In 1743, a geometric method for approximating Equal Temperament was published in the proceedings of the Royal Swedish Academy of Sciences. It had been devised by the Swedish organ builder Daniel P. Strähle, who reported that he had hit upon it after much experimentation. The article appeared again in a German translation of the proceedings published in 1751, and was later referred to by Marpurg and Chladni, among others. Strähle’s method was improved by subsequent authors, bringing the approximation much closer to a true ET, and the improved version appeared in several important works on piano design and construction published in Italy, Germany and England as late as 1916.

Strahle

The lengths for one octave of Equal Temperament are found along the line MR as well as any of the other lines parallel to it, which Strähle referred to as his “musical line”. Of much more interest to us, however, is how Strähler applied his invention in order to realize Equal Temperament in practice:

“According to this contrivance, I have made a divided monochord, such that it has altogether 13 strings, and could rightly be called a Tredekachordium, but since the strings are all of a common gauge, length and tone, I retain the old-fashioned name.

“Connected to these 13 strings there is a normal keyboard of one octave, but underneath each of the strings, which are all tuned to the same tone, I place movable bridges on those points, or at those distances from the nut pins[?], which are indicated by the musical line I have just now described, by which the strings acquire their proper tone.

“The harpsichord which I have made accordingly is constructed precisely according to my musical line for the upper three octaves, in regards to the lengths and differences of the strings. And in order that the tuning can be done without difficulty, my monochord is made in such a way that it can be placed on top of the harpsichord, and thereby one octave of the harpsichord can be tuned note for note against the proper notes on the monochord, from which all the other notes of the harpsichord can be tuned by octaves; the tuning is also very easy to accomplish with the ear, since [the notes] must be free of all wavering.”
(translation: PP)

The original 1751 German translation by Abraham Gotthelf Kästner:

Nach dieser Erfindung habe ich ein Monochord hergestalt abgetheilet, daß es volkommen 13 Saiten hat, und mit größern Rechte Tredekachordium heißen könnte, aber wie all Saiten von einer Nummer, Länge und Ton sind, so behalte ich den alten Namen.

An diese 13 Saiten ist ein gewöhnliches Manual von einer Octave gefüget, unter jede Saite aber, nachdem solche genau alle auf einen Ton gestimmet sind, feße ich bewegliche Stege an die Puncte, oder in den Entfernungen von den Crepinen, welche meine itzo beschreibene musikalische Linie erfordert, worauf jede Seite ihren gehörigen Tone bekömmt.

Das Klavier, das ich hiezu verfertiget habe, ist ebenfalls in den drey höhern Octtaven genau nach meiner musikalischen Linie eingerichtet, was die Längen und Untershiede der Saiten betrifft. Und damit das Stimmen ohne Beschwerlichkeit geschehen kann, so ist mein Monochord so gemacht, daß es oben auf das Clavier kann gefeßt werden, da denn eine Octave auf dem Clavier Ton für Ton gegen ihre zugehörigen Töne auf dem Monochord gestimmet wird, worauf alle andere Töne auf dem Clavier octavenweise gestimmet werden, die Stimmung ist auch für das Ohr am leichtsten zu bewerkstelligen, weil sie des Schwebens wegen frey seyn muß.