source file: mills2.txt Date: Sat, 23 Dec 1995 08:01:07 -0800 Subject: Paul Rapoport's article From: COUL@ezh.nl (Manuel Op de Coul) For those who want to know the formulas used to calculate the numbers in the big file I've put together, or to know more about the theory of equal temperaments in general I'd like to make mention of Paul Rapoport's very interesting article "The Structural Relationships of Fifths and Thirds in Equal Temperaments", Journal of Music Theory vol. 37 no. 2, pp. 351-389, Fall 1993. It has three sections about respectively the perfect fifth, the major third and the relationships between the two. It must be said however that it's somewhat difficult material, but anyone with some mathematical background should be able to follow it. The intervals figurating in the article are: p Pythagorean comma a perfect octave k syntonic comma v perfect fifth, structurally just d diesis t major third, structurally just s schisma Theta major third, structurally Pythagorean q diaschisma I have found it practical to have an overview of the equations in question while studying the article, which is this: Theta = 4v - 2a p = 12v - 7a k = Theta - t d = a - 3t s = p - k q = 2k - p Schismatic major third = 5a - 8v. Two errata that should be mentioned: page 379, one line above the bottom line, replace "Basic t is best t." with "Best t=5." page 387, on the bottom line the second expression must be 19v-11a. Manuel Op de Coul coul@ezh.nl Received: from eartha.mills.edu [144.91.3.20] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Sat, 23 Dec 1995 18:42 +0100 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id JAA16412; Sat, 23 Dec 1995 09:42:18 -0800 Date: Sat, 23 Dec 1995 09:42:18 -0800 Message-Id: Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu