The Pythagoreans noticed that nice simple ratios of string length made nice sounds together.
Using an understanding that 1:2 and 2:3 were good ratios, start with a length and keep reducing it to 2/3 of itself. Each time that took the length under 1/2 they doubled it to get back within range.
The scale on a piano does something clever : the ratio (interval) between any adjacent points on the scale is equal. If you play any note, twelve points higher will be exactly an octave on.
Why is the modern piano tuned using an equal tempered scale and what has this got to do with logarithms?
Use Euclid's algorithm to get a rational approximation to the number of major thirds in an octave.
Show that it is rare for a ratio of ratios to be rational.