A Real Time Piano Model Including Longitudinal Modes

Transcription

1 Introduction String Modeling Implementation Issues A Real Time Piano Model Including Longitudinal Modes Stefano Zambon and Federico Fontana Dipartimento di Informatica Università degli Studi di Verona 26 ottobre 2007

2 Introduction String Modeling Implementation Issues Summary 1 Introduction Physical Models 2 String Modeling Transversal vibration Longitudinal Vibration 3 Implementation Issues Real-time prototype Results

3 Introduction String Modeling Implementation Issues Physical Models Physical Modeling of the Piano Source-based instead than signal-based approach to the synthesis of piano sounds Advantages: higher fidelity in the interaction, meaningful synthesis parameters Drawbacks: computational cost, static sound quality Main research topics in the last 10 years: 1 Accurate string simulation (Chaigne-Askenfelt 1994, Bensa-Bilbao 2003) 2 Geometric nonlinearities (Bank 2005) 3 Real-time models (Borin et al. 1997, Pianoteq 2006)

4 Introduction String Modeling Implementation Issues Physical Models Physical Modeling of the Piano Source-based instead than signal-based approach to the synthesis of piano sounds Advantages: higher fidelity in the interaction, meaningful synthesis parameters Drawbacks: computational cost, static sound quality Main research topics in the last 10 years: 1 Accurate string simulation (Chaigne-Askenfelt 1994, Bensa-Bilbao 2003) 2 Geometric nonlinearities (Bank 2005) 3 Real-time models (Borin et al. 1997, Pianoteq 2006)

5 Introduction String Modeling Implementation Issues Physical Models Physical Modeling of the Piano Source-based instead than signal-based approach to the synthesis of piano sounds Advantages: higher fidelity in the interaction, meaningful synthesis parameters Drawbacks: computational cost, static sound quality Main research topics in the last 10 years: 1 Accurate string simulation (Chaigne-Askenfelt 1994, Bensa-Bilbao 2003) 2 Geometric nonlinearities (Bank 2005) 3 Real-time models (Borin et al. 1997, Pianoteq 2006)

6 Introduction String Modeling Implementation Issues Physical Models Piano Structure Struck string instrument Large extension and high dynamic range Sound production mechanism: key hammer string bridge soundboard

7 Introduction String Modeling Implementation Issues Physical Models Piano Structure Struck string instrument Large extension and high dynamic range Sound production mechanism: key hammer string bridge soundboard

8 Introduction String Modeling Implementation Issues Physical Models Model Structure Principal modeling blocks: Bridge / Body Dynamics F h y(x ) 0 F b,t Hammer Transversal vibr. Bridge Soundboard F b p Long. exc. force Longitudinal vibr. F b,l H l linear nonlinear Secondary modeling blocks, mostly signal-based: Coupled twin string, dampers, impact knock noise

9 Introduction String Modeling Implementation Issues Physical Models Model Structure Principal modeling blocks: Bridge / Body Dynamics F h y(x ) 0 F b,t Hammer Transversal vibr. Bridge Soundboard F b p Long. exc. force Longitudinal vibr. F b,l H l linear nonlinear Secondary modeling blocks, mostly signal-based: Coupled twin string, dampers, impact knock noise

10 Introduction String Modeling Implementation Issues Transversal vibration Longitudinal Vibration String Modeling µ 2 y t 2 = T 0 2 y x 2 1-D wave equation with additional terms for: Stiffness Frequency-dependent losses External force density Hammer is treated as a lumped system made up of a mass connected in series with a nonlinear string.

11 Introduction String Modeling Implementation Issues Transversal vibration Longitudinal Vibration String Modeling µ 2 y t 2 = T 0 2 y x 2 ESκ 2 4 y x 4 1-D wave equation with additional terms for: Stiffness Frequency-dependent losses External force density Hammer is treated as a lumped system made up of a mass connected in series with a nonlinear string.

12 Introduction String Modeling Implementation Issues Transversal vibration Longitudinal Vibration String Modeling µ 2 y t 2 = T 2 y 0 ESκ 2 4 y 2R(ω)µ y x 2 x 4 t 1-D wave equation with additional terms for: Stiffness Frequency-dependent losses External force density Hammer is treated as a lumped system made up of a mass connected in series with a nonlinear string.

13 Introduction String Modeling Implementation Issues Transversal vibration Longitudinal Vibration String Modeling µ 2 y t 2 = T 0 2 y x 2 ESκ 2 4 y x 4 2R(ω)µ y t + d y (x, t) 1-D wave equation with additional terms for: Stiffness Frequency-dependent losses External force density Hammer is treated as a lumped system made up of a mass connected in series with a nonlinear string.

14 Introduction String Modeling Implementation Issues Transversal vibration Longitudinal Vibration String Modeling µ 2 y t 2 = T 0 2 y x 2 ESκ 2 4 y x 4 2R(ω)µ y t + d y (x, t) 1-D wave equation with additional terms for: Stiffness Frequency-dependent losses External force density Hammer is treated as a lumped system made up of a mass connected in series with a nonlinear string.

15 Introduction String Modeling Implementation Issues Transversal vibration Longitudinal Vibration Discretization: Modal Synthesis Idea: discretization of the modal solution to the string equation y(x, t) = w k (x) y k (t), k=1 where w k (x) are space-dependent weights and y k (t) are the temporal component of the normal modes The motion of the normal modes is described by a set of ODEs: d 2 y k dt 2 + a dy k 1,k dt which are discretized using the Impulse Invariance Transform + a 0,k = b 0,k F y,k,

16 Introduction String Modeling Implementation Issues Transversal vibration Longitudinal Vibration Discretization: Modal Synthesis Idea: discretization of the modal solution to the string equation y(x, t) = w k (x) y k (t), k=1 where w k (x) are space-dependent weights and y k (t) are the temporal component of the normal modes The motion of the normal modes is described by a set of ODEs: d 2 y k dt 2 + a dy k 1,k dt which are discretized using the Impulse Invariance Transform + a 0,k = b 0,k F y,k,

17 Introduction String Modeling Implementation Issues Transversal vibration Longitudinal Vibration Block Diagram for String Simulation W in, 1 b 1,1 y 1 W out, 1 a 1,1 z 1 a 2,1 z 1 F h z 1 W in, k b 1,k y k W out, k F b a 1,k z 1 a 2,k z 1 W in, N b 1,N y N W out, N a 1,N z 1 a 2,N z 1

18 Introduction String Modeling Implementation Issues Transversal vibration Longitudinal Vibration Why Modal Synthesis? Powerful control in the frequency domain High accuracy in dispersion and losses modeling Allows many extensions to the basic model, for example longitudinal modes Parallelizable and suitable for efficient DSP implementation

19 Introduction String Modeling Implementation Issues Transversal vibration Longitudinal Vibration Longitudinal Vibration Perceptually important for low notes at high dynamics Example Equation for the motion of the longitudinal component ξ: 2 ξ t 2 = ES 2 ξ x 2 2R ξ(ω) + 1 ( y/ x) 2 2 x It is a 1-d lossy wave equation with an additional nonlinear forcing term depending on the transversal slope

20 Introduction String Modeling Implementation Issues Transversal vibration Longitudinal Vibration Sound Synthesis of Longitudinal Modes Basic assumption: there is energy transfer only from the transversal to the longitudinal motion Longitudinal modes are simulated by a set of few resonators excited by a nonlinear force: Magnitude [db] 60 f min f max (a) Magnitude [db] 60 (b) f min 40 f max Magnitude [db] 60 (c) Frequency [Hz]

21 Introduction String Modeling Implementation Issues Real-time prototype Results Real-time prototype Written in C++, runs at full polyphony on current hardware Allows run-time change of a set of meaningful parameters: string and hammer mass, stiffness coefficients, etc. Midi Control Change Midi NoteOn / NoteOff Controller + Handles MIDI and initializes state change Parameters (runs at control rate) initialize Hammer / Damper State AudioSynth PostProcessing String_Param[88] Strings/Hammers parameters update Coeffs String_State[88] Strings and Hammers state read State + Compute force at send F b + Soundboard filter audio output (i.e. mass, inharmonicity..) (i.e., filters coefficients/states) update State bridge junction + Post processing (runs at audio rate)

22 Introduction String Modeling Implementation Issues Real-time prototype Results Results Demo