The Encyclopaedia Britannic describes aleatory music thus:
Aleatory music, also called chance music, (aleatory from Latin alea, “dice”), 20th-century music in which chance or indeterminate elements are left for the performer to realize. The term is a loose one, describing compositions with strictly demarcated areas for improvisation according to specific directions and also unstructured pieces consisting of vague directives, such as “Play for five minutes.”
The indeterminate portion of aleatory music commonly occurs in two areas. The performers may be told to arrange the structure of the piece—e.g., by reordering its sections or by playing sections simultaneously as they wish. The musical score may also indicate points where performers are to improvise or even to include quasi-theatrical gestures. Such requirements may give rise to inventive notation, including brackets enclosing a blacked-out space, suggesting pitch area and duration of the improvisation. Among notable aleatory works are Music of Changes (1951) for piano and Concert for Piano and Orchestra (1958), by the American composer John Cage, and Klavierstück 11 (1956; Keyboard Piece XI), by Karlheinz Stockhausen.
Music is as much about what is not present as what is: it is about the gaps between the notes as well as the notes themselves. I once heard a quotation from a famous sculptor (can't remember who it was) in which he said if he's sculpting a lion, his job is to starts with a block of stone and remove all the bits which aren't the lion. Music is perhaps what you get when you start with some noise and chip away all the bits which aren't music. This is why introducing absolute randomness does not therefore produce something most people credit as music.
Without some form of shape to turn it into music it is simply sound. Noise has in itself become an artform as a kind of anti-music but even the act of producing it involves shaping, selecting sources and deliberately creating some kind of texture. But let's not get into that now, eh? Otherwise we'll end up with that tree in the forest argument.
The most famous example of aleatory composition is the musical dice game attributed to Mozart, published a few months after his death in 1791. It was released by his publisher Nikolaus Simrock apparently based on notes left by the composer though nobody knows for sure if it was actually Mozart's work. It doesn't particularly matter whether it really was was just a clever marketing ploy by Simrock: the idea of the musical dice game had been around for many years and Mozart's is by no means the only example. A roll of dice is used to select small sections of written music which can then be patched together to create a single musical piece. Another version also attributed to Mozart uses letters instead of numbers and invites the “player” to produce the composition derived from the spelling of his or her name. It is therefore theoretically possible to translate the script of an edition of “Homes Under the Hammer” into a composition by Mozart.
So "chance music" is not truly random. To be random is surprisingly difficult: even things which may appear completely random can have a traceable influence. If you think it is possible for a computer to generate randomness, bear in mind the fact that a computer can only operate on the basis of its programming – which means therefore that its randomness is explicable. In the same way the advanced science can appear to be magic, most of what appears random can be explained if only you have enough data and the inclination to go into things in mind-numbing detail and abandon all pretence at having any kind of life whatsoever. In order to generate true randomness, computers need to be linked to some external physical source, for example background noise (but not the noise of a machine as it's too regular).
The dice game does not generate an infinite number of potential compositions. It is rather huge but a fixed number (Cataloguers do not credit each of these as individual compositions so if it was an attempt by Mozart to claim to be more prolific than Bach it didn't work). What is important about aleatory music is that it is not open to infinite possibilities. It's not completely random: there is still a person who can be credited as the composer. Where the apparent randomness comes in is in the fact that the composer surrenders some of the control of the piece either to a random factor or to the performers, above and beyond what is usually brought in by interpretation. This is often achieved by the piece being composed in sections and then allowing the random factor to shape the order in which the sections are performed. In this way the music can still incorporate traditional concepts such as melody and harmony while retaining a degree of unpredictability, albeit a finite one.
A more random example is Terry Riley's “In C” in which the performers are given 53 short sequences in the key of C. All of those playing (and there can be as many players as you wish) can interpret and even repeat a sequence as they wish before proceeding to the next. Each performance is therefore unique although it will still be recognisably derived from the same compositional source. In the piece one power Riley has not chosen to give up to the performer is the order in which the sections are played: each must move sequentially from 1 to 53.
A still more open form such as Stockhausen's Klavierstück 11 allows the sections to be played in any order determined by chance or performer choice.
These are still, however, examples of “closed form” because of the amount of music which is predetermined by the composer. An “open form” piece of music in which not even the sections are fully defined relies further on the performer: by leaving it deliberately incomplete. It is not however completely open to chance. The performer is invited to improvise and this means making choices informed by musical traditions, personal taste, interaction with the other performers and whether the audience is looking bored by the thought of yet another tedious guitar solo. These ideas are by no means unique to Western music nor have they simply grown from the intellectual pursuits of composers. They are if anything, more rooted in the origins of music as a communal activity not exclusive to the trained writer or performer.
You could therefore argue that Eric Morecambe's confrontation with Andre Previn is a perfect summary of the tension between traditional rule-bound classical music and modern experimentalism: "I'm playing all the right notes but not necessarily in the right order.”