the note G is 392.0 Hz. The ratio of E to C
is about 5/4, or every fifth frequency wave
of E matches with every fourth wave of C;
the ratio of G to E is also about 5/4; and
the ratio of G to C is about 3/2. Because
the notes’ frequencies match with the
other notes’ frequencies, they all sound
harmonious to a person’s ear.
It is interesting to note that none of
the ratios for the familiar “western-style”
scale are truly exact, but rather are
approximations. This is because when this
scale was put together, the creators want-
ed to make the notes go up in equal inter-
vals and the ratios to be in tune. The only
way to do this was to compromise and use
“inexact” ratios.
What is the “music of the spheres”?
Not only was the Greek mathematician
and philosopher Pythagoras of Samos
(c. 582–c. 507 BCE) credited as the first to prove the Pythagorean theorem, he also dis-
covered the “music of the spheres.” He found that the pitch of a musical note depends
on the length of the string producing the sound, enabling him to develop intervals of
the musical scale with simple numerical ratios. When a stringed instrument is played,
if the musician puts pressure halfway along the string’s length, he or she produces a
note that is one octave above the string’s note—in other words, the same quality of
sound but at a higher pitch. Octaves increase by one step each time a string vibrates at
twice the frequency of the previous note; this is expressed mathematically as a fre-
quency ratio of 1:2 (string:octave). Pythagoras recognized other ratios, too, such as
the perfect fifth (ratio 2:3) and the perfect fourth (3:4), thus developing the mathe-
matical basics of musical harmony.
Pythagoras took music and mathematics a bit further, believing that the musical
octave was the simplest and most profound expression of the relationship between
spirit and matter (for more about Pythagoras, see below and “History of Mathemat-
ics”). He also taught that each of the known planets produced a particular note (gen-
erated by its motion) according to the planet’s distance from the Earth, calling this
Musica mundana,or the “music of the spheres,” it was music no one could really
hear. He and his followers, called the Pythagoreans, further used music to heal the
body and to elevate the soul, yet they believed earthly music was just a faint echo of
the universal notes. Although today it may seem more “magic” than hard science and 379
MATH IN THE HUMANITIES
The music of the brilliant composer Wolfgang
Amadeus Mozart is said by some to increase intelli-
gence in babies who are exposed to regular doses of
his melodies. This is known as the “Mozart Effect.”
Library of Congress.