Number Harmony
It is easy to recognize octaves because the frequency of an octave above a certain pitch is exactly twice the frequency of that pitch. Octaves harmonize so well that they almost sound identical, so we call these notes by the same name: an octave above or below middle C is another C; an octave above or below concert A, 440 Hz, is another A (880 or 220 Hz). Mathematically, if a certain note H has frequency f then a note with frequency 2nf, where n is an integer, is n octaves above H (if n is negative, it is a positive power of 1/2 and represents |n| octaves below H).
Not alone in their ability to harmonize well, octaves are joined by all the intervals that make up a major or minor scale (in the Western music system), notably including perfect fifths (fifth note of a scale, 3/2 times the frequency of the starting note) and major or minor thirds (third note of a scale, respectively 5/4 or 6/5 times the frequency of the starting note). All these frequencies are ratios of relatively small whole numbers - this contributes to the harmony of the notes, just like the ratio 2/1 does for octaves. The simpler the frequency ratio, the higher the quality of harmony achieved by an interval when played out loud. The only requirement is for the ratio to be a (positive) rational number, able to be written with whole numbers for the numerator and denominator.
However, suppose you tuned a piano perfectly according to one of the scales. Then you can play that scale and it would be perfectly in tune - but the harmony of all the other scales get thrown off! For example, E is both the third note of a C major scale and the second note of a D major scale. By tuning the piano to the C major scale, you guarantee that an E has frequency 5/4 times the frequency of a C (C to E is a major third). In a perfect C scale, D has frequency 9/8 times that of a C. Call these frequencies fC, fD, and fE.
fE = (5/4)fC
fD = (9/8)fC
⇒ fE = (5/4)(8/9)fD = (10/9)fDThis is still a relatively simple rational number ratio, but it’s the wrong ratio. In a perfect D major scale, E has frequency 9/8 that of D. The relative error when tuning to C is
|10/9 - 9/8| = |80-81|/72 = 1/72.
In the first days of the harpsichord and piano (keyboard instruments), tuners chose one scale to tune to, sacrificing the harmony of the other scales. Interestingly, some of the music from that era took that into account; on one hand some scales were considered “sweeter” than others based on common tuning practices, and on the other some songs were purposely written in one of the sour-sounding scales for their dissonant harmonies.
Today’s most common tuning, or temperament, is called equal temperament. Each scale sounds equally good (or equally bad, depending on your tolerance of imperfection), and the only interval which is perfectly preserved is octaves. Since, in the Western music system, there are 12 semitones from octave to octave (12 white and black keys from a note to an octave above the note), each of those keys is assigned the frequency of exactly the twelfth root of 2 times the key preceding it. What’s great about that, of course, is that this is a completely egalitarian system: no scale is sweeter or sourer sounding that any other. Yet the cost is the complete destruction of the rational number harmonies: the twelfth root of 2 is as irrational as they come, and could never in any number theorist’s wildest dreams be written as a ratio of whole numbers.
—
Further reading:
”Why you’ve never really heard the Moonlight Sonata,” Jan Swafford, Slate Magazine
UNLIMITED ♥ 
