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IEEE ICASSP99, Phoenix, Arizona, March 1999 IEEE ICASSP99, Phoenix, Arizona, March 1999 HELSINKI UNIVERSITY OF TECHNOLOGY Plucked-String Synthesis Algorithms Plucked-String Synthesis Algorithms with Tension Modulation Nonlinearity with

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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

Vesa Välimäki, Tero Tolonen & Matti Karjalainen Helsinki University of Technology Laboratory of Acoustics and Audio Signal Processing (Espoo, Finland)

http://www.acoustics.hut.fi/

IEEE ICASSP’99, Phoenix, Arizona, March 1999 IEEE ICASSP’99, Phoenix, Arizona, March 1999

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Välimäki, Tolonen, and Karjalainen 1999 2

HELSINKI UNIVERSITY OF TECHNOLOGY

Outline

➤ Introduction ➤ Tension Modulation ➤ Synthesis Algorithms with Tension Modulation ➤ Synthesis Examples ➤ Conclusions

Plucked-String Synthesis Algorithms Plucked-String Synthesis Algorithms with Tension Modulation Nonlinearity with Tension Modulation Nonlinearity

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Välimäki, Tolonen, and Karjalainen 1999 3

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FD FD Delay line Delay line H Hl

l(

(z z) ) In In Out Out Loop filter Loop filter

Linear Plucked-String Synthesis Model Linear Plucked-String Synthesis Model

Fractional delay filter Fractional delay filter

  • Originally developed at CCRMA, Stanford University

(Smith, 1997; Karjalainen et al., 1998) Fundamental frequency Decay rate

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HELSINKI UNIVERSITY OF TECHNOLOGY

  • 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 Time-Varying Fractional Delay Filters (Välimäki et al., 1998)

Introduction to Nonlinear String Models Introduction to Nonlinear String Models

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  • Elongation caused by string vibration affects tension
  • Tension modulation affects transversal wave speed c:

Tension Modulation Tension Modulation

where Ft 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!

c F

t

= ρ

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General Signal-Dependent Delay Line General Signal-Dependent Delay Line

FD FD d d( (n n) ) Signal in Signal in Signal out Signal out Delay line Delay line G G

  • 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)

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HELSINKI UNIVERSITY OF TECHNOLOGY

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 248 249 250 251

Frequency (Hz)

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 149.5 150 150.5

Time (s)

Effects of Tension Modulation: part 1 Effects of Tension Modulation: part 1

  • Variation of fundamental frequency

Variation of fundamental frequency in plucked-string tones

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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

0.2 0.4 0.6 0.8 1

  • 60
  • 50
  • 40
  • 30
  • 20
  • 10

Magnitude (dB) Time (s) #1 #2 #3 #4

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Dual-Delay-Line (DDL) String Model Dual-Delay-Line (DDL) String Model with Tension Modulation with Tension Modulation

  • Elongation approximation = sum of squared sums
  • I(z) is a leaky integrator with phase inversion

d d( (n n) ) Out Out I I( (z z)/2 )/2 Elongation Elongation approximation approximation R Rb

b(

(z z) ) FD FD Delay line Delay line Delay line Delay line R Rf

f(

(z z) ) FD FD Initial slope/2 Initial slope/2 Initial slope/2 Initial slope/2

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Välimäki, Tolonen, and Karjalainen 1999 10

HELSINKI UNIVERSITY OF TECHNOLOGY

Single-Delay-Loop Single-Delay-Loop (SDL) String Model (SDL) String Model with Tension Modulation with Tension Modulation

FD FD Elongation Elongation approximation approximation Delay line Delay line d d( (n n) ) H Hl

l(

(z z) ) In In In In Out Out I I( (z z) )

  • Pitch variation and coupling of harmonics can be

controlled by the coefficients of the leaky integrator:

I z g a a z ( ) = − + +

− p p p

1 1

1

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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: ap = -0.999 3) tension modulation model: ap = -0.99 4) tension modulation model: ap = -0.97 5) linear synthesis model

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  • Originally every 3rd

harmonic missing

  • The generation of

the missing harmonics is controlled by ap

Comparison of Comparison of Spectra Spectra

1 2 3 4 5

  • 50

50 50 1 2 3 4 5

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50

Magnitude (dB)

1 2 3 4 5

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50 1 2 3 4 5

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50

Frequency (kHz) Linear string model Nonlinear string model ap = -0.999 ap = -0.99 ap = -0.97

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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: gp = 200 3) tension modulation model: gp = 1000 4) tension modulation model: gp = 10000 5) linear synthesis model

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Sound Example 3 Sound Example 3

  • Elongation approximation

Elongation approximation is the most time-consuming part in the new synthesis model – Sum of Lnom squared sums!

  • We propose to compute only every Mth sum (Lnom = 45)

1) M = 1 2) M = 6 3) M = 12 4) M = 24 5) M = 1

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0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 147 147.5 148

Frequency (Hz)

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 147 148 149 150

Frequency (Hz) Time (s)

Pitch Variation in Synthetic Tones Pitch Variation in Synthetic Tones

DDL SDL SDL with M = 6 Method: Displacement 2 mm Displacement 4 mm

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HELSINKI UNIVERSITY OF TECHNOLOGY 0.1 0.2 0.3 0.4 0.5 0.6

  • 60
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Magnitude (dB)

0.1 0.2 0.3 0.4 0.5 0.6

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Magnitude (dB)

0.1 0.2 0.3 0.4 0.5 0.6

  • 60
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Magnitude (dB)

Time (s)

Envelopes of Harmonics Envelopes of Harmonics

Linear model Nonlinear SDL model (gp = 10) …with sparse squared sum (M = 6) #1 #2 #3 Harmonic: Method:

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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/