source file: m1456.txt Date: Tue, 23 Jun 1998 15:58:35 -0500 Subject: Re: Double Equal Temperament From: "Benjamin Tubb" On Mon, 22 Jun 1998 14:21:02 -0400, Paul H. Erlich wrote: >Double Equal Temperament, or 144-tET, does not provide for anything resembling traditional "mean temperament" (presumably meaning meantone temperament). The fifths of meantone temperament are typically around 696-698 cents, while 144-tET contains no intervals between 692 and 700 cents. 288-tET would be required to unify meantone with these other systems of tuning. The tuning scale steps in Double Equal Temperament available for around the 5th are: Name Ratio Cents G- 2^(83/144) 691.667 G 2^(7/12) 700 G+ 2^(85/144) 708.333 In "On the Sensations of Tone" by Helmholtz, page 484, meantone temperament "specimens of tuning" ranges for G are 697-733 cents. Likewise for Just Intonation, on page 486, G is usually tuned to 702 cents (a ratio of 3/2). In either case, Schillinger only claims that "the micro-units are the best averages for all differences between units of the 12th root of 2 and just intonation (natural scale)". There are further comments in "Tuning" by Owen Jorgensen concerning the tuners used by Alexander Ellis (the editor of Helmhotz' work) which indicate their minor innacuracies but they are still within the above cents constraints of "best averages". ------------- Benjamin Tubb brtubb@cybertron.com http://home.cybertron.com/~brtubb