Physical Modeling Synthesis of Recorder Sound

Hiroko Shiraiwa, CCRMA Stanford University email: [email protected] Kenshi Kishi, University of Electro-Communications Isao Nakamura, Athena. Co. Ltd.

Abstract Verge [8]. He developed a unique model of sound generation by means of a rigorous fluid dynamics approach, which This paper describes time domain simulation of the so- is based upon Navier-Stokes equations. Unfortunately, his prano baroque recorder based on digital waveguide model method is not practical for the simulation because of the (DWM) and an air reed model, in particular, negative acous- rough approximations and many uncertain parameters, in tic displacement model (NADM). Two models are proposed addition to the lack of computational stability. to couple DWM and NADM. Moreover, the jet amplifica- A spectrum of the recorder sound is shown in Fig. 2. The tion coefficient is remodeled for the application of NADM recorder sound typically has a spectrum of larger amplitudes which was originally built for organ flue pipes to the recorder, at the odd harmonics [9][10]. regarding the recent experimental reports by Yoshikawa [3]. The simulation results are presented in terms of the mode pipe transient characteristics and the spectral characteristics of 12.5mm the sounds. They indicate that the NADM is not sufficient to describe the realistic mode transient characteristics of the 6cm recorder, while the synthesized sounds maintain almost re- semble spectral characteristics to the recorder sounds. 23cm

1 Introduction

This paper reports the synthesis of soprano baroque recorder sound by a time domain physical simulation. The aim is to generate a realistic recorder sound that is controllable by physical parameters, such as blowing pressure and ﬁnger po- 8mm mouth sition. The ultimate goal of this research is to simulate the whole Figure 1: The recorder structure instrumental sound generation system by combining of the

instrument model and the physiological and psychoacous-

¡ tical model of the player. For this purpose, the instrument £

model must have acoustic accuracy and good quality syn- ¡

£ ¡

thesized timbre at the same time.

¥ ¡

The structure of the recorder is shown in Fig. 1. When a

'*+

)

§ ¡

(

recorder is blown, an air jet emerges from the ﬂue slit. The '

#$ % &

jet travels across the mouth opening to the edge. At the edge,

¢ ¡ ¡

the jet ﬂows into and out of the resonator and oscillates the

¢ £ ¡

air column.

¢ ¥ ¡

¢ ¡ ¡ ¡ £ ¡ ¡ ¡ ¤ ¡ ¡ ¡ ¥ ¡ ¡ ¡ ¦ ¡ ¡ ¡ § ¡ ¡ ¡ ¨ ¡ ¡ ¡ © ¡ ¡ ¡

The simulation system consists of a jet oscillation model ¡

and a resonator model. The jet oscillation model follows the simulation method of organ flue pipe by Adachi [2] which is Figure 2: Spectrum of the recorder sound based on the semi-empirical model of jet drive by Fletcher [1]. However, this application requires a slight modification of the air jet amplification coefficient [3][4], considering the structural and dimensional differences between the recorder and the organ flue pipe. The resonator model was devised based on the digital waveguide modeling (DWM) [5]. Another simulation on the recorder has been done by

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2 Simulation system P

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N O V U

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2.1 Resonator model N

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The resonator model is realized by delay line (a.k.a. dig- W

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ital waveguide modeling) [5]. The wall losses are imple- N

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mented with an FIR ﬁlter, which is designed according to N

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the reference [7]. The tone hole model which was proposed N

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in [6] is adopted either.

N N O P N O Q N O R N O S T T O P T O Q T O R XCY

2.2 Jet oscillation model

Figure 4: The jet amplification coefficient £ The jet oscillation model is based on the negative acoustic displacement model proposed by Fletcher and Thwaites [1]. A time domain simulation of the organ flue pipe was

done by Adachi [2]. The simulation follows the method by Table 1: Dimensions of the recorder mouth c

Adachi except for the approximation of the jet amplification Pipe cross-sectional area ^`_¥ab_7 1.3 cm = coefficient , which was redone according to recent refer- Cut-up length 3 mm ences [3][4]. Mouth width : 10 mm The structure of the recorder mouth is shown in Fig. 3. Flue slit thickness d¡ 1.5 mm In the simulation, voicing (rounding off the right angled cor- Edge offset ¡ JK K -0.75 mm ner at the flue exit) is neglected. The air jet which emerges Mouth end correction egf 2.5 cm

from the flue exit, travels across the mouth opening to the £¥¤ edge. Let ¢¡ be the flue slit thickness and the half thickness of the jet at the flue exit. Then the half thickness of the

and the phase velocity of the jet ﬂuctuation )_h i¥jk7 . The ¦©¨

travelling jet £§¦©¨ and the center velocity of the jet are

qp l;srt0 1 phase velocity is given as )_Ah i¥j?7.mlon , where given by in the simulation. The ratio of the jet displacement 3 to the

acoustic particle velocity M is described in the frequency do-

main as

£ ¨¢

£§¦©¨ (1)

¤ ¤

¦©¨ £ £§¦ ¨!

(2)

B B 3¥uv¦©¨

aCu}| ~¥ >

@CBDy¦kB ¦ ¨ kzd{

wGx (4)

u L

(' - -

)¤* "+,¤

where £"¤#%$&¡ , . Here, is the density

1&2

of the air and .%/)0 is the spreading angle of the jet. In addition, is modeled with an analytic function, such 465798 Given the jet displacement 3 , the jet volume ﬂow as into the pipe is obtained by integrating the jet velocity proﬁle

inside the pipe. {kk¥G ¢ £;)£"z (5)

where § (wave number). This function approx-

3)¦©=G HDI¡,JK K imately satisﬁes the calculation by Drazin and Howard [1]

5798

;:<£§¦>=> ? ¦©=> A@CBDE¢)F

4 (3)

£§¦©=G L when `B . However, the function had to be remodeled because the recorder has the longer ﬂue than a organ ﬂue

where ¡ JK K is the offset of the edge position. pipe [3]. According to recent calculation[4], it is appropri-

The jet displacement 3 is determined by the particle ve-

ate to utilize instead. This approximation is shown

locity inside the pipe M , the jet ampliﬁcation coefﬁcient , in Fig.4.

Determining the inverse Fourier transform (IFT) of Eq.

¦ ? 3

x = 0 (4) as kernel function , the simulation of in time do-

¦ ? main is given by the convolution of M ¦©? and . 2h ξ(t)

offset c

? ¦> kM ¦©vDI k¡ 3)¦ ¨ (6)

Ujet(t)

| ¦©? The kernel function is given as Um (t) Upipe(t)

vm (t)

¤§c ¤¥D

M P

| { ª«¢¬

¦ ? ¢£ ¦ ? k£ ¦©¤¥D? z

(7) ¨

cut−up ∆ x ¨

B/&¦ §

m£ ¦©°A _h i¥jk7A }_Ah i¥j?7 ±B Figure 3: The mouth structure where ¤®¯¨ , , and . During the simulation by OFLM (one feedback loop model),c

¡ ¢ ¤ p 0

¤ Vphase Vc

£ ¡ ¢

¡ ¢

¦ §

conv. Ujet

¡ ¢ ¥

f (t, x) 1/Apipe f (t+ ∆ t, x+∆ x) £

¥ n n

£ ¡ ¢

¡ ¢ £ £ ¡ ¢ ¤

¨ © ¨

∆ ∆

bn (t+ t, x) bn (t, x+ x)

| ¦ ? Figure 5: The kernel function Figure 6: One feedback loop model (OFLM)

which is explained in section 2.3, it is difﬁcult to obtain a p 0 steady state oscillation. For this reason, the coefﬁcient Vphase Vc

was set to three exceptionally in the OFLM simulation. c

| ¦©? An example of the kernel function appears in Fig. 5. Table 1 gives the dimensions of the recorder mouth. conv. Ujet 1/Apipe 2.3 Coupling model f (t, x) −1 f (t+ ∆ t, x+∆ x) In this section, the coupling of the jet oscillation model n n and the resonator model is discussed. In the simulation, the volume ﬂow part of jet drive is ∆ ∆ taken into account. The momentum part of jet drive is ne- bn (t+ t, x) bn (t, x+ x) glected for simplicity. Considering Fig. 3, we assume that

the total volume ﬂow at the plain M and P are conserved; Figure 7: Three feedback loop model (TFLM)

_Aa _7§¦©? 4 ¦©? 4 5798¥¦©?

4 (8)

The mouth impedance is generally modeled as Fig. 4, and the choice of is crucial to obtaining a steady

state oscillation. For example, the simulation with

- f

e does not generate a steady state.

(9)

_Aa _7 ^ Based on an experiment [3] on the relationship between

the jet velocity proﬁle and the ﬂue length, it is more appro-

_Aab_A7 ^

where egf is the mouth end correction and is the

pipe cross sectional area. Therefore, a digital waveguide of priate to approximate £ with , considering the ﬂue

length of the recorder. With this approximation, the simula-

_Aab_7 egf ^ and was applied as the mouth part. The simulation system consists of three parts; the mouth, tion generates a steady state oscillation, and the approxima- the resonator, and the jet ﬂow part, which connects the mouth tion seems reasonable according to the experiments and the

and resonator. calculations reported in [4]. That report mentions that the

The OFLM, shown in Fig. 6, is a simplified coupling long flue affects the jet amplification coefficient , resulting

in the stability of the sound generation.

5798 M

model. 4 takes only the forward going part of as in- 5798

put, and the output 4 is fed to only the forward going part _Aab_7 of 4 . 3.2 Mode transition characteristics The three feedback loop model (TFLM), shown in Fig.

7, considers the backward going wave as well. The input of Mode transition characteristics of the simulation are shown

in Fig. 8. In the ﬁgure, the fundamental frequency of the

57 8 volume ﬂow 4 contains the backward going wave of the

synthesized sound is plotted as a function of blowing pres-

5798 4

M , and the is fed to the mouth part too.

¤ A

The simulation input is constant blowing pressure, while sure + by solid lines. The half fundamental frequency,

¥ 1

the volume velocity at the pipe end was taken as the synthe- is plotted by dash-dot lines. Similarly, is plotted by

A $ sized sound. dash lines, and appears again as a solid line. Both OFLM and THLM generate the overblown states, i.e., the recorder sounds with a higher fundamental frequency 3 Simulation results when it is played with high blowing pressures. OFLM generates steady state oscillation with any blowing pressure in- 3.1 Jet ampliﬁcation coefﬁcient put. During the transition, it generates sounds that have

two dominant frequencies. On the other hand, TFLM does

In the simulation, the jet ampliﬁcation coefﬁcient plays not generate the steady state oscillation in several regions

an important role to synthesize sustained timbres. The coef- between two different mode. Since a real recorder makes

£ ﬁcient in Eq. 5, determines the shape of as shown in

! " #%$"& '!( "%

¡ ¡

¦ 4 Conclusion

¥ ¥ ¡

In the simulation, the jet ampliﬁcation coefﬁcient plays

¥ ¡ ¡

¨ )

§ This qualitatively supports the reports by Yoshikawa [3] and

) * +

) * ,

¥ ¡

¤ Verge [4], such that the windway proﬁle of the air reed in-

¡ ¢ ¡ £

struments affects the jet velocity proﬁle and the jet ampliﬁ-

- ."/

" ! " #%$"& '!( "%

¡ ¡ ¦ cation coefﬁcient , resulting in stability of the sound gen-

eration.

¥ ¥ ¡

The two coupling models, OFLM and TFLM, provide

the different characteristics, although their basis is same.

¥ ¡ ¡

) ©

) * + ¨

§ OFLM provides more realistic mode transient characteris-

) * ,

) * ¤

¥ ¡

¤ tics, while TFLM produces the more natural timbres in syn-

¢ £

¡ ¡ " thesis. It seems more reasonable to consider the backward- going waves for input and output of the jet volume ﬂow, Figure 8: Mode transition characteristics although it deteriorates the mode transient characteristics. above: OFLM, below: TFLM The result suggests that the sound generation mechanisms should be remodeled to obtain more realistic simulation in

terms of sound quality and acoustic characteristics.

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References

PST

5 0

P Q

LM N O

0 0

1 [1] Fletcher and Rossing: The Physics of Musical Instru-

8 J

I ments 2nd. ed., Springer-Verlag (2000).

1 5 0

0 1 0 0 0 2 0 0 0 3 0 0 0 4 0 0 0 5 0 0 0 6 0 0 0 7 0 0 0

: ;=<=>?;=@=ABDC EFHG

9 [2] Adachi: Time-Domain Modeling and Computer Simu-

: ;=;V9 ;=;=W=X?Y=AZD[

^ _

U?U=\ lation of an Organ Flue Pipe, Proc. of ISMA 1997, pp.

2 0

0 251-260 (1997).

2 0

4 0

8 [3] Yoshikawa and Arimoto: Measurement of the Jet Half-

6 0

P Q

8 Thickness and Discussion on the Jet-Wave Ampliﬁca-

L N

M O

8 ]

K tion in Organ Pipe Models, Proc. of ISMA 2001, pp.

1 0 0

1 2 0

8 309-312 (2001).

0 1 0 0 0 2 0 0 0 3 0 0 0 4 0 0 0 5 0 0 0 6 0 0 0 7 0 0 0

: ;=<=>?;=@=ABDC EFHG 9 [4] Segoufin, Fabre, Verge and Hirschberg: Recorder Windway Profile: Influence On Sound Production, Figure 9: Spectra of the synthesized sounds Proc. of ISMA 1998, pp. 203-208 (1998). above: OFLM, below: TFLM [5] Smith: Physical Modeling Synthesis Update, Com- puter Music J., 20:2, pp. 44-56 (1996). sounds at any blowing pressure, OFLM better approximates [6] Valim¨ aki,¨ Karjalainen and Laakso: Modeling of Wood- the real instrument with respect to the mode transition char- wind Bores with Finger Holes, ICMC proceedings acteristics. 1993, pp. 32-39 (1993). [7] Benade: On the Propagation of Sound Waves in a 3.3 Spectral characteristics Cylindrical Conduit, J. Acoust. Soc. Am. 44, pp. 616- 623 (1968). The sounds synthesized by OFLM and TFLM have different timbres. The spectra of the synthesized sounds ap- [8] Verge, Hirschberg and Causse:´ Sound production in pear in Fig. 9. Let us compare these spectra to that of the recorderlike instruments. II. A simulation model, J. real instrumental sound (Fig. 2). The spectrum of TFLM Acoust. Soc. Am. 101(5), pp. 2925-2939 (1997). represents the dominative odd harmonics, which is the typ- [9] Andou: Acoustics of Musical Instruments, New Ed., ical characteristics for the real instrumental sounds, while Ongaku-no-Tomo-Sha Corp. (1996), (in Japanese). OFLM does not represent them. Therefore, TFLM is con- sidered the better quality system with respect to the sound [10] Nakamura, Naganuma, Iwaoka, Maruyama: On the synthesis. difference of recorder tones depend on instruments and players (Research on the analysis of wind instruments part 2.), Proceedings of Acoust. Soc. Japan 1993 March, pp. 551-552 (1992), (in Japanese).