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## 'Rarity' printed from http://nrich.maths.org/

Show that there are no integer solutions $m, n$ of the equation
$$\left({5\over 4}\right)^m = \left({2\over 1}\right)^n $$ which
gives the number of major thirds in an octave on a musical
scale.

Given integers $a, b, c $ and $d$ where $a$ and $b$ are coprime and
$c$ and $d$ are coprime, find necessary and sufficient conditions
for there to exist positive integers $m$ and $n$ such that
$$\left({a\over b}\right)^m = \left({c\over d}\right)^n.$$

[The set of three problems

Tuning and Ratio ,

Euclid's Algorithm and Musical Intervals and this problem
Rarity were devised by Benjamin Wardaugh who used to be a member of
the NRICH team. Benjamin is now doing research on the history of
music and mathematics at Oxford University. Read Benjamin's article

Music
and Euclid's Algorithm for more on this subject.]