It think deep is when the composer is getting beyond music itself, when the ideas are breaking new ground or pointing toward the future of music history. In Beethoven's case, I think it's where the piano and the music are just a path toward something else, like a religious experience. Deep can also be when a piece is complex and you can find more and more logical and interesting patterns each time you study it.
i think it is the chord progressions, harder to find melodies, and forms that start being expanded as if he has all the time in the world to play the piece.
he says "Beethoven appears to be intent on creating an area in which maximum differences in texture, melody, and rhythm are juxtaposed closely while still giving an appearance of flow from one to another." This is spoken about the Eroica symphony, but it is a style he kept on teasing until the 9th symphony and his late piano sonatas. In the Eroica you can readily split it up into it's 'components' as with earlier sonatas, in the ninth symphony and the later sonatas (the last three) you have less detection of where the line stops and starts.
He makes groups so large and different in components, and lacking in repetition that "he cannot allow it to take part in his new dramatic development - he can only use it to fulfill the recapitulatory principle. i suppose what applies to the eroica can also start being applied the farther you go in the sonatas. certain material is not developed and expanded at the expense of other material. There are "relationships, alliances, perils, and threats - that are not immediately or remotely suspected until they turn up."
leaping bass - a rosseau-esque figure - helps communicate a purpose.
"Beethoven's new way integrates the two essential elements of the temporal (dramatic, discursive,narriative) and the spatial (formal, recapitulatory) sometimes the bass line falls further in the recap than the exposition (lacking will?) and yet 'heroism lies in the rational adjustment of that which is out of phase, and the result is security."
Beethoven was something of a contradiction - he takes us along for a discourse as he 'overcomes himself.'
maybe fear of death is replaced with a sort of understanding about the world with music deeper than philosophy.
no one has ever told me in a class to favor any composer (or one over another). even my piano teacher , who probably has some personal favorites, is very open to all composers. despite my saying that i like mozart and beethoven and play them more than some other composers, i have never told anyone that their music is deeper or more profound than say Chopin, Scriabin, Rachmaninov or Shostakovich. Many of the russian composers lived very difficult lives, too. I think from difficulty one learns to cope in whatever ways are most helpful. For Beethoven, this was moving into composition seriously (as he probably wished to perform well his Emperor Concerto - but it was not well received at the time - if i remember correctly).Deepness and profoundness to me is when you see a composer progress from the first compositions he writes to a certain fluency of his or her style that opens up the door of curiosity for those who study their works. you can see patterns develop. you can hear things that you didn't hear before - and with analysis you start to hear different layers in music. the top layer is what your impression is without studying details. the next might be going and getting the score, reading about the composers intentions, and listening while looking at the score - making note of the form and allthe third layer is probably a better understanding of why the composer did certain things. why did he choose to do this over that?in haydn, mozart, and beethoven's later works, they are 'profound' in that they almost throw out form and allow emotion to enter more fully. with haydn's creation oratorio you hear the profoundness of the first section Chaos represented musically (following the text and not form as much). in fact, non of the pieces sung in the creation have an exact form, only a similarity to the forms. In Beethoven, also, you have a desire to express certain ideas (whether spiritual or philosophical) and he almost becomes like Einstein in his last sonatas. if you hear a younger student play them - you say - wow that was impressive. but if you hear an older person (say in their 60's or older) play them, you say 'that was actually moving - you can almost hear his thoughts progress from chaos to sublime, from temptation to submission of will to 'fate' or 'providence' and good overcoming evil. Beethoven claimed at one point that he overcame himself. he did not want to stay static and unimaginative. he wanted to be a sort of guide for people who start out with humble beginnings and seriously study to the point that when they get to the end of their lives, they can trace the journey and be satisfied with the results of all their work. to me this is deep.