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Hey, I didn't know you had done this analysis. For some reason Technorati didn't show this link. This is a very good analysis, especially the comments about the motives. I do offer an alternative reading:
Start in E minor rather than A minor. While tonal music traditionally starts and ends in the same key, there are instances of starting off-tonic in Romantic music. This is sometimes called "Auxiliary cadences" in Schenkerian circles, because the first cadence is in an auxiliary or secondary key.
Starting in e minor, the key of the minor dominant, the chords are:
i - III64 - V/III - III
Then the b minor section sounds like the minor dominant of e minor.
m. 9-10 echo m. 4-5, so rather than A major, I would label this section in D major, with the expected cadence at m. 11 avoided with the half-diminished chord. This chord leads us to the final key of a minor as you show, though it shows up as a very weird common-tone half-diminished seventh chord.
Why the implied Neapolitan in measure 18? There are no Bb's anywhere around. Call it instead a iv.
Wow. Your analysis is so much clearer than mine. The long-range V - ii - I that I used in measures 1 through 10 are definitely more justified as the key changes you describe (E min - B min).
The mysterious Bb I heard in measure 18 was a parallel to the function of the melodic note in previous phrases. What I had labeled as motif B always ends with the third of the chord in the melody. I had originally tried to hear it as a vi, but it never really worked in my ears.
Thanks for the input.
Okay, I can see the Bb possibility from the previous motives, as if the VI is a V/bII. I think Chopin avoided that both to keep the ambiguous monophony going and to avoid the tritone required to get back to the i64. Or he felt that it would have been too repetitive, especially for such a short piece.
Could the opening be IV of B Harmonic Minor & be ahead of it's time?
The d naturals in m.13/14 definitely have to be d-sharps...
Looking at IMSLP here: http://imslp.org/wiki/Preludes_Op.28_(Chopin%2C_Frederic), you're correct. Not sure how that natural got in there. Thanks.
These are all very interesting analyses but they all somehow fail the most important test of all: can any of us truly hear A minor as the "ruling" tonality until bar 14? And if we can't then is it incumbent upon us to force the preceding progressions into this unheard tonality by, for example, naming the opening harmony the "minor dominant"? I suggest a different reading: this piece begins in one key and ends with another, making this prelude one of the only instances of atonal, or quasi-tonal music in the early-mid 19th century. It is not without cause that this piece failed to be understood by many a great musicians as far as 1883 when the pianist Jan Kleczynski declared that it should never be played because it is "too bizzare".
In reply to ysilver, it could be dangerous to apply the term "atonal" to music like this, and as early as this, especially as in every single bar here there is a sense of key or urge towards one! Just look at the repeating bass notes reinforcing this, whatever the chromatics of the inner lines.
As for the final bars: I'd suggest the pedal E is sustained, whether explicitly or not, from bar 15-21, as part of a clouded A minor 64 harmony: the melody lands, with decreasing tension, first on F (very dissonant), then on D (more like a suspension), then on B, resolution only coming with the E major triad. The rectified D sharps in bar 14 over the F bass make for a good old-fashioned augmented 6 chord (with C double-sharp as chromatic auxiliary). This emphasises the move towards A minor.
You've all bored me to tears - dissection is not analysis. The piece is obviously a unique paraphrase of Dies irae. This is a useful observation, one that helps musically comprehend and perform the piece.
@Wolfman0: Interesting suggestion. I've added the Dies irae in the main post above.
I don't think, and I doubt most others involved here think, that analyzing a piece's structure and harmonic progressions is the only step in understanding it. It is one step as is recognizing its melodic quotations or (as Scott Spiegelberg pointed out) its compositional quirks related to the time period. These are just all tools to help inform the performer.
For me analysis deepens rather than, as you seem to imply, cheapen the appreciation of a work.
Here's an alternative analysis:
1) the piece definitely does not begin in a- minor. It begins in e-minor, but quickly we hear e-minor as vi of G major, on which there is an IAC at m. 6 after the V6/4-5/3 dominant of G in m. 4-5.
2) This opening phrase appears to begin to be repeated in b-minor (though our memory now tells us that b-minor functions as vi of D-major). Everything proceeds as before, transposed up a fifth until the expected cadence in D-major in m. 11, which is replaced by a half-diminished seventh chord on D#.
3) From m. 11 until the end of the piece, there are really only 3 harmonic functions: II, V, and I. M. 11-14 function as II, transformed chromatically. The first half-diminished chord can be seen as a B-chord in a-minor, missing its root and with a sharp 3rd, 7th and sharp 9th. The fully-diminished chord that follows, simply lowers the 9th by a semitone. Then the chord is revoiced to have F-sharp in the bass, and then the F-sharp is lowered to F-natural (i.e. the fifth of the II chord is flattened) and then the 9th is flattened once again to achieve a French augmented-sixth chord. Functionally, all of these chromatic chords serve the same purpose: dominant preparation. M. 15-20 are V6/4 which resolve to V 5/3 and then 7/5/3 in 21, finally resolving to I in the last bar.
I think another interesting thing to think about is the overall form of this prelude. Is it a period with an expanded consequent because of the deceptive cadence? Or some type of hybrid antecedent + continuation becomes cadential? It has a circle of fifths statement-response type phrasing from e (to G) and b (to D) e and b being fifths and G to D being fifths and then going to the true key of a being a fifth to e and a fifth to D.