There are a few musical notes that always go well with each other. They call it 1-4-5. In the key of A, the notes are A-D-E. This is the fundamental progression of 90% of western music. But why? Why are there 12 notes in an octave? Why can't there be 17? We'll ignore sharps and flats. Sharps and flats are an exercise in chaos theory dreamed up by an ergotine-crazed madman. Why do some notes have half steps and some whole steps between notes? Whoa! One thing at a time. We'll cover that another day. 

So what is it about these three notes that makes them sound good together? 

Simple. Convert to frequency. Let's use A=440hz. The 'D' below it is 293.66 hz. What's magic about this? The ratio 0.667470909. It's the closest note in the European scale to (2/3f) where f = 440. What about E? 329.63hz. Or a ratio of 0.74915909. It's the closest note in the European scale to (3/4f).  So the entire reason why notes sound good together is because they have simple frequency ratios. What about (1/4f, 1/2f, and 2f)? Those my friend are octaves.  They always sound good together.

Now let's ask the next logical question - so then why are there 12 notes in the European scale instead of 17 or 22 or something else? Go back to 7th grade math. What's the lowest common denominator of our magic good sounding ratios 2/3 and 3/4?

Bingo! 12 (2/3 -> 8/12 and 3/4 -> 9/12). Therefore, if you want to have a musical system which includes both 2/3f and 3/4f (our good sounding frequencies), the minimum number of unique steps it needs to have (if the steps are assumed to be equal) is 12. In fact, you need to have twelve steps between 1/2f and 1f (or 1f and 2f).

[...An interesting side note is that if you have a digital frequency source, 293.33333 makes a better sound with A than concert 'D'. Likewise 330hz is a more perfect E than E itself. That's because these are perfect ratios, unlike the European scale which is based on centuries of compromise and rounding errors.]