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IEEE ICASSP’99, Phoenix, Arizona, March 1999 IEEE ICASSP’99, Phoenix, Arizona, March 1999 HELSINKI UNIVERSITY OF TECHNOLOGY Plucked-String Synthesis Algorithms Plucked-String Synthesis Algorithms with Tension Modulation Nonlinearity with Tension Modulation Nonlinearity Vesa Välimäki, Tero Tolonen & Matti Karjalainen Helsinki University of Technology Laboratory of Acoustics and Audio Signal Processing (Espoo, Finland) http://www.acoustics.hut.fi/ Välimäki, Tolonen, and Karjalainen 1999 1

HELSINKI UNIVERSITY OF TECHNOLOGY Plucked-String Synthesis Algorithms Plucked-String Synthesis Algorithms with Tension Modulation Nonlinearity with Tension Modulation Nonlinearity Outline ➤ Introduction ➤ Tension Modulation ➤ Synthesis Algorithms with Tension Modulation ➤ Synthesis Examples ➤ Conclusions Välimäki, Tolonen, and Karjalainen 1999 2

HELSINKI UNIVERSITY OF TECHNOLOGY Linear Plucked-String Synthesis Model Linear Plucked-String Synthesis Model • Originally developed at CCRMA, Stanford University (Smith, 1997; Karjalainen et al. , 1998) In In Out Out FD FD Delay line H l ( z z ) ) Delay line H l ( Fractional delay filter Fractional delay filter Loop filter Loop filter Fundamental frequency Decay rate Välimäki, Tolonen, and Karjalainen 1999 3

HELSINKI UNIVERSITY OF TECHNOLOGY Introduction to Nonlinear String Models Introduction to Nonlinear String Models • In practice, a vibrating string is always nonlinear • Tension modulation Tension modulation caused by elongation of the string • (Carrier, 1945; Morse, 1948; Legge & Fletcher, 1984) • Passive nonlinearity Passive nonlinearity (Pierce and Van Duyne, 1997) • • Passive nonlinearities can be generalized by using Time-Varying Fractional Delay Filters (Välimäki et al. , Time-Varying Fractional Delay Filters 1998) Välimäki, Tolonen, and Karjalainen 1999 4

HELSINKI UNIVERSITY OF TECHNOLOGY Tension Modulation Tension Modulation • Elongation caused by string vibration affects tension • Tension modulation affects transversal wave speed c : F = t c ρ where F t is tension and ρ is linear mass density • Change of c affects the wave propagation delay ⇒ Length of delay line must be modulated! ⇒ Length of delay line must be modulated! Välimäki, Tolonen, and Karjalainen 1999 5

HELSINKI UNIVERSITY OF TECHNOLOGY General Signal-Dependent Delay Line General Signal-Dependent Delay Line Signal in Signal in Delay line Signal out Delay line FD Signal out FD G G d ( ( n n ) ) d • Input signal controls the fractional-delay (FD) filter fractional-delay (FD) filter (Välimäki et al. , ICMC’98) • Function Function G G computes delay parameter d ( n ) • Välimäki, Tolonen, and Karjalainen 1999 6

HELSINKI UNIVERSITY OF TECHNOLOGY Effects of Tension Modulation: part 1 Effects of Tension Modulation: part 1 • Variation of fundamental frequency Variation of fundamental frequency in • plucked-string tones 251 Frequency (Hz) 250 249 248 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 150.5 150 149.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Time (s) Välimäki, Tolonen, and Karjalainen 1999 7

HELSINKI UNIVERSITY OF TECHNOLOGY Effects of Tension Modulation: part 2 Effects of Tension Modulation: part 2 • Coupling of partials in plucked-string tones, e.g., generation of missing harmonics generation of missing harmonics 0 #1 Magnitude (dB) -10 #2 -20 #3 -30 -40 #4 -50 -60 0 0.2 0.4 0.6 0.8 1 Time (s) Välimäki, Tolonen, and Karjalainen 1999 8

HELSINKI UNIVERSITY OF TECHNOLOGY Dual-Delay-Line (DDL) String Model Dual-Delay-Line (DDL) String Model with Tension Modulation with Tension Modulation Initial slope/2 Initial slope/2 Out Out FD Delay line FD Delay line d ( ( n n ) ) d Elongation Elongation R b ( z z ) ) I ( ( z z )/2 )/2 R f ( z z ) ) R b ( I R f ( approximation approximation FD Delay line FD Delay line Initial slope/2 Initial slope/2 • Elongation approximation = sum of squared sums • I ( z ) is a leaky integrator with phase inversion Välimäki, Tolonen, and Karjalainen 1999 9

HELSINKI UNIVERSITY OF TECHNOLOGY Single-Delay-Loop (SDL) String Model (SDL) String Model Single-Delay-Loop with Tension Modulation with Tension Modulation In In In In Out Out FD Delay line H l ( z z ) ) FD Delay line H l ( d ( ( n n ) ) d Elongation Elongation I ( ( z z ) ) I approximation approximation • Pitch variation and coupling of harmonics can be controlled by the coefficients of the leaky integrator: + 1 a ( ) = − p I z g − + p 1 1 a z p Välimäki, Tolonen, and Karjalainen 1999 10

HELSINKI UNIVERSITY OF TECHNOLOGY Sound Example 1 Sound Example 1 • Plucked-string synthesis including tension modulation varying the coupling of harmonics coupling of harmonics: 1) linear synthesis model 2) tension modulation model: a p = -0.999 3) tension modulation model: a p = -0.99 4) tension modulation model: a p = -0.97 5) linear synthesis model Välimäki, Tolonen, and Karjalainen 1999 11

HELSINKI UNIVERSITY OF TECHNOLOGY Comparison of Comparison of Linear string model 50 Spectra Spectra 0 -50 0 1 2 3 4 5 Magnitude (dB) 50 • Originally every 3rd Nonlinear string model 50 harmonic missing a p = -0.999 0 • The generation of -50 0 1 2 3 4 5 50 the missing a p = -0.99 0 harmonics is -50 0 1 2 3 4 5 controlled by a p 50 a p = -0.97 0 -50 0 1 2 3 4 5 Frequency (kHz) Välimäki, Tolonen, and Karjalainen 1999 12

HELSINKI UNIVERSITY OF TECHNOLOGY Sound Example 2 Sound Example 2 • Plucked-string synthesis including tension modulation varying the extent of pitch variation pitch variation: 1) linear synthesis model 2) tension modulation model: g p = 200 3) tension modulation model: g p = 1000 4) tension modulation model: g p = 10000 5) linear synthesis model Välimäki, Tolonen, and Karjalainen 1999 13

HELSINKI UNIVERSITY OF TECHNOLOGY Sound Example 3 Sound Example 3 • Elongation approximation Elongation approximation is the most time-consuming • part in the new synthesis model – Sum of L nom squared sums! • We propose to compute only every M th sum ( L nom = 45) 1) M = 1 2) M = 6 3) M = 12 4) M = 24 5) M = 1 Välimäki, Tolonen, and Karjalainen 1999 14

HELSINKI UNIVERSITY OF TECHNOLOGY Pitch Variation in Synthetic Tones Pitch Variation in Synthetic Tones Frequency (Hz) 148 Displacement 2 mm 147.5 Method: DDL 147 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Frequency (Hz) SDL 150 Displacement 4 mm SDL with 149 M = 6 148 147 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time (s) Välimäki, Tolonen, and Karjalainen 1999 15

HELSINKI UNIVERSITY OF TECHNOLOGY Envelopes of Harmonics Envelopes of Harmonics Method: Magnitude (dB) -40 Linear model Harmonic: -60 #1 0 0.1 0.2 0.3 0.4 0.5 0.6 Magnitude (dB) Nonlinear -40 #2 SDL model -60 ( g p = 10) #3 0 0.1 0.2 0.3 0.4 0.5 0.6 Magnitude (dB) …with sparse -40 squared sum -60 ( M = 6) 0 0.1 0.2 0.3 0.4 0.5 0.6 Time (s) Välimäki, Tolonen, and Karjalainen 1999 16

HELSINKI UNIVERSITY OF TECHNOLOGY Conclusions and Future Work Conclusions and Future Work • Computationally efficient nonlinear plucked-string model that accounts for the tension modulation tension modulation • Length of delay line is changed continuously using a time-varying fractional delay FIR filter time-varying fractional delay FIR filter • Extended work including parameter estimation of tension modulation models submitted to a journal • Sound examples available at our Web site: http://www.acoustics.hut.fi/~ttolonen/sounddemos/tmstr/ Välimäki, Tolonen, and Karjalainen 1999 17