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.
The perfect fifth is formed when the two tones are vibrating in the proportion 3:2.
Early chant was performed in a style called "organum," in which the melodies
were often harmonized in the haunting, hollow sound of perfect fifths.
The interval of a major third is formed when the two tones are vibrating in the proportion 5:4.
A melody harmonized in thirds sounds warmer and less austere than one harmonized in perfect fifths.
Combining a fundamental tone with both its third and fifth produces a triad--the harmonic foundation of Baroque and Classical music.
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.
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.
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:
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
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.
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.
Harpsichord courtesy of Ed Brewer