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. It was first published in the 1743 Proceedings of the Royal Swedish Academy of Sciences by Swedish master organ maker Daniel Stråhle (1700–1746). 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, who have praised it as a unique and remarkably elegant solution developed by an unschooled craftsman.
The name "Strähle" used in recent English language works appears to be due to a transcription error in Marpurg's text, where the old-fashioned diacritic raised "e" was substituted for the raised ring.Wilhelm Friedrich Marpurg Versuch über die musikalische Temperatur 1776 p.167-168
Background Daniel P. Stråhle was active as an organ builder in central Sweden in the second quarter of the eighteenth century. He had worked as a journeyman for the important Stockholm organ builder Johan Niclas Cahman and, in 1741, four years after Cahman's death, Stråhle was granted his privilege for organ making. According to the system in force in Sweden at the time a privilege, a granted monopoly which was held by only a few of the most established makers of each type of musical instruments, gave him the legal right to build and repair organs, as well as to train and examine workers, and it also served as a guarantee of the quality of the work and education of the maker.Eva Helenius-Öberg "Svensk Instrumentmakeri 1720-1800 - En Preliminär Översikt" Svensk Tidskrift för Musikforskning, 1977 p.22 An organ by him from 1743 is preserved in its original condition at the chapel at Strömsholm Palace; History: Strömsholm Palace The Royal Palaces - The Royal Court (accessed October 14, 2009) 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.Edward L. Kottick and George Lucktenberg, Early Keyboard Instruments in European Museums Indiana University Press 1997 p.196Daniel Stråhle, 1738 Klavikord, Inv. nr. N145765 - Stockholm Music Museum (accessed August 28, 2009) His apprentices included his nephew Petter Stråhle and Jonas Gren, partners in the famous Stockholm organ builders Gren & Stråhle,"Jonas Gren" Nordisk Familjebok femte band, Gernandts boktryckeri-aktiebolag, Stockholm. 1882 p.1554 and according to Abraham Abrahamsson Hülphers in his book Historisk Afhandling om Musik och Instrumenter published in 1773, Stråhle himself had studied mechanics (which has been assumed to have included mathematicsKerala J. Snyder The Organ as a Mirror of its Time: North European Reflections, 1600-2000 Oxford University Press, 2002 p.8, 15) with Swedish Academy of Science founding member Christopher Polhem.Tobias Norlind Svensk Musikhistoria Helsingborgs Typografiska Anstalt, Helsingborg 1901. p. 132 He died in 1746 at Lövstabruk in northern Uppland.
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. According to organologist Eva Helenius musical tuning was a subject of intense debate in the Academy during the 1740s,Eva Helenius-Öberg "Cembalon i Sverige samt Frågan om det Svenska Klavikordets Uppkomst" Svensk Tidskrift för Musikforskning 1979 p.31-42 and though Stråhle himself was not a member his was the third article on practical musical topics published by the Academy—the first two were by amateur musical instrument maker, minister, and Academy member Nils Brelin"Nils Brelin" Nordisk Familjebok fjärde bandet Nordisk Familjeboks förlags aktiebolag Stockholm 1905 p.86 which related inventions applicable to harpsichords and clavichords.Nils Brelin "Et Påfund at Storligen öka Claviers och Cymbalers godhet" Kongliga Swenska Wetenskaps Academiens Handlingar, för Månaderne Julius, August och September 1739 vol. 1, p.81; Nils Brelin "Beskrifning öfver det nyt Påfund af en uprät-stående dubbel Clavesin, inrättad met forte och Piano til 8. graders förändring, som wid 1741. års Riksdag för Riksens Höglofl. Ständer blifwit upwist, af des inventor Nicol. Brelin Kongliga Swenska Wetenskaps Academiens Handlingar, för Månaderne Julius, August och September vol.3 p.218
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. His instructions produce an irregular tuning with a range of tempered intervals similar to better known tunings published during the same period, but he provided no further comments or description about the tuning itself; today it is generally considered to be an approximation of equal temperament.Barbour (1951) p.65 He also did not elaborate upon any advantages of his construction, which can produce accurate and repeatable results without calculations or measurement with only a straightedge and dividers; he described the construction in only five steps, and it is less iterative than arithmetic methods described by Dom Bédos de Celles method for determining organ pipe lengths in just intonation or Vincenzo Galilei for determining string fret positions in approximate equal temperament, and geometrical methods such as those described by Gioseffo Zarlino and Marin Mersenne—all of which are much better known than Stråhle's. 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.
Construction Stråhle instructed first to draw a line segment QR of a convenient length divided in twelve equal parts, with points labeled I through XIII. QR is then used as the base of an isosceles triangle with sides OQ and OR twice as long as QR, and rays drawn from vertex O through each of the numbered points on the base. Finally a line is drawn from vertex R at an angle through a point P on the opposite leg of the triangle seven units from Q to a point M, located at twice the distance from R as P. The length of MR gives the length of the lowest sounding pitch, and the length of MP the highest of the string lengths generated by the construction, and the sounding lengths between them are determined by the distances from M to the intersections of MR with lines O I through O XII, at points labeled 1 through 12.
Stråhle wrote that he had named the line PR "Linea Musica", which Helenius noted was a term Polhem had used in an undated but earlier manuscript now located at the Linköping Stifts- och Landsbibliotek and which is accompanied by notes from composer and geometer Harald Vallerius (1646–1716) and Stråhle's former employer J. N. Cahman.
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. He only described an indirect method of setting its tuning, however, requiring that he first establish reference pitches by transferring the corresponding string lengths to the movable bridges on a keyed thirteen string monochord whose open strings had been previously tuned in unison.
Faggot's numerical representation
The article following Stråhle's was a mathematical treatment of it by Jacob Faggot (1699–1777), then secretary of the Academy of Sciences and future director of the Surveying Office, who in the same volume also contributed articles on a weight measure for lye and methods for calculating the volume of barrels. Faggot was one of the first members of the Academy, and had also been member of a special commission on weights and measures."Jakob Faggot" Nordisk Familjebok Band 7, Nordisk familjeboks förlags aktiebolag, Stockholm 1907 p.1259 He apparently was not a musician, though Helenius described he was interested in musical topics from a mathematical perspective and documented that he periodically came in contact with musical instrument makers through the Academy.Eva Helenius-Öberg "Kyrkoherden Nils Brelin—Mechanicus och Klaverälskare" Svensk Tidskrift för Musikforskning 1987 p.112 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.The instruments—a harpsichord and clavichord, both in the Stockholm Music Museum collection—are notable for their extreme large size and long tenor strings, where string length doubling extends down five octaves, and three octaves and a fourth respectively. Both incorporate many of the ideas described by Brelin in an article the Academy's 1739 Proceedings; Helenius (1979) Johannes Broman, harpsichord, Stockholm 1756 Museum no. 83118 Stockholm Music Museum; Johannes Broman, clavichord, Stockholm 1756 Museum no. N57244 Stockholm Music Museum (accessed September 13, 2009)
+Faggot's and Stråhle's string lengths and their relative pitches note Faggot N. 3 StråhleBarbour (1951) p.68 note length cents length cents ratio C 10000 0 10000 0 1/1 c ♯ 9379 111 9432 101 211/199 d 8811 219 8899 202 109/97 d ♯ 8290 325 8400 302 25/21 e 7809 428 7930 401 29/23 f 7365 529 7490 500 239/179 f ♯ 6953 629 7073 600 41/29 g 6570 727 6676 699 253/169 g ♯ 6213 824 6308 798 65/41 a 5881 919 5955 897 89/53 a ♯ 5568 1014 5621 997 137/77 b 5274 1108 5303 1098 281/149 c 5000 1200 5000 1200 2/1