Music's most "agreeable" harmonies are formed by the simplest mathematical relationships, as demonstrated in the following audio examples, as performed on harpsichord by Ed Brewer. You will need the free RealPlayer Basic to listen.

The octave--the span from Do to Do or Re to Re--is sounded when the higher tone is vibrating twice as fast as the lower one.
Listen to an example

The perfect fifth is formed when the two tones are vibrating in the proportion 3:2.
Listen to an example

Early chant was performed in a style called "organum," in which the melodies were often harmonized in the haunting, hollow sound of perfect fifths.
Listen to an example

The interval of a major third is formed when the two tones are vibrating in the proportion 5:4.
Listen to an example

A melody harmonized in thirds sounds warmer and less austere than one harmonized in perfect fifths.
Listen to an example

Combining a fundamental tone with both its third and fifth produces a triad--the harmonic foundation of Baroque and Classical music.
Listen to an example

Unfortunately, these proportional relationships cannot be maintained between all the notes of a keyboard instrument; when some tones are placed in tune with each other, others will necessarily be out of tune with each other. In the following example, the harpsichord is in Just Intonation, a tuning that preserves "pure" thirds and "pure" fifths for some keys. When the lovely triad we just heard is shifted across the keyboard so that it begins on La Sharp rather than on Do, the result sounds awful.
Listen to an example

Thus, a keyboard instrument on which the thirds and fifths are in their purest proportional relationships will sound beautiful in certain keys, as can be heard in the opening Aria of Bach1s Goldberg Variations.
Listen to an example

However, in Just Intonation, shifting the Bach piece to a different key by beginning it one half-step lower will bring about the following calamity:
Listen to an example

Equal Temperament, a tuning in which the proportions forming music's concordances are "tempered," or compromised from their purist form, in order to eliminate this problem, allows us to begin the Bach piece on any note at all.
Listen to an example

This ability to play in any key without fear of producing ugly, "wolf" tones, is crucial in Romantic music, where the harmonic "center of gravity" of a piece is constantly shifting. In the following example, a piano-like synthesizer renders Chopin's Prelude in E minor using equal temperament. The result is trouble free.
Listen to an example

When the synthesizer is placed in a tuning that preserves the ideal proportions for thirds and fifths between some notes, the same Chopin piece quickly loses its charm.
Listen to an example

Harpsichord courtesy of Ed Brewer