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Lets take Fur Elise as an example. Would we class the left hand accompaniment as a melody in its own right?

If this is so, would this be considered contrapuntal writing?

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The left hand of this is basically arpeggiated chords. So it's just putting an accompaniment with the appropriate harmony under the melody.

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It certainly can be. Historically though, the bass part (whether played by a bass or not) was often considered more of an arrangement detail than a composition detail. In "figured bass" notation, the bass part, while written, is considered to be largely improvisational.

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A piece of classical music will almost always have, at any given moment, a functional bass note that allows the current harmony to be heard in root position (root in the bass) or an inversion (some other chord tone in the bass). In this example, the bass line consists of the low notes at the downbeat of each measure, beginning A, E, A, which are also the root notes of the first three chords.

Styles vary widely as to the melodic importance of the bass. In Bach's works it frequently rivals the melody line in intricacy and importance. However, even in a simple melody-and-accompaniment texture, as in Für Elise, composers usually give the bass line a basic level of contrapuntal independence by avoiding parallel fifths and octaves with the melody, avoiding unmelodic leaps (e.g. tritones), and often including scalewise contours (e.g. at the three chords leading into the F major middle section).

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Uh, is this a trick question? "Für Elise" is (at least regarding the well-known beginning) rather particular in its "accompaniment" in that accompaniment and melody line alternate (which makes it a nice beginners' piece since you don't have to focus on both hands at once) with the overlap provided by pedalling and with the arpeggiated line running from bass to treble starting as a "static" arpeggiation of the harmony (containing in-chord notes) and transition into a melodic line start at the end of the arpeggiation.

So for this particular case, we have a rather clever meld of accompaniment and melodic line. The left hand material is not an "accompaniment in its own right" since it is playing catch with the right hand, and it also is not really contrapuntal since both are not competing in melodic and harmonic content but are more or less sequentially combined.

Somewhat similar in design is that rather well-known Prelude #1 (C major) in Bach's "Wohltemperirtes Clavier" but its arpeggiation line does not end in a similarly melodic element as "Für Elise" does, so it's arguably "only" accompaniment and is used in that function in Gounod's "Ave Maria".

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