Alex Ross has a very good article on traditions of improvisation (ornamentation and cadenzas especially) in classical music. The article is subtitled "reviving the art of classical improvision" and Ross means it literally, as in bringing back the dead. And there really is a sense that the improvisatory tradition is not only moribund, but was murdered: Ross quotes conductor Will Crutchfield's characterization of a Caruso cadenza so widely duplicated as to have become the canonical cadenza for the aria into which it is inserted as the “death-of-tradition” and Ross himself describes Beethoven's written-out cadenza for the Mozart d minor Concerto as helping to "kill" it.
I'm of two minds about improvisatory elements in music. I agree that they can make a performance more fresh, more lively and, in effect, open up the musical text, but that doesn't remove the composer's responsibility to compose a score that is, on its own terms, fresh, lively, and rewarding of repeated play and listening. Also, the simple inclusion of improvisatory elements does not automatically make the performer an interesting or musically convincing improvisor. Further, it is one thing to consider improvisatory practices which are part and parcel of a musical style, in which the particular turns and figures chosen will be understood rhetorically in terms of that style, and it is quite another to consider improvisatory elements in the context of new music, in which the stylistic background radiation is highly diffused.
Nevertheless, the project of re-opening the musical work to the extemporaneous has been an important part of the radical music. The examples of music which invite or require improvisatory elements — Christian Wolff's cuing pieces, the variable forms introduced in Feldman's Intermission 6 and widely expanded upon, particularly in the European avant-garde, or the animation of small cells of music common to many pieces in the West Coast experimental tradition, or Richard Maxfield's concert works using soloists improvising against tape works based on their own recorded improvisation, for example — continue to be rich in potential for new music. There is nothing (yet) like the thick tradition of French baroque agréments, ornaments for which a composer can appeal to a body of figures and their shorthand notation as well as a tradition for their appropriate placement within a piece of music which will be understood by a broad community of musicians as the point of departure for improvisation, but there are still recognizeable elements of a tradition in the works in which, for example, the cues of Wolff scores from the 1950's are echoed in the game-structure works of John Zorn or in the networked improvisations of small computer-based ensembles.
The project of recovering historical examples of improvisation is musicologically interesting and musically useful if, at the very least, it brings alternative cadenzas and ornamentations into the concert hall. But performances of these revived examples are still not a restoration of improvisation to classical music, and the repetition, from a notated transcription of a historical example of improvisation is definitely not improvisation either. Early music performers are, in general, further along this route than mainstream classical players. The best recorder and gamba soloists today are gifted, inventive improvisers as well and when they play a set of divisions their fidelity to style is so high that it is often very difficult to know where composition ends and improvisation begins. One is clearly hearing "the piece", but "the piece" has also been made anew through the extemporaneous elements.
A parallel project, of recovering, through transcription, landmarks of more recent improvised music, raises lots of questions. Again, this is musicologically interesting and a player can learn a lot from it, but as successful as a particular improvisation may have been, the composer/improviser is fallable, and more than likely to harbor some doubts about some or all of it. But more critically, isn't simply reproducing the transcription out of the spirit of the initial enterprise? It would be entirely possible, for example, to play a transcription of a single performance of La Monte Young's The Well-Tuned Piano, but a performance of the transcription would not be responding to the particular set and setting in which the original performance unfolded and would not be open to the possibilities for alteration that the composer always allows himself. The better way, it seems to me, is to learn the piece as the composer prescribes, rehearsing with him directly until such a point that one has the confidence (one's own as well as that of the composer) to make one's own realization. Even more so with works of music in which the composer's own open notation is available: while it would be possible to learn to play a Christian Wolff piano piece by transcription of a David Tudor recording, the composer's notation was specifically designed to create an indefinite number of realizations, so freezing the piece around an old Tudor recording is introducing an unwarranted restriction on the work itself, the avant-garde version of the "death of tradition." The notational tools for a very precise, closed musical text are readily available to composers and when a composer makes a deliberate decision for a score in which elements are not all precisely or decisively described or are to be defined in real time by the performer, then it is a plain misreading of the score's notation not to reserve these elements for the improvisational domain.