Stråhle's Equal-Tempered Triumph

Abstract

In 1743, the Swedish organ builder Daniel P. Stråhle gave an elegant geometric construction for determining the precise pitches of musical notes—for example, for locating frets along the neck of a guitar. We analyze how closely Stråhle's construction approximates equal temperament as a function of the pitch gamut, that is, the frequency range covered, and the tilt ratio of the perspectivity that locates the frets. With a pitch gamut of one octave, the tilt ratio then needing to be roughly , we show that Stråhle's choice of the ratio 24/17 gives better fretboards than the continued-fraction convergent 17/12. We also compare the accuracy of Stråhle's construction to that of its main competitor, a saddle-shifting variant of Vincenzo Galilei's rule of 18. Galilei is more accurate on a gamut of one octave, but Stråhle is more accurate on the quarter-octave gamut that would tempt a compass-equipped perfectionist.