In "Two Kinds", why were "Pleading Child" and "Perfectly Contented" referred to as two halves of the same song?

This is a great question! In this short story we have
observed the conflict between Jing-Mei and her mother from its highs to its lows. This
of course finds its climax in the piano recital and the bitter argument that happens
afterwards. At the end of the story we advance forward a few years to Jing-Mei as an
adult, after her, as she puts it, "failing her mother so many times", but each time
"asserting my own will, my right to fall short of expectations." It is when her mother
gives her the piano that Jing-Mei begins to change in her attitude. She describes the
piano using an interesting metaphor, "a shiny trophy", because she had won it on her own
terms and not her mother's.

At the end, Jing-Mei receives
the piano and she beings to play "Pleading Child" again. The last paragraph is worthy of
some serious analysis:

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And for the first time, or so it seemed, I
noticed the piece on the right-hand side. It was called "Perfectly Contented." I tried
to play this one as well. It had a lighter melody but the same flowing rhythm and turned
out to be quite easy. "Pleading Child" was shorter but slower; "Perfectly Contented" was
longer but faster. And after I played them both a few times, I realised they were two
halves of the same
song.

Jing-Mei realises that
just as these two pieces of music go together inseparably, being "two halves of the same
song", so in her life, the stage of "Pleading Child", which interestingly is described
as short but slow, is inextricably linked to "Perfectly Contented", which was longer and
faster. Jing-Mei, through her childhood was the "Pleading Child", wanting her mother's
attention and praise, and now, as an adult, she has reached the stage of being
"Perfectly Contented", knowing who she is as an adult and being happy in her identity.
However, what she realises is that she can't have one without the other - both are
irreplaceable parts of life's journey.