The Ditonic Comma - Look and Listen
The main idea here is that (3/2)^k = 2^n has no positive integer solutions, so we consider the approximation where k=12 and n=7.
Well, let's compare two notes. We apply the label F#4 to 371.25 Hz and Gb4 to about 366.25 Hz; they satisfy the ratio described above. We then graph sine functions with these periods. Below, we can see F#4 as the function f, in red and Gb4 as the function g, in blue. Since these frequencies are quite high, we'll need to zoom in very far.
What are the units? The x-axis is labeled in milliseconds. Now, the frequency of f is 371.25 Hz, so we should see 371.25 oscillations per second; how many per 10 milliseconds? What other evidence is there that we're seeing accurate representations of the waves we're trying to illustrate? For example, which graph should have smaller wavelength?
Now that we're convinced we're seeing representations of F#4 and Gb4, we'll need to zoom out. We want to see what happens when we hear f and g played at the same time.What is the period of the resultant function f+g? Click 'show f+g' to display the pointwise sum of the functions.
Well that's surprising! From two functions with high frequencies, we get this! So, what does it sound like? We can use another free program to play sine waves with the desired frequencies. First is F#4 in the left channel, then Gb4, then both together. The graph below shows a bit of what will be played; take a look to be sure we're listening to the right thing (units are in seconds).
Click on the graph to play the .mp3 file.
Well, what do we hear? Given that the pitches are played simultaneously for exactly 2 seconds, what is the frequency of the 'beats?' Does this correspond with the graphical representation above?
Tim Cieplowski, August 8, 2009, Created with GeoGebra