Duffing’s Equation as an Excitation Mechanism for Plucked String Instrument Models by Justo A. Gutierrez Master’s Research Project Music Engineering Technology University of Miami School of Music December 1, 1999

Purpose The objective of this study is to provide the basis for a new excitation mechanism for plucked string instrument models which utilizes the classical nonlinear system described in Duffing’s Equation.

Advantages Using Duffing’s Equation provides a means to use a nonlinear oscillator as an excitation A mathematical model lends itself to user control Removes the need for saving samples in a wavetable

Overview Plucked String Instrument Modeling Excitation Modeling with Duffing’s Equation Model Performance and Analysis

Wavetable Synthesis Method of synthesis that uses tables of waveforms that are finely sampled Desired waveform is chosen and repeated over and over producing a purely periodic signal Algorithm written as: Y t = Y t-p p is periodicity parameter frequency of the tone is f s /p

The String Model z -L is delay line of length L H(z) is the loop filter F(z) is the allpass filter x(n) and y(n) are the excitation and output signals respectively

Length of String Effective delay length determines fundamental frequency of output signal Delay line length (in samples) is L = f s /f 0

The Comb Filter Works by adding, at each sample time, a delayed and attenuated version of the past output

Standing Wave Analogy Poles of the comb filter occur in the z-plane at 2n  /L This is the same as the natural resonant frequencies for a string tied at both ends Does not sound like a vibrating string because it is a perfectly periodic waveform Does not take into account that high frequencies decay much faster than slow ones for vibrating strings

The Loop Filter Idea is to insert a lowpass filter into the feedback loop of the comb filter so that high-frequency components are diminished relative to low-frequency components every time the past output signal returns Original Karplus-Strong algorithm used a two-tap averager that was simple and effective

Loop Filter (continued) Valimaki et al proposed using an IIR lowpass filter to simulate the damping characteristics of a physical string Loop filter coefficients can be changed as a function of string length and other parameters H 1 (z) = g(1+a 1 )/(1+a 1 z -1 )

Loop Filter Signal Flowchart

Loop Filter Magnitude Response and Group Delay

Loop Filter Impulse Responses

The Allpass Filter Used to fine-tune the pitch of the string model If feedback loop were only to contain a delay line and lowpass filter, total delay would be the sum of integer delay line plus the delay of the lowpass filter Fundamental frequency of f s /D is usually not an integer number of samples

Allpass Filter (continued) Fundamental frequency is then given by f 1 = f s /(D+  ) where  is fractional delay Allpass filters introduce delay but pass frequencies with equal weight Transfer function is H(z) = (z -1 +a)/(1+az -1 ) a = (1-  /(1+ 

Allpass Phase Response

Allpass Delay Response

Inverse Filtering KS algorithm used a white noise burst as excitation for plucked string because it provided high-frequency content as a real pluck would provide Valimaki et al found a pluck signal by filtering the output through the inverted transfer function of the string system

Inverse Filtering (continued) The transfer function for the general string model can be given as S(z) = 1/[1-z -L F(z)H(z)] The inverse filter is simply S -1 (z) = 1/S(z)

Inverse Filtering Procedure Obtain residual by inverse filtering Truncate the first ms of the residual Use the truncated signal as the excitation to the string model Run the string model

Steel-string Guitar Sample

Residual After Inverse Filtering

Truncated Residual Signal

Resynthesized Guitar

Duffing’s Equation In 1918, Duffing introduced a nonlinear oscillator with a cubic stiffness term to describe the hardening spring effect in many mechanical problems It is one of the most common examples in the study of nonlinear oscillations

Duffing’s Equation (continued) The form used for this study is from Moon and Holmes, which is one in which the linear stiffness term is negative so that x” +  x - x + x3 =  cos  t. This model was used to describe the forced oscillations of a ferromagnetic beam buckled between the nonuniform field of two permanent magnets

Experimental Apparatus (Moon and Holmes)

Modeling the Excitation For this experiment, the coefficients in Moon and Holmes’ modification of Duffing’s Equation were adjusted to produce the desired residuals The Runge-Kutta method was the numerical method used to calculate Duffing’s Equation

Procedure for manipulating Duffing’s Equation Generate a waveform of desired frequency with (x, y). f  10y is a good rule of thumb for starters. Adjust the damping coefficient so that its envelope resembles the desired waveform’s Adjust , , and  to shape the waveform, holding one constant to change the other Normalize the waveform to digital maximum

Guitar Residual Synthesized by Duffing’s Equation

Synthesizing the Plucked String

Synthesized Guitar Using Duffing’s Equation as the Excitation

Timbral Characteristics Synthesized guitar from Duffing’s Equation very similar to that from inverse filtering Frequency of both residuals different from pitch of synthesized strings  inharmonicity Sonograms of both residuals also very similar

Sonogram for Guitar (Inverse Filtering)

Sonogram for Guitar (Duffing’s Equation)

Tuning Performance (Harmony) For individual pitches, the algorithm played fairly close to being in tune (perhaps slightly sharp). The allpass filter parameters can be adjusted to remedy this. The C major chord played very well in tune, sounding very consonant with no apparent beats.

Tuning Performance (Range) To test effective range of the algorithm, the lowest and highest pitches in a guitar’s range were synthesized. Low E played in tune by itself. High E was flat. This was more readily apparent when sounded together.

Summary of Tuning Performance Algorithm performed as expected; it performed like Karplus-Strong; high frequencies tend to go flat, and this would have to be accounted for in the overall system.

Changing Damping Coefficient Changing the damping coefficient can have pronounced effect on timbre of sound, specifically difference between type of pick used and type of string The damping coefficient was adjusted to attempt to produce different sounds

Synthesized Residual (  = 0.2)

Synthesized Guitar (  = 0.2)

Synthesized Residual (  = 0.5)

Summary of Damping Coefficient Adjustments For  = 0.2, contribution of residual made for a very hard attack, as if picked For  = 0.5, guitar tone had much softer attack, as if finger-picked Sonograms confirm that the latter had more high-frequency content

Sonogram for  = 0.2

Sonogram for  = 0.5

Production of Other Waveforms Duffing’s Equation can be used to form a variety of waveforms User has some control over its behavior if properties of the oscillator can be controlled to obtain the desired waveform

Residual with Damping Only

Residual with Beta Only

Residual with Forcing Function Only

Residual with Strong High- Frequency Forcing Function

Algorithm Speed For 200 MHz Pentium Pro, Karplus-Strong with an inverse filtered residual took s. with approximately 2500 samples saved on a wavetable With synthesized residual, Duffing’s Equation added only s; total computation time increased by only about 5% with no saved samples

Conclusion Plucked string sounds were successfully produced Model plays in tune Different plucked string sounds can be produced by changing the damping coefficient Algorithm is fast