Although much of the harmony here is triadic, few of the chords function in a conventional way. It is possible to give each chord a "name"; for instance, the first 13 bars could be notated as these chords (with a little enharmonic licence):
Em / / / | B7sus4 / B7 / | B7b5 / Bm7b5 Dm7b5 | E7 / Em7 Edim7 |
Am7 / F#m7b5 ?? | F#m7b5 / D#dim7 ?? | D7 / / / | Dm7 / Bm7b5 Bdim7 |
Cmaj7+ Am / / | B7sus4 B7 F#m7b5 (Am) | B7 / F#m7b5 (Am) | B7 / / / (implied) | Em …
At this point, we could try to analyse these chords in terms of key, however this quickly becomes problematic, for two main reasons:
- few of these chords feel strongly associated with the tonic, E Minor; it makes little sense to analyse them in terms of this key.
- although there are a number of dominant 7th chords, we never cadence in any key (except the final return to E Minor in bars 12-13), so no modulations are effected, making it unhelpful to analyse chords in terms of any other key, either.
So, although we may be able to describe each of the chords in this harmonic progression, and even relate them to one or more keys (using roman-numeral notation, for instance), this would neither describe how we perceive this progression, or how it is constructed. Instead, we need to view this chord sequence as the result of a series of simultaneous (mostly) chromatic descending "voices". (@Pat Muchmore gives a good explanation of this in his answer.)
However, we do feel some "harmonic-signposts" along the way. Essentially the first 13 bars function as a single i-iv-V-i cadence, embellished by many chromatic passing notes. Bar 1 is clearly chord i; with some decoration, bars 9-10 are basically chord iv and bars 11-12 are chord V; we cadence strongly back to i in E Minor at bar 13. (Possibly the D7, bVII7, chord in bars 7-8 could be considered pre-cadential too…)
This article, which I already mentioned in comments above, clearly shows the chromatic and step-wise movement of the "voices" in this harmonic progression. Although it is a rather "tongue-in-cheek" article, suggesting an approach to "create-your-own-Chopin-Prelude", it does show that the harmonic movement in this Prelude can be viewed at a variety of different background "rates", which progressively become closer to the actual Prelude, as further embellishments are made.