STATEMENT OF PROBLEM

The pitch of a musical note is determined by the frequency
of the vibration which causes it. Middle C on the piano,
for example, corresponds to a vibration of 263 Hz (cycles per
second). A note one octave above middle C vibrates at 526 Hz, and a note two octaves above middle C vibrates at 1052 Hz.

Below is a table which shows how the frequency changes as a function of the number, n, of octaves above middle C.

Input := 

MatrixForm[Table[{{" n", "freq(n)"}, {"----", "--------"},
      {-3, 32.875}, {-2, 65.75}, {-1, 131.5},
      {0, 263}, {1, 526}, {2, 1052}, {3, 2104}, {4, 4208}}]]

Output =

 n         freq(n)

----       --------

-3         32.875

-2         65.75

-1         131.5

0          263

1          526

2          1052

3          2104

4          4208

1) Graph the frequency as a function of n.

2) The data graphed in #1 should appear exponential. The general form of an exponential function is:
F(x) = a bx
where b>0 is the base of the exponent and a is an arbitrary constant. Exponential data has the property that the ratio of the output for equally spaced input is a constant. Show that freq(n) is an exponential function of n and find the function.

3) Graph the function and the data on the same graph.

The following problems can only be done if you have a machine that will play audible frequencies such as any modern computer running Mathematica using the Play[] command. The problems that follow are modeled for Mathematica.

4) Use Mathematica's Play[] command to verify that the frequencies given in the table do increase by musical octaves. Note that frequencies are given in cycles per second, so you'll have to make sure you're using the correct frequency. For example, the following command plays a tone at 500 cycles per second for 3 seconds since Sin[t] has a frequency of (1 cycle)/(2 Pi seconds).

Input := 

Play[Sin[2*Pi*500*t], {t, 0, 3}]

Output =

-Sound-

5) Choose any audible frequency that you haven't played yet. Using the same base b that you found in #2, use Mathematica's Play[] command to see if the octaves still increase by n.

The next problem is more of a trig/frequency review than an exponential problem.

6) Make Mathematica play a varying range of frequencies between 200 and 400 Hz over a 1 second period.