Piano Synthesis

The main elements of the piano we need to model are the

• hammer (§9.3.2),
• string (Chapter 6, §6.6),
• bridge (§9.2.1), and
• soundboard, enclosure, acoustic space, mic placement, etc.

Since piano strings are audibly stiff, we need to include a dispersion allpass filter in the string model, as introduced in (§6.11.3). The next section provides additional details specific to piano strings.

From the bridge (which runs along the soundboard) to each ``virtual microphone'' (or ``virtual ear''), the soundboard, enclosure, and listening space can be well modeled as a high-order LTI filter characterized by its impulse response (Chapter 8). Such long impulse responses can be converted to more efficient recursive digital filters by various means (Chapter 3,§8.6.2).

A straightforward piano model along the lines indicated above turns out to be relatively expensive computationally. Therefore, we also discuss, in §9.4.4 below, the more specialized commuted piano synthesis technique [467], which is capable of high quality piano synthesis at low cost relative to other model-based approaches. It can be seen as a hybrid method in which the string is physically modeled, the hammer is represented by a signal model, and the acoustic resonators (soundboard, enclosure, etc.) are handled by ordinary sampling (of their impulse response).

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Subsections

• Stiff Piano Strings
  • Piano String Wave Equation
  • Damping-Filter Design
  • Dispersion Filter-Design
  
  
• Nonlinear Piano Strings
  • Nonlinear Piano-String Synthesis
  • Regimes of Piano-String Vibration
  • Efficient Waveguide Synthesis of Nonlinear Piano Strings
  • Checking the Approximations
  
  
• High-Accuracy Piano-String Modeling
  • A Stiff Mass-Spring String Model
  • Nonlinear Piano-String Equations of Motion in State-Space Form
  • Finite Difference Implementation
  
  
• Commuted Piano Synthesis
  • Force-Pulse Synthesis
  • Multiple Force-Pulse Synthesis
  • Commuted Piano Synthesis Architecture
  • Progressive Filtering
  • Excitation Factoring
  • Excitation Synthesis
  • Coupled Piano Strings
  • Commuted Synthesis of String Reverberation
  • High Piano Key Numbers
  • Force-Pulse Filter Design
  
  
• Literature on Piano Acoustics and Synthesis

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Previous: Feathering
Next: Stiff Piano Strings

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.