I've always had problems with tuplets (3/2, 4/3, etc), until I was taught this nifty trick:
- subdivide the total duration (4 quartuplets, 3 triplets, etc) by the least common multiple (12, for 4/3)
- place the tuplets along the appropriate subdivisions (4/3: quartuplets = 12/4 = every 3 subdivisions, triplets = 12/3 = every 4 subdivisions)
- tap out, and learn, the rhythm
For example, for 4/3:
| | | | | o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o o---o---o---o---o---o---o---o---o---o---o---o---o | | | | |
- Rhythms shift to quickly (4/7 -> 2/4 -> 2/6 -> 4/2.5 -> 3/2.5 -> 8/5 -> 12/10)
- Rhythms are too plentiful and exotic to be learned using the above method
Right now I'm just guessing when to play the notes by their positions on page. Unfortunately, there's just something about this approach that hinders (muscle) memorization and practice. For example, while I have already learned and am consistently playing - at a slower tempo - sections of equal or greater technically difficulty with straightforward (or simple, 3/2, 4/3, 5/2) rhythm, I am still stumbling through sections with unusual rhythms, despite having practiced them more.
In situations such as these, with much more practice than I'd prefer, over time I become sufficiently familiar these poly-rhythmic sections that eventually my hands intuitively, subconsciously converge towards a "good-enough" estimation of the actual rhythm. Then, I learn the section "normally", as if were a mono-rhythmic passage. So you see, this "two-pass" learning procedure is very inefficient. Eventually, with sufficient mastery, I can adjust the speed/rhythms of each hand independently.
I've been told, "just practice each hand separately, then put them together". First, I've never been able to, or just never understood, this. Of course it helps to practice each hand independently, but I've never had a piece "magically" come together with both hands, after each has achieved independent fluidity. Perhaps this is because I'm a notorious sheet/sight reader - I'm always looking at the page, never my hands - and I (almost) never memorize music. Secondly, even if I was able to "put them together", the rhythms are just to exotic to practice independently in the first place. As I mentioned near the beginning, I'm used to practicing rhythms against each other - for example, triplets against the steady beat of quartuplets.
Anyway, I'm curious about your techniques for efficiently & quickly learning difficult poly-rhythmic passages, such as from Ornstein's 4th sonata (in particular, PDF pp 6-7 or numbered pp 4-5, starting with "Tempo I°").
P.S. If you do recommend "practicing each hand independently", please calibrate to my shortcomings and experience with that method, as I've described.
I've settled on practicing each hand separately, with a metronome, for measures 45 - 52 (starting with Tempo I). The hands are sufficiently independent and even-rhythmed that I think I do a decent job when practicing them together. Since the left hand is arpeggiated with a strong tempo while the right hand plays a simple, repetitive, and catchy melody, I find it simpler to just "fit/squish" the few single-beat occurrences of 3/7 or 4/7 into the overall beat. For longer repetitions of the more complicated rhythms (more than just a beat) I would practice as suggested by Scott Wallace. I'm impressed that he's already on 7/4 - I've just mastered 4/3. However, I don't think anyone realizes how chaotic measures 54 and 56 are - how dynamic the poly-rhythms are. I've inscribed those measures in Denemo to produce a training PDF. Each measure is preceded by a duplicate identical in all respects, except the notes have been normalized per-staff (much as in drum notation). I've then recorded the resulting MIDI using the Salamander Grand Piano V3 soundfont, at 15bpm and 50bpm (published tempo is 76bpm). Some friends commented, "so basically, it's a play-whatever-the-#$@&%*! you want measure". To contrast: the least-common-multiple-method that Scott Wallace and I mention would require dividing each measure into 120(!!) subdivisions (60 excluding the 16th-triplets) to ensure that each note falls evenly on an interval. Anyway, I plan on leaving the question unanswered for a while longer - I was hoping to generate a wider range of answers.
I've inscribed the full first movement into MuseScore. It is available for online listening at Ornstein, Leo: Piano Sonata No.4, SO 360: I. In comparison, notice how - especially in measures 54 and 56 - the official Poon Hill Press edition on IMSLP actually places the noteheads incorrectly or even out-of-order for the more involved poly-rhythmic sections. It goes to show how little you can depend on note positions on the page.