A syllable-initial consonant is semi-vowelized before a semivowel?

Solution 1:

To answer your question "What's the IPA pronunciation of the word music?"

[mʲuzɪk] is not an accurate transcription in any dialect of English*. I have also used square brackets instead of slashes as palatalization is not contrastive in English, and so brackets are properly used to indicate a level of detail beyond phonemic detail.

The superscript "j" ([◌ʲ]) indicates palatalization (which does occur in English), not "semi-vowelization." From Wikipedia:

Stops are palatalized before the front vowel /i/ and not palatalized in other cases.

However, /m/ is a nasal (sometimes called a nasal stop), but I don't think this claim is meant to include nasal stops, however, I'm not a phonetician. You can hear [mʲ] pronounced aloud here, where you can tell that this isn't a native English* sound.

The Oxford English Dictionary gives the palatal approximant ([j]) as the correct transcription.

Brit. /ˈmjuːzɪk/ U.S. /ˈmjuzɪk/

Tl;dr: /ˈmju(ː)zɪk/ is correct, [mʲ] is not an English sound*.

* - That I am aware of. I would love to hear about one, though.