Doug's Mathematics Page
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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:

  •   B -- 16/15 of C's length

  •  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 September 07, 2015

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