 Pa/s. Asthis regime does not appear for higher slopes, ithighlights that a modification of the blowing pres-sure dynamics can allow the musician to avoid (orconversely to favor) a given oscillation regime.

3.3 Influence of the pressure dynamics

before the static threshold

To better understand the mechanisms involved inthe case of a dynamic bifurcation between tworegisters, this section focuses on the influence, onthe regime change thresholds, of the evolution ofPm(t) before the static threshold Pstat has beenreached. In other words, the aim is to determinewhether the way Pm(t) evolves before the staticthreshold is reached impacts the dynamic regimechange threshold.

To investigate this issue, different piecewise lin-

8

100 200 400 500

400

500

600

700

800

900

1000

1100

Pm

(Pa)

Osc

illat

ion

freq

uenc

y of

Vac

(H

z)

stable solutionsunstable solutionstime−domain simulation a=40 Pa/stime−domain simulation a=400 Pa/stime−domain simulation a=800 Pa/s

Pstat

1−−>2

Pstat

2−−>1

Figure 11: Bifurcation diagram of the G finger-ing, superimposed with time-domain simulationsof increasing linear ramps of the blowing pres-sure, for different values of the slope a: rep-resentation of the oscillation frequency with re-spect to the blowing pressure Pm. For the bi-furcation diagram, the two branches correspondto the first and the second register, solid anddashed lines represent stable and unstable partsof the branches, respectively. The vertical dottedlines highlight the static regime change thresholdsPstat 1→2 and Pstat 2→1

0 100 200 300 400 500 6000

100

200

300

400

500

600

700

800

900

osci

llatio

n am

plitu

de (

Pa)

Pm

(Pa)

a = 20 Pa/sa = 100 Pa/sa = 280 Pa/sa = 340 Pa/s

1st register

2nd register

quasiperiodicregime

transient

Figure 12: Increasing linear ramps of the blowingpressure, with different slopes a, achieved with anartificial mouth: oscillation amplitude of the A4

fingering of an alto recorder, with respect to theblowing pressure.

ear ramps have been achieved both with the arti-ficial mouth and in time-domain simulation. Forrising pressures, these profiles are defined such

as dPm

dt = a1 for Pm < Pknee and dPm

dt = a2 forPm > Pknee (where a1 and a2 are constants) andPknee is a constant that may be adjusted. For di-minishing pressure, they are such as dPm

dt = a1 for

Pm > Pknee and dPm

dt = a2 for Pm < Pknee.

3.3.1 Experimental results

Experimentally, blowing pressure profiles consti-tuted by two linear ramps with different slopes(a1 = 350 Pa/s, a2 = 40 Pa/s) have been achievedfor the G4 fingering. The pressure Pknee at whichthe knee break occurs varies between the differentrealisations.

Figure 13 presents these experimental resultsin terms of Pdyn 1→2 and Pdyn 2→1, with respectto Pknee − Pstat. Thereby, a zero abscissa cor-responds to a change of slope from a1 to a2 ata pressure equal to Pstat 1→2 for rising pressure,and equal to Pstat 2→1 for diminishing pressure.It highlights that for rising pressure, Pdyn 1→2 re-mains constant as long as Pknee < Pstat 1→2 (i.e.for negative values of the abscissa), and that thisconstant value (about 258 Pa) corresponds to thevalue of Pdyn 1→2 previously observed for a linearramp with constant slope a2 = 40 Pa/s (see fig-ure 10). Conversely, once Pknee > Pstat 1→2, thevalue of Pdyn 1→2 gradually increases to reach 295Pa, which corresponds to the value observed for alinear ramp with a contant slope a1 = 350 Pa/s.

The same behaviour is observed for the decreas-ing threshold: as long as Pknee > Pstat 2→1, thevalue of Pdyn 2→1 is almost contant and close tothat observed previously for a linear ramp of con-stant slope a2 = 40 Pa/s (see figure 10). How-ever, for Pknee < Pstat, the value of Pdyn 2→1 pro-gressively decreases to about 142 Pa, which cor-responds to that observed for a linear ramp ofconstant slope a1 = 350 Pa/s (see figure 10).

As a conclusion, as long as the slope break oc-curs before the static threshold has been reached,the dynamic threshold is driven by the slope ofthe second part of the blowing pressure profile.If it occurs just after the static threshold hasbeen reached, the dynamic threshold lies betweenthe dynamic thresholds corresponding to the twoslopes of the blowing pressure profile. Finally, ifthe slope break occurs, for rising pressure, at apresure sufficiently higher (respectively lower fordiminishing pressure) than the static threshold,the dynamic threshold is driven, as expected, bythe slope of the first part of the blowing pressure

9

−100 −50 0 50 100

260

270

280

290

300

(Pknee

− Pstat 1 −−> 2

) (Pa)

Pdy

n 1

−−

> 2

(P

a)

−100 −50 0 50 100

145

150

155

160

165

( Pknee

− Pstat 2 −−> 1

) (Pa)

Pdy

n 2

−−

> 1

(P

a)

Pdyn 1−>2

(a1)

Pdyn 1−>2

(a2)

Pdyn 2−>1

(a1)

Pdyn 2−>1

(a2)

Figure 13: Piecewise linear ramps of the blow-ing pressure (a1 = 350 Pa/s and a2 = 40 Pa/s),achieved on the G4 fingering of an alto recorderplayed by an atificial mouth. Ordinate: dynamicthreshold Pdyn 1→2 (up) and Pdyn 2→1 (down).Abscissa: difference between the pressure Pknee

at which the knee occurs and the static regimechange threshold Pstat. Dashed lines representthe dynamic regime change thresholds observedpreviously for linear ramps of constant slope a1and a2 respectively.

profile.

3.3.2 Results of time-domain simulations

These experimentally observed behaviours arealso observed on numerical simulations of themodel. For modal coefficients corresponding tothe G4 fingering, the comparison has been madebetween the dynamic thresholds obtained forthree different cases:

• linear increasing ramps of Pm(t), with slopea2.

• a first piecewise linear increasing ramp, witha slope change at Pknee = 250Pa, and a fixedvalue of a1 = 500 Pa/s.

• a second piecewise linear increasing ramp,with a slope change at Pknee = 250Pa, and afixed value of a1 = 200 Pa/s.

It is worth noting that for the two kinds of piece-wise linear ramps, Pknee is lower than Pstat 1→2,predicted by a bifurcation diagram at 311.5 Pa(see figure 11). For each case, various simulationswere achieved, for different values of a2.

0 200 400 600 800 1000

330

340

350

360

370

380

390

400

410

a2 (Pa/s)

Pdy

n 1

−−

> 2

(P

a)

Pknee = 270 Pa; a

1 = 500 Pa/s

Pknee = 270 Pa; a1 = 200 Pa/s

linear ramp

Figure 14: Time-domain simulations of piece-wise linear ramps of the blowing pressure withPknee = 270Pa (a1 = 500 Pa/s for squares anda1 = 200 Pa/s for circles) and of linear ramps ofthe blowing pressure (crosses). Representation ofthe increasing dynamic regime change thresholdPdyn 1→2 for the G4 fingering, as a function of a2(slope of the second part of the blowing pressureprofile for piecewise linear ramps, and slope of thelinear ramps).

Figure 14 provides the comparison of value ofPdyn 1→2 obtained for these three kinds of blow-ing pressure profiles as a function of a2. With amaximum relative difference of 3.5%, the thresh-olds obtained for the piecewise linear profiles arestrongly similar to those obtained with linearramps. As for the experimental results, if Pknee <Pstat 1→2, the dynamic threshold Pdyn 1→2 is thusdriven by the second slope a2 of the profile.

For the particular profile in which a1 = 500Pa/s and a2 = 830 Pa/s, the influence of the valueof Pknee on Pdyn 1→2 has been studied. The re-sults are represented in figure 15 in the same wayas the experimental results in figure 13. As ex-perimentally, if Pknee < Pstat 1→2, the value ofPdyn 1→2 is driven by a2, and a constant thresh-old of about 385 Pa is observed, corresponding tothe value obtained for a linear ramp with a slopeequal to a2 = 830 Pa (see figure 14). Conversely,when Pknee > Pstat1→2, Pdyn1→2 gradually shiftsto finally achieve the value of 369 Pa, equal tothat oberved for a linear ramp with a slope equalto a1 = 500 Pa/s. The dynamic threshold is thendriven by a1.

10

−100 −50 0 50 100

368

370

372

374

376

378

380

382

384

386

388

Pknee

− Pstat 1−−>2

Pdy

n 1

−−

>2

Pdyn 1−>2

(a1)

Pdyn 1−>2

(a2)

Figure 15: Time-domain simulations of piecewiselinear ramps of the blowing pressure (a1 = 500Pa/s and a2 = 830 Pa/s), for the G4 fingering.Ordinate: dynamic threshold Pdyn 1→2. Abscissa:difference between the pressure Pknee at which theknee occurs and the static regime change thresh-old Pstat 1→2. Dashed lines represent the valuesof Pdyn 1→2 previously observed for linear rampsof slope a1 and a2.

3.4 Comparison with the results of an

experienced musician

The strong influence of the dynamics of Pm(t) onthresholds and hysteresis suggests, by comparisonwith results presented in section 1, that musiciansuse this property to access to a wider control interms of nuances and timbre. However, the com-parison between the musician and the artificialmouth (see figures 2, 3 and 10) shows that the val-ues of Pdyn 1→2 obtained by the musician remains,for the different fingerings studied, between 61 %and 134 % higher than the maximal thresholdsobtained with the artificial mouth for high valuesof the slope a. In the same way, the hysteresis ob-tained by the musician remains between 26% and102% wider than the maximal hysteresis observedwith the artifical mouth for the F4, G4, A4 andBb

4 fingerings, and up to 404 % wider for the B4

fingering.

3.5 Discussion

These results bring out the strong influence of thedynamics of the blowing pressure on the oscilla-tion regime thresholds in flute-like instruments.Comparisons between experimental and numer-ical results show that the substantial simplifi-

cations involved in the state-of-the-art physicalmodel of the instrument do not prevent it to faith-fully reproduce the phenomena observed exper-imentally. Suprisingly enough, different resultsshow good agreement not only qualitatively butalso quantitatively. Moreover, both the exper-imental and numerical results show that the dy-namic threshold does not depend on the dynamicsof the blowing pressure before the static thresholdhas been reached.

Although the system studied here is mathemat-ically very different from one that models reedinstruments (see for example [2, 44, 9]), and al-though focus is set here on bifurcations of peri-odic solutions, results can be compared with somephenomena highlighted by Bergeot et al. on thedynamic oscillation threshold of reed instruments[14, 13]. As in the work of Bergeot, phenomenahighlihted are not predicted by the static bifurca-tion theory, often involved in the study of musicalinstruments.

Moreover, the comparison between the resultsobtained with the artificial mouth and with an ex-perienced flutist suggests that the musicians com-bine the dynamics of the blowing pressure withother control parameters in order to enlarge thehysteresis associated to regime change. Indeed,other works on flute-like instruments [45, 46], to-gether with different studies on other wind instru-ments [47, 48, 49] suggests that the vocal tract canalso influence the regime change thresholds.

4 Toward a phenomenological

model of register change

The different properties of the register changephenomenon, observed both experimentally andin simulations in the previous part, allow to pro-pose a preliminary phenomenological modelling ofthis phenomenon.

4.1 Proposed model

Starting from the results presented in figures 13,14, and 15, which lead to the conclusion that Pdyn

only depends on the dynamics of the blowing pres-sure after the static threshold has been reached,this modelling is based on the following hypothe-sis:

• The regime change starts when Pm(t) =Pstat.

11

• The regime change is not instantaneous, andhas a duration tdyn during which the blowingpressure evolves from Pstat to Pdyn.

We thus write Pdyn as the sum of the staticthreshold Pstat and a correction term Pcorr relatedto the dynamics of the blowing pressure:

Pdyn = Pstat + Pcorr. (2)

Based on the two hypothesis cited above, weintroduce a new dimensionless quantity, the frac-tion of regime change ζ(t). By definition, ζ = 0when the regime change has not started (i.e.when Pm(t) < Pstat for rising pressure and whenPm(t) > Pstat for diminishing pressure), and ζ =1 when the regime change is completed (i.e. whenPm(t) = Pdyn, which corresponds to the change offundamental frequency, as defined in the previoussection). ζ is consequently defined as:

ζ(t) =

∫ t

tstat

∂ζ

∂tdt (3)

where tstat is the instant at which Pm(t) = Pstat.Defining the origin of time at tstat leads to t =t− tstat, and thus gives:

ζ(t) =

∫ t

0

∂ζ

∂tdt (4)

As a simplifiying assumption, we consider thatthe rate of change ∂ζ

∂tof the variable ζ(t) only de-

pends on the gap ∆P (t) = Pm(t)−Pstat betweenthe mouth pressure Pm(t) and the static regimechange threshold:

∂ζ

∂t= f(∆P ), (5)

where f is an unknown monotonous and continu-ous function.

According to the latest hypothesis, function fcan be estimated at different points through therealisation of ”steps” profiles of Pm(t), from avalue lower than Pstat 1→2, to a value larger thanPstat 1→2 (see figure 16). Indeed, in such a case,for a step occuring at t = 0, ∆P (t) correspondsto the difference between the pressure at the topof the step and Pstat 1→2, and is thus constant fort > 0. Consequently, f(∆P ) is constant with re-spect to time. From equations 4 and 5, one thusobtains for blowing pressure steps:

ζ(t) =

∫ t

0f(∆P )dt

=f(∆P )

∫ t

0dt

=f(∆P ) · t

(6)

Recalling that tdyn is the instant at which Pm(t) =Pdyn, we have by definition ζ(tdyn) = 1, and fi-nally obtain for blowing pressure steps:

f(∆P ) =1

tdyn(7)

For each value of the step amplitude, a differ-ent value of tdyn is obviously measured through afrequency detection: tdyn is defined as the timeafter which the oscillation frequency varies nomore than two times the frequency resolution.Therefore, successive time-domain simulations ofPm steps (see figure 16) with different amplitudesare carried out, to determine the function f(∆P )through equation 7. Such simulations have beenachieved for the two fingerings F4 and G4, in bothcases for transitions from the first to the secondregister. The results are represented in figure 17with respect to ∆P .

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7240

280

Pstat

340

time (s)

Blo

win

g pr

essu

re P

m

0 tdyn 0,4 0,6

200

400

600

800

1000

time (s)

Osc

illat

ion

freq

uenc

y

∆ P

regime change

duration

Figure 16: Illustration of the step profiles of theblowing pressure (up) achieved in time-domainsimulations, and of the detection of the transientduration tdyn (down).

In the two cases, the results follow a squareroot function: the linear correlation coefficients

between ∆P and(

1tch

)2are of 0.96 for the F4

12

0 20 40 60 80 1000

1

2

3

4

5

6

7

8

9

∆ P (Pa)

1/t dy

n (in

vers

e of

the

tran

sien

t dur

atio

n)

0 10 20 30 400

1

2

3

4

5

6

∆ P (Pa)

1/t dy

n (in

vers

e of

the

tran

sien

t dur

atio

n)

simulations − F4 fingering

fit sqrt

simulations − G4 fingering

fit sqrt

Figure 17: Estimation of the function f(∆P ):representation of the inverse of the transient du-ration for step profiles of the blowing pressure,for both the F4 and G4 fingerings (left and rightrespectively), with respect to the difference ∆Pbetween the target pressure of the steps and thestatic threshold Pstat 1→2. Dashed lines representfit of the data with square root functions.

fingering and 0.97 for the G4 fingering. Such re-sults thus suggest to approximate the function fthrough:

f(∆P ) = α√

(∆P ); (8)

where the coefficient α depends on the consideredfingering.

4.2 Assessment of the model

To check the validity of this modelling, the caseof the linear pressure ramps studied in the previ-ous section is now examined. In such a case, thedifference between the blowing pressure and thestatic threshold is defined through ∆P (t) = a · t,where a is the slope of the ramp in Pa/s. Recall-ing that ζ(tdyn) = 1 and injecting equations 5 and8 in equation 4 leads to:

∫ tdyn

0f(∆P (t))dt = 1

∫ tdyn

0α

√∆P (t)dt = 1

∫ tdyn

0α√

at · dt = 1

∫ tdyn

0

√tdt =

1

α√a

tdyn =

(3

2α√a

) 2

3

(9)

Moreover, due to the expression of ∆P (t) in thecase of linear ramps, one can write from equations2 and 9:

Pcorr =Pdyn − Pstat

=∆P (tdyn)

=a · tdyn

=

(3

2αa

) 2

3

(10)

According to this modelling, the value of Pcorr

obtained with linear ramps should be propor-tional to the slope a to the power 2/3. Time-domain simulations for linear ramps of Pm(t) withslope a are performed for two fingerings (F4 andG4). Figure 18 represents the threshold Pdyn cor-responding to the end of the transition from thefirst to the second register with respect to theslope a power 2/3. The results are correctly fit-

0 20 40 60 80 100 120 140280

300

320

340

360

380

400

420

440

a2/3 ((Pa/s)2/3)

Pdy

n (P

a)

F4 fingering

G4 fingering

linear fitlinear fit

Figure 18: Time-domain simulations of linear in-creasing ramps of the blowing pressure, for boththe F4 fingering (+) and the G4 fingering (x): rep-resentation of the dynamic regime change thresh-old Pdyn 1→2 with respect to the power 2/3 of theslope a. Solid and dashed lines represent linear fitof the data, which both present linear correlationcoefficients higher than 0.99.

ted by straight lines, with correlation coefficientshigher than 0.99. This good agreement with themodel prediction (equation 10) thus allows to val-idate the proposed modelling of the phenomenonof regime change. Moreover, on such a repre-sentation, the intercept of the fit with the y-axisprovides a prediction of the static regime changethreshold, which can not be exactly determined,strictly speaking, with linear ramps of the blowingpressure. The static thresholds thereby obtained

13

are 294 Pa and 314 Pa for the F4 and G4 fin-gering respectively. These values present relativedifferences of 0.1% and 0.8% with the thresholdsof 294.3 Pa and 311.5 Pa predicted by the bi-furcation diagrams computed through numericalcontinuation (see figure 11 for the bifurcation di-agram of the G4 fingering), which supports thevalidity of the proposed modelisation.

4.3 Case of experimental data

The experimental thresholds displayed in figure10 for the five fingerings studied are representedin figure 19 with respect to a2/3. Similarly to fig-ure 18, the different curves are correctly fitted bystraight lines, with linear correlation coefficientsbetween 0.88 and 0.99. The fact that these coeffi-cients are, in some cases, lower than those of simu-lations can be explained by the presence of noiseand of small fluctuations of the mouth pressureduring the experiment, which sometimes preventsa threshold detection as accurate and systematicas in the case of numerical results. However, thegood agreement of the experimental results withequation 10 also allows to validate the proposedphenomenological modelling of regime change.

−60 −40 −20 0 20 40 600

100

200

300

400

500

600

a2/3 ((Pa/s)2/3)

Pdy

n (P

a)

G4 fingering

A4 fingering

B4 flat fingering

B4 fingering

F4 fingering

linear fit

Figure 19: Same data as in figure 10: represen-tation of the dynamic thresholds Pdyn 1→2 andPdyn 2→1, for five fingerings of an alto recorderplayed by an artificial mouth, with respect to thepower 2/3 of the slope a of linear ramps of theblowing pressure. Solid lines represent linear fitof the data. Data present linear correlation coef-ficients between 0.88 and 0.99.

4.4 Influence of the regime of arrival

In the case of time-domain simulations, for the G4

fingering, starting from the second register andachieving linear decreasing ramps of Pm(t) leadsto a particular behaviour. As shown in figure 20,Pdyn does not appear, at least in a first stage,to be proportional to the power 2/3 of the slope.However, this case is particular in the sense thatdifferent oscillation regimes are reached, depend-ing on the slope a of the ramp. Thereby, as high-lighted with circles in figure 20, low values of theslope (|a| < 20 Pa/s) lead to a transition from thesecond to the first register, whereas higher valuesof the slope lead to a transition from the secondregister to an aeolian regime, as represented withcrosses in figure 20. In flute-like instruments, aeo-lian regimes corresponds to particular sounds, oc-curing at low values of the blowing pressure, andoriginating from the coupling between a mode ofthe resonator (here the 5th) and an hydrodynamicmode of the jet of order higher than 1 [6, 50, 3].As highlighted in the same figure, considering thetwo different transitions separately allows to find,as previously, the linear dependance between Pdyn

and a2/3. Indeed, linear correlation coefficients of0.98 for |a| < 20 Pa/s, and of 0.95 for |a| > 20Pa/s are found. Since the corresponding slope isthe inverse of 2

3α to the power 2/3 (see equation10), such results suggest that α does not only de-pend on the fingering, but also on the oscillationregimes involved in the transition.

The study of the Floquet exponents ρm of thesystem supports this hypothesis. The Floquetexponents, computed for the system linearisedaround one of its periodic solutions, allow to es-timate the (local) stability properties of the con-sidered periodic solution [51, 35]. More precisely,they provide information on whether a small per-turbation superimposed on the solution will beamplified or attenuated with time. If all the Flo-quet exponents have negative real parts, any per-turbation will be attenuated with time, and theconsidered solution is thus stable. Conversely, ifat least one of the Floquet exponents has a posi-tive real part, any perturbation will be amplifiedin the ”direction” of the phase space correspond-ing to the eigenvector associated to this exponent,and the solution is thus unstable.

The real part of the Floquet exponents of theconsidered system, linearised around the periodicsolution corresponding to the second register (i.e.

14

−100 −80 −60 −40 −20 0

180

190

200

210

Pcross

240

a2/3 ((Pa/s).2/3)

Pdy

n (P

a)

arrival regime: 1st register

arrival regime: aeolian (5th resonance mode)linear fitlinear fit

Figure 20: Time-domain simulations of linear de-creasing ramps of the blowing pressure, for the G4

fingering: representation of the dynamic regimechange threshold Pdyn 2→1 with respect to thepower 2/3 of the slope a. Circles and crossesrepresent transitions from the second register tothe first register and to an aeolian regime, respec-tively. Solid and dashed lines represent linear fitof the data, which present linear correlation co-efficients of 0.98 and 0.95, respectively. The dot-dashed line indicates the pressure at which theFloquet exponents of the starting regime cross infigure 21.

to the ”starting” regime of the decreasing blowingpressure ramps considered here), are representedin figure 21 with respect to the blowing pressurePm. It highlights that the second register is stablefor all values of Pm between 300 Pa and 259 Pa.A first Floquet exponent introduces an instabilityat Pm = 259 Pa, wich corresponds to the desta-bilisation of the second register (see figure 11).As highlighted in [9], such a destabilisation, cor-responding to a bifurcation of the second register,causes the regime change. This point is thus thestatic threshold Pstat 2→1, already highlighted infigure 11. A second Floquet exponent reaches apositive real part at Pm = 229Pa. Moreover, thereal part of the latest exponent becomes higherthan the first one for Pm < Pcross, with Pcross =224 Pa.

Comparison of results presented in figure 21with those of figure 20 suggests that, in the caseof a regime change, the ”arrival” regime is drivenby the Floquet exponent of the starting regimewith the highest real part: indeed, as highlightedin figures 20 and 21, until the dynamics of Pm(t)induces a regime change threshold higher than the

pressure Pcross for which the Floquet exponentsintersect, one observes a transition to the first reg-ister. On the other hand, once the dynamics ofPm(t) induces a threshold lower than Pcross, thetransition leads to the aeolian regime.This interpretation seems furthermore to be

consistent with the slope change observed in fig-ure 20 and with the physical meaning of the realpart of the Floquet exponents. Indeed, as thevalue of the real part of a Floquet exponent is re-lated to the amplification of a perturbation withtime, a high value of ℜ(ρm) should correspondto a small duration tdyn of the regime change,whereas a small value of ℜ(ρm) should correspondto a high value of tdyn. Therefore, by analogywith equations 7 and 8, coefficient α can be re-lated to the evolution of ℜ(ρm) with Pm. Thereby,a greater evolution of ℜ(ρm) with respect to ∆Pshould correspond to a higher value of α. Due toequation 10, valuable for linear ramps of Pm(t),it finally corresponds to a smaller rate of changeof the straight line linking Pdyn and a2/3. Thisproperty is here verified by the comparison be-tween figures 20 and 21: the real part of the ”sec-ond” floquet exponent (in bold black dashed linein figure 21), related to a regime change to theaeolian regime, presents a greater evolution with∆P = Pm − Pstat than the floquet exponent in-ducing a transition to the first register (in boldblue line in figure 21). In the same way, the rateof change of straight line related in figure 20 to theregime change toward the aeolian regime (dashedline) is smaller than that of the straight line re-lated to the transition from the second to the firstregister (in solid line).Surprisingly enough, these results thus high-

light that bifurcation diagrams and associatedFloquet stability analysis provide valuable infor-mation in the dynamic case, despite the fact thatthey involve the static bifurcation theory and alinearisation of the studied system around the”starting” periodic solution. In the dynamic case,they remain instructive on the following charac-teristics:

• the arrival regime resulting from the regimechange.

• a qualitative indication on the duration ofthe regime change, through an estimation ofthe parameter α. It thus informs on boththe dynamic threshold and its evolution withrespect to the difference ∆P between the

15

mouth pressure and the static regime changethreshold.

• as highlighted in the previous section, theevolution of the oscillation amplitude andfrequency with respect to the mouth pres-sure, even after the static threshold has beencrossed.

200 210 240 259 280 300

−50

0

50

100

150

200

250

Pm

(Pa)

Re(

ρ m)

transition to the 1st register

2nd register stabletransition to theaeolian regime

Pstat 2−>1

Pcross

= 224

Figure 21: G4 fingering: real parts of the Flo-quet exponents of the system linearised aroundthe periodic solution corresponding to the sec-ond register, with respect to the blowing pressurePm. Floquet exponents provide information onthe stability properties of the considered regime.

5 Conclusion

Recent studies in the field of musical acousticshave demonstrated that musicians are able tomodify strongly the behaviour of the instrumentconsidered alone (see for example [47, 49]), andthus argue for a wider consideration of the differ-ent control parameters.

A comparison between an experienced flutist, anon musician and an artificial mouth, in terms ofregime change thresholds between the two firstregisters, and associated hysteresis, shows thatthe experienced musician seems to have devel-oped strategies allowing him to significantly shiftthe regime change thresholds, and thus to enlargethe hysteresis, which presents an obvious musi-cal interest. Conversely, for most fingerings stud-ied, the behaviour observed when the recorderis played by a non musician and by an artifi-ical mouth do not present significant differencesin terms of regime change thresholds.

The experimental and numerical results pre-sented in this article highlight that the slopeof linear increasing and decreasing ramps of theblowing pressure strongly affects the pressureregime change thresholds, and thus the hysteresis.Moreover, it appears that the important criterionlies only in the dynamics of the blowing pressureafter the static regime change threshold has beenreached. The modification of the dynamics of theblowing pressure can thus allow, in some cases,to avoid or conversely to favor a given oscillationregime, and thereby to select the ”arrival” regimeresulting from a regime change.The phenomenological model proposed accord-

ing to these observations allows to predict the dy-namic regime change threshold from the knowl-edge of the temporal evolution of the blowingpressure. It highlights that the bifurcation dia-grams and the associated Floquet stability anal-ysis provide valuable information in the dynamiccase, despite the fact that they involve a statichypothesis and a linearisation of the studied sys-tem.However, taking into account the dynamics of

the mouth pressure does not allow to shift thethresholds and to enlarge the hysteresis as muchas the experienced flutist does. It thus suggeststhat flutists develop strategies to combine the ef-fects of the dynamics with those of other controlparameters, such as for example the vocal tract,whose influence on regime change thresholds hasbeen recently studied [46]. Moreover, the studypresented here focuses on linear profiles of themouth pressure. As such a temporal evolutiondoes not seem realistic in a musical context (seefor example [52, 11]), it would be interesting toconsider the effect of more complex temporal evo-lutions of the blowing pressure. Finally, it wouldbe interesting to study more widely step profilesof the mouth pressure, whose importance is cru-cial in the playing of winf instruments.

Aknowledgment

The authors would like to thank MarineSablonniere and Etienne Thoret for their partici-pation in experiments.

A Table of notation

16

Symbol Associated variable

Pm (Pa) Blowing pressure

a (Pa/s) Slope of linear ramps of theblowing pressure

Static pressure thresholdPstat 1→2 (Pa) from the first to the second

register (case of risingblowing pressure)

Static pressure thresholdPstat 2→1 (Pa) from the second to the first

register (case of diminishingblowing pressure)

Dynamic pressure thresholdPdyn 1→2 (Pa) from the first to the second

register (case of risingblowing pressure)

Dynamic pressure thresholdPdyn 2→1 (Pa) from the second to the first

register (case of diminishingblowing pressure)

Slope of the first part ofa1 (Pa/s) piecewise linear ramps

of the blowing pressure

Slope of the second part ofa2 (Pa/s) piecewise linear ramps

of the blowing pressure

Pressure at which the slopePknee (Pa) break occurs in the case of

piecewise linear rampsof the blowing pressure

Difference between thePcorr (Pa) static regime change

threshold and the dynamicregime change threshold

ζ (dimensionless) Fraction of regime change

tstat (s) Time at which Pm = Pstat

time variable, whose origin

t(s) is defined at tstatTime at which Pm = Pdyn

tdyn (s) (end of the regime change)

Difference between the∆P (t) (Pa) blowing pressure Pm(t) and

the static regime changethreshold Pstat

ρm Floquet exponents

Table 1: Table of notations used throughout thearticle.

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[2] R Auvray, B Fabre, and P Y Lagree. Regimechange and oscillation thresholds in recorder-like instruments. Journal of the AcousticalSociety of America, 131(4):1574–1585, 2012.

[3] S Terrien, Ch Vergez, and B Fabre. Flute-like musical instruments: A toy model in-vestigated through numerical continuation.Journal of sound and vibration, 332(1):3833–3848, 2013.

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