It is easy to find approximate ratios between 2 numbers by using the Euclidean algorithm to calculate continued fractions. However I can not find a method to do this for 3 numbers.
I have tried a shared Euclidean algorithm (dividing by the lowest of the 3 remainders each time) and it did not come close to any approximate ratios found by simply testing each denominator in turn.
The first problem is that I do not know what a canonical measure of error would be, as there are 3 pairs of ratios with differing errors that can be calculated.
After that, I do not know how to adjust the euclidean algorithm to successivly approximate 3 ratios simultaneously. The perfect fit for a particular ratio (found by the Euclidean algorithm) often has a high error for the other ratios.
I have found published convergents of 3 or more ratios as integer sequences at oeis.org/A060528 related to the even/equal tempering of musical instruments and the desired ratios of muical intervals. However in the description the author explains that this was done through testing nearly 7 million successive possibilities, rather than an algorithm as is used for simple continued fractions.
If there is no solution, is this known to be impossible, a known "hard" problem, or just something mathematicians haven't cared about?
