I haven't yet read William McNeill's book Keeping Together in Time, but the basic idea is that shared rhythmic motion is a powerful way to bond individuals within a social group, be it a church, military unit, or sports team. Just think of something so simple and effective as, doon doon CHA, doon doon CHA, We Will, We Will, Rock You!
As modern populations become more cerebral, and less corporeal, books like this are sorely needed to remind ourselves of the power of ritual, and the relative weakness of beliefs, in holding a group together.
I should probably wait until reading the book before saying much more on that topic, but there seems to be another, less important application of the idea, namely to human-animal bonding. My impression is that the closest bonds are between horse-riders and their horses, then cat-owners and their little purr-balls. Riding a horse is the highest degree of shared rhythmic motion that a person and an animal can reach. Petting a purring cat is not so striking of an example, but still above all other cases of normal human-animal interaction. You generally don't see dog-owners stroking their dog's fur rhythmically for a decent stretch of time, and the dog's happy panting isn't as resonatingly rhythmic as a cat's purring. And dog-owners maintain more emotional distance from their dogs, regarding the cat-owners as too closely connected for the good of their own independence.
Depending on how they're being used, cattle, sheep, and goats may also elicit strong feelings of attachment from their owners. Generally, if they're being driven by pastoralists, an intense bond forms -- this is kind of a looser form of riding the animal from point A to point B, with the owner walking alongside the herd, their trotters all striking the ground with more or less the same rhythm. If they're kept as more sedentary barnyard animals, though, with little or no shared rhythmic motion, you don't find their owners composing songs to them, using their eyes as the ideal to compare the beauty of human eyes to, and so on.
The same seems to be true with pigs: if they're being driven hither and thither by their young male owners, they get real attached to the little snub-noses. Being kept as barnyard animals doesn't lower the affection as much as it does for cattle or sheep, perhaps because the driving-around case is not as intense as it is for herd animals. My mom grew up on a "farm" -- there was little agriculture up there in the Appalachian hills, but they did keep lots of domesticated animals and relied on them somewhat for subsistence. At least in her case, she said that slaughtering the pigs always felt like the greatest loss of a social bond.
Shallow bonds are the most you can hope for between people and birds or fish, and sure enough they don't share any rhythmic motion, or even get in synch with little or no motion, as when cats purr. You can bet, though, that if we ever domesticate a large bird of flight, and feel the repeated flapping of its wings as we rode it for transportation, we'd develop a real soft spot for them.
At the very bottom are the creepy-crawlies -- insects, arachnids, snakes, and so on -- that we not only do not share any rhythmic motion with, but have typically been a threat to our health.
I'm no expert on human-animal relationships, but the basic idea seems to be borne out, not just across species but even within species, like what emotions cattle stir up in pastoralist vs. sedentary societies, or how connected people feel to dogs that just loaf around the house vs. go out jogging with their owners.