You may know the famous Greek mathematician Pythagoras as the fellow that
put forth the basic principles of geometry. He
was also very interested in the science of music, among many other things. He knew,
thousands of years ago, that a musical tone is created when something vibrates - a
string, for example. He knew that if the same string vibrates two (2) times as fast
(twice the frequency), the
pitch of the note will be exactly an octave higher. He also knew that if the string
vibrates three (3) times as fast (three times the frequency), the pitch will be an octave plus a fifth higher. He
determined these things from experiments that he performed with strings. So
he concluded that if the string vibrates one and a half (1½) times faster, the pitch will
be just exactly a musical fifth higher.
Pythagoras was a very clever fellow, and he knew about
The Circle of Fifths
all those years ago. Being a good mathematician, he also realized
that multiplying the frequency of a given starting note by 2 seven times will not
produce exactly the same result
as multiplying the same starting frequency by 1½ twelve times. He calculated that the frequency
reached by ascending a perfect fifth twelve times is about 1% higher than the frequency reached
by ascending seven octaves from the same starting note.
If you are a whiz at arithmetic, try working out
the exact numbers.
Back in the eighteenth century piano tuners learned to tune the fifths on a piano
(or on a harpsichord back in those days) about 1% small so that everything comes out
just about right. This tuning trick is called
Equal Temperament,
because it allows
the instrument to play equally well (and an equally tiny bit out of tune) in every key.
The famous Baroque composer
Johann Sebastian Bach
loved this capability, and set about writing music in every possible key
to take advantage of it. One very well-known set that you may be familiar with is usually
called
"The Well Tempered Clavier"
in English. Bach's original German title is "Das Wohltemperierte Klavier".
You may know the famous Greek mathematician Pythagoras as the fellow that
put forth the basic principles of geometry. He
was also very interested in the science of music, among many other things. He knew,
thousands of years ago, that a musical tone is created when something vibrates - a
string, for example. He knew that if the same string vibrates two (2) times as fast
(twice the frequency), the
pitch of the note will be exactly an octave higher. He also knew that if the string
vibrates three (3) times as fast (three times the frequency), the pitch will be an octave plus a fifth higher. He
determined these things from experiments that he performed with strings. So
he concluded that if the string vibrates one and a half (1½) times faster, the pitch will
be just exactly a musical fifth higher.
Pythagoras was a very clever fellow, and he knew about
The Circle of Fifths
all those years ago. Being a good mathematician, he also realized
that multiplying the frequency of a given starting note by 2 seven times will not
produce exactly the same result
as multiplying the same starting frequency by 1½ twelve times. He calculated that the frequency
reached by ascending a perfect fifth twelve times is about 1% higher than the frequency reached
by ascending seven octaves from the same starting note.
If you are a whiz at arithmetic, try working out
the exact numbers.
Back in the eighteenth century piano tuners learned to tune the fifths on a piano
(or on a harpsichord back in those days) about 1% small so that everything comes out
just about right. This tuning trick is called
Equal Temperament,
because it allows
the instrument to play equally well (and an equally tiny bit out of tune) in every key.
The famous Baroque composer
Johann Sebastian Bach
loved this capability, and set about writing music in every possible key
to take advantage of it. One very well-known set that you may be familiar with is usually
called
"The Well Tempered Clavier"
in English. Bach's original German title is "Das Wohltemperierte Klavier".