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All about Ratios

A new problem posed by Lyndon Baker who has devised many NRICH problems over the years.

Tuning and Ratio

Age 16 to 18 Challenge Level:

You don't need ANY knowledge of music to solve this problem. All the maths you need to know and understand is the definition of logarithms.

Don't be put off by the fact that there is a lot to read and don't be put off if you have not enjoyed music theory in the past. Read the problem and you'll find it tells you all you need to know about music to solve it.

To find how many tones there are in an octave, trial and error is perfectly adequate, but you can also get an exact answer using logarithms.

Similarly to find the number of thirds in an octave you can get an exact answer using logarithms.

To relate the different scales to the linear equal tempered scale in cents you need to calculate the power of ${(2)}^{1/12}$ corresponding to the ratio for each note and then multiply by 100 to convert the measure to cents.

For example, to find the measures in cents corresponding to the note A, the calculations are, for the just intonation: $$1200\times {\log (5/3)\over \log 2}$$ and for the Pythagorean scale: $$1200\times {\log(27/16)\over \log 2}.$$