Here's a
geekier take on it.
An octave is an exact doubling of
pitch, a.k.a.
frequency.
A above middle C is
canonically defined as
440 Hz1 (two
octaves above the A string on a
guitar which is 110
Hz). What that "
440 Hz" means in
plain English2 is that the thing producing the sound is making four hundred and forty complete back-and-forth motions per second
3, and pushing the air back and forth at the same rate.
Within an
octave, the
frequency of each note (
semitone, I should say; we're using both black and white keys) is greater than the next lower one by a factor of 1.0594631
(4), the 12th root of 2. Why? Because that way, you double your
frequency every twelve
semitones, and the
pitches you get happen to be at useful ratios from each other. Two
pitches will sound nice together
5 if the
ratio of their
frequencies is relatively simple: An
octave (A + A, 2:1) sounds very pleasant, or "
consonant"; a "
perfect fifth" (A + E, 2:3) sounds nice, and a
perfect fourth (A + D, 3:4) also sounds nice. Weird or "
dissonant" ones like the
augmented fourth (A + D#, 32:45) sound like crap, but they can be useful as flavoring: You wouldn't eat a handful of
salt, but a little bit is a good thing.
The important thing here is that the
ratio between two notes in one
octave (say, A 110 and E ~164.8) will be the same as the
ratio between the same two notes in a different
octave (like A 440 and E ~659.3). And if A 110 sounds good with E ~164.8, you can safely assume that it'll sound good with E ~659.3 because that's the way
fractions work.
I've left out some of the most interesting stuff, like
harmonics, but this is probably enough for one day. Anyhow I'm not gonna try to explain
harmonics without illustrations. Maybe when I hit level 6 I'll put a
bitmap with some
waveforms in my home node. Aren't you excited?!
1 Hz ==
Herz ==
cycles per second
2 There's no such thing.
3 This assumes that you've got it
tuned perfectly, and in the real world that never quite happens, but it should be reasonably close. If it's a stringed instrument and you tune it much too high, you'll wreck the thing. The crucial thing is to agree with the other strings on the same instrument, and with any other
instruments you may be playing along with.
4 This is
tempered intonation. The first volunteer to explain
just intonation vs.
tempered intonation gets a cupcake and my
undying admiration. In brief,
tempered intonation is a
pragmatic fudge factor thing: The
ratios between notes are all made to come out
slightly wrong, instead of some being perfect and others being lousy.
This page seems moderately informative:
http://home.earthlink.net/~kgann/tuning.html
5 von Helmholtz spent hundreds of pages on the
gory details, but "they sound nice together" will do fine :)
themusic is right about subjectivity and
harmony:
Music is full of math, but let us not forget that you can't
reduce it to math. There's a lot of squishy stuff in there too, which I tend do discuss only with words like "whoa" and "cool". I'm just
a fuckin' punk with a guitar; he's a
musician. :)
That bit about F# and Gb scares me. With
frets, it's the same note, so I can avoid thinking about it. And I will, too.