CIS 110 Summer 2012 - Introduction to Computer Programming

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CETS Answers
Plucking a Guitar String
Programming Assignment

Write a program to simulate plucking a guitar string using the Karplus-Strong algorithm. This algorithm played a seminal role in the emergence of physically modeled sound synthesis (where a physical description of a musical instrument is used to synthesize sound electronically).

Digital audio. Before reading this assignment, review the material in the textbook on digital audio (pp. 147–151, 202–206).

Simulate the plucking of a guitar string. When a guitar string is plucked, the string vibrates and creates sound. The length of the string determines its fundamental frequency of vibration. We model a guitar string by sampling its displacement (a real number between -1/2 and +1/2) at N equally spaced points (in time), where N equals the sampling rate (44,100) divided by the fundamental frequency (rounding the quotient to the nearest integer).

      Sampling from Karplus-Strong

  • Plucking the string. The excitation of the string can contain energy at any frequency. We simulate the excitation with white noise: set each of the N displacements to a random real number between -1/2 and +1/2.

    White noise

  • The resulting vibrations. After the string is plucked, the string vibrates. The pluck causes a displacement which spreads wave-like over time. The Karplus-Strong algorithm simulates this vibration by maintaining a ring buffer of the N samples: the algorithm repeatedly deletes the first sample from the buffer and adds to the end of the buffer the average of the first two samples, scaled by an energy decay factor of 0.996.

    the Karplus-Strong update

Why it works? The two primary components that make the Karplus-Strong algorithm work are the ring buffer feedback mechanism and the averaging operation.

  • The ring buffer feedback mechanism. The ring buffer models the medium (a string tied down at both ends) in which the energy travels back and forth. The length of the ring buffer determines the fundamental frequency of the resulting sound. Sonically, the feedback mechanism reinforces only the fundamental frequency and its harmonics (frequencies at integer multiples of the fundamental). The energy decay factor (.996 in this case) models the slight dissipation in energy as the wave makes a roundtrip through the string.

  • The averaging operation. The averaging operation serves as a gentle low-pass filter (which removes higher frequencies while allowing lower frequencies to pass, hence the name). Because it is in the path of the feedback, this has the effect of gradually attenuating the higher harmonics while keeping the lower ones, which corresponds closely with how a plucked guitar string sounds.
From a mathematical physics viewpoint, the Karplus-Strong algorithm approximately solves the 1D wave equation, which describes the transverse motion of the string as a function of time.

Ring buffer. Your first task is to create a data type to model the ring buffer. Write a class named RingBuffer that implements the following API:

public class RingBuffer
        RingBuffer(int capacity)  // create an empty ring buffer, with given max capacity
    int size()                    // return number of items currently in the buffer
boolean isEmpty()                 // is the buffer empty (size equals zero)?
boolean isFull()                  // is the buffer full  (size equals capacity)?
   void enqueue(double x)         // add item x to the end
 double dequeue()                 // delete and return item from the front
 double peek()                    // return (but do not delete) item from the front
Since the ring buffer has a known maximum capacity, implement it using a double array of that length. For efficiency, use cyclic wrap-around: Maintain one integer instance variable first that stores the index of the least recently inserted item; maintain a second integer instance variable last that stores the index one beyond the most recently inserted item. To insert an item, put it at index last and increment last. To remove an item, take it from index first and increment first. When either index equals capacity, make it wrap-around by changing the index to 0.

Implement RingBuffer to throw an exception if the client attempts to dequeue() from an empty buffer or enqueue() into a full buffer.

Ring buffer

Guitar string. Next, create a data type to model a vibrating guitar string. Write a class named GuitarString that implements the following API:

public class GuitarString
       GuitarString(double frequency)  // create a guitar string of the given frequency, using a sampling rate of 44,100
       GuitarString(double[] init)     // create a guitar string whose size and initial values are given by the array
  void pluck()                         // set the buffer to white noise
  void tic()                           // advance the simulation one time step
double sample()                        // return the current sample
   int time()                          // return number of tics

  • Constructors. There are two ways to create a GuitarString object.

    • The first constructor creates a RingBuffer of the desired capacity N (sampling rate 44,100 divided by frequency, rounded to the nearest integer), and initializes it to represent a guitar string at rest by enqueueing N zeros.

    • The second constructor creates a RingBuffer of capacity equal to the size of the array, and initializes the contents of the buffer to the values in the array. On this assignment, its main purpose is for debugging and grading.

  • Pluck. Replace the N items in the ring buffer with N random values between -0.5 and +0.5.

  • Tic. Apply the Karplus-Strong update: delete the sample at the front of the ring buffer and add to the end of the ring buffer the average of the first two samples, multiplied by the energy decay factor.

  • Sample. Return the value of the item at the front of the ring buffer.

  • Time. Return the total number of times tic() was called.

Interactive guitar player. is a sample GuitarString client that plays the guitar in real-time, using the keyboard to input notes. When the user types the lowercase letter 'a' or 'c', the program plucks the corresponding string. Since the combined result of several sound waves is the superposition of the individual sound waves, we play the sum of all string samples.

  public class GuitarHeroLite {
      public static void main(String[] args) {

          // create two guitar strings, for concert A and C
          double CONCERT_A = 440.0;
          double CONCERT_C = CONCERT_A * Math.pow(2, 3.0/12.0);  
          GuitarString stringA = new GuitarString(CONCERT_A);
          GuitarString stringC = new GuitarString(CONCERT_C);

          while (true) {

              // check if the user has typed a key; if so, process it   
              if (StdDraw.hasNextKeyTyped()) {
                  char key = StdDraw.nextKeyTyped();
                  if      (key == 'a') { stringA.pluck(); }
                  else if (key == 'c') { stringC.pluck(); }

              // compute the superposition of samples
              double sample = stringA.sample() + stringC.sample();
              // play the sample on standard audio
              // advance the simulation of each guitar string by one step   

Write a program GuitarHero that is similar to GuitarHeroLite, but supports a total of 37 notes on the chromatic scale from 110Hz to 880Hz. In general, make the ith character of the string
String keyboard = "q2we4r5ty7u8i9op-[=zxdcfvgbnjmk,.;/' ";
This keyboard arrangement imitates a piano keyboard: The "white keys" are on the qwerty and zxcv rows and the "black keys" on the 12345 and asdf rows of the keyboard.
Piano keyboard
The ith character of the string corresponds to a frequency of 440 × 2(i - 24) / 12, so that the character 'q' is 110Hz, 'i' is 220Hz, 'v' is 440Hz, and ' ' is 880Hz. Don't even think of including 37 individual GuitarString variables or a 37-way if statement! Instead, create an array of 37 GuitarString objects and use keyboard.indexOf(key) to figure out which key was typed. Make sure your program does not crash if a key is played that is not one of your 37 notes.

Checklist.   Don't forget to read the checklist. Pay especially close attention to the Debugging section.

Submission.   Submit,, and Finally, submit a readme_guitar.txt file and answer the questions.

Extra credit 1. Write a program (by modifying that plots the sound wave in real-time, as the user is playing the keyboard guitar. The output should look something like this, but change over time.

  Sampling from Karplus-Strong
You should not redraw the wave on every sample. Instead, draw the wave of the last n samples every n timesteps for an appropriate value of n. Experiment with different values of n to find one that you think looks good and draws smoothly.

Extra credit 2. Bring your laptop to recitation and perform a piece for your classmates.

Challenge for the bored. Modify the Karplus-Strong algorithm to synthesize a different instrument. Consider changing the excitation of the string (from white-noise to something more structured) or changing the averaging formula (from the average of the first two samples to a more complicated rule) or anything else you might imagine.

This assignment was developed by Andrew Appel, Jeff Bernstein, Maia Ginsburg, Ken Steiglitz, Ge Wang, and Kevin Wayne.
Copyright © 2005