Physical modelling of trombone mutes, the pedal note issue

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Physical modelling of trombone mutes, the pedal note

issue

Lionel Velut, Christophe Vergez, Joël Gilbert

To cite this version:

Lionel Velut, Christophe Vergez, Joël Gilbert. Physical modelling of trombone mutes, the pedal note

issue. Acta Acustica united with Acustica, Hirzel Verlag, 2017, 103 (4), pp.668-675. �hal-01385524v2�

Lionel Velut 1

, Christophe Vergez 1

, and Joël Gilbert 2 2

1

LMA, CNRS, UPR 7051, Aix-Marseille Univ., Centrale Marseille, F-13453 3

Marseille edex 13, Fran e. 4

2

Laboratoire d'A oustique de l'Université du Maine, UMR CNRS-6613, Avenue 5

Olivier Messiaen, 72085 Le Mans edex 9, Fran e 6

February 9, 2017 7

Abstra t 8

Brass players use a variety of mutes to hange the sound of their instrument for artisti 9

expression. However, mutes an alsomodify the intonation and theplayabilityof themuted 10

instrument. An example is the use of a straight mute on a trombone, whi h makes it very 11

di ult toplay stablepedal notes. 12

Previous studies have shownthat using a straight mute establishes a subsidiary a ousti 13

resonan einthetrombone. To an elthismodi ation,ana tive ontroldevi ewasdeveloped 14

andintegratedinto amute,withsatisfyingexperimentalresults [Meurisseet al.,2015 ℄. With 15

this devi e,the perturbedpedalnotes an easily be played again. 16

This paper investigates the ability of a physi al model of brass instrument to reprodu e 17

the behaviour of the trombone pedal

B♭

without mute, or with an "a tive" or a "passive" 18

straight mute. Linear stabilityanalysisand time-domain simulations areusedto analysethe 19

behaviour of the model inthe parameter range orresponding to thepedal note. Numeri al 20

results are ompared for dierent models of instruments: a trombone, a trombone with a 21

straight (passive) mute, an a trombone with an a tive mute. It is shown that the simple 22

physi al model onsidered behaves rather qualitatively similarly to what is experien edwith 23

real instruments: the playing of the pedal note is perturbed with a passive mute, whereas 24

themodel of trombone with theexperimental a tive mute givesresults very similar to those 25

obtained withanopen trombone. 26

I Introdu tion 27

Ausual solutionfor hangingthetimbreofabrass instrument onsistsinusing amute,whi hisa 28

devi e pluggedtotheopening oftheinstrumentbell,orheldby handjustinfrontofthebell. The 29

the emitted sound [Campbelland Greated, 1994, p.398℄. As a side ee t, introdu ing anobsta le 31

inthe bell,or lose to it,alsomodies the a ousti al properties of the instrument [Ba kus, 1976℄. 32

This has various onsequen es, in luding modi ation of the instrument tuning and of reported 33

instrument-player intera tion. 34

Pedal notes are the lowest playable notes on a trombone. When the slide is fully losed, 35

the note played is a

B♭1

, orresponding to a playing frequen y of

58Hz

in equal temper-36

ament. In brass instruments, most regimes of os illation of the instrument have a playing 37

frequen y slightly above the a ousti al resonan e supporting the os illation. However, the 38

pedal note has a playing frequen y unusually far above the resonan e frequen y of the a ous-39

ti al mode supporting the os illation, making it a parti ular regime of os illation as detailed 40

in[Gilbertand Aumond,2008, Velut etal.,2017℄. Furthermore, the pedal note os illationis sup-41

ported by the lowest mode of the trombone, whi h is inharmoni with the other modes of the 42

trombone as shown in Fig. 2. When a straight mute is inserted in the trombone, playing sta-43

ble pedal notes on the three rst slides positions -

B♭1

,

A

1

and

A♭

1

- is uneasy and results in 44

a rolling, unstable sound [Slu hin and Caussé, 1991, Meurisse et al.,2015℄. Measurements of the 45

input impedan e of a trombone with a mute [Meurisseet al.,2015, Velut et al.,2016b℄ show the 46

o urren e of a subsidiary a ousti al mode between the rst and the se ond modes. 47

This paper will parti ularly fo us on the pedal

B♭1

, orresponding to the slide in the short-48

est position. It will hereinafter be referred to as "the pedal note". An a tive ontrol devi e has 49

been previously developed to remove this subsidiary mode [Meurisseet al.,2015℄, whi hmakesit 50

possible toplay the pedal note with a straight mute. It onsists of an "a tive mute", a ommer-51

ial straight mute equipped with an a tive ontrol devi e whi h an els the aforesaid subsidiary 52

resonan e mode. 53

Thepurposeofthispaperistoinvestigatetowhatextentasimpletrombonephysi almodel an 54

predi tthe ee tofa trombonestraightmute onthepedalnoteand the ee tivenessof thea tive 55

mute. The physi al modelof a brass instrumentis rst presented. Then, linear stability analysis 56

(LSA) and time-domain simulations are used to analyse the behaviour of the pedal note in this 57

model. Analyses are ondu ted on an "open trombone" onguration (tenor trombone without 58

any mute), a "passive mute" onguration(the same trombone with a ommer ial straight mute) 59

and an "a tive mute" onguration (the same trombone with an identi al straight mute and the 60

des ribed a tive ontrol loop). Results of this model are ompared with the experimental results 61

from [Meurisseet al.,2015℄. 62

II.A Brass instrument model 64

A physi almodel of trombone, suitable fora large lass of musi instruments,is presented in this 65

arti le. Following Helmholtzpioneering work [vonHelmholtz,1870℄,the trombone ismodelledas 66

a losed-loop system onsisting of an ex iter and a resonator whi h are oupled, as illustrated in 67

Fig. 1. Su h asystem an produ e auto-os illationondierentregimes. 68

69

Figure1: Closed-loopmodelin freeos illation,suitablefor the des ription of most self-sustained musi alinstruments,in ludingtrombones. Self-sustainedos illationsaregeneratedbythelo alised non-linear oupling (here the airow between lips) between a linear ex iter (here the lips) and a linear resonator (here the air olumn inside the instrumentbore).

70

Forabrassinstrument,theex iteristhelipsofthemusi ian,whi ha tasavalve: these tionof 71

the hannelbetweenthelipsdependsonthepressuredieren ethroughtheselipsaswellasontheir 72

me hani al hara teristi s. Multiplemodelsof thelipreedhavebeen proposed andused,with one 73

degreeoffreedom[Eliott and Bowsher, 1982,Flet her, 1993,Cullenet al.,2000,Silvaet al.,2007℄ 74

or 2 DOF [Ada hi and Sato, 1996, Campbell,2004, Lopez et al.,2006, Newton etal., 2008℄. The 75

modelusedforthispaperistheone-DOFvalvemodel,usuallyreferredtoasthe"outward-striking" 76

model, also alled

(+, −)

swinging-doormodelinthe literature: 77

d

2

h

dt

2

+

ω

l

Q

l

dh

dt

+ ω

2

l

(h − h

0

) =

1

µ

(p

b

− p(t)),

(1)

where

h

is the height of the lip hannel (m);

p

is the pressure at the input of the instrument, in 78

the mouthpie e (Pa);

p

b

is the onstant blowing pressure in the mouth (Pa);

ω

l

= 2πf

l

(rad · s

−1

₎

79

isthe lipresonan e angularfrequen y;

Q

l

isthe (dimensionless)qualityfa torof thelips;

h

0

isthe 80

value of

h(t)

at rest;

µ

is anequivalent surfa e mass

(kg · m

−2

₎

. 81

Althoughitdoesnotfullyreprodu ealltheobservedbehaviours ofhumanorarti iallips,this 82

model issu ient for reprodu ing the normal playing situations[Yoshikawa,1995℄, in luding the 83

pedal note of the trombone [Velut etal.,2017℄ and multiphoni sounds [Velut et al.,2016a℄. As a 84

limitation, this model is known toos illate at higher frequen ies than those at whi h a musi ian 85

would play on the same a ousti al mode. Even for this relatively simple model, hoosing the lip 86

parameters is hallenging and requires a thorough bibliographi al review. This was ondu ted in 87

[Velut etal.,2017℄. The resultingset of parameters isgiven in table I. 88

h

0

(m)

W (m)

1/µ (m

2

· kg

−1

_{) Q}

l

5 × 10

−4

_{12 × 10}

−3

0.11 7 89

Table I: Lip parametersretained in this study. 90

The resonator is the air olumn ontained in the bore of the instrument. Given the low 91

playing amplitude onsidered in this arti le, the brassiness phenomenon, related to non-linear 92

propagationintheinstrument[Myers etal., 2012℄isnottakenintoa ount. Underthishypothesis, 93

the resonator anbe fullydes ribed by itsinput impedan e,whi hisby denition the ratioof the 94

pressure

P (ω)

to the ow

U(ω)

atthe input of the instrument,in the frequen y domain: 95

Z(ω) =

P (ω)

U(ω)

.

(2)

This value an be measured using the sensor des ribed in [Ma alusoand Dalmont,2011℄. In 96

this paper, three input impedan e measurements are used: the impedan e of an open trombone 97

(without any mute), the impedan e of the same trombone with a "passive mute" (mute without 98

a tive ontrol) and the impedan e of this trombone with an "a tive mute", with the feedba k 99

a tive ontrol devi e enabled. 100

The input impedan e an be onsidered as a sum of peaks, ea h peak orresponding to a 101

resonan e mode of the air olumn inside the instrument. Thus, it an be tted with a sum of 102

omplex modes, orresponding toa sum of poles-residues fun tions: 103

Z(ω) = Z

c

·

N

m

X

n=1



C

n

jω − s

n

+

C

∗

n

jω − s

∗

n



,

(3)

C

n

and

s

n

being the dimensionless omplex residues and poles of the omplex modes of the 104

tted impedan e, respe tively.

Z

c

=

ρ.c

π.r

2

is the hara teristi impedan e of the resonator,

ρ

is the 105

airdensity,

c

the elerityofa ousti wavesintheairand

r

the inputradiusof themouthpie e.

N

m

106

is the number of modes used to t the impedan e, xed to

N

m

= 13

in this arti le. Translation 107

of eq. (3) in the time domain leads to an ordinary dierential equation for ea h omplex modal 108

omponent

p

n

of the pressure

p(t)

: 109

dp

n

dt

= s

n

p

n

(t) + Z

c

C

n

.u(t)

∀n ∈ [1..N

m

],

(4)

where

u(t)

is the time-domain expression of the ow at the input of the instrument. F urther-110

more,

p(t) = 2

P

N

m

n=1

ℜ[p

n

(t)]

. Details of this modal formulation of the pressure, already used in 111

[Velut etal.,2017℄, an befound in[Silva,2009℄. 112

The t is optimised by a least mean squares algorithm. This results in a very good mat h 113

between the measured impedan e and the t, as shown in Fig.2. 114

0

100

200

300

0

1

2

3

4

5

6

7

x 10

7

_{open trombone}

Frequency (Hz)

Magnitude (Pa.m

−3

.s)

0

100

200

300

0

1

2

3

4

5

6

7

x 10

7

_{passive mute}

Frequency (Hz)

0

100

200

300

0

1

2

3

4

5

6

7

x 10

7

_{active mute}

Frequency (Hz)

0

100

200

300

−3

−2

−1

0

1

2

3

Frequency (Hz)

phase (rad)

0

100

200

300

−3

−2

−1

0

1

2

3

Frequency (Hz)

phase (rad)

0

100

200

300

−3

−2

−1

0

1

2

3

Frequency (Hz)

phase (rad)

38 Hz

112.2

Hz

171.4

Hz

228.4

Hz

289.9

Hz

290.5

Hz

172.5

Hz

228.7

Hz

37.8 Hz

38 Hz

113.6

Hz

113 Hz

174.5

Hz

231 Hz

290.2

Hz

115

Figure 2: ( olour online) Comparison of the measured impedan es (blue, dash-dotted) and their modalts (red,plain)with 13 omplexmodes. Magnitudes (topplots) andphases (bottomplots) of the impedan es for the three situations - open trombone (left), passive straight mute (middle) and a tive mute (right) - are displayed. The dash-dotted line at

65.7

Hz indi ates the subsidiary resonan e. The resonan e frequen ies of the other modes are written near the amplitude peaks. 116

Thelipsandtheresonatorare oupledthroughtheexpressionoftheow

u(t)

oftheairjetthrough 117

the lip hannel: 118

u(t) = W.h(t).

s

2.|p

b

− p(t)|

ρ

.sign(p

b

− p(t)).θ(h),

(5)

where

W

is the width of the lip hannel and

ρ

the air density,

sign

is the sign fun tion and 119

θ(h)

is the Heaviside step fun tion. This non-linear expression of the ow was proposed in 120

[Wilson and Beavers, 1974, Eliott and Bowsher, 1982℄ and has been used in almost every publi-121

ation about brasswind and woodwind physi al models sin e. 122

The whole model an therefore be written: 123







































d

2

h

dt

2

+

ω

l

Q

l

dh

dt

+ ω

2

l

(h − h

0

) =

1

µ

(p

b

− p(t))

u(t) = W.h(t).

r 2.|p

b

− p(t)|

ρ

.sign(p

b

− p(t)).θ(h)

dp

n

dt

= s

n

p

n

(t) + Z

c

C

n

.u(t)

∀n ∈ [1..N

m

]

p(t) = 2

P

N

m

n=1

ℜ[p

n

(t)]

(6)

The model des ribed above has avarietyof possiblebehaviours. One of themis a stati solution, 125

allvariablesbeing onstant. The stabilityof thisstati solutionisauseful pie eof information,as 126

instability ofthe stati solutionindi atespossible emergen eof os illatingsolutionsthrough Hopf 127

bifur ations. Thisstabilityanalysis anbe arriedout onalinearisedmodel: non-linearequations 128

are linearised in the vi inity of the stati solution. Then, the stability of this stati solution is 129

assessed through omputationofthe eigenvaluesof theJa obianmatrix. Ifatleastone eigenvalue 130

has apositive real part, any perturbation of the stati solution willgrow exponentially, whi h by 131

denition meansthe solutionis unstable. Detailsonthe method appliedtobrass instruments an 132

be found in [Velut et al.,2017℄. 133

Thismethodisusedtondthelowestblowingpressurevalueleadingtoanunstablestati solution. 134

This

p

b

value is hereafter alled

p

thresh

. The imaginary part of the same eigenvalue indi ates the 135

os illationangularfrequen yfor

p

b

= p

thresh

,providedthattheos illatingsolutionisperiodi . The 136

orresponding frequen y is noted

f

thresh

. 137

LSA has been used for ute-like instruments [Auvray et al.,2012, Terrien etal., 2014℄ as 138

well as reed woodwinds [Wilson and Beavers, 1974, Chang,1994, Silvaetal., 2008℄ and brass-139

winds [Cullen etal., 2000, Velut etal.,2017℄. This method does not provide information about 140

the stability ofthe os illatingsolutionwhi hresultsfromthedestabilisationofthe stati solution. 141

The only pie e of information about the resulting waveform is

f

thresh

, whi h is only valid if said 142

solutionis periodi . 143

An example of results is given in Fig. 3:

p

thresh

(a) and

f

thresh

(b) are plotted against the lip 144

resonan e frequen y

f

l

, whi h is a ontrol parameter used by the musi ian to hange the note 145

played with the trombone. As observed in[Velut etal., 2017℄, the plots an bedivided inseveral 146

f

l

ranges orrespondingtoU-shapedse tionsofthe

p

thresh

urvesandverylightlygrowingplateaus 147

of

f

thresh

just above the a ousti resonan e frequen iesof the resonator. 148

50

100

150

200

250

300

350

400

450

500

0

1

2

3

x 10

4

lip resonance frequency (Hz)

oscillation threshold (Pa)

50

100

150

200

250

300

350

400

450

500

0

200

400

600

800

lip resonance frequency (Hz)

freq. at threshold (Hz)

open trombone

passive mute

active mute

a)

b)

149

Figure 3: ( olour online). Linear stability analysis results:

p

thresh

(a) and

f

thresh

(b) are plotted against

f

l

. Results for the open trombone (blue, dashed), the passive mute (red, solid) and the a tivemute(bla k, dotted)are displayed. Bla k dottedlinesof the bottomplot are the resonan e frequen ies of the open trombone (horizontal) and the bise tor of the axes (

f

thresh

= f

l

). The qualitativebehaviourof the open trombone,the passivemuteand a tive mute are very similarat this s ale.

150

II.C Time-domain simulation 151

Toget moreinformationaboutthe natureofos illatingsolutionsof theinstrumentmodel, solving 152

the non-linearequationsystem Eq.(6) isrequired. Numeri aldierentialequationsolversprovide 153

simulated values of the system variables. Simulated values of the pressure at the input of the 154

instrument

p

havebeenobtainedwiththeopen-sour ePythonlibrary alledMoReeSC[Mor, 2016℄, 155

whi hhas beendeveloped spe iallyfortime-domainsimulationofself-os illatingreedandlipvalve 156

instrumentmodels[Silva etal.,2014℄. 157

Toillustratetheadditionalinformationprovidedby time-domainsimulation,waveforms and spe -158

tra of two simulated pressure signals are given in Figure 4. The simulation in Fig. 4 (a) and ( ) 159

was omputedwith

f

l

= 90

HzwhiletheoneinFig.4(b)and(d) was omputedwith

f

l

= 110

Hz, 160

ea hone onanopen trombone,with ablowingpressure

10%

higherthan theos illationthreshold. 161

While LSA results for these two situationsare very lose to one another, numeri al resolution of 162

periodi for

f

l

= 90

Hz,it appears tobe quasi-periodi for

f

l

= 110

Hz. 164 a)

0

5

10

15

−4000

−3000

−2000

−1000

0

1000

2000

time (s)

p(t) (Pa)

b)

0

5

10

15

−1500

−1000

−500

0

500

1000

time (s)

p(t) (Pa)

165 )

0

200

400

600

800

1000

40

60

80

100

120

140

160

20log(|FFT(p)|)

frequency (Hz)

d)

0

200

400

600

800

1000

40

60

80

100

120

140

160

frequency (Hz)

20log(|FFT(p)|)

166

Figure 4: Waveforms of simulated

p

signals for

f

l

= 90

Hz (a) and

f

l

= 110

Hz (b) with zooms on some periods, along with spe tra of their respe tive sustained regime in ) and d). For ea h simulation

p

b

is set to

1.1 · p

thresh

.

f

l

= 90

Hz results in a periodi os illation while

f

l

= 110

Hz results ina quasi-periodi os illationwith welldened se ondary peaks.

167

The

f

l

and

p

b

values for simulations are hosen thanks to LSA, avoiding a long and umber-168

some sear h forthe os illationthresholdwith multiple simulations. The omplementarityof these 169

methods qui kly provides alot of informationabout relevantpoints of the os illationregime. 170

III Results 171

III.A LSA 172

Linearstabilityanalysiswasperformedonthethree ongurationsstudied: opentrombone,passive 173

mute, and a tive mute. Choosing a onguration involves hoosing

C

n

and

s

n

values among 174

the three sets obtained by tting, all other parameters of the model remaining the same. Lip 175

parameters were taken fromTable I. LSAwasperformed withinthe pedalnote range, for

f

l

from 176

30

Hz to

65

Hz. This resultsin

f

thresh

values orrespondingtoanos illationsustained by the rst 177

a ousti al mode of the open trombone. Figure 5 is a zoom on Fig 3 in the onsidered

f

l

range. 178

Fig.5a)showingthethresholdpressures

p

thresh

,whileFig.5b)isthe frequen yatthreshold

f

thresh

. 179

30

35

40

45

50

55

60

65

0

500

1000

lip resonance frequency f

l

(Hz)

p

thresh

(Pa)

30

35

40

45

50

55

60

65

50

60

70

lip resonance frequency f

l

(Hz)

f

thresh

(Hz)

open trombone

active mute

passive mute

b)

a)

f=58Hz

f=65.7Hz

180

Figure 5: Results of LSA in the vi inity of the pedal note (zoom of Fig.3). Results with an open trombone (dashed line), apassive mute (solid line) andthe a tivemute (dotted) are plotted together. (a)istheos illationthresholdpressure

p

thresh

,(b)istheos illationfrequen yatthreshold

f

thresh

, against

f

l

. Horizontal dash-dotted lines in (b) indi ate

58

Hz (playing frequen y of the pedal

B♭

) and

65.7

Hz (resonan e frequen y of the subsidiary mode of the passive mute). While open trombone and a tive mute have very similarbehaviours, the os illation regime expe ted for the trombonewith the passivemutebe omesdierent above

f

l

= 55Hz

with asudden in rease in the

p

thresh

and

f

thresh

values.

181

The open trombone and the a tive mute behaviours are similar: the

p

thresh

= F (f

l

)

plot is U-182

shaped.

f

thresh

is abovethe trombone'srst a ousti al resonan e frequen y (

39

Hz)and in reases 183

monotoni allywith

f

l

. Withinthis

f

l

range, theos illationthreshold ofthe a tivemute trombone 184

is about

75

Pa higherthan that of the open trombone, and

f

thresh

is also

0.5

to

1.5

Hz higher. 185

For

f

l

≤ 54

Hz, theresults forthe trombonewith apassivemuteare similartothose forthe other 186

ongurations. But from

f

l

= 55

Hz, both the pressure threshold and the expe ted playing fre-187

quen yin rease signi antly:

p

thresh

suddenlyjumps from

198.6

to

536.9

Pa,while

f

thresh

in reases 188

by

8.3

Hz (

13%

, i.e. slightly more than a tone), to rea h

66.8

Hz. This value is just above the 189

resonan e frequen y of the subsidiarypeakindu ed by the passive mute. 190

f

thresh

overs a range of frequen ies around the expe ted playing frequen y of a pedal

B♭ = 58

191

Hz. The results for the open trombone and the a tive mute ongurations are very lose to one 192

another, the only dieren e being a rather small oset in

p

thresh

and

f

thresh

. In ontrast, the 193

passive mute results stand out from the two other ongurations: for

f

l

values above

55

Hz, 194

p

thresh

and

f

thresh

in rease suddenly. The

f

thresh

value obtained is above the a ousti resonan e 195

frequen y of the subsidiary mode related to the passive mute, and so the regeneration ondi-196

tion [Eliott and Bowsher,1982, Campbell,2004℄ is satised for an os illation supported by this 197

subsidiary mode. 198

Theseresults ana ount forthe di ultyof playinga stablepedal notewitha passive mute: the 199

LSAindi atesaperturbationof theos illationfrequen yatthreshold, forparameterswhi h ould 200

be those used for the pedal note. However, experimental results shown in [Meurisse et al.,2015℄ 201

suggest anon-periodi os illationwhen amusi ian triesto playa pedal note with apassivemute. 202

As LSA annot predi t the nature of the os illation, further investigation on the omplete non-203

linearmodelisneeded. Thisisthepurposeof thenumeri alsimulationspresented inthe following 204

se tion. 205

III.B Time-domain simulations 206

Time-domain simulationswere arriedout within the same range of

f

l

as for LSA,in 1 Hz steps, 207

for ea h onguration: trombone alone, trombone with apassive mute and nallytrombone with 208

a tive mute. Theblowingpressure wasset to

p

b

= 1.1 · p

thresh

asin[Velut et al.,2017℄ inorderto 209

keepmanageabletransienttimes. Thisvalueis loseenoughto

p

thresh

sothat autious omparisons 210

an be arried out between these simulationsand LSA. 211

Simulatedpressure signalswere separatedintoatransientandasustained regimewith thehelp of 212

the "mironsets"fun tionfromMIRtoolbox [Lartillotand Toiviainen, 2007℄. The spe traof allthe 213

sustained regimes were omputed. Figure6 plots spe tra of p(t) for representative values of

f

l

. 214

0

50

100

150

200

40

60

80

100

120

140

frequency (Hz)

Amplitude (dB)

l

0

50

100

150

200

40

60

80

100

120

140

frequency (Hz)

Amplitude (dB)

l

0

50

100

150

200

40

60

80

100

120

140

frequency (Hz)

Amplitude (dB)

0

50

100

150

200

40

60

80

100

120

140

frequency (Hz)

Amplitude (dB)

open trombone

passive mute

active mute

b)

c)

a)

d)

f

l

= 55 Hz

f

l

= 53 Hz

f

l

= 56 Hz

f

l

= 58 Hz

216

Figure6: (Coloursonline) Spe tra of the simulated

p(t)

signals, for

f

l

= 53

Hz (a),

55

Hz (b),

56

Hz( ) and

58

Hz(d).

p

b

isset to

110%

of theos illationthreshold. Forea h

f

l

value, resultswith the open trombone (blue),the passive mute (bla k) and the a tive mute (red) are displayed. The results for the open trombone and the a tivemute are noti eablysimilar.

217

For

f

l

< 55

Hz, the three ongurations - open trombone, passive mute, a tive mute - lead to a 218

periodi os illation, as illustrated for

f

l

= 53

Hz by (Fig. 6a). The os illation frequen y is a bit 219

higher than

f

thresh

:

7.5%

for open trombone and a tive mute, and

2.5%

higher for the passive 220

mute. Os illationfrequen ies higher than

f

thresh

when

p

b

> p

thresh

is oherent with the fa t that 221

a musi ian's playing frequen y gets higher when the blowing pressure in reases. The trombone 222

with a passive mute has a lower os illationfrequen y than the open trombone, whi h has itself a 223

slightlyloweros illationfrequen ythanthatofthetrombonewiththea tivemute. Theos illation 224

frequen ies range from

60

to

64

Hz, a bit higher than

B♭1 = 58

Hz. This is sensible sin e this 225

modelis known to os illateat higherfrequen ies than those at whi h amusi ianplays. 226

At

f

l

= 55

Hz (Fig. 6b) the os illation frequen y of the passive mute onguration suddenly 227

jumps from

59.4

to

69.6

Hz, making it play sharper (nearly a minor third) than the two other 228

ongurations. This is onsistent with the LSA results, where

f

thresh

suddenly in reases by

8

Hz 229

for this

f

l

value. The os illationsare still periodi and above the a ousti al resonan e frequen y 230

of therst mode; but for

f

l

= 56

Hz and above(illustratedby Fig. 6 )the fundamentalfrequen y 231

ofthe passivemute fallsto

34.8

Hz. Thisisnearly halfitsformervalue, andunderthe trombone's 232

rst a ousti resonan e frequen y (

39

Hz). Finally, for

f

l

≥ 58Hz

(Fig. 6d), all ongurations 233

resultin fundamentalos illationfrequen iesabout half,ora quarter,of theos illationfrequen ies 234

obtained for lower

f

l

values. 235

Simulation and LSA results are onsistent: when

f

l

rea hes

55

Hz, the os illation frequen y of 236

the trombone with a passive mute suddenly in reases. This is related to a regime hange in the 237

instrument: for

f

l

< 55

Hz, the os illationis mainlysupported by the trombone's rst a ousti al 238

mode whi hresonan e frequen y is

38

Hz. Theunusal gapbetween thetrombone's rstmode and 239

the pedal noteisstudiedin[Velut et al.,2017℄. For

f

l

= 55

Hz andabove, thesubsidiary modeat 240

65.7

Hz ausedby the mute be omes the mainsupporting mode of the os illation,whi h explains 241

the in rease inthe os illatingfrequen y. 242

Above

f

l

= 56

Hz,however,theos illationfrequen yofthepassivemutede reasestohalfofits for-243

mervalue. Astheos illationfrequen yisunderthetrombone'srsta ousti alresonan efrequen y, 244

theregeneration onditionofamodelwithoutward-strikingvalveisnot satised[Campbell,2004℄. 245

This situation suggests a period-doubling phenomenon [Bergé et al.,1995℄. When in reasing

f

l

246

again, the three ongurations appear to undergo period doubling, whi h is further doubled for 247

theopentrombone withafundamentalfrequen y of

16.2

Hz. Sub-harmoni as ades havealready 248

been observed for trombones [Gibiat and Castellengo,2000℄, and simulated in a previous study 249

with the very same modeland parameters [Velut etal.,2017℄. 250

These results onrm the existen e of a subsidiary regime of os illation for the passive mute 251

onguration, whi h ould explain why musi ians experien e di ulties when trying to play the 252

B♭

pedal inthis situation. This subsidiaryregime is sustained by the subsidiarya ousti al mode 253

introdu ed by the mute. Furthermore, in a ordan e with the experimental results published 254

in[Meurisse et al.,2015℄, the simulationresults are qualitativelythe same for the open trombone 255

and the a tive mute, with very lose os illation frequen ies. The range of

f

l

leading to periodi 256

os illationsnearthepedalnotefrequen yisnoti eablywiderfortheopen tromboneandthea tive 257

mute than for the passive mute. 258

IV Con lusion 259

Playingastable

B♭1

onatrombonewithastraightmuteisverydi ult. Ana tivemutehasbeen 260

developed [Meurisseet al.,2015℄ to deal with this issue. When applied to a trombone equipped 261

with this a tive mute and to an open trombone, LSA and time-domain simulation give nearly 262

identi al results. The modelis therefore able to predi t the e ien y of the a tive ontrol devi e 263

whi h makes the pedal note easily playable again. Results of the modelof a trombone equipped 264

with a passive mute, however, are learly dierent from those of the open trombone model: the 265

mode added by the mute. Hen e, even a "small" perturbation of the input impedan e, su h as a 267

peak 20 times smaller inamplitude than surrounding peaks, an strongly ae t the behaviour of 268

a resonator. 269

As in a previous paper [Velut etal.,2017℄, this study shows a rather good agreement between 270

LSA results and time-domain simulations, within the limits of the LSA method. This study on 271

mutes also shows the relevan e of the hosen brass instrument model, whi h is able to predi t a 272

number of behaviours of the trombone, in luding parti ular playing regimes [Velut et al.,2017, 273

Velut etal., 2016a℄and,inthe present ase, theinuen e of modi ationsof the instrument bore. 274

Beginning a study with LSA very qui kly gives an overview of the potential behaviour of the 275

system under given onditions. This fast omputation already provides interesting results, whi h 276

anbeinterpretedalone. However, iffurther explorationofthe os illationregimeisrequired,LSA 277

results give hintsfor hoosing

f

l

and

p

b

values for initialisingother analysis methods. 278

V A knowledgements 279

The authors would parti ularly like to thank Thibault Meurisse and Adrien Mamou-Mani for 280

providingthe input impedan e measurementsused here. 281

This work was done in the frameworks of the Labex MEC (ANR-10-LABX-0092) and of the 282

A*MIDEX proje t (ANR-11-IDEX-0001-02), funded by the Fren h National Resear h Agen y 283

(ANR). 284

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