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Doug's Mathematics Page
contains miscellaneous mathematical discoveries and problems.
Mathematical Theory of Music:
Pythagoras was a Greek Mathematician born in
569 B.C. who studied math, music,
and astronomy. He was a fine musician, playing the lyre, and he used music as
a means to help those who were ill. Pythagoras and the Pythagoreans were the
first to connect music and mathematics and they made remarkable contributions
to the mathematical theory of music. Pythagoras noticed that vibrating strings
produce harmonious tones when the ratios of the lengths of the strings are
whole numbers, and that these ratios could be extended to other instruments.
He is credited with discovering that the
interval of an octave is rooted in the ratio 2:1, that of the fifth in 3:2,
that of the fourth in 4:3, and that of the whole tone in 9:8. Followers
of Pythagoras applied these ratios to lengths of a string on an instrument
called a canon, or monochord, and thereby were able to determine
mathematically the intonation of an entire musical system.
The higher the frequency (number of
vibrations per second) of a plucked string, the higher its pitch. Specific
notes are dependent upon the length of the string. A
string that is twice as long as another string will vibrate half as much as
the shorter string. These two notes will produce the same sound since
they are
an "octave" apart. In other words, "two notes are an octave apart if the
frequency of one is double the frequency of the other note.... [The
Pythagoreans] found that an entire scale could be produced by taking integral
ratios of a string's length." (More Joy of Mathematics by Theoni
Pappas, Wide World Publishing, 1991, p. 190-92).
With reference a string length for a note C,
the other lower notes in the octave are:
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B -- 16/15 of C's length
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A -- 6/5
of C's length
-
G -- 4/3
of C's length
-
F -- 3/2
of C's length
-
E -- 8/5
of C's length
-
D -- 16/9
of C's length
-
C --
2/1 of C's length
The following are good external links that
further describe Pythagoras's discovery:
Coin problem:
How many pennies are required to perfectly
surround a central penny?
Each penny would touch the central penny and its two neighbors.
How many nickels are required to perfectly
surround a central nickel?
How many quarters are required to perfectly
surround a central quarter?
You may have solved this coin problem by
taking out some coins and physically relocating the coins, and now you know
that the answer is the same for each question. Can you mathematically
prove that this is the case for all sized coins? As a hint, try drawing
same sized circles/coins and connect the centers to form a type of triangle.
Feel free to contact me for the intriguing solution!
More intriguing discoveries and problems will
be added to this site! Check back soon!
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Page was last updated on
June 26, 2004
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