If the Muses represented multiple disciplines of the arts, how and why did the word music come to represent only, well, music. Why aren’t poetry, dance or comedy considered music. Would Calliope and Clio be pissed at this turn of events?
From a brief look at dictionaries: The ancient greeks did use it for anything inspired by the muses, but especially music. The Romans then borrowed the word for that specific meaning, and the rest is history.
Details left to any actual scholars.
Yep, I found the basics. It’s the details I’m after, if they’re known.
OED may be of help.
A museum doesn’t necessarily have music, and it might not even have art.
The muses did not represent art. They represented inspiration, of both art and science. For example, Clio was the Muse of History, and Urania the Muse of Astronomy. The term “museum” was initially used for a place to worship the muses.
The Romans narrowed the muses down to three, one for the motion of water, one for the sound made by striking the air, and the third for the human voice. This shows how the Romans focused specifically on music over the other kinds of inspiration represented by the muses. I’m not sure what led the Romans to do that except to say that they copied and adapted many concepts from Greek mythology and culture and altered them slightly (see how many Greek gods changed their names when they became Roman gods).
Unfortunately the muses did not have antlers or hang out with flying squirrels.
Dance certainly was considered at least allied; Terpsichorewas the Greek muse of both dance and chorus.
(She’s also the muse which was portrayed by Olivia Newton-John in “Xanadu”. 
)
And people wonder why they’ve fallen into obscurity.
Have roller skates been around since ancient Greece?
Hermes was close, but in his case they were little wings not wheels.
OP question was OP for millennia.
Professional etymologists eventually derived from water, or the nymphs thereby, although the word root in Greek, and later Latin, is obscure. I grabbed this article, but undoubtedly later work elaborates— “Musica Dicitur A Moys, Quod Est Aqua,” Noel Swerdlow, Journal of the American Musicological Society, XX:1, (Spring, 1967):
From Isidore of Seville’s ca. 600 Etymologies (which was Europe’s Wiki for about a thousand years):
Et dicta Musica per derivationem a Musis. Musae autem appellate [greek not iPad-ed], id est a quaerendo, quod per eas, sicut antiqui voluerunt, vis et vocis modulate quaereretur. (And Music is so called by deriva Muses. The Muses are named [ditto], that is, “from seeking,” the ancients said, the essence of songs [or poems] and the melody of is sought through them.)
…
*Ipsas autem dicunt et Musas quas et nymphas, nec inmerito. Nam aqu musician efficit. *(They say the Muses themselves are also nymphs, not without cause. For music is made by the movement of water).
Cites 4th-century Servius commentary of Virgil; not unrelated (not in Servius or Isidore, however) Egyptian “Moses” cognate.
Correct, and this is the precursor of the above. I do not know why “water” won the competition–I’ll check Aristoxenus to see what kind of pride of place the killer Greek water organ the hydraulis gets; the name itself because it is both water (driver) and air. Too bad it didn’t sing. And, to screw things around with a muse or two, Calliope got dragged in as the name of its rinky-dink successor.
And the Aristoxenus tradition is the one that OP is probably familiar with: the stuff we hear. Remember, “music” swings both ways.
I feel honor bound to note, since classical music theory is afoot, and its revival and elaboration was a hallmark of humanism, that the word used by Atamasama and so readily understood by us, has strong roots in 2nd century Galenic physiology–which ruled the roost in the West until the Renaissance-- of “pneuma”–breath/inspiration, which was integrated into a rich philosophical and physiological model for hearing (including music and its effects ) by Ficino and the neo-Platonism of the 15th century. (Marsilio Ficino (Stanford Encyclopedia of Philosophy))
FWIW, in at least one regulation book in an early medieval scriptorium, a fart was not to interrupt work or be adversely commented on, as it is a sign that the divine inflatus was literally “inspiring” the servant of God.
FWIW #2, perhaps I should say this post is “self-honor bound,” because I nearly died writing a graduate seminar paper showing the progression of the idea and its elaboration in Renaissance thought.
You suffered for your wisdom; now it’s our turn?
Kidding, kidding. Thanks for the post - interesting.