Continuation of my last post, a proof that music BY ITSELF is sexy squared:
The tone A440hz is one octave below A880hz. This is because an octave represents a doubling in frequency... an octave is also the most consonant (uniform-sounding) interval, thanks to the fact that the waveforms of the two notes meet in the maximum number of places - every other crest of the lower note. You've added 1/1 of the lower frequency to get the upper.
A fifth is the second most consonant interval, the waveforms meet at every third crest of the lower note, and the frequency of the higher is 50% larger than the lower. A fifth above A440hz is E660hz. You've added 1/2 of the lower frequency to get the upper.
A fourth is next in line, being the third most consonant interval, the two waveforms meeting at every fourth crest of the lower note. The frequency of the higher is 33.33...% larger than the lower. A fourth above E660hz is A880hz (yay, fifth plus fourth equals octave), and you've added 1/3 of the lower frequency to get the upper.
And so on, ad infinitum. Our western system of logarithmic frequencies is limited in its precision, though - the smallest interval being the minor second, or one semitone. An octave is 12 semitones, a fifth is 7 semitones, a fourth is 5 semitones. Some civilizations (notably India) use quarter-tones as well, and their music may sound unusual to us, but it has twice the capacity for precision, since a quarter-tone is logarithmically twice as small as a semitone.
Pianos used to be tuned to "mean-temperament," which was sloppier than modern methods, but produced a slightly darker/brighter set of tones in each "diatonic" scale - the D-flat scale was very dark-sounding, while the A scale was very bright-sounding. We now use "equal-temperament," which is more precise along the logarithmic curve, but produces a set of "diatonic" scales which sound exactly the same except for their relative pitches - there are no differences between the proportions of notes from one scale to the next. All scales sound middle-of-the-road on brightness. If you are familiar with a modern piano's sound and you ever have the chance to play on a piano that has been tuned to mean-temperament (or possibly a harpsichord), you will immediately notice a difference.
I could then go into what makes a "diatonic" scale diatonic, the differences between diatonic and arabic scales, mode translation, and possibly concert pitch and transposition, but the above is probably enough to freak out some people's "nerd cool-meters." I geek out a bit when I talk about this side of music to people! 8)
(An aside: A440hz is what orchestras and other groups use as a tuning standard for instruments. I work at a store which has a key letter and several key numbers for each department; A is the health and beauty care department, and one of A's key numbers is 440. No one at work got why I was so excited.)
