Resolving Pythagorean Major Thirds

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?



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> I had never heard
> the word "wandering tonics" and it seems a good way to name the concept.
> (Do you know where, when and who coined this term ?).

Sadly, I'm afraid I don't recall where I heard the term. I doubt if I
invented it, because I think I remember somebody else suggesting it. But I
suppose it's not impossible that I did coin the term myself, and I'm being
subconsciously modest!

"Free JI" strikes me also as a decent, but perhaps a bit nonspecific, term
for the concept as well.


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