Music!

Music can be thought of (rather unartistically) as collections of frequencies. Here are the ways that we get notes.

In Western music, the standard is (generally) that a "Concert A" is tuned to 440 Hz. All other notes can be tuned with respect to that. Some orchestras tune to different standards, but we won't worry about those now.

An octave "above" a note is defined as twice the frequency of the note. Similarly, an octave "below" is taking a note and dividing it by two.

The tougher concept is - how do we get from one note on a piano (or any instrument) to the next note (and beyond)? The answer is wrapped up in the "equal tempered scale", a scale such that the ratio of one note to the previous note is always the same.

In short - to get from one note to its octave, multiply by two. But since there are 12 "jumps" or "semi-tones" or "half-steps" from one note to its octave, we ask ourselves (in an equal-tempered scale), what number multiplied by itself 12 times gives us two? The answer?

The twelfth root of 2, or 2 to the 1/12 power --- around 1.0594

This number to the 12th power is 2.

But wait, what did I mean by 12 "jumps"?

A A# B C C# D D# E F F# G G# A

That's 13 notes, but 12 "jumps" or piano keys from A to the next A.

To go from A to A#, multiply the frequency of A by 1.0594.

If you wanted to get to B instead, multiply A's frequency by 1.0594 twice (or by 1.0594^2). To get to C, multiply A by 1.0594^3.



In gory detail:

http://en.wikipedia.org/wiki/Equal_temperament