Here are a variety of attempts at definitions of music, especially modern and contemporary music...
Gunther Stent thought that his argument for the self-limiting nature of science was supported by looking at what had happened to the creative arts. Like many others of his generation and European cultural background he was mystified by the direction that the creative arts had taken. He noticed that many commentators (even some within the artistic community) thought that art was no longer 'real' art, but merely some form of spilt emotion. Taking a longer look at the situation, Stent tried to interpret the present state of affairs as the end result of an evolutionary process which has steadily relaxed the compositional constraints placed upon the artist. Over the centuries we have seen new materials and media appearing to enlarge now creativity may be expressed. At the same time the traditional restrictions on what may (or may not) be portrayed, and how it may be done, have been steadily eroded. As the restraints imposed by convention, technology or individual preference have been relaxed, so the resulting structure is less formally patterned, closer to the random, and harder to distinguish from the work of others working under a similar freedom from constraint.
One of the characteristic features of appreciated music in all cultures is the way that it combines sequences of sounds to produce an optimum balance of surprise and predictability. Too much surprise and we have unengaging random noise; too much predictability and our minds are soon bored. Somewhere in between lies the happy medium. This intuition can be put on a firmer footing. Some years ago, two physicists at Berkeley, Richard Voss and John Clarke, discovered that human music has a characteristic spectral form. The spectrum of a sequence of sounds is a way of engaging how the sound intensity is distributed over different frequencies. What Voss and Clarke discovered was that all the musical forms they examined had a characteristic spectral form called '1/f noise' (pronounced 'one-over-eff noise') by engineers, which is precisely the optimal balance between unpredictability and predictability: there are correlations over all time intervals in the sound sequence.
We can add something to this characterization of music by applying it to the style of the composition. When a musical composition is in a style that is highly constrained by its rules of composition and performance, it will be far more predictable than if its style is free from constraints. The listener does not receive very much new information, over and above that present to establish the stylistic framework, from listening to the music. Conversely, if the style has too few constraints, the unpredictabilities in the sequences of sounds can be too great. An instant appreciation of the weak probabilistic patterns of sounds will be hard to make, and the result will be perceived as less attractive than the optimal 1/f spectral pattern.
Stent argued that music must evolve in the direction of greater stylistic freedom. Because of the cumulative nature of previously created works in each genre and the growing sophistication of the listeners' appreciation, it is the only place left to go. Starting with the maximal rigidity of rhythmic drumming in ancient times, music has exhausted the scope of each level of constraint for its listeners, before relaxing them and moving down to a new level of freedom of expression. At each stage, from ancient to medieval, renaissance, baroque, romantic, to the atonal and modern periods, evolution has proceeded down a staircase of ever-loosening constraints, the next step down provoked by the exhaustion of the previous level's repertoire of novel patterns.
This evolution is one of increasing sophistication in information-processing with time. The invention of musical notations and new media for recording and replaying music privately greatly accelerated the sophistication process, giving many more ways in which to develop away from the constraints of average tastes. The culmination of this evolutionary process in the 1960s saw composers like John Cage relinquish all constraints, leaving the listeners to create what they would from what they heard: an acoustic version of the Rorschach inkblot test. Instead of communicating satisfying patterns, they sought to evoke trascendental experiences. Their music does not invite interpretation as correlated temporal sequence of sounds; it just is. Distinguishing music from noise depends entirely on context; it is sometimes impossible, and even undesirable.
From John D Barrow, Impossibility, the Limits of Science and the Science of Limits, OUP, 1998
When I feel well and in a good humour, or when I am taking a drive or walking after a good meal, or in the night when I cannot sleep, thoughts crowd into my mind as easily as you could wish. Whence and how do they come? I do not know and I have nothing to do with it. Those which please me I keep in my head and hum them; at least others have told me that I do so. Once I have my theme, another melody comes, linking itself with the first one, in accordance with the needs of the composition as a whole: the counterpoint, the part of each instrument and all the melodic fragments at last produce the complete work. Then my soul is on fire with inspiration. The work grows; I keep expanding it, conceiving it more and more clearly until I have the entire composition finished in my head although it may be long. Then my mind seizes it as a glance of my eye a beautiful picture or a handsome youth. It does not come to me successively, with various parts worked out in detail, as they will later on, but in its entirety that my imagination lets me hear it.
Mozart, as quoted in Hadamard, 1945 (p16) The psychology of invention in the mathematical field, Princeton University Press.
NB According to Dennett, 1995, this story is not correct, although he suggests that this doesn't affect the validity of the idea expressed.
An extreme example is Mozart's ability to 'seize as a glance' an entire composition 'though it may be long' (see above). One must assume, from Mozart's description, that this 'glance' contains the essentials of the entire composition, yet that the actual external time-span, in ordinary physical terms, of this conscious act of perception, could be in no way comparable with the time that the composition would take to perform. One might imagine that Mozart's perception would have taken a different form altogether, perhaps spatially distributed like a visual scene or an entire musical score laid out. But even a musical score would take a considerable time to peruse - and I would very much doubt that Mozart's perception of his compositions could have initially taken this form (or surely he would have said so!). The visual scene would seem closer to his descriptions, but (as with the most common of mathematical imagery that is more familiar to me personally I would greatly doubt that there would be anything like a direct translation of music into visual terms. It seems to me to be much more likely that the best interpretation of Mozart's 'glance' must indeed be taken purely musically, with the distinctly temporal connotations that the hearing (or performing) of a piece of music would have. Music consists of sounds that take a definite time to perform, the time that in Mazart's atual description allows '...that my imagination lets me hear it.'
Listen to the quadruple fugue in the final part of J.S.Bach's Art of Fugue. Noone with a feeling for Bach's music can help being moved as the music stops after ten minutes of performance, just after the third theme enters. The composition as a whole still seems somehow to be 'there', but now it has faded from us in an instant. Bach died before he was able to complete the work, and his musical sore simply stops at that point, with no written indication as to how he intended it to continue. Yet it starts with such an assurance and total mastery that one cannot imagine that it was done this way. Like Mozart, he must somehow have been able to conceive the work in its entirety, with the intricate complication and artistry that fugal writing demands, all conjured up together. Yet, the temporal quality of music is one of its essential ingredients. How is it that music can remin music if it is not being performed in 'real time'?
Penrose, Roger, 1989 (p575) The Emperor's New Mind, Oxford University Press