I think I probably learn something from every crossword puzzle I do. One thing I learned from this one is that if the clue is "Prefix meaning half" and you've _EMI, you've still got a long way to go.
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Middle 50% is by Pete Muller. He tells you what he's cooked up in 56A: Partially obscured (and a hint to this puzzle's theme) (HALF HIDDEN). I'm about to show my ignorance yet again, but I was done with the puzzle before I went back and figured out what the theme was all about. And then I got SEMI right away in the middle of CRUISEMISSILE (51: Modern war weapon), but hemi, from 23A: 1982 Michael Palin movie (THE MISSIONARY), and demi, from 18A: Car part with a warning (SIDEMIRROR) left me scratching my head for a minute. I thought hemi was whatever the hell it is Chrysler is bragging about in those commercials (I know as much about cars as I do prefixes) and that Demi was Mrs. Ashton Kucher. (I did like the 18A clue a lot, you'd think a car part with a warning would be something more dangerous than a mirror, but "objects are closer than they appear" is definitely a warning.)
So, it turns out that "semi" is the most common prefix for half, as in semicircle, semifinal, semiconscious, et cetera. But "demi" will work too, as in demigod, the offspring of a deity and a mortal so he's a half-god, or demitasse, which literally means half a cup. Most of the other demi- words are archaic or obscure, including demicircle, which means semicircle. There are a bunch of hemi-words, but most of them I've never heard of but might be useful, like hemicrania, a pain in one side of the head. And of course, hemisphere. There's also hemidemisemiquaver, which means a 64th note. (I've lost track here, is that a half of a half of a half note?) It's a great word, not much chance I'll be able to use it in conversation.
Hey, did you know that the logo of the St. Louis Blues hockey team is a hemidemisemiquaver? (Maybe it won't be as hard as I thought.)
Wait a minute, I just noticed that SPHERE (14D: Globe) actually intersects with the HEMI portion of 23A, so that must mean -- Yep, there is GOD (6D: Mars, e.g.) crossing DEMI. And, let's see, okay it must be TONE (37D Pitch) to make "semitone." But again since semi is the more common prefix, semisecession, semiripens and semigem at least look like they might mean something.
Other entries of note:
15A: BP gas brand (AMOCO) Hard to believe that British Petroleum owns American Oil Company.
55A: Pope of the mid 100s (PIUSI) Nobody likes these obscure pope clues, but with this one, since he's from early on in the Christian church, you know there won't eb any V's or X's.
67A: Part of POSSLQ (SEX) POSSLQ was a term coined by the US Census Bureau for Person of Opposite Sex Sharing Living Quarters. It was briefly in vogue during the late 70's, but you don't see it much anymore.
1D: Get a wife? (WED) Not really sure why this clue has a question mark. That's literally how you get a wife. Maybe because it sounds semi-similar to "Get a life."
13A: Progeny of Aphrodite (EROS) Back in those days humans and gods had of interaction. Aphrodite was actually jealous of a mortal woman named Psyche and she sent her son Eros to make Psyche fall in love with the ugliest man on earth, but Eros either disobeyed or pricked himself with one of his own arrows. At any rate he fell in love with Psyche, who got his buddy Zephyrus (the West Wind) to blow her to this cave where he used to go and make love with her all night. He would not allow her to turn on the lights because he didn't want her to know who he was, and the wings were a dead give away. But her sisters convinced her to take a peek and Eros flew off. Psyche went to Aphrodite to find Eros, and Mom gave her a series of impossible tasks to perform culminating with a trip to Hades. But everywhere she went, gods were willing to help her out, even guys like Charon, and she eventually married Eros and became immortal. Even more unlikely she became friends with Aphrodite.
62A: River through western Kazaakhstan (URAL) No, I haven't researched the difference between Ural and Aral yet, but it's on my (extensive) list of things to do.
Speaking of SECESSION (29D: Breakaway) Everybody knows that people in favor of this are called seceshes, and during the Civil War many were called Johnny Secesh. (A little joke for everybody who was at Stamford this year.)
46D: Dr. Seuss's real last name (GEISEL) He actually wrote several books (including under the name Theo LeSieg (Geisel spelled backwards) before he became famous as Dr. Seuss.
8D: They're untitled (COMMONERS) Not the same thing as being nameless.
45A: Use a blinker (SIGNAL) This could also have been clued as "something people driving in front of me seem to have forgotten how to do nowadays."
Have a semiprecious Tuesday.