As a composer, this is an issue that has resonated with me since my days as a student. When someone would ask me, “What do you do,” I would invariably reply, “I compose … ‘Contemporary classical music’.” This would feel odd to me, especially since the classical period included the early period of Beethoven, the works of Mozart, Haydn, and many others, none of whom I emulate when I compose.
Most recently, I began using a different term to describe my work, or more precisely, two terms: Post-tonal contemporary music (for those wanting a more technical description … which usually leads to a discussion regarding decophonic modal usage in my work), or a composer of Artmusic (or Artmusik). An advantage of the term “Artmusic” is that there is no doubt as to what it refers to: music which is intended as entertainment, be it chamber, concerto, symphonic, or even operatic (or other choral work). Artmusic is a simple way of saying that this “composed” music is an extension of the literature, a work that has evolved from the history of the tradition from which the composer is a member and, as such, studied their craft in order to become a practitioner of an art form that dates back to the time of the ancients.
It is important to stress the importance, as you did when discussing “Composed music”, of the literature of music, for composing is a language; composers are as literate as any writer of novels or poetic works. We use melodies, notes, and rests as the foundations of our phrases, which then serve as the building blocks of larger structures, but it is all part of a literature, which has a long, rich history. All composers (dare I say, all composers worth their salt) take joy in delving into the literature, especially as students, for it is how we learn our craft. I have never understood how someone can be involved in the “field of music” — earning their living from music, and be willfully ignorant of the language of music. Being incapable of reading the lingua franca — the notes on the page — which anyone can learn with a few short lessons is artistic illiteracy and the definition of ignorance in practice. People refusing to learn, claiming that it “robs them of spontaneity” have no idea what it means to be a composer of Artmusic.
There is a huge difference between “writing a song” with a few chords and composing a piece of music with actual notation, setting the notes on a page, hearing the piece in your head (audiating the piece as it takes shape). That is composing, and that is what separates “Composed” music, or Artmusic from other genres.
Using Artmusic it is unnecessary to differentiate between “Jazz” or other notated genres: they have their own adherents who do not require to defend their works or redefine them or define them from other historical periods. Composers of Artmusic, on the other hand, are in the unique position of composing works that are distinct from other historical periods (the caveat being composers of pastiche works, which would be classified as “Artmusic pastiche” — should the clarification be necessary).