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Bach: INTP 9w1 I don't know much about his life, but his mastery of counterpoint and craftsmanship suggest t-dom. Counterpoint requires entertaining multiple melodies at once (ne works), and he was a traditionalist (si). He had a big family, which suggests fe.
Mozart: ESTP 3w4 I see se in his hedonism, aestheticism, and focus on the present. He was ti-fe; he produced all his music from his head fully formed, never amending or correcting (ti), and he was social and emotionally open (fe).
Beethoven: INTJ 4w5 A really obvious fi-te; he had no regard for politeness, social standards, or the feelings of others, but he had a very rich inner world of emotion (fi). Unlike Mozart, he would constantly rework his scores until he had them just right (te, implied xxxj). Like House, his inferior se can make him look like a perceiver; he was definitely a slob. But he was also directive, driven, arrogant, single-minded, selfish, uncompromising and controlling. He was an introvert who preferred to focus on his private work rather than form social connections. And he was a Ni user with a strong appreciation for metaphor and archetype (fate knocking at the door...). Of the three big composers, he was the only major innovator (ushering in the romantic movement), and he was certainly ahead of his time. It makes strong intuitive sense for Bach to be a t-dom, Mozart to be an s-dom, and Beethoven to be an n-dom. And I see little ne, he was single-minded and obsessive. INTJ 4s are rare, which is why he's often typed as a fi-dom. Not every tortured artist is fp.
I agree with Beethoven, INTJ 4w5 fits well for him. INTP sounds good for Bach, but I can't say about his Enneagram type. Mozart was definitely an ESxP (contrary to my OP). His life and records of personality show Type 7, most likely 7w6.
I think the question of whether Bach (the best music composer ever and my adopted grandfather) was more of a Te type or a Ti type is highly debatable. His music is procedural and rule-based, and much of it is structured using external templates and principles. On the other hand, he had an exceptional talent for drawing out the contrapuntal potential in a melodic passage, and seems to have done so partly as a hobby, judging by the fact that some of his most exemplary work in that regard had little financial potential.
We do have a little bit of information about Bach outside of his music. Going by what I've read about his personality, he seems to have been much more comfortable with controversy than any INTP I've ever encountered. I would say that he was probably a T, though, since he seems to have been largely devoid of interpersonal finesse (people will point to his technical excellence as a sign of him being a T, but that's nothing to go by. There are plenty of Feeler artists who have a strong ability and desire to make their works technically rich, and in Bach's time, technical skill in a composer was much more highly valued than it is today).
Hmm, it's a difficult one, but more likely Ti, because of how rigorously logical his music is. It doesn't say much, but I think if he had been a Fi user, his music might have been a little less "cold" and technical, and more emotionally passionate. Of course, that might just be from the time he was in.