This word has a little bit of irony in it. The Y chromosome is the male chromosome, and polygyny has three Y’s, but polygyny means having multiple women – specifically multiple wives. Perhaps it’s because a guy who marries multiple women is sure to find himself saying “Why, why, why?” (It may be because he’s been jailed for it, or it may just be that they all want to go on vacation at the same time to completely different places.) You could see the y‘s as funnels, and the guy’s life is going down the drain – hell hath not too many furies like a woman two-timed, but one of the furies greater might be that of a woman three-timed, four-timed, or what have you.
It may be more appropriate that this word has a bit of an echo of religion, since a man’s practice of polygyny is usually dictated by what religion he adheres to. It also has a bit of a sound of appalling, but no more than any other poly word has. It also rhymes with aborigine, which works OK in that Australian Aborigines have a cultural history (if not present) of polygyny, though of course they’re far from the only ones. It also sounds like pigeon, which again is ironic, as most pigeons bond in pairs for life.
The more common term used for multiple marriage is polygamy, but that actually refers to any multiple marriage – a woman could have multiple husbands and be a polygamist; where the law allows people of the same sex to marry, any combination of sexes would still be polygamy as long as a given person is married to more than one other person. But usually the terms polygamy and polygyny are used in reference to societies where the practice is accepted; in cultures such as ours where it’s illegal, you’re more likely to hear of bigamy, in part because it’s exponentially more difficult to hide each additional marriage, and in part just because of the flavour common usage has given the words.
There is also a word for having multiple husbands, by the way: polyandry. It is less often seen, perhaps because a woman only has so much time and a lot of it is eaten up by cleaning up after even one man, let alone several, but probably more likely for various other biological and cultural factors that deserve more space than I can reasonably give here. William James is credited with one speculative proposal: “Higgamus, hoggamus, woman’s monogamous – hoggamus, higgamus, man is polygamous.”
Polyandry differs from polygyny and polygamy in an important way: while the latter two are secundus paeons, the former is a ditrochee. By which I mean that the latter two have one stress, on the second syllable (like impossible and adorable and impeccable and a Cadillac and so on), whereas polyandry has two, a simple two-beat rhythm, as though it were a woman’s name: “This is Polly Andry, and these are her husbands!”
The g in polygyny, by the way, has the same sound as the g‘s in frigid and rigid and syzygy. That’s a deviation from the source, of course: in Classical Greek, it was always the sound you hear in polygamy. I find the alveopalatal affricate (as in polygyny) has a more delicate sound than the velar stop (as in polygamy), and it does give the word an additional echo of Ginny, which might be the name of the second wife (the first being Polly, of course).
You probably know well enough that the poly in these words means “many”. The gyn, for its part, may look very familiar by itself as the beginning of gynecologist (which has the velar stop [g], just to keep you dizzy). The andr shows up in various references to men, as for instance on a sign you might see in a hospital near the gynecology sign: andrology. (It also appears in android.) The gam just means you’re game. No it doesn’t! Well, it does, but that’s not the origin; it comes from a Greek word, gamos, meaning “marriage”.
As for the newer term polyamory, by the way (which really does sound like a person’s name, and may well be, Amory being a real surname), it’s macaronic: it mixes Greek and Latin, amory coming from the Latin for “love”. The term is generally used by people who want to have multiple boyfriends or girlfriends; it’s not restricted to spouses. The term was first documented in the early 1990s. Polygyny, on the other hand, dates from the 18th century, polyandry from the 16th, and polygamy also from the 16th century. The words, I mean; the practices have been well established in some cultures for rather longer, and only in some cases are fading.