he got fairly close, sometimes.
zappa was famously of sicilian background, and himself often mistaken for arab. and, my father would have identified as italian before he identified as phoenician - probably as much due to the identification with frank as with identification with his mother's side.
"how do you identify, ethnically?"
"i'm going to have to go with frank zappa."
his mother, my grandmother, is actually phenotypically quite white; she comes off more as a white jew than an italian woman, and i do believe probably is, although the adoption makes things hard to guess at. my father's darker skin comes from his father, not his italian mother.
i'm a jazz fan myself, as well, as you should know. i've gone through phases where i've picked up old classic jazz records second hand. all kinds of stuff. as i was listening to some davis record or something, my father would sometimes come in and point out that "holy shit, i haven't heard that in years. your grandfather liked that record". that may seem like a weak ethnic marker; certainly, you don't have to be black to like jazz, or create it. but, his formative years were before the 60s. it's suggestive, anyways.
it's certainly possible - and i've pointed this out before - that the missing great-grandfather may have been mixed, himself. in addition to what looks like a jewish or mixed african male, there are rumours of native ancestry on that side, as well.
i don't want to pay for a dna test, but it's probably the only way to figure it out.