need help with problem 1 and 2
CPSC 231 - Fall 2018 - University of Calgary cpsc 231 assignment #6 Guitar Heroine The goal of your final assignment in cpsc 231 is to explore compu- tational algorithms for digital synthesis of sounds, instruments, and music. In the process, you’ll practice using and creating data types in Python, and even spend some time writing a little recursive function. Audio is encoded digitally on computers as a discrete signal, or a sequence of discrete samples, that represents the sound pressure level over time. In other words, it’s no more than a big, one-dimensional array of numbers. Periodic repetitions and patterns in the numbers give the sound its characteristic tone and timbre. For example, the sound of a strum on an acoustic guitar looks like this when plotted as a graph: -1 -0.5 0 0.5 1 If we can come up with an algorithm that generates these numbers so that the result has a meaningful tone and timbre, we can synthe- size the sound of musical instruments! In 1983, Kevin Karplus and Alex Strong described a very simple algorithm that can generate the sound of plucking a string on a guitar.1 Despite its simplicity, it pro- 1 Kevin Karplus and Alex Strong. Digital synthesis of plucked-string and drum timbres. Computer Music Journal, 7 (2):43–55, 1983 duces an incredibly rich and realistic sound. You’ll end your work in cpsc 231 this semester by implementing the Karplus-Strong algo- rithm to turn your computer into a live acoustic guitar synthesizer that you can play music on. Due Dates Individual component Friday, November 30, 11:59 pm Paired component Friday, December 7, 11:59 pm 2 cpsc 231 - fall 2018 - university of calgary Individual Component As with all your previous assignments, we’ll use the individual com- ponent to “warm up” a little bit before going about the live synthesis of the sound of an acoustic guitar. Here, we’ll first use a recursive al- gorithm to generate a one-dimensional fractal noise signal. This kind of noise can create patterns that are more similar to natural phenom- ena than noise consisting of uniformly-distributed random numbers (i.e. white noise). We will stream this noise to the audio output of your computer to synthesize the sound of ocean waves crashing on the shore. Before you begin, it may be worthwhile to review the sections in the course text that pertain to digital audio and music (pp. 171–175 and 231–235). You can use the exact same programming environment you set up for previous assignments to complete this assignment. Create and submit a separate Python program for each problem. Problem 1: Brownian Bridge A Brownian bridge is a stochastic process (think of this as a random function over time) that models a random walk connecting two points. In our case, the bridge is a function of time, and if we fix its value at an initial and a final time point, b(t0) and b(t1), we can compute the in-between values with the following recursive process: 1. Compute the midpoint of the interval, ym = 12 (b(t0) + b(t1)). 2. Add to the midpoint value a random value δ, drawn from a nor- mal distribution2 with zero mean and given variance, and assign 2 You can use random.gauss() or random.normalvariate() to generate such a random value, providing the mean and standard deviation (square root of variance) as arguments. the sum to the bridge function: b(tm)← ym + δ. 3. Recur on the subintervals (t0, tm) and (tm, t1), dividing the vari- ance for the next level by a given scale factor. The shape of the function is controlled by two parameters: the volatility and the Hurst exponent. Volatility is the initial value of the variance, and controls how far the random walk strays from the straight line connecting the fixed endpoints. The Hurst exponent, H, controls the smoothness of the result. The variance of the distribution from which the random value δ is drawn is divided by 22H at each recursive level. When H = 0.5, the variance is halved after each recur- sion, and the result is a true Brownian bridge that models Brownian motion between the endpoints. Note that Program 2.3.5 in the course text uses a recursive function to plot a Brownian bridge on the screen. Rather than drawing a graph, this problem requires you to fill an array with values of the function. Studying that example would certainly be a good idea, and you may complete this problem by starting from the textbook example and modifying it if you like. Write a recursive Python function to fill an array with numerical values that form a Brownian bridge between two specified endpoints. The function should take a reference to an array, a first and last in- dex, a variance, and a scale factor as parameters. These would allow the function to fill the array according to the recurrence described. For example, your function may have a definition that looks like this: assignment #6: guitar heroine 3 def fill_brownian(a, i0, i1, variance, scale): """ Recursive function that fills an array with values forming a Brownian bridge between indices i0 and i1 (exclusive). Assumes that a[i0] and a[i1] are values of the fixed endpoints. :param a: Array to fill with Brownian bridge. :param i0: Index of first endpoint. :param i1: Index of second endpoint :param variance: Variance of the normal distribution from which to sample a displacement. :param scale: Scale factor to reduce variance for next level. """ When you’ve completed the function, write a program to test it out. Call your recursive function to fill an array with 129 elements, the first and last of which are anchored at a value of 0.0. Use a volatility of 0.05 and allow the Hurst exponent to be specified as a command line argument. Then visualize the contents of your array by drawing a graphical line plot of the values to the screen. You may either write your own code, use some of the code we wrote in class or that’s found in the course text, or use the stdstats module from the booksite library to draw the plot. Figure 1: Line plots of 129 values of a Brownian bridge with Hurst exponents of 1.0, 0.5, and 0.2, respectively from top to bottom. Inputs: The Hurst exponent for your Brownian bridge function, spec- ified as a command line argument. For example, you might run your program with the following invocation: $ python3 my-p1.py 0.5 Outputs: A line plot of your randomly-generated Brownian bridge between the two zero endpoints. It should resemble those shown on the right when run with the corresponding Hurst exponent. Problem 2: Ocean Waves If you stream the Brownian bridge function values to your com- puter’s audio output,3 the result sounds like a pleasant, rumbling 3 You can use the booksite library function stdaudio.playSamples(a) to stream a sequence of samples in the array a to your computer’s default audio output device. The values in a should be floats between -1.0 and +1.0. kind of noise. In fact, if you generate the values with a Hurst expo- nent of 0.5, the result is what audio engineers call “brown noise” or “red noise”. This is distinct from “pink noise” and the higher-pitched and more commonly found “white noise”, which is simply a stream of uniformly distributed random numbers. Both kinds of noise are useful, and we can actually synthesize a half-decent sound of ocean waves crashing on the shore by oscillating between brown noise and white noise at a very low frequency. If nb(t) represents the sequence of brown noise values over time, and nw(t) represents white noise, we can achieve the ocean wave effect by 4 cpsc 231 - fall 2018 - university of calgary blending them with a time-varying blend factor, s(t), as follows: y(t) = (1− s(t)) nb(t) + s(t) nw(t), with s(t) = sin6(π f t). In this equation, t represents the time (in seconds), and f is the fre- quency at which the ocean waves are crashing on the shore. Write a program that generates 20 seconds of crashing ocean waves using the method described above, then plays the sound through the computer’s audio output. If you use the stdaudio mod- ule, the output sample rate is fixed to 44 100 Hz (samples per sec- ond), which means you will need to generate a total of 20 × 44 100 = 882 000 data values. Use your Brownian bridge function from Prob- lem 1 to generate values for nb(t).4 You can adjust any parameters 4 You will want to re-anchor your bridge to a value of 0.0 several times a second so that the values don’t drift too far from the neutral zero- point. In other words, rather than generating all 882 000 samples with a single function call, you should generate a few thousand samples at a time, setting the two endpoints to 0.0 each time. within these equations that you’d like in order to synthesize a more realistic or more pleasant ocean sound, but perhaps we can suggest that you start with the following: • Use a Hurst exponent of H = 0.5. Increase it slightly if you want lower-frequency noise, and decrease it for higher-frequency noise. • A frequency of f = 0.25 will give you a wave crash every four seconds. Adjust as you like. • The exponent of 6 on the blend factor, s(t), controls the balance of the two noise types during each cycle. Increase the exponent for quicker wave crashes, but use even exponents to keep the blend factor non-negative. • An amplitude of about 0.25 works well for the white noise compo- nent. In other words, generate random values between -0.25 and +0.25 for nw(t). Adjust as you like as well. You may also want to introduce one more slowly-oscillating function to modulate the amplitude (volume) of your output sound to achieve the ultimate ocean wave simulator, but that’s purely optional. Inputs: None. Outputs: The sound of synthetic ocean waves crashing to sooth and relax you for 20 seconds when you run your program. Paired Component For once, the paired component of the assignment might actually be easier than the individual component! We know it’s the last week of the term, and you’re both busy and exhausted with all your other end-of-semester work, so this is our gift to you for making it all the way to the finish line. The end result is quite fantastic, especially for the amount of time required to complete it. And if you’ve really assignment #6: guitar heroine 5 got time and energy to spare at this point, we’ve got many bonus opportunities worth taking up. You may still work with a partner of your own choosing to com- plete the remainder of this assignment, though remember that it may not be the same