A series of posts beginning with this one will examine the problem of the minor key from a mostly traditional Schenkerian point of view and in that small repertoire of compositions that includes 17th and early 18th century music relying on the Dorian octave, two remarkable pieces by François Couperin, and two anomalous 19th century compositions by Beethoven and Hugo Wolf.
The ascending Urlinie is most likely from ^5, and what I should perhaps call the "mirror Urlinie" takes the form ^8 down to ^5 then up again to ^8. Here are versions in both major and minor keys:
The simplest ascending form in the minor key uses the raised ^6 and ^7. With I-V-I only, ^#6 is either part of a ^5-^6 figure over I (as in "a" below) or a passing tone over V (as in "c" below). Example "b" shows how S chords can be introduced by elaboration of the version in "a".
At "x" the augmented second is prominent; although one can find such scale figures in compositions, positioning in a structural cadence is unlikely. At "y" the augmented second is mitigated by one of the variant forms I discuss in my JMT article (1987), but as "z" shows it is very hard to get rid of the sense of the natural ^6 as a neighbor note rather than a scale step rising to ^#7.
A graphic with fourteen ascending figures may be found here: link. The series will follow the sequence of these figures, with examples from compositions, though I admit that only a few of them are consequential [the majority are hypothetical--I haven't found them in pieces] and in fact the first set will be counter-examples from Schubert. At the end of the series, I will add posts with a couple additional counter-examples (from Beethoven and Offenbach) and other posts with historical context about the Dorian octave, the examples coming mainly from Praetorius and Eyck.
