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Euclid's Algorithm and Musical Intervals

Age 16 to 18 Challenge Level:

How many major thirds are there in an octave on a musical scale?

Going back in history, before the discovery of logarithms, some theorists used Euclid's algorithm to find the answer to this question.

A rational approximation ${m\over n}$ for the relationship between 5/4, the musical interval called the major third, and the octave 2/1, is given by $$ \left({5\over 4}\right)^m \approx \left({2\over 1}\right)^n, $$ where $m$ and $n$ are integers. Using Euclid's algorithm show that ${m\over n}={28\over 9}$ gives a first approximation and find three closer rational approximations.

In the articles Euclid's Algorithm and Approximations, Euclid's Algorithm and Continued Fractions you can find out about this method and also that Euclid's algorithm can be used not only for integers but for any numbers.

[See also the problems Tuning and Ratio and Rarity. The set of three problems on mathematics and music was devised by Benjamin Wardaugh who used to be a member of the NRICH team. Benjamin is now doing research into the history of mathematics and music at Oxford University and his article Music and Euclid's Algorithm should help you with this problem.]