source file: mills2.txt Date: Mon, 25 Sep 1995 08:21:28 -0700 Subject: Bach's Tuning From: Johnny Reinhard Re: Kirnberger's saying to Marpaug that "Bach tune each of his major thirds sharp." This is true in both equal temperament _and_ Werckmeister III. The Werckmeister III contains 4 sizes of major third, one of which is equivilent to 402 cents, one that is Pythagorian at 408 cents, one at 396 cents, and the smallest one at 390 - all sharper than the just 386 cents. Further, Kirnberger himself gave directions for the tempered fifth whose ratio (10935:16384) is almost exactly 1/12 of a comma smaller than the pure fifth in several publications, including his *Von der Temperatur*. The method by which he arrived was to take seven pure fifths in succession plus one pure major third. This gives the tempered fourth 8192:10935, which is larger than the pure fourth by 1/12 of a comma, its inversion is the tempered fifth 10935:16384. According to Kirnberger he showed this discovery to Leonhard Euler in 1766 and remarked to him that this fifth could be used to determine all ratios of equal temperament, an idea subsequently developed by Johann Heinrich Lambert and published in the *Memoires de l'academie royale des sciences et belle lettres* (Berlin 1774) and used by Marpaug against Kirnberger. (see p. 20 of the *Art of Strict Musical Composition* by Kirnberger, Yale University Press) There is great bitterness against Kirnberger, chief violinist and keyboardist at the court in Berlin by Marpaug, the Lottery King Of Berlin. Clearly, this argument of fifths over thirds is an aesthetic battle of historic proportions. Politics enters at every level and success is oft measured by clout. What intrigues me most about Prof. Rasch's perspective is that the Netherlands has been a leader in performing "early music" in historic intonation and as its main musical intellectual in this area, Rasch bucked the trend. Perhaps if he were in any one of the other countries that had been resistant to historic intonation systems, he might have headed the other way. Johnny Reinhard American Festival of Microtonal Music 318 East 70th Street, Suite 5FW New York, New York 10021 USA (212)517-3550/fax (212) 517-5495 reinhard@ios.com p.s. Manuel please forward as appropriate. Thank you. Received: from eartha.mills.edu [144.91.3.20] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Mon, 25 Sep 1995 23:48 +0100 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id OAA25337; Mon, 25 Sep 1995 14:48:01 -0700 Date: Mon, 25 Sep 1995 14:48:01 -0700 Message-Id: Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu