I've memorized the second verse of The Entertainer for the right hand and have begun playing with the left hand part, which is, as was the case with the first verse, a lot more difficult to commit to memory. The "home base" seems to be an inverted C chord though, which is fairly simple, but you get to this C chord from several different positions which is tricky. Plus Joplin had a great love for all these weird Chord inversions off all sorts, which sound beautiful when played but for an apprentice such as myself, are dificult to memorize out of the blue.
But I did it for the first verse, and I have confidence I can do it for the second.
Interestingly enough, Joplin modified his strategy for the second verse melody. In the first verse, the melody is an octave (the same not played n octave apart) with an added third. In the first verse, this added third note is at the bottom of the octave. In the second verse, the added third is at the top of the octave. This creates an interesting dilemma when you reach the notes G and A--in order to keep the melody harmonious, they have to share the same third note, in this case, E.
I know people hear this piece and think "Well, that's pretty." But when you study it as a piece of composition, it's fascinating and complex. Which is why I jumped into this to begin with 9well and to keep my brain moist and spongy). I wanted to understand the music I was listening to, and appreciate the composer's intention.
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