You are all familiar with the idea of an optical illusion: something that tricks your eyes. The animation below will run an auditory illusion: something that tricks your ears.
A keyboard appears below. Click on the keys to play them; only the white keys work. Now do the following: play up the scale from the low C to the high C. Each note is higher than the last, right?
Now, play the high C and then the low C. These two are the same note! Ok, now what is going on here?
A very good question, indeed. Did you notice that the tones played by the keys sound sort of funny? They are computer-generated tones that are a mix of different pure tones. The frequencies of the tones that you will hear as a note "C" are as follows (actually, they are a tiny bit different from this, but we chose this set of numbers because the math was easy for someone who does a lot of computer stuff):
...8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16,384, 32,768...
Your can only hear notes in the region between about 20 and 20,000 cycles per second (Hertz), and so the bottom few notes and the top one are out of range of your hearing.
If the computer makes a tone that is 256 Hertz, you will hear it as a middle C. Now, what we did to make the tones on the keyboard above was to make a mix of all possible tones that are C's: all the ones from the list above. So, you know that it is a C, but you don't know how high or how low the note is.
We made the "D" on the keyboard by mixing all of the pure tones that you will hear as a "D": all of the high tones and all of the low tones: ...9, 18, 36, 72, 144, 287, 575, 1149, 2299, 4598, 9196, 18,391, 36,782...
Now, when you play this note, you can't tell which D it is. But when you play it right after the C, you will hear it as a little bit higher than the C, since all of the tones are just a little bit higher than the ones we put together for the C: 8 to 9, 16 to 18, etc.
We did the same thing for each note in the scale. As long as you play up the scale, each note sounds higher than the one before it. But when you get to the high C, it is the same as the low C, since both are a mix of all of the different possible C tones. So you are back to where you started from!
Now, go back to the keyboard, and try this: Play the low C, and then the high A. This should sound like a step down. How come? Well, the A tones are high enough that each is just below the next C tone in the sequence. This means that the shortest distance from the C to the A is to go two steps down, rather than 5 steps up. So you hear it as a step down!
This is a really good question. What happens is this: some people hear it as a step up, and others hear it as a step down. The animation below is a little program that plays two tones in quick succession. Do you hear it as a step up, or a step down? Now ask someone else. Do they agree?
(Above animation courtesy of Pixelwings.)
At some point, we will put on a form that allows you to say what you hear, and submit your vote. It turns out that different people hear it differently: some hear a step up, some a step down. It's always the same for each person, but different people hear it different ways!
There is a very nice, accessible article about all of this in the journal Scientific American. The reference is:
"Paradoxes of Musical Pitch", Diana Deutsch, Scientific American, August, 1992, page 88