|
MIT OCW 6.006 talks about this (in much higher resolution than when I last saw!): https://youtu.be/TDo3r5M1LNo?t=2806 An optimal fingering for n notes played one at a time by an F fingered hand can be found in nF^2 time, or I believe n(F^F)^2 if you allow F note chords. "Optimal" in the sense of minimizing a cost function defined in terms of state transitions: c(t, f, t', f') is the cost of playing note t with finger f, followed by note t' with finger f'. E.g. c(a3, 1, b3, 2) < c(a3, 1, b3, 5) because it's unpleasant to scrunch your pinky (5) that close to your thumb. (Notably, t/t' do not mean t_i and t_j, two notes in the piece.) There are papers quantifying such cost functions, apparently. Side rant - Having grown up on violin, but learning piano as an adult, (and as a programmer), it kills me to index fingers from 1 (thumb) to 5 (pinky). Violin doesn't use the thumb, so the pointer to pinky are 1->4! Worse, as a Suzuki violinist, I hear (to some approximation) the number of the finger I'm thinking about while playing. Worse yet, I read bass by "adding two" notes to treble, so I get a nice off-by-two to think about with my off-by-one. I should, uh, probably get a teacher. |