Skip to main content
Log in

3d Modeling and simulation of a harpsichord

  • Original Article
  • Published:
Computing and Visualization in Science

Abstract

The mathematical characterization of the sound of a musical instrument still follows Schumann’s laws (Schumann in Physik der klangfarben, Leipzig, 1929). According to this theory, the resonances of the instrument body, “the formants”, filter the oscillations of the sound generator (e.g., strings) and produce the characteristic “timbre” of an instrument. This is a strong simplification of the actual situation. It applies to a point source and can be easily performed by a loudspeaker, disregarding the three dimensional structure of music instruments. To describe the effect of geometry and material of the instruments, we set up a 3d model and simulate it using the simulation system UG4 (Vogel et al. in Comput Vis Sci 16(4):165–179, 2013; Reiter et al. in Comput Vis Sci 16(4):151–164, 2014). We aim to capture the oscillation behavior of eigenfrequencies of a harpsichord soundboard and investigate how well a model for the oscillation behavior of the soundboard approximates the oscillation behavior of the whole instrument. We resolve the complicated geometry by several unstructured 3d grids and take into account the anisotropy of wood. The oscillation behavior of the soundboard is modeled following the laws of linear orthotropic elasticity with homogenous boundary conditions. The associated eigenproblem is discretized using FEM and solved with the iterative PINVIT method using an efficient GMG preconditioner (Neymeyr in A hierarchy of preconditioned eigensolvers for elliptic differential operators. Habilitation dissertation, University of Tübingen, 2001). The latter allows us to resolve the harpsichord with a high resolution hybrid grid, which is required to capture fine modes of the simulated eigenfrequencies. We computed the first 16 eigenmodes and eigenfrequencies with a resolution of 1.8 billion unknowns each on Shaheen II supercomputer (https://www.hpc.kaust.edu.sa/content/shaheen-ii). To verify our results, we compare them with measurement data obtained from an experimental modal analysis of a real reference harpsichord.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  1. Schumann, E.K.: Physik der klangfarben. Leipzig (1929)

  2. Vogel, A., Reiter, S., Rupp, M., Nägel, A., Wittum, G.: Ug 4: a novel flexible software system for simulating PDE based models on high performance computers. Comput. Vis. Sci. 16(4), 165–179 (2013)

    Article  Google Scholar 

  3. Reiter, S., Vogel, A., Heppner, I., Wittum, G.: A massively parallel geometric multigrid solver on hierarchically distributed grids. Comput. Vis. Sci. 16(4), 151–164 (2014)

    Article  Google Scholar 

  4. Neymeyr, K.: A hierarchy of preconditioned eigensolvers for elliptic differential operators. Habilitation dissertation, University of Tübingen (2001)

  5. Shaheen II. Shaheen II. https://www.hpc.kaust.edu.sa/content/shaheen-ii

  6. Hackbusch, W.: Elliptic Differential Equations: Theory and Numerical Treatment. Volume 18 of Springer Series in Computational Mathematics. Springer, Berlin (2003)

    MATH  Google Scholar 

  7. Johnson, C.A., Bilhuber, P.H.: The influence of the soundboard on piano tone quality. J. Acoust. Soc. Am. 11(3), 311–320 (1940)

    Article  Google Scholar 

  8. Wogram, K.: Acoustical research on pianos: vibrational characteristics of the soundboard. Das Musikinstrument 24, 694–702 (1980)

    Google Scholar 

  9. Wang, I.-C., Kindel, J.: Modal analysis and finite element analysis of a piano soundboard. In: IMAC, pp. 1545–1549 (1987)

  10. Giordano, N.: Mechanical impedance of a piano soundboard. J. Acoust. Soc. Am. 103(4), 2128–2133 (1998)

    Article  Google Scholar 

  11. Chabassier, J., Chaigne, A., Joly, P.: Modeling and simulation of a grand piano. J. Acoust. Soc. Am. 134, 648–665 (2013)

    Article  Google Scholar 

  12. Polytec: http://www.polytec.com

  13. Martin, D.: The Art of Making a Harpsichord. Robert Hale, London (2012)

  14. Joly, P.: The mathematical model for elastic wave propagation. In: Kampanis, N.A., Dougalis, V.A., Ekaterinaris, J.A. (eds.) Effective Computational Methods for Wave Propagation. CRC Press, New York (2008)

    Google Scholar 

  15. Kollmann, F., Côté, W.A.: Principles of Wood Science and Technology. Volume 1: Solid Wood. Springer, Berlin (1968)

    Book  Google Scholar 

  16. Rupp, M..: Ein filterndes algebraisches Mehrgitterverfahren mit Anwendungen in der Strukturmechanik. Dissertation, Universität Frankfurt (2017) (Unpublished manuscript)

  17. Blender. Blender: https://www.blender.org

  18. Reiter, S.: Effiziente Algorithmen und Datenstrukturen für die Realisierung von adaptiven, hierarchischen Gittern auf massiv parallelen Systemen. Dissertation, Universität Frankfurt (2014)

  19. Reiter, S.: Promesh3d, meshing of unstructured grids in 1, 2, and 3 dimensions. http://promesh3d.com/. (Accessed July 2019)

  20. Piperkova, R.: Diplomarbeit. Dissertation, Universität Frankfurt (2014)

  21. Polytec: Psv-400 scanning vibrometer. http://www.polytec.com/us/products/vibration-sensors/vibrometer-accessories/scanning-vibrometer-accessories/psv-400/. (Accessed July 2019)

  22. Giordano, N.: Sound production by a vibrating piano soundboard: experiment. J. Acoust. Soc. Am. 104(3), 1648–1653 (1998)

    Article  Google Scholar 

  23. Braess, D.: Finite Elemente. Springer, Berlin (2003)

    Book  Google Scholar 

  24. Corradi, R., Miccoli, S., Squicciarini, G., Fazioli, P.: Modal analysis of a grand piano soundboard at successive manufacturing stages. Appl. Acoust. 125, 113–127 (2017)

    Article  Google Scholar 

Download references

Acknowledgements

The support by Merzdorf GmbH and Polytec GmbH is gratefully acknowledged. For computing time, this research used the resources of the Supercomputing Laboratory at King Abdullah University of Science and Technology (KAUST) in Thuwal, Saudi Arabia. Furthermore, we would like to thank Rossitza Piperkova for setting up an initial geometry of the soundboard.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Lukas Larisch.

Additional information

Communicated by Michael Heisig.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix

Appendix

See Figs. 9, 10, 11, 12, 13, 14, 15, 16, 17 and 18.

Fig. 9
figure 9

Setup of the experimental modal analysis

Fig. 10
figure 10

Eigenmodes for 5th and 16th Eigenfrequency on models \(M_{1}\) and \(M_{2}\). Left: 77.9 \({\mathrm{Hz}}\), 113.3 \({\mathrm{Hz}}\) , Right: 175.8, 224.5 \({\mathrm{Hz}}\)

Fig. 11
figure 11

Simulation on \(M_{4}\) versus Experiment. Left: EF1 @ 81.9 \({\mathrm{Hz}}\) versus 68.5 \({\mathrm{Hz}}\) , Right: EF2 @ 86.6 \({\mathrm{Hz}}\) versus 75.5 \({\mathrm{Hz}}\)

Fig. 12
figure 12

Simulation on \(M_{4}\) versus Experiment. Left: EF3 @ 95.5 \({\mathrm{Hz}}\) versus 97.5 \({\mathrm{Hz}}\) , Right: EF4 @ 101.6 \({\mathrm{Hz}}\) versus 104 \({\mathrm{Hz}}\)

Fig. 13
figure 13

Simulation on \(M_{4}\) versus Experiment. Left: EF5 @ 115.3 \({\mathrm{Hz}}\) versus 110 \({\mathrm{Hz}}\) , Right: EF6 @134.7 \({\mathrm{Hz}}\) versus 146.5 \({\mathrm{Hz}}\)

Fig. 14
figure 14

Simulation on \(M_{4}\) versus Experiment. Left: EF9 @ 178 \({\mathrm{Hz}}\) versus 198 , Right: EF10 @ 184 \({\mathrm{Hz}}\) versus 202.5 \({\mathrm{Hz}}\)

Fig. 15
figure 15

Simulation on \(M_{4}\) versus Experiment. Left: EF11 @ 192.2 \({\mathrm{Hz}}\) versus 203.5 , Right: EF16 @226.11 \({\mathrm{Hz}}\) versus 234.5 \({\mathrm{Hz}}\)

Fig. 16
figure 16

Eigenmodes for 15th and 16th Eigenfrequency on models \(M_{3}\) and \(M_{4}\). Left: 223.9 \({\mathrm{Hz}}\), 226.05 \({\mathrm{Hz}}\), Right: 228.1 \({\mathrm{Hz}}\), 226.11 \({\mathrm{Hz}}\)

Fig. 17
figure 17

Eigenmodes for 9th and 10th Eigenfrequency on models \(M_{2}\) and \(M_{3}\). Left: 115.1 \({\mathrm{Hz}}\), 175.5 \({\mathrm{Hz}}\), Right: 125.1 \({\mathrm{Hz}}\), 177.8 \({\mathrm{Hz}}\)

Fig. 18
figure 18

Eigenmodes for 12th and 16th Eigenfrequency on models \(M_{2}\) and \(M_{3}\). Left: 193.1 \({\mathrm{Hz}}\), 198.8 \({\mathrm{Hz}}\), Right: 224.5 \({\mathrm{Hz}}\), 228.1 \({\mathrm{Hz}}\)

1.1 Eigenmodes from experimental modal analysis

See Figs. 19, 20, 21 and 22.

Fig. 19
figure 19

68.5 \({\mathrm{Hz}}\), 75.5 \({\mathrm{Hz}}\), 97.5 \({\mathrm{Hz}}\), 104 \({\mathrm{Hz}}\)

Fig. 20
figure 20

110 \({\mathrm{Hz}}\), 146.5 \({\mathrm{Hz}}\), 157.5 \({\mathrm{Hz}}\), 167.5 \({\mathrm{Hz}}\)

Fig. 21
figure 21

198 \({\mathrm{Hz}}\), 201.5 \({\mathrm{Hz}}\), 203.5 \({\mathrm{Hz}}\), 112.5 \({\mathrm{Hz}}\)

Fig. 22
figure 22

229.5 \({\mathrm{Hz}}\), 230.5 \({\mathrm{Hz}}\), 234.5 \({\mathrm{Hz}}\)

1.2 Eigenmodes from simulation

See Figs. 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37 and 38.

Fig. 23
figure 23

EF1: 28.5 \({\mathrm{Hz}}\), 81.2 \({\mathrm{Hz}}\), 81.1 \({\mathrm{Hz}}\), 81.9 \({\mathrm{Hz}}\)

Fig. 24
figure 24

EF2: 43.7 \({\mathrm{Hz}}\), 86.5 \({\mathrm{Hz}}\), 86.6 \({\mathrm{Hz}}\), 86.6 \({\mathrm{Hz}}\)

Fig. 25
figure 25

EF3: 57.9 \({\mathrm{Hz}}\), 94.7 \({\mathrm{Hz}}\), 95.4 \({\mathrm{Hz}}\), 95.5 \({\mathrm{Hz}}\)

Fig. 26
figure 26

EF4: 63.8 \({\mathrm{Hz}}\), 98.6 \({\mathrm{Hz}}\), 101.4 \({\mathrm{Hz}}\), 101.6 \({\mathrm{Hz}}\)

Fig. 27
figure 27

EF5: 77.9 \({\mathrm{Hz}}\), 113.3 \({\mathrm{Hz}}\), 114.7 \({\mathrm{Hz}}\), 115.3 \({\mathrm{Hz}}\)

Fig. 28
figure 28

EF6: 85.7 \({\mathrm{Hz}}\), 134.3 \({\mathrm{Hz}}\), 135.3 \({\mathrm{Hz}}\), 134.7 \({\mathrm{Hz}}\)

Fig. 29
figure 29

EF7: 99.9 \({\mathrm{Hz}}\), 145.8 \({\mathrm{Hz}}\), 147.9 \({\mathrm{Hz}}\), 146.9 \({\mathrm{Hz}}\)

Fig. 30
figure 30

EF8: 101.4 \({\mathrm{Hz}}\), 159.7 \({\mathrm{Hz}}\), 160.5 \({\mathrm{Hz}}\),159.3 \({\mathrm{Hz}}\)

Fig. 31
figure 31

EF9: 115.1 \({\mathrm{Hz}}\), 175.5 \({\mathrm{Hz}}\), 178.5 \({\mathrm{Hz}}\), 178 \({\mathrm{Hz}}\)

Fig. 32
figure 32

EF10: 125.1 \({\mathrm{Hz}}\), 177.8 \({\mathrm{Hz}}\), 184.3 \({\mathrm{Hz}}\), 184 \({\mathrm{Hz}}\)

Fig. 33
figure 33

EF11: 127.3 \({\mathrm{Hz}}\), 190.4 \({\mathrm{Hz}}\), 194.1 \({\mathrm{Hz}}\), 192.2 \({\mathrm{Hz}}\)

Fig. 34
figure 34

EF12: 143.1 \({\mathrm{Hz}}\), 193.1 \({\mathrm{Hz}}\), 198.8 \({\mathrm{Hz}}\), 197.2 \({\mathrm{Hz}}\)

Fig. 35
figure 35

EF13: 152.5 \({\mathrm{Hz}}\), 212.1 \({\mathrm{Hz}}\), 217.9 \({\mathrm{Hz}}\), 217.1 \({\mathrm{Hz}}\)

Fig. 36
figure 36

EF14: 156.1 \({\mathrm{Hz}}\), 218.6 \({\mathrm{Hz}}\), 222.9 \({\mathrm{Hz}}\), 222.8 \({\mathrm{Hz}}\)

Fig. 37
figure 37

EF15: 165.5 \({\mathrm{Hz}}\), 222.1 \({\mathrm{Hz}}\), 223.9 \({\mathrm{Hz}}\), 226.05 \({\mathrm{Hz}}\)

Fig. 38
figure 38

EF16: 175.8 \({\mathrm{Hz}}\), 224.5 \({\mathrm{Hz}}\), 228.1 \({\mathrm{Hz}}\), 226.11 \({\mathrm{Hz}}\)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Larisch, L., Lemke, B. & Wittum, G. 3d Modeling and simulation of a harpsichord. Comput. Visual Sci. 23, 6 (2020). https://doi.org/10.1007/s00791-020-00326-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00791-020-00326-1

Navigation