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Show that it is rare for a ratio of ratios to be rational.

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:

Robert managed to solve this toughnut:

Find the ratio corresponding to B. Also compare the D-A ratio with the ideal fifth (3/2):

"B's ratio is calculating by use of the fact that the interval from E to B is a perfect fifth, and a perfect fifth's ratio is 3/2. Also E's ratio is 5/4. Therefore B = 15/8, and:


The ideal fifth ratio=81/54.Therefore if we were to tune two adjacent keys to two different A's, the first being in the ratio A/D, and the second a perfect fifth above D (in the ratio 3/2). The second A would be higher than the first by a ratio of 81/80."

Find also exactly how many major tones/thirds there are in an octave.

To do this for the tonesRobert took logs of the equation.
We are trying to find $x$ such that $\left({9\over 8}\right)^x = 2$ then $$x \log{9\over 8} = \log 2,\ x= {\log 2\over \log 9/8} = 5.8849492$$ to 8 significant figures.

Similarly for thirds:
To find the number of thirds in an octave we are looking for the value of $y$ such that $\left({5\over 4}\right)^y = 2$. Observe that: $$\left({5\over 4}\right)^2 = 1.5625,\ \left({5\over 4}\right)^3 = 1.953125,\ \left({5\over 4}\right)^4 = 2.4414063$$ and hence $3< y< 4$ and $y\approx 3.1$. Using logs $$y\log{5\over 4} = \log 2,\ y= {\log 2\over \log 5/4} = 3.1062837$$ to 8 significant figures.

In order to convert the Pythagorean & Just Intonation ratios for intervals, into the standard units of tuning ie cents, One must solve the following equation and then multiply the solution by 100 to end up with cents:
$(2^{1/12})^n$=R, where R = The Pythagorean or Just Intonation ratio for an interval.

Doing this, Robert obtained the solution:

Equal tempered scale 0 200 400 500 700 900 1100 1200
Pythagorean scale 0 204 408 498 702 906 1110 1200
Just intonation 0 204 386 498 702 884 1088 1200

Well Done!