Math 106: Project I

Can You Hear the Difference?

Click to download the TeX file for this project.

Guidelines: You should consider your project as a technical paper (like an English paper only different). Part of your grade will be based on the quality of your written presentation. The paper you turn in should have a mix of equations, formulas and prose. Graphs may be copied directly from your calculator, but should be clearly labelled. Use complete sentences, good grammar and correct punctuation. Spelling is also important. You should aim to write your report in such a way that it can be read and understood by anyone who knows the material for this course. Cordelia McFlute is in her lab one day, measuring the output from her synthesiser. She's playing the A below middle C (which has a frequency of 440 cycles per second).

A single note on the synthesiser comes out as a sinusoidal wave-which is why the tones sound rather "colorless". Cordelia is trying to make harmonies and more complicated sound waves, which will give her music a richer tone.

Draw a graph of the output, taking care to label it carefully.

On a second channel she begins to play the note one octave higher (that is, with twice the frequency). On the same scale, draw a graph of the output from the second channel, and also one of the output from playing the two notes together.

Next, she tries pressing the second key a little after the first. Sketch some graphs to show some possible outputs from the combined channels.

Think about what really happens when Cordelia presses the two keys together, and decide if the last question makes you want to revise your answer to the second "".

Cordelia experiments with varying the relative loudness of the two channels, and with striking the keys at different times. Try some (mathematical) experiments yourselves---what are the effects on the synthesiser's output? How different can you make your outputs? What features do they all have in common?

What would you hear if you were in the lab? What changes can Cordelia make which affect what you hear, and what changes make no difference?

The last part of the project concerns the function SYNTH that you should have been given in Recitation class. If you plot SYNTH as a function of x, it will give you the output of a particular setting of the synthesiser, when x is measured in milliseconds (ms). (To plot the function, just enter y1=SYNTH.)

Draw a graph of the output.

Is this the graph of a trigonometric function? Take care to justify your answer.

What two keys on the keyboard could Cordelia play, and at what amplitudes, to get her best approximation to this output? Draw a graph which shows the accuracy of this approximation.

Will the output she gets from doing this always look like the graph of SYNTH? Does that matter?

Verify Explain Justify