Schubert's "Riemannian Hand"

The modulation to C# major is an LP transformation (Hook 139): A major moves to c# minor moves to C# major. Here is that change from the first to second strain, from A major to C# major, as a direct move in the left hand (thanks to Steve Rings for pointing this out):


While thinking about improvisation, about Schubert sitting at the piano playing while his friends danced, I realized that the piano permitted the sound of the waltz that would have been most familiar to people in Vienna about 1800 -- two violins and bass -- to be transferred from tavern or restaurant to the home. The three-layer texture of melody (first violin), bass, and accompanimental chords (second violin) became right hand, left-hand accents, and the offbeat "oompahs", respectively. (Link to D790n3 played apparently on a period instrument: note the timbral differences in the three registers.) In the heat of improvisation, the latter could serve Schubert well as voice leading stabilizers -- and, as in this case, enablers of modulations. Indeed, we might speak of his "Riemannian hand" and visualize it, as below, where a simple shift of thumb, middle finger, or pinky would effect a particular transformation.

Reference:

Julian L. Hook. "Signature Transformations." In Jack Douthett, Martha Hyde, and Charles J. Smith, eds.Music Theory and Mathematics: Chords, Collections, and Transformations, pp. 137-160. University of Rochester Press, 2008. Also see these posts: (1); (2).