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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, and is praised by them 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.196 Daniel 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 mathematics〔Kerala 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Strähle construction」の詳細全文を読む スポンサード リンク
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