source file: mills2.txt Date: Wed, 20 Nov 1996 02:07:50 -0800 Subject: Resolving Pythagorean Major Thirds From: Daniel Wolf <106232.3266@compuserve.com> Answering bits and pieces of Alves, Erlich, and others: (1) I had to catch this one. A Pythagorean Major third will will _resolve_ in one of the following ways, only one of which is to a perfect fifth: Given the series of fifths/fourths F through B natural, the third c1/e1 can resolve directly to the octave f/f1, the fifth g/d1, or the fourths b/e1 or c1/f1. The third f1/a1 will not resolve to the fifth e1/b1 because of the cross relation f1/b1, and likewise for the third g1/b1 to f1/c. Resolution being a contextual matter, the diminished fifth, 1024/729, b/f1 _resolves_ to 81/64 c1/e1! The best practical guide to counterpoint that I know is Diether de la Motte´s _Kontrapunkt_ (dtv/Baerenreiter 1981, 3rd ed. 1988), sadly only available in the German original. (I would happily translate this if a textbook firm wants to make me an offer). What recommends his approach to early counterpoint is his insistance upon the collection of pitches having priority over any particular scalar arrangement, all theorists to the contrary, but verified by the actual ranges of part writing in the repertoire. I personally have always heard the collection to have priority over mode, and that modal descriptions are most useful in describing local melodic figures. (In this, Harry Powers* _brilliant_ article on **Mode and Melodic Type** in the New Grove is recommended, although most intonation theorists drop it for lack of tuning information). (2) Once again, Paul Erlich has rediscovered the wheel: that Pythagorean is terrific for melodic and octaval/quintal/quartal vertical structures, but that once a triad is introduced, triadic just intonation is needed, and a temperament is called for when these come into conflict. Erv Wilson*s paper _Some Patterns Underlying Genus 17 (12+5)_ (1981?), suggests alternative, just intonation and schismatic, solutions to this conflict and definitely needs wider circulation. (3) Taking Yasser´s age and era into account, I find it terribly unfair to characterize him as ethnocentric. In fact, I defy anyone to tell me exactly what ethnicity was central to Yasser. (He was a Russian-born scholar of Jewish Music). His Darwinian streak, is, however, fair game, but can someone suggest a more P.C. term than _evolution_ to describe transitions towards more complex tonal systems? Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Wed, 20 Nov 1996 12:18 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA00440; Wed, 20 Nov 1996 12:20:10 +0100 Received: from eartha.mills.edu by ns (smtpxd); id XA00438 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id DAA04310; Wed, 20 Nov 1996 03:20:07 -0800 Date: Wed, 20 Nov 1996 03:20:07 -0800 Message-Id: <199611201002.HAA26657@chasque.apc.org> Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu