The Construction of the Equitempered Scale

I recently got a file from Alain Busser, who combines two of my programs (Z.u.L. and Euler) to attack and explain a construction by Daniel Strähle, a Swedish from 1743. The problem is the construction of the equitempered musical scale. Vincenzo Galilei (the father of Galileo Galilei) used 18/17 as an approximation for 2^(1/12), which is fairly good, being the first continued fraction expansion. The next best would have been 107/101.

Strähle came up with the following construction, which should be almost obvious from the image. You can also download this construction.

The segment CB is divided into 12 parts, lines g and a are parallel, and AB is divided by D in two equal parts. g goes through the 7th point F on BC.

If the blue line is the chord, we can take the intersections with the 12 black lines as positions for the frets. By the construction the 8th (the octave) at the point D will be exact.

The trick is that the direction of the point A is chosen by this construction in such a way, that the fifth (the quint) at E is exact. It happens that all other fret positions are very close to the exact values, marked in blue in the above image. This is not the best solution in terms of the maximal error, but very close to the best one.

On the Euler homepage you find computations to this construction under Examples / Straehle.

R. Grothmann