Modeling of the piano

The piano is one of the most popular instruments in western music. It has a wide dynamic range and playing range more than seven octaves. Its popularity probably arises from its versatility and its relatively simple control mechanism. When the player depresses the key, the damper laying on the string is lifted and the hammer is thrown towards the string. The kinetic energy of the hammer is stored into the normal modes of the string, from which the energy is leaking into the soundboard through the bridge. The soundboard colors and amplifies the audible sound.

In the modeling procedure, some features characteristic to the piano sound must be taken into account. For example, the stiffness of the strings makes the sound inharmonic. In addition, the decay process of the sound is relatively complicated due to different decay rates of the partials and the two-stage decay. Piano synthesis has become commercially very popular due to the success of digital pianos and synthesizers using sampling technique. Some research has been done on physics-based piano modeling, but it is not yet able to challenge sampling technique in commercial use, because the sound quality is not high enough at the moment. However, it seems that this might change in the future.

The following table lists our piano-related research in inverse chronological order. For some publications, the PDF file and a companion web page containing sound examples are provided. These can be found in the leftmost column.

Publication Short description
Rauhala, J., "The Beating Equalizer and its Application to the Synthesis and Modification of Piano Tones," in Proceedings of the 10th International Conference on Digital Audio Effects (DAFx-07) Bordeaux, France, Sept. 10-15, 2007, pp. 181-188.
The beating effect is implemented with a second-order equalizing filter by modulating its gain value with a single parameter. Thus, the method offers a direct relation between the filter parameters and the beating effect. The method can also be used increase or even cancel the beating effect from recorded tones.

Lehtonen, H.-M., Penttinen, H., Rauhala, J., and Välimäki V., "Analysis and Modeling of Piano Sustain-Pedal Effects," Journal of the Acoustical Society of America, vol. 122, no. 3, pp. 1787-1797, September 2007.
The features of the piano sustain pedal are presented through signal analysis that is performed on recorded piano tones. Based on this information, a reverberation algorithm is designed for synthesizing the effect.
Rauhala, J., Lehtonen, H.-M., and Välimäki V., "Fast Automatic Inharmonicity Estimation Algorithm," Journal of the Acoustical Society of America, vol. 121, no. 5, pp. EL184-189, May 2007.
A new computationally efficient method for estimating the inharmonicity coefficient is presented. The idea is to form a partial frequency curve, which indicates whether the current inharmonicity coefficient estimate is higher or lower than the accurate value. This process is iterated with an adaptive step size. Also the fundamental frequency estimate can be refined, and this can be used to improve the inharmonicity coefficient estimate.

Rauhala, J., Lehtonen, H.-M., and Välimäki V., "Toward Next-Generation Digital Keyboard Instruments," IEEE Signal Processing Magazine, vol. 24, no. 2, pp. 12-20, March 2007.
This paper presents an overview of the acoustics of three keyboard instruments: the clavichord, the harpsichord, and the piano. Parametric synthesis of these instruments is discussed. In addition, the paper presents a new beating model, which uses amplitude modulation for producing the effect.

Rauhala, J. and Välimäki V., "Dispersion Modeling in Waveguide Piano Synthesis Using Tunable Allpass Filters," in Proceedings of the 9th International Conference on Digital Audio Effects (DAFx-06) Montreal, Canada, Sept. 18-20, 2006, pp. 71-76.
The previously proposed tunable allpass filter method to design first-order dispersion filters for piano synthesis is extended in this paper. Moreover, the number of filters in cascade is parameterized, and, hence, a first-order dispersion filter consisting of an arbitrary number of filters in cascade can be designed with a closed-form formula.
Rauhala, J. and Välimäki V., "Tunable Dispersion Filter Design for Piano Synthesis," IEEE Signal Processing Letters, vol. 13, no. 5, pp. 253-256, May 2006.
This article introduces a new closed-form approach to design dispersion filters. Closed-form design formulae are determined based on the Thiran fractional delay filter design method.
Rauhala, J. and Välimäki V., "Parametric excitation model for waveguide piano synthesis," in Proceedings of 2006 IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP 2006), Toulouse, France, May 14-19, 2006, pp. 157-160.
A new excitation model is proposed in this paper. In the proposed model, the excitation signal is first produced by using additive synthesis and a noise generator. Then, it is windowed with a shaping window. Finally, the signal is filtered with velocity-dependant equalizing filter before it is fed to the string model.
N/A Lehtonen, H.-M., Rauhala, J., and Välimäki V., "Sparse Multi-Stage Loss Filter Design for Waveguide Piano Synthesis," in Proceedings of IEEE Workshop on Applications of Signal Processing to Audio and Acoustics (WASPAA'05), pp. 331-334, New Paltz, NY, USA, October 16-19, 2005.
A new method for the loss filter design for digital waveguide piano synthesis purposes is presented. The method is also applicable to the synthesis of other struck or plucked string instruments. The structure of the loss filter consists of a cascade of sparse FIR filters which are designed on subbands and upsampled for implementation. In addition, with this method it is possible to exactly match the decay rate of a finite number lowest-order partials.
N/A Rauhala, J., Lehtonen, H.-M., and Välimäki V., "Multi-ripple loss filter for waveguide piano synthesis," in Proceedings of the International Computer Music Conference (ICMC 2005), pp. 729-732, Barcelona, Spain, September 5-9, 2005.
The multi-ripple loss filter, a combination of a single one-pole IIR filter and an arbitrary-order sparse FIR filter, is able to powerfully model the differences in the partial decay rates needed in waveguide piano synthesis. The FIR filter can be efficiently designed by using the frequency-sampling method. In addition, a presented data smoothing technique improves the performance at lower filter orders.
N/A Bank, B., and Välimäki V., "Robust loss filter design for digital waveguide synthesis of string tones," IEEE Signal Processing Letters, vol. 10, no. 1, pp. 18-20, January 2003.
A robust loss filter design method, which aims at minimizing the decay time error in partials of a synthetic tone, is presented. The method is applicable to digital waveguide synthesis of string tones. The key idea in the design is to use a new weighting function based on the first-order Taylor series approximation of the decay time errors. In order to facilitate the stability of the process, it is proposed that the design is required to be minimum-phase.
Bank, B., Välimäki V., Sujbert, L., and Karjalainen, M., "Efficient Physics-Based Sound Synthesis of the Piano Using DSP Methods," in Proceedings of the European Signal Processing Conference (EUSIPCO 2000), vol. 4, pp. 2225-2228, Tampere, Finland, September 5-8, 2000.
A new structure of the digital waveguide for string synthesis is presented. Especially, the structure can be used for efficient modeling of two important features of the piano sound, the beating and the two-stage decay. In addition to the computational efficiency of the model, the structure enables flexibility in controlling the envelopes of individual harmonics. As a result, a more natural piano sound is achieved.
N/A Bank, B., "Nonlinear interaction in the digital waveguide with the application to piano sound synthesis," in Proceedings of the International Computer Music Conference (ICMC00), pp. 54-58, Berlin, Germany, August 27 - September 1, 2000.
The interaction between the hammer model and the digital waveguide model of the piano is discussed. A solution to the discontinuity problem occurring when feeding the interaction force into the digital waveguide model is proposed. To overcome the stability problem of the hammer model, a novel multi-rate implementation is presented.

Bank, B., "Physics-Based Sound Synthesis of the Piano," Report no. 54, Helsinki University of Technology, Laboratory of Acoustics and Audio Signal Processing, Espoo, Finland, June 2000.
An extensive description of the physics based sound synthesis of the piano is given. The work includes a careful analysis of the piano sound as well as many new solutions that can be used as a part of the digital waveguide model of the piano.

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