Strähle construction
Strähle’s construction is a simple geometric recipe for choosing the lengths of a set of vibrating strings with the same thickness and tension so they sound in a carefully tempered musical scale. It was created by Daniel Stråhle, an 18th‑century Swedish organ maker, and published in 1743 in the Proceedings of the Royal Swedish Academy of Sciences. The method uses a few straightedge-and-compass steps to produce a string scale, which Stråhle claimed would give a pleasant, evenly spaced tuning for keyboards and similar instruments.
Stråhle was active in central Sweden. He had trained with the Stockholm organ builder Cahman and, after Cahman’s death, obtained a royal privilege to build and repair organs and to train others. An example of his work from 1743 survives at Strömsholm Palace. He also made clavichords, and one notable instrument with an unusual string scale is in the Stockholm Music Museum. Stråhle wrote that his method could be used for clavichords and other instruments, though he described it for a clavichord in his article. He described the construction briefly, saying it could be done with a straightedge and dividers and did not require heavy calculations.
In the same volume, Jacob Faggot, the Academy secretary, gave a mathematical treatment of the pitches produced by Stråhle’s method. His table of string lengths was later published by others, including Friedrich Wilhelm Marpurg in 1776. However, the figures Faggot produced were not exactly what Stråhle’s instructions would generate. The discrepancy was first clearly pointed out by Christlieb Benedikt Funk in 1779, who noted that Faggot had used a tangent-based calculation that led to incorrect lengths. Ernst Chladni later discussed the matter, and Swedish writers of the time did not publish similar corrections.
The tuning produced by Stråhle’s method is not a perfect equal temperament, but an approximation that keeps many intervals close to equal steps. In practice, it gives a practical, repeatable way to lay out string lengths with a ruler and compass, producing a tempered scale where fifths, thirds, and other intervals fall in a useful, music‑friendly range.
In the 20th century, J. Murray Barbour revived interest in Stråhle’s construction. He showed that the method can be understood as an approximation to equal temperament and related it to a simple geometric process. Barbour also generalized the approach, showing how a mean-proportional construction could replace some of the original angles and lengths and still produce very good results for small changes in pitch. He examined how close the method comes to the ideal and how the errors behave as the number of notes grows.
Two later mathematicians, Ian Stewart and Isaac Schoenberg, explored Stråhle’s idea from the perspective of projective geometry. They showed that the construction can be viewed as a fractional linear transformation, and that it naturally links to how frets on guitars and other fretted instruments are placed. Stewart, in particular, highlighted how the construction uses a fraction like 41/29 as a near half‑octave, a notable mathematical curiosity that arises from the method’s geometry.
Over the years, similar geometric ideas appeared in other instrument design literature, including work on fretting guitars and pianos. Some 19th‑ and early 20th‑century piano makers and theorists described related methods for determining string lengths so that tensions would be similar across an octave, or in ways that could be adapted to different tunings. These later discussions show how Stråhle’s geometric spirit influenced practical instrument making beyond his own era.
Stråhle’s name is sometimes written as “Strähle” in English texts, a result of a transcription error in an old source. He did not live to see wide recognition of his method, but today it is remembered as a clever, elegant solution devised by a craftsman who worked with tools rather than equations. The Linea Musica, a term Stråhle used for his key line in the construction, echoes ideas from earlier Swedish mechanics and geometry, linking musical tuning to the broader science of shapes and proportions.
In short, Stråhle’s construction is a neat, historical example of using straight lines and simple geometry to lay out a tuned string scale. It offered a practical way to approach tempered tuning long before modern theory, and its later analysis helped readers understand how geometry can approximate musical ratios with striking accuracy.
This page was last edited on 2 February 2026, at 05:12 (CET).