Strähle construction

Strähle's construction is a geometric method for determining the lengths for a series of vibrating strings with uniform diameters and tensions to sound pitches in a specific rational tempered musical tuning.

The Academy's secretary Jacob Faggot appended a miscalculated set of pitches to the article, and these figures were reproduced by Friedrich Wilhelm Marpurg in Versuch über die musikalische Temperatur in 1776.

Several German textbooks published about 1800 reported that the mistake was first identified by Christlieb Benedikt Funk in 1779, but the construction itself appears to have received little notice until the middle of the twentieth century when tuning theorist J. Murray Barbour presented it as a good method for approximating equal temperament and similar exponentials of small roots, and generalized its underlying mathematical principles.

It has become known as a device for building fretted musical instruments through articles by mathematicians Ian Stewart and Isaac Jacob Schoenberg, and is praised by them as a unique and remarkably elegant solution developed by an unschooled craftsman.

[2] An organ by him from 1743 is preserved in its original condition at the chapel at Strömsholm Palace;[3] he is also known to have made clavichords, and a notable example with an unusual string scale and construction signed by him and dated 1738 is owned by the Stockholm Music Museum.

Stråhle published his construction as a "new invention, to determine the Temperament in tuning, for the pitches of the clavichord and similar instruments" in an article that appeared in the fourth volume of the proceedings of the newly formed Royal Swedish Academy of Sciences, which included articles by prominent scholars and Academy members Polhem, Carl Linnaeus, Carl Fredrik Mennander, Augustin Ehrensvärd, and Samuel Klingenstierna.

[10] Stråhle wrote in his article that he had developed the method with "some thought and a great number of attempts" for the purpose of creating a gauge for the lengths of the strings in the temperament which he described as that which made the tempering ("sväfningar") mildest for the ear, as well comprising as the most useful and even arrangement of the pitches.

Stråhle concluded by stating that he had applied the system to a clavichord, although the tuning as well as the method of determining a set of sounding lengths can be used for many other musical instruments, but there is little evidence showing whether it was put into more widespread practice other than the two examples described in the article, and whose whereabouts today are unknown.

Stråhle instructed first to draw a line segment QR of a convenient length divided in twelve equal parts, with points labeled I through XIII.

[8] Stråhle also showed line segments parallel to MR through points NHS, LYT, and KZV in order to illustrate how once created the construction could be scaled to accommodate different starting pitches.

Stråhle stated at the conclusion of the article that he had implemented the string scale in the highest three octaves of a clavichord, although it is unclear whether this section would have been strung all with the same gauge wire under equal tension like the monochord which he wrote it resembled, and whose construction he described in more detail.

[13] Helenius also presented a theory that Faggot had a more active, if indirect and posthumous influence on the construction of musical instruments in Sweden, claiming that he may have suggested the long tenor strings used in two experimental instruments built by Johan Broman in 1756 which she proposed influenced the type of clavichord built in Sweden in the late eighteenth and early nineteenth centuries.

"b Both articles were reproduced in a German edition of the Academy's proceedings published in 1751,[16] and a table of Faggot's calculated string lengths was subsequently included by Marpurg on his 1776 Versuch über die musikalische Temperatur,[1] who wrote that he accepted their accuracy but that rather than accomplishing "Strähle"'s stated goal, the tuning represented an unequal temperament "not even of the tolerable type.

These intervals fall within what is considered to have been acceptable but there is no distribution of better thirds to more frequently used keys that characterize what are today the most popular of the tunings published in the seventeenth and eighteenth centuries, which are known as well temperaments.

He also demonstrated how close Stråhle's construction was to the best approximation the method could provide, which reduces the maximum errors in major thirds and fifths by about half a cent and is accomplished by substituting 7.028 for the length of QP.

Barbour presented a more complete analysis of the construction in "A Geometrical Approximation to the Roots of Numbers" published six years later in American Mathematical Monthly.

For musical applications it is simpler and its results are slightly more uniform than Stråhle's, and it has the advantage of producing the desired string lengths without additional scaling.

Barbour concluded with a discussion of the pattern and magnitude of the errors produced by the generalized construction when used to approximate exponentials of different roots, stating that his method "is simple and works exceedingly well for small numbers".

Schoenberg also noted that Barbour's equation could be viewed as an interpolation of the exponential curve through the three points m=0, m=1/2, and m=1, which he expanded upon in a short paper titled "On the Location of the Frets on the Guitar" published in American Mathematical Monthly in 1976.

Stewart considered the construction from the standpoint of projective geometry, and derived the same formulas as Barbour by treating it from the start as a fractional linear function, of the form

The geometric and arithmetic methods for dividing monochords as well as musical instrument fretboards compiled by Barbour were for the stated purpose of illustrating the different tunings each represents or implies, and Schoenberg's and Stewart's works retained similar focus and references.

Three textbooks on piano building that are not included by them show similar constructions to Stråhle's for designing new instruments but treat the tuning of their pitches independently; both constructions employ a non-perpendicular form as suggested by Schoenberg's observation in Barbour's "A Geometrical approximation to the Roots of Numbers", and one achieves optimal results while the other demonstrates an application with a root other than 2.

Carl Kützing, an organ and piano maker in Bern during the middle of the 19th century wrote in his first book on piano design, Theoretisch-praktisches Handbuch der Fortepiano-Baukunst from 1833, that he devised a simple method of determining the sounding lengths in an octave after reading of the different geometric constructions described in an issue of Marpurg's Historisch-kritischen Beitragen zur Aufnahme der Musik; he stated that the divisions would be very accurate and that the construction could be used for fretting guitars.

[27] English piano maker Samuel Wolfenden presented a construction for determining all but the lowest sounding plain string lengths in a piano in A Treatise on the Art of Pianoforte Construction published in 1916; like Sievers, he did not explain whether this was an original procedure or one in common use, commenting only that it was "a very practical method of determining string lengths, and in past years I used it altogether".

"Enligit detta påfund, har jag bygt et Monochordium, i så måtto, at det fullan hafver 13 strängar, ock skulle dy snarare heta Tredekachordium, men som alla strängarna, äro af en nummer, längd ock thon; så behåller jag det gamla namnet.

"Det Claver, som jag här til förfärdigat är jämnväl i de tre högre Octaverne, noga rättadt efter min Linea Musica, til strängarnes längd ock skilnad : ock på det stämningen, må utan besvär, kunna ske; så är mit Monochordium så giordt, at det kan ställas ofvan på Claveret, då en Octav på Claveret stämmes, thon för thon, mot sina tillhöriga thoner på Monochordium, derefter alla de andra thonerne, å Claveret, stämmas Octavs-vis; den stamningen, är ock för örat lättast at värkställa, emedan den bör vara fri för svängningar."

"Huruvida thonernes stämning, efter förut beskrefne Påfund, förnöger hörsten, med behageligare ljud, ock med bättre likstämmighet, i de Musikaliska thonerne å et Claver, än de gamla ock härtils bekanta stämnings sätt, derom lärer förståndet bättre kunna döma, när ögat får se det örat hörer.

Jacob Faggot, durch eine sehr mühsame trigonometrische Berechnung der Strählischen Linien, gefunden Zahlen voellig überzeuget bin.