The "dialectic" that Schopenhauer seems to present is the WILL. The Will is the actualizing drive that produces objects; all individual objectification is really a matter of will, on his account. From this idea, he perpetuates the idea that the world has two parts: 1) the inner part of the Will, and 2) the world as the Will and the world as representation.
We are able to know the representational world by the principle of sufficient reason: our sense, motivation, judgment. This principle provides the compete scientific explanation of what something is, for Schopenhauer the what something is, is a representation.
To continue further, from the Will, ideas individuate, or become represented by the actual world. The result of the individuation of the Will is that the world in-and-of-itself is striving against itself and is thus: frustrating, meaningless, and void of knowledge. Additionally, within this continuos strife there is us, humans, and within us there is also present this unyielding strife in the form of a desire cycle. In this cycle we will to get something/desire something, but because one lacks what you desire, you suffer. However, once you obtain something you possess a brief moment of fulfillment, and then ennui sets in, which is, again, suffering. So, desire -> satisfaction-> ennui-> desire = suffering.
How is it, then, that we can escape this vicious cycle? This is where the aesthetic applies. By way of art and aesthetic experience, we are able to achieve a more tranquil state of consciousness, and the genius is able to make manifest the ideas of platonic forms. Aesthetic perception thus raises a person into a pure will-less , painless, and timeless state of knowledge (Will-lessness is the negation of the Will). So, to explain how this works: suppose that you are outside and you see an amazing sunset, and during that brief moment you experience a time in which you forget all particulars and desires - that is the state of will-less-ness. We appreciate works of art (of the artistic genius) so as to have better knowledge of the Platonic ideas, and the will-less state leads us closer to the these ideas.
" Only through the pure contemplation described above, which ends entirely in the object, can ideas be comprehended; and the nature of genius consist in preeminent capacity for such contemplation" and "But what kind of knowledge is concerned with that which is outside and independent of all relations, that which alone is really essential to the world, the true content of its phenomena that which is subject too no chance and therefore is known with equal truth for all time and in a word the ideas, which are the direct and adequate objectivity of the thing in itself, the will?" (The World as Will and Idea)
(So the genius is the capacity to tap into these ideas and represent them via art. And music is able to bypass and know the will directly -music is unique)
The idea that one is able to come to an apprehension of certain Platonic forms by way of art is basically taking Plato's argument upside-down. Plato has claimed that art is down with the reflections and shadows of the physical world, and thus, draws us away from the truth -the forms. Nevertheless, Schopenhauer holds that this is not the case, rather we come closer to them by art.
I think that it is worthy to note that the story of Ixion ( a Greek myth) provides an interesting analogy of how Schopehauer cashes-out his idea of the Will and how one escapes it. According to the myth, Ixion was sentenced to be tied to a burning wheel by Zeus. Ixion received this punishment for lusting after Zeus' wife Hera, and the only time that Ixion is able to feel any form of relief from this punishment is when Orepheus played his Lyre.
Sources and web pages to be visited for further info:
2) Certain lectures