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I am trying to analyze a Bortkiewicz prelude, op. 40 n.4 in F# major, but I have some doubts.
I divided the prelude into three parts:
Moreover I don't understand the harmonies of the last part, the coda (bars 30-41). Do you think there are any modulations?
One last question: Is polyrhythm (used in part B) intended to accelerate the progress of musical discourse?
I'll get the shorter discussion point out of the way first: as far as I can interpret, the polyrhythm used in Part B is not intended to accelerate the progress of musical discourse. That 2-against-3 polyrhythm is already used in Bars 1-4 in Part A, so Part B merely takes that use further. In fact, right before Part B, there is an 8th-note triplet in the melody. The melody, in fact, slows down its note rate right when Part B starts instead of speeding it up because it now uses regular 8th notes instead of 8th-note triplets. It is the animando expression marking that does all the work of accelerating the progress of musical discourse there - given the rit. later in Part B, the animando is implied to indicate an increase in tempo.
Now for the harmonic analysis of the coda:
The entirety of the coda can be explained as being in F sharp major with modal mixture and tonicization of IV.
Bars 30-38 all have a F♯ pedal note at the start of each measure in the left hand.
Bar 30 roughly uses only I - the tonic chord.
Bar 31 is easier to explain as V13/IV by taking the pedal F♯ note into account along with the right hand's E-G♯-A♯-D♯ chords, even if the right hand never plays F♯.
Bar 32 is similarly easier to explain as IV13 by taking in the left hand's F♯ and B along with the right hand's D♯-G♯-A♯-C♯ chords.
Bar 33 starts with vii°7 (E♯-G♯-B-D) and changes to V13 on Beat 3. Note that vii°7 is borrowed from the tonic minor - F sharp minor.
So, the chord progression of Bars 30-33 is I-V13/IV-IV13-vii°7-V13.
Note that Bars 34-37 repeat the same chord progression as Bars 30-33.
Bar 38 is an almost exact quote of Bar 30 that uses the same chord: I.
Bar 39 starts with a D7/C chord, which can be interpreted as a re-spelled German Aug. 6th of F sharp major (or F sharp minor) or as V7/♮II, except it resolves like neither of them. It proceeds to a more bizarre A♯7 chord on Beat 3, which at least can be explained as a chromatic mediant of F sharp major and shares A♯ in common with the tonic chord.
Bars 40-41 all use only I (barring non-chord tones) and end the piece with the tonic chord.
Given how often the coda returns to the home key of F sharp major (even if the beginning of Part A obscures that home key quite hard), taking only 1-measure breaks at longest, I cannot say that the coda modulates.
