This is an update to the post on the "Riemannian hand" and the post on closed-position voicings. I have reconstructed the graphic from that post in two respects below: (1) we now see the hand from the top, as a pianist would see it, not in the "Guidonian position" of the singer putting his palm out in order to remember the sol-fa; and (2) I have charted the two stages of the transformation--the first hand for L that takes A major (A+) to C# minor (C#-) and the second hand for the P that changes minor to major (C#+).
The point of this graphic is to show that the "Hand" isn't stationary. Although L shifts the pinky from A+ to C#-, the subsequent thumb move isn't the thumb's "proper" R (on the first hand), but P (on the second hand). In other words, the abstract LPR hand-group has rotated while the physical hand has not. The separation of these two is something Schubert certainly would have noticed while playing, and that realization would very plausibly have facilitated the notion of modulation rather than "just" chord change (the latter would have led to C# as V/F#-, or V/vi, the move in many of Schubert's dances). The difference is that of harmonic thinking (in the familiar nineteenth-century sense) rather than figured-bass or continuo thinking (eighteenth-century).
Here is further explanation of the rotations. I am grateful to Steve Rings for working this out. The text is his.
It seems that the specific assignment of LPR to the fingers is based on both mode and triadic position/inversion. The table below illustrates:5/3 6/3 6/4M LPR PRL RLPm RPL PLR LRPThe LPR triples here are to be read from left to right in a registral fashion as "low-mid-high," which allows one to apply them to either right or left hand. One toggles back and forth between major and minor by using neo-Riemannian letters that occupy the same registral "slot" in the triples. Thus, for example, if we begin with a major triad in 6/3 position and apply L to it, we will be moving our "highest" finger, as the given triple is PRL. This will then take us to a minor triad in 5/3 position, as it is the only minor entry in the table with L in the rightmost slot. Note the dualist patterning: the neo-R triples for the major triads are all rotational permutations of LPR, while those for the minor triads are rotational permutations of RPL. LPR and RPL are of course retrogrades of each other (a result of Riemann's dualist conceptions of major and minor).
Postscript (1): I can't show the piece for copyright reasons, but there is one instance of a mirror to the A+-C#+ transformation in D779n13: the beginning of the second strain of D980d, a waltz published in January 1828. The main key is C major, where the first strain ends. The second strain drops to A minor and repeats that chord for three bars, followed by an F minor 6/3 in the fourth bar, and G7 in the fifth. The left-hand voicing is an imperfect wedge: E-F-G in the left-hand thumb, A-Ab-G in the pinky. A- to F- is also an LP transformation.
Postscript (2): Guy Capuzzo notes a transformation series that fits the guitarist's hand in relation to the frets (183). The "Guitarist's Hand" and the "Riemannian Hand" are related to, but distinct from, the keyboard topology of Minturn and Jones (the latter according to email correspondence from Neil Minturn, 26 December 2009).
Postscript (3) [added 1-05-10]: a detail from Joseph Kupelwieser's watercolor: see this post for more information:
References:
Capuzzo, Guy. "Neo-Riemannian Theory and the Analysis of Pop-Rock Music." Music Theory Spectrum 26/ 2 (2004):177-199.
Minturn, Neil, and M. Rusty Jones. "Toward a Theory of Keyboard Topology." Paper read at the annual meeting of the Society for Music Theory, Montreal, 31 October 2009.
