July  2022, 9(3): 465-482. doi: 10.3934/jcd.2022011

Emergence of quasiperiodic regimes in a neutral delay model of flute-like instruments: Influence of the detuning between resonance frequencies

1. 

Laboratoire d'Acoustique de l'Université du Mans (LAUM), UMR 6613, Institut d'Acoustique - Graduate School (IA-GS), CNRS, Le Mans Université, Le Mans, France

2. 

Aix Marseille Univ, CNRS, Centrale Marseille, LMA UMR 7031, Marseille, France

3. 

Sorbonne Université, CNRS, Institut Jean Le Rond d'Alembert, UMR 7190, Paris, France

4. 

Escuela de Ingeniería-Instituto de Música, Pontificia Universidad Católica de Chile, Santiago, Chile

* Corresponding author: Soizic Terrien (soizic.terrien@univ-lemans.fr)

Received  October 2021 Revised  January 2022 Published  July 2022 Early access  April 2022

Fund Project: The last author is supported by ANID through FONDECYT Project No. 1201551

Musical instruments display a wealth of dynamics, from equilibria (where no sound is produced) to a wide diversity of periodic and non-periodic sound regimes. We focus here on two types of flute-like instruments, namely a recorder and a pre-hispanic Chilean flute. A recent experimental study showed that they both produce quasiperiodic sound regimes which are avoided or played on purpose depending on the instrument. We investigate the generic model of sound production in flute-like musical instruments, a system of neutral delay-differential equations. Using time-domain simulations, we show that it produces stable quasiperiodic oscillations in good agreement with experimental observations. A numerical bifurcation analysis is performed, where both the delay time (related to a control parameter) and the detuning between the resonance frequencies of the instrument – a key parameter for instrument makers – are considered as bifurcation parameters. This demonstrates that the large detuning that is characteristic of prehispanic Chilean flutes plays a crucial role in the emergence of stable quasiperiodic oscillations.

Citation: Soizic Terrien, Christophe Vergez, Benoît Fabre, Patricio de la Cuadra. Emergence of quasiperiodic regimes in a neutral delay model of flute-like instruments: Influence of the detuning between resonance frequencies. Journal of Computational Dynamics, 2022, 9 (3) : 465-482. doi: 10.3934/jcd.2022011
References:
[1]

H. D. I. Abarbanel, Analysis of Observed Chaotic Data, Institute for Nonlinear Science, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-0763-4.

[2]

R. AuvrayB. Fabre and P.-Y. Lagrée, Regime change and oscillation thresholds in recorder-like instruments, The Journal of the Acoustical Society of America, 131 (2012), 1574-1585.  doi: 10.1121/1.3672815.

[3]

D. A. W. BartonB. Krauskopf and R. E. Wilson, Collocation schemes for periodic solutions of neutral delay differential equations, J. Difference Equ. Appl., 12 (2006), 1087-1101.  doi: 10.1080/10236190601045663.

[4]

D. A. W. BartonB. Krauskopf and R. E. Wilson, Bifurcation analysis tools for neutral delay equations: A case study, IFAC Proceedings Volumes, 39 (2006), 36-41.  doi: 10.3182/20060710-3-IT-4901.00007.

[5]

F. Blanc, P. de la Cuadra, B. Fabre, G. Castillo and C. Vergez, Acoustics of the flautas de chinos, Proceeding of 20th International Symposium on Music Acoustics, (2010).

[6]

F. BlancV. FrançoisB. FabreP. de la Cuadra and P.-Y. Lagrée, Modeling the receptivity of an air jet to transverse acoustic disturbance with application to musical instruments, The Journal of the Acoustical Society of America, 135 (2014), 3221-3230.  doi: 10.1121/1.4874598.

[7]

A. Chaigne and J. Kergomard, Acoustics of Musical Instruments, Springer New York, 2016. doi: 10.1007/978-1-4939-3679-3.

[8]

J.-P. Dalmont, Acoustic impedance measurement, Part Ⅰ: A review, Journal of Sound and Vibration, 243 (2001), 427-439.  doi: 10.1006/jsvi.2000.3428.

[9]

J.-P. Dalmont, Acoustic impedance measurement, Part Ⅱ: A new calibration method, Journal of Sound and Vibration, 243 (2001), 441-459.  doi: 10.1006/jsvi.2000.3429.

[10]

J.-P. DalmontB. GazengelJ. Gilbert and J. Kergomard, Some aspects of tuning and clean intonation in reed instruments, Applied Acoustics, 46 (1995), 19-60.  doi: 10.1016/0003-682X(95)93950-M.

[11]

P. de la Cuadra, The Sound of Oscillating Air Jets: Physics, Modeling and Simulation in Flute-Like Instruments, Ph.D thesis, Stanford University, 2006.

[12]

P. de la Cuadra, C. Vergez and B. Fabre, Visualization and analysis of jet oscillation under transverse acoustic perturbation, Journal of Flow Visualization and Image Processing, 14 (2007).

[13]

O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H.-O. Walther, Delay Equations: Functional, Complex-, and Nonlinear Analysis, Applied Mathematical Sciences, 110. Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4206-2.

[14]

J.-B. Doc and C. Vergez, Oscillation regimes produced by an alto saxophone: Influence of the control parameters and the bore inharmonicity, The Journal of the Acoustical Society of America, 137 (2015), 1756-1765.  doi: 10.1121/1.4916197.

[15]

J.-B. DocC. Vergez and S. Missoum, A minimal model of a single-reed instrument producing quasi-periodic sounds, Acta Acustica united with Acustica, 100 (2014), 543-554.  doi: 10.3813/AAA.918734.

[16] P. G. Drazin, Introduction to Hydrodynamic Stability, Cambridge University Press, 2002.  doi: 10.1017/CBO9780511809064.
[17]

K. EngelborghsT. Luzyanina and D. Roose, Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL, ACM Trans. Math. Softw., 28 (2002), 1-21.  doi: 10.1145/513001.513002.

[18]

K. Engelborghs, T. Luzyanina and G. Samaey, DDE-BIFTOOL v. 2.00: A Matlab Package for Bifurcation Analysis of Delay Differential Equations, Technical Report TW-330, Department of Computer Science, K.U. Leuven, Belgium, 2001.

[19]

K. Engelborghs and D. Roose, Smoothness loss of periodic solutions of a neutral functional-differential equation: On a bifurcation of the essential spectrum, Dynamics and Stability of Systems, 14 (1999), 255-273.  doi: 10.1080/026811199281994.

[20]

B. Fabre, J. Gilbert and A. Hirschberg, Modeling of wind instruments, Springer Handbook of Systematic Musicology, (2018), 121–139. doi: 10.1007/978-3-662-55004-5_7.

[21]

B. Fabre and A. Hirschberg, Physical modeling of flue instruments: A review of lumped models, Acta Acustica united with Acustica, 86 (2000), 599-610. 

[22]

B. FabreA. Hirschberg and A. P. J. Wijnands, Vortex shedding in steady oscillation of a flue organ pipe, Acta Acustica united with Acustica, 82 (1996), 863-877. 

[23] G. Falkovich, Fluid Mechanics: A Short Course for Physicists, Cambridge university press, Cambridge, 2011.  doi: 10.1017/CBO9780511794353.
[24]

N. H. Fletcher, Nonlinear dynamics and chaos in musical instruments, Complexity International, 1 (1994), 106-117. 

[25]

J. GilbertS. Maugeais and and C. Vergez, Minimal blowing pressure allowing periodic oscillations in a simplified reed musical instrument model: Bouasse-Benade prescription assessed through numerical continuation, Acta Acustica, 4 (2020), 27.  doi: 10.1051/aacus/2020026.

[26]

J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Springer, 2013.

[27]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Third edition, Applied Mathematical Sciences, 112. Springer-Verlag, New York, 2004. doi: 10.1007/978-1-4757-3978-7.

[28]

A. Lefebvre, G. Goudou and G. Scavone, The wind instrument acoustic toolkit, Available from: http://www.music.mcgill.ca/caml/wiat/.

[29]

D. H. Lyons, Resonance frequencies of the recorder (English flute), The Journal of the Acoustical Society of America, 70 (1981), 1239-1247.  doi: 10.1121/1.387136.

[30]

C. MaganzaR. Caussé and F. Laloë, Bifurcations, period doublings and chaos in clarinetlike systems, Europhysics Letters, 1 (1986), 295. 

[31]

M. E. McIntyreR. T. Schumacher and J. Woodhouse, On the oscillations of musical instruments, The Journal of the Acoustical Society of America, 74 (1983), 1325-1345.  doi: 10.1121/1.390157.

[32]

A. W. Nolle, Sinuous instability of a planar air jet: Propagation parameters and acoustic excitation, The Journal of the Acoustical Society of America, 103 (1998), 3690-3705.  doi: 10.1121/1.423089.

[33]

D. Roose and R. Szalai, Continuation and bifurcation analysis of delay differential equations, Numerical Continuation Methods for Dynamical Systems, Underst. Complex Syst., Springer, Dordrecht, (2007). 359–399. doi: 10.1007/978-1-4020-6356-5_12.

[34]

L. F. Shampine, Dissipative approximations to neutral DDEs, Appl. Math. Comput., 203 (2008), 641-648.  doi: 10.1016/j.amc.2008.05.010.

[35]

J. Sieber, K. Engelborghs, T. Luzyanina, G. Samaey and D. Roose, Manual - Bifurcation analysis of delay differential equations, Available from: arXiv.org/abs/1406.7144.

[36]

P.-A. TaillardF. SilvaP. Guillemain and J. Kergomard, Modal analysis of the input impedance of wind instruments. Application to the sound synthesis of a clarinet, Applied Acoustics, 141 (2018), 271-280.  doi: 10.1016/j.apacoust.2018.07.018.

[37]

S. TerrienR. BlandinC. Vergez and B. Fabre, Regime change thresholds in recorder-like instruments: Influence of the mouth pressure dynamics, Acta Acustica united with Acustica, 101 (2015), 300-316.  doi: 10.3813/AAA.918828.

[38]

S. TerrienC. VergezP. de la Cuadra and B. Fabre, Experimental analysis of non-periodic sound regimes in flute-like musical instruments, The Journal of the Acoustical Society of America, 149 (2021), 2100-2108.  doi: 10.1121/10.0003758.

[39]

S. TerrienC. Vergez and B. Fabre, Flute-like musical instruments: A toy model investigated through numerical continuation, Journal of Sound and Vibration, 332 (2013), 3833-3848.  doi: 10.1016/j.jsv.2013.01.041.

[40]

S. TerrienC. VergezB. Fabre and D. A. W. Barton, Calculation of the steady-state oscillations of a flute model using the orthogonal collocation method, Acta Acustica united with Acustica, 100 (2014), 690-704.  doi: 10.3813/AAA.918748.

[41]

C. VauthrinB. Fabre and I. Cossette, How does a flute player adapt his breathing and playing to musical tasks?, Acta Acustica united with Acustica, 101 (2015), 224-237.  doi: 10.3813/AAA.918821.

[42]

M. P. Verge, Aeroacoustics of Confined Jets: With Applications to the Physical Modeling of Recorder-Like Instruments, Ph.D thesis, Technische Universiteit Eindhoven, 1995.

[43]

M. P. VergeB. FabreW. E. A. MahuA. HirschbergR. R. van HasselA. P. J. WijnandsJ. J. de Vries and C. J. Hogendoorn, Jet formation and jet velocity fluctuations in a flue organ pipe, The Journal of the Acoustical Society of America, 95 (1994), 1119-1132.  doi: 10.1121/1.408460.

[44]

H. A. K. Wright and D. M. Campbell, Analysis of the sound of chilean pifilca flutes, The Galpin Society Journal, 51 (1998), 51-63.  doi: 10.2307/842760.

show all references

References:
[1]

H. D. I. Abarbanel, Analysis of Observed Chaotic Data, Institute for Nonlinear Science, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-0763-4.

[2]

R. AuvrayB. Fabre and P.-Y. Lagrée, Regime change and oscillation thresholds in recorder-like instruments, The Journal of the Acoustical Society of America, 131 (2012), 1574-1585.  doi: 10.1121/1.3672815.

[3]

D. A. W. BartonB. Krauskopf and R. E. Wilson, Collocation schemes for periodic solutions of neutral delay differential equations, J. Difference Equ. Appl., 12 (2006), 1087-1101.  doi: 10.1080/10236190601045663.

[4]

D. A. W. BartonB. Krauskopf and R. E. Wilson, Bifurcation analysis tools for neutral delay equations: A case study, IFAC Proceedings Volumes, 39 (2006), 36-41.  doi: 10.3182/20060710-3-IT-4901.00007.

[5]

F. Blanc, P. de la Cuadra, B. Fabre, G. Castillo and C. Vergez, Acoustics of the flautas de chinos, Proceeding of 20th International Symposium on Music Acoustics, (2010).

[6]

F. BlancV. FrançoisB. FabreP. de la Cuadra and P.-Y. Lagrée, Modeling the receptivity of an air jet to transverse acoustic disturbance with application to musical instruments, The Journal of the Acoustical Society of America, 135 (2014), 3221-3230.  doi: 10.1121/1.4874598.

[7]

A. Chaigne and J. Kergomard, Acoustics of Musical Instruments, Springer New York, 2016. doi: 10.1007/978-1-4939-3679-3.

[8]

J.-P. Dalmont, Acoustic impedance measurement, Part Ⅰ: A review, Journal of Sound and Vibration, 243 (2001), 427-439.  doi: 10.1006/jsvi.2000.3428.

[9]

J.-P. Dalmont, Acoustic impedance measurement, Part Ⅱ: A new calibration method, Journal of Sound and Vibration, 243 (2001), 441-459.  doi: 10.1006/jsvi.2000.3429.

[10]

J.-P. DalmontB. GazengelJ. Gilbert and J. Kergomard, Some aspects of tuning and clean intonation in reed instruments, Applied Acoustics, 46 (1995), 19-60.  doi: 10.1016/0003-682X(95)93950-M.

[11]

P. de la Cuadra, The Sound of Oscillating Air Jets: Physics, Modeling and Simulation in Flute-Like Instruments, Ph.D thesis, Stanford University, 2006.

[12]

P. de la Cuadra, C. Vergez and B. Fabre, Visualization and analysis of jet oscillation under transverse acoustic perturbation, Journal of Flow Visualization and Image Processing, 14 (2007).

[13]

O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H.-O. Walther, Delay Equations: Functional, Complex-, and Nonlinear Analysis, Applied Mathematical Sciences, 110. Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4206-2.

[14]

J.-B. Doc and C. Vergez, Oscillation regimes produced by an alto saxophone: Influence of the control parameters and the bore inharmonicity, The Journal of the Acoustical Society of America, 137 (2015), 1756-1765.  doi: 10.1121/1.4916197.

[15]

J.-B. DocC. Vergez and S. Missoum, A minimal model of a single-reed instrument producing quasi-periodic sounds, Acta Acustica united with Acustica, 100 (2014), 543-554.  doi: 10.3813/AAA.918734.

[16] P. G. Drazin, Introduction to Hydrodynamic Stability, Cambridge University Press, 2002.  doi: 10.1017/CBO9780511809064.
[17]

K. EngelborghsT. Luzyanina and D. Roose, Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL, ACM Trans. Math. Softw., 28 (2002), 1-21.  doi: 10.1145/513001.513002.

[18]

K. Engelborghs, T. Luzyanina and G. Samaey, DDE-BIFTOOL v. 2.00: A Matlab Package for Bifurcation Analysis of Delay Differential Equations, Technical Report TW-330, Department of Computer Science, K.U. Leuven, Belgium, 2001.

[19]

K. Engelborghs and D. Roose, Smoothness loss of periodic solutions of a neutral functional-differential equation: On a bifurcation of the essential spectrum, Dynamics and Stability of Systems, 14 (1999), 255-273.  doi: 10.1080/026811199281994.

[20]

B. Fabre, J. Gilbert and A. Hirschberg, Modeling of wind instruments, Springer Handbook of Systematic Musicology, (2018), 121–139. doi: 10.1007/978-3-662-55004-5_7.

[21]

B. Fabre and A. Hirschberg, Physical modeling of flue instruments: A review of lumped models, Acta Acustica united with Acustica, 86 (2000), 599-610. 

[22]

B. FabreA. Hirschberg and A. P. J. Wijnands, Vortex shedding in steady oscillation of a flue organ pipe, Acta Acustica united with Acustica, 82 (1996), 863-877. 

[23] G. Falkovich, Fluid Mechanics: A Short Course for Physicists, Cambridge university press, Cambridge, 2011.  doi: 10.1017/CBO9780511794353.
[24]

N. H. Fletcher, Nonlinear dynamics and chaos in musical instruments, Complexity International, 1 (1994), 106-117. 

[25]

J. GilbertS. Maugeais and and C. Vergez, Minimal blowing pressure allowing periodic oscillations in a simplified reed musical instrument model: Bouasse-Benade prescription assessed through numerical continuation, Acta Acustica, 4 (2020), 27.  doi: 10.1051/aacus/2020026.

[26]

J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Springer, 2013.

[27]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Third edition, Applied Mathematical Sciences, 112. Springer-Verlag, New York, 2004. doi: 10.1007/978-1-4757-3978-7.

[28]

A. Lefebvre, G. Goudou and G. Scavone, The wind instrument acoustic toolkit, Available from: http://www.music.mcgill.ca/caml/wiat/.

[29]

D. H. Lyons, Resonance frequencies of the recorder (English flute), The Journal of the Acoustical Society of America, 70 (1981), 1239-1247.  doi: 10.1121/1.387136.

[30]

C. MaganzaR. Caussé and F. Laloë, Bifurcations, period doublings and chaos in clarinetlike systems, Europhysics Letters, 1 (1986), 295. 

[31]

M. E. McIntyreR. T. Schumacher and J. Woodhouse, On the oscillations of musical instruments, The Journal of the Acoustical Society of America, 74 (1983), 1325-1345.  doi: 10.1121/1.390157.

[32]

A. W. Nolle, Sinuous instability of a planar air jet: Propagation parameters and acoustic excitation, The Journal of the Acoustical Society of America, 103 (1998), 3690-3705.  doi: 10.1121/1.423089.

[33]

D. Roose and R. Szalai, Continuation and bifurcation analysis of delay differential equations, Numerical Continuation Methods for Dynamical Systems, Underst. Complex Syst., Springer, Dordrecht, (2007). 359–399. doi: 10.1007/978-1-4020-6356-5_12.

[34]

L. F. Shampine, Dissipative approximations to neutral DDEs, Appl. Math. Comput., 203 (2008), 641-648.  doi: 10.1016/j.amc.2008.05.010.

[35]

J. Sieber, K. Engelborghs, T. Luzyanina, G. Samaey and D. Roose, Manual - Bifurcation analysis of delay differential equations, Available from: arXiv.org/abs/1406.7144.

[36]

P.-A. TaillardF. SilvaP. Guillemain and J. Kergomard, Modal analysis of the input impedance of wind instruments. Application to the sound synthesis of a clarinet, Applied Acoustics, 141 (2018), 271-280.  doi: 10.1016/j.apacoust.2018.07.018.

[37]

S. TerrienR. BlandinC. Vergez and B. Fabre, Regime change thresholds in recorder-like instruments: Influence of the mouth pressure dynamics, Acta Acustica united with Acustica, 101 (2015), 300-316.  doi: 10.3813/AAA.918828.

[38]

S. TerrienC. VergezP. de la Cuadra and B. Fabre, Experimental analysis of non-periodic sound regimes in flute-like musical instruments, The Journal of the Acoustical Society of America, 149 (2021), 2100-2108.  doi: 10.1121/10.0003758.

[39]

S. TerrienC. Vergez and B. Fabre, Flute-like musical instruments: A toy model investigated through numerical continuation, Journal of Sound and Vibration, 332 (2013), 3833-3848.  doi: 10.1016/j.jsv.2013.01.041.

[40]

S. TerrienC. VergezB. Fabre and D. A. W. Barton, Calculation of the steady-state oscillations of a flute model using the orthogonal collocation method, Acta Acustica united with Acustica, 100 (2014), 690-704.  doi: 10.3813/AAA.918748.

[41]

C. VauthrinB. Fabre and I. Cossette, How does a flute player adapt his breathing and playing to musical tasks?, Acta Acustica united with Acustica, 101 (2015), 224-237.  doi: 10.3813/AAA.918821.

[42]

M. P. Verge, Aeroacoustics of Confined Jets: With Applications to the Physical Modeling of Recorder-Like Instruments, Ph.D thesis, Technische Universiteit Eindhoven, 1995.

[43]

M. P. VergeB. FabreW. E. A. MahuA. HirschbergR. R. van HasselA. P. J. WijnandsJ. J. de Vries and C. J. Hogendoorn, Jet formation and jet velocity fluctuations in a flue organ pipe, The Journal of the Acoustical Society of America, 95 (1994), 1119-1132.  doi: 10.1121/1.408460.

[44]

H. A. K. Wright and D. M. Campbell, Analysis of the sound of chilean pifilca flutes, The Galpin Society Journal, 51 (1998), 51-63.  doi: 10.2307/842760.

Figure 1.  Radiated sound produced by a Chilean flute (left) played by an experienced player, and by an alto recorder (right) played by a pressure-controlled artificial mouth. The blowing pressure is around 1260 Pa for the Chilean flute and fixed to 650 Pa for the recorder. The inset shows an enlargement on the first 0.01 seconds of the signal (solid blue line), and displays a comparison with the waveform of the periodic sound (dashed red line) obtained using the same instrument and same fingering, but for a lower blowing pressure $ P_m $ = 350 Pa. The corresponding sounds are available online as supplementary material of [38]
Figure 2.  Schematic representation of a Chilean flute, highlighting the shape of the resonator. The air jet oscillating around the labium is represented in blue. $ Q_{in} $ and $ Q_{out} $ represent the alternate flow injection inside and outside of the resonator, which constitutes the source
Figure 3.  Schematic representation of the cross section of the upper part of a recorder flute. The naturally unstable air jet blown by the musician is perturbed by the internal acoustic field $ v_{ac} $ at the channel exit, leading to a transversal deflection $ \eta_0 $ of the jet. This perturbation is subsequently convected and amplified along the jet
Figure 4.  Modulus (top, log scale) and phase (bottom) of the admittance $ Y $ of a Chilean flute (left) and of an alto recorder (right). The experimentally measured admittance of the Chilean flute and the admittance determined analytically from the precise pipe geometry of the recorder are shown in blue. The corresponding fitted admittances written as a sum of resonance modes are shown in red
Figure 5.  Simulated sound regimes: dimensionless acoustical velocity $ v_{ac} $ with respect to time, for the Chilean flute (left) and recorder parameters (right). The rescaled delay time $ \tilde \tau $ is 0.74 (left) and 0.295 (right), corresponding to pressure $ P_m = 915 $ Pa and $ P_m = 4900 $ Pa in the musician's mouth, respectively
Figure 6.  Projection of the Poincaré sections of the two simulated time series shown in Figure 5, for the Chilean flute parameters (left) and the recorder parameters (right), and for a rescaled delay $ \tilde \tau = $ 0.74 (left) and $ \tilde \tau = $0.295 (right). The delay $ \tau_e $ is estimated at the first zero of the autocorrelation of $ v_{ac}(t) $
Figure 7.  Modulus of the spectrum (log scale) of the simulated time series shown in Figure 5 (left), for the Chilean flute parameters and for a dimensionless delay of $ \omega_1 \tau = $ 0.74. The right panel shows an enlargement of the spectrum shown in the left panel over a low frequency range. Shown are the two base frequencies $ f_1 $ and $ f_2 $ of the quasiperiodic regime, as well as some of their linear combinations
Figure 8.  Modulus of the spectrum (log scale) of the simulated time series shown in Figure 5 (right), for the recorder parameters and for a dimensionless delay of $ \omega_1 \tau = $ 0.295. Shown are the two base frequencies $ f_1 $ and $ f_2 $ of the quasiperiodic regime, as well as some of their linear combinations
Figure 9.  One-parameter bifurcation diagram representing the local maxima of the simulated acoustic velocity $ v_{ac} $ with respect to the delay time $ \tilde \tau $, for an increasing (left) and a decreasing (right) delay $ \tilde \tau $. The shaded area indicates the parameter range of the spectrogram in Figure 10
Figure 10.  Left: Spectrogram of the simulated acoustic velocity $ v_{ac} $ shown in Figure 9 (left, shaded area), for a increasing value of the delay $ \tilde \tau $. Right: modulus of the fitted input admittance $ Y(\omega) $ of the Chilean flute, already shown in Figure 4 (top, left)
Figure 11.  One-parameter bifurcation diagram of (7)–(8) in the dimensionless delay $ \tilde \tau $, showing the maximum of the acoustic velocity $ v_{ac} $ (left) and the frequency (right) along branches of periodic solutions. The stable and unstable solutions are represented in blue and red, respectively. Shown are Hopf bifurcation points H (blue dots), torus bifurcation points T (green dots) and point S of saddle-node bifurcation of periodic orbits (red dot). In the right panel, the light gray lines show the resonance frequencies of the instrument. The insets show enlargements of the branch of periodic solution corresponding to the first register on the range of $ \tilde \tau $ represented in Figure 9
Figure 12.  Two-parameter bifurcation diagram of (7)–(8) in the dimensionless delay $ \tilde \tau $ and the inharmonicity parameter $ \Gamma $. Shown are the Hopf bifurcation curves H (blue), and numbers indicate the register emerging at each Hopf bifurcation curve. The horizontal grey line at $ \Gamma = 1 $ highlights the inharmonicity corresponding to the case of the Chilean flute. The inset shows an enlargement of the bifurcation diagram in the framed parameter region, showing curves T of torus bifurcation (green), curve S of saddle-node bifurcation of periodic solutions (brown) and curves PD of period doubling bifurcations (pink). The black dots highlight 1:1 and 1:2 resonances
Table 1.  Values of the modal parameter considered for the admittance of both the Chilean flute and the recorder
$a_0$ $b_0$ $c_0$ $a_1$ $\omega_1$ $Q_1$ $a_2$ $\omega_2$ $Q_2$
Chilean flute $0$ $1.60$ $3.31$ $11.39$ $1157$ $26.07$ $7.04$ $2343$ $34.39$
recorder $11.22$ $1.60$ $3.31$ $22.36$ $2510$ $44.9$ $16.39$ $5113$ $59.65$
$a_3$ $\omega_3$ $Q_3$ $a_4$ $\omega_4$ $Q_4$ $a_5$ $\omega_5$ $Q_5$
Chilean flute $9.55$ $4796$ $50.67$ $8.12$ $5943$ $52.87$ $12.94$ $8419$ $58.00$
recorder $12.64$ $7569$ $67.2$ $10.55$ $9719$ $73.57$ $10.32$ $11909$ $79.98$
$a_0$ $b_0$ $c_0$ $a_1$ $\omega_1$ $Q_1$ $a_2$ $\omega_2$ $Q_2$
Chilean flute $0$ $1.60$ $3.31$ $11.39$ $1157$ $26.07$ $7.04$ $2343$ $34.39$
recorder $11.22$ $1.60$ $3.31$ $22.36$ $2510$ $44.9$ $16.39$ $5113$ $59.65$
$a_3$ $\omega_3$ $Q_3$ $a_4$ $\omega_4$ $Q_4$ $a_5$ $\omega_5$ $Q_5$
Chilean flute $9.55$ $4796$ $50.67$ $8.12$ $5943$ $52.87$ $12.94$ $8419$ $58.00$
recorder $12.64$ $7569$ $67.2$ $10.55$ $9719$ $73.57$ $10.32$ $11909$ $79.98$
Table 2.  Values of the parameters associated with the exciter, for both the Chilean flute and the recorder
h W $ y_0 $ $ \rho $ $ \alpha_{vc} $
Chilean flute $ 10^{-3} $ $ 10^{-2} $ $ 2 \cdot 10^{-4} $ $ 1.2 $ $ 0.6 $
recorder $ 10^{-3} $ $ 4.25\cdot 10^{-3} $ $ 10^{-4} $ $ 1.2 $ $ 0.6 $
h W $ y_0 $ $ \rho $ $ \alpha_{vc} $
Chilean flute $ 10^{-3} $ $ 10^{-2} $ $ 2 \cdot 10^{-4} $ $ 1.2 $ $ 0.6 $
recorder $ 10^{-3} $ $ 4.25\cdot 10^{-3} $ $ 10^{-4} $ $ 1.2 $ $ 0.6 $
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