Although Aristotle states a theory of beauty in the Metaphysics: “The chief forms of beauty are order and symmetry and definiteness, which the mathematical sciences demonstrate in a special degree” (The Complete Works of Aristotle Barnes ed., volume 2, 1705, 1078a36), the theory he offers in the Poetics is quite different, and quite a bit more original. The Metaphysics comment is preceded by a statement that the mathematical sciences say a great deal about beauty.
The Metaphysics theory can be found as far back as the fragments and lore of Pythagoras. Pythagoras gave us the terms "harmony" and "symmetry" both of which seem to be central to his (or the Pythagoreans') theory of beauty. The reference to mathematics also connects both the Pythagorean and the Platonic notions of beauty. Aristotle, by contrast to Plato, was generally more inspired by biology than mathematics, and this comes out in his discussion of beauty in the Poetics.
It is also noteworthy that in The Organon Topics Book 3 Part 3 Aristotle says, "if one thing be desirable for itself, and the other for the look of it, the former is more desirable, as (e.g.) health than beauty. A thing is defined as being desired for the look of it if, supposing no one knew of it, you would not care to have it. Also, it is more desirable both for itself and for the look of it, while the other thing is desirable on the one ground alone. Also, whichever is the more precious for itself, is also better and more desirable. A thing may be taken to be more precious in itself which we choose rather for itself, without anything else being likely to come of it." [taken from R. B. Jones]
But as I have said, the most interesting contribution to a theory of beauty is to be found in his Poetics. There, he develops the notion of beauty as related to organic wholes. Yet, his initial discussion of beauty does not actually mention the term. It is in Chapter 3. There, he talks about how imitation is natural to man, how we are the most imitative creatures, and how we delight in works of imitation. (I am working here with The Oxford Translation of Aristotle edited by W. D. Ross and appearing in Art in its Significance by Stephen David Ross, 3rd. ed., 1994). It seems that this delight is an aesthetic delight. He observes that we even delight in realistic representations of "the lowest animals" and of dead bodies. (This sort of thing is accounted for by some 20th century philosophers in terms of "taking an aesthetic attitude." See, for example, Paul Ziff "Anything Viewed.") He further observes that we delight in a picture because we learn from it at the same time, for example we learn that a man falls into a certain category. Even things we have not seen before can, when represented, give us delight in the execution or coloring. We know he is thinking of beauty here since he also mentions that harmony and rhythm are natural to us.
When Aristotle gets around to defining tragedy in Chapter 6 of Poetics he doesn't explicitly mention beauty and, since the purpose of tragedy is catharsis of pity and fear, it seems that beauty is not central to tragedy. But there are two reasons to question this conclusion. First, one could say that there is a kind of beauty in that which causes catharsis. Second, his account of beauty is essential to evaluating tragedy. I will discuss that later. Note also that tragedy is often full of depictions of painful things...and these were recently mentioned in the section on the value of imitation. In the definition of tragedy he does explicitly mention "language with pleasurable accessories" by which he means "with rhythm and harmony superadded." This may imply that these things provide a kind of beauty, although perhaps only as an add-on to the core experience of catharsis. So the key issue is whether the play in its central purpose can be understood in terms of beauty.
The central discussion of beauty comes in the second part of Chapter 6 when Aristotle talks about the proper construction of a Plot. We find that a tragedy is an "imitation of an action complete in itself, a whole of some magnitude" and that the whole should have a beginning, middle and end, the beginning and end being non-arbitrary. I take it that "magnitude" does not just mean "size" but rather "appropriate size," and maybe even more than that, i.e. "appropriate size to be considered beautiful." For then he says "Again: to be beautiful, a living creature, and every whole made up of parts, must not only present a certain order in its arrangements of parts, but also a certain definite magnitude." The analogy to the living creature will be picked up later as he expands on the notion of an organic whole. Clearly it is not enough for something to be beautiful to be made up of ordered parts, but that the parts need to be arranged in a way similar to the way they appear in an organism. Seemingly reverting to the Pythagorean view, Aristotle follows this by saying that "Beauty is a matter of size and order." Still, he understands "order" now in a different way because of the animal analogy.
But, before we go into the animal analogy, we need to consider his idea that beauty is impossible in a minute creature, or in one of vast size. In the first case, the creature is not beautiful because it is indistinct, and in the second case it is not beautiful because it cannot be seen "all at once" and "the unity and wholeness ... is lost to the beholder." So beauty has something to do not only with harmony, symmetry and order but also with clarity, unity and wholeness.
When Aristotle applies this idea to poetry, he introduces a psychological dimension. The beauty of a poem depends on our memory's ability to take it all in, much like the beauty of a large object depending on our perceptual ability to do so. So, he holds, the longer a story is the more beautiful it is based on its magnitude. Here, he is taking magnitude to mean "the right size." However, he then gives another account of magnitude. It is a length that "by a series of probable or necessary stages [the hero passes] from misfortune to happiness or from happiness to misfortune." He says that this "may suffice as a limit for the magnitude of the story." That is, the story has good magnitude if it has the right length for this series of probable or necessary stages to be worked through, i.e. making a good plot (and thus bringing catharsis to the audience.)
Note that in the very next sentence, but the first sentence of Chapter 9, he also uses the phrase "probable or necessary," although in this case referring to the function of the poet, which is to describe "not the thing that has happened, but a kind of thing that might happen, i.e. what is possible as being probable or necessary." So, part of what is meant by "magnitude" is the sense of probable or necessary movement within the poem with respect to the plot. This is the very thing that distinguishes poetry from history, which is only concerned with what "has been," not with what "might be." Thus, Aristotle's defense of poetry's cognitive power against Plato, follows from his theory of beauty as magnitude, where magnitude is seen as a quality of an organic whole in which the parts are intimately related.
The key quote in this is to be found in Chapter 8, which is mainly about the unity of a plot (unity being necessary for beauty). That a plot is about one man is not enough to make it unified: there must be one story. So here is the quote: "so in poetry the story, as an imitation of action, must represent one action, a complete whole, with its several incidents so closely connected that the transposal or withdrawal of any one of them will disjoin and dislocate the whole. For that which makes no perceptible difference by its presence or absence is no real part of the whole." So, the organic whole is like the body of an organism: if you take away an organ you will severely diminish the capacity for the organism to function. Similarly the parts of the beautiful play are closely intertwined that if you take away one you will destroy the beauty of it. This is Aristotle's original idea about beauty.
There is one qualification to this. Aristotle also holds that something can be made more beautiful if it goes beyond just imitating an organic whole. The painter or poet can make something true-to-life and yet make it more beautiful, by preserving the type and yet also making it nobler: "Again, since Tragedy is an imitation of persons who are above the common level, the example of good portrait painters should be followed. They, while reproducing the distinctive form of the original, make a likeness which is true to life and yet more beautiful. So too the poet, in representing men...[who have] defects of character, should preserve the type and yet ennoble it." (Chapter 15)
It may not be that all forms of beauty need to meet Aristotle's standard as set forth here. At the same time, it is arguable that organic beauty is a particularly impressive form of beauty, and may be distinguished from that which, although called beautiful, may be a simple beauty, or even merely charming or pretty. The concept of "necessity" may be of value here: there is a kind of necessity (not mathematical or causal) which reveals a compelling dynamic energy, that (coupled with such things as order, harmony and symmetry) gives us beauty in this sense. Thus it would make sense to apply the term beauty paradigmatically to a great tragic play despite the painfulness of the events portrayed.