Welcome to the world of Barbershop music (one of my primary hobbies) ;) Unlike a piano, which must be tuned and which has a temperament that is fixed from chord to chord in a piece, the human voice can make minute adjustments to come as close as possible to those nice integer ratios (both in the fundamental and in the upper partials produced by their voices). When we do, we are rewarded with (sometimes screaming loud) overtones caused by the constructive interference between the sounds being produced by each of the four parts.
As this article points out, it's mathematically impossible to perfectly tune some of these intervals, but depending on the relationships between the notes being sung, you can tune to one singer or the other. It takes a lot of practice and a good ear, but the resulting effect is pretty darned cool.
I went looking for examples of what you were talking about and discovered that, amazingly enough, a single human voice can even reliably produce overtones! I'm guessing these are very different, though, than what you're talking about https://www.youtube.com/watch?v=2i61_JNc_Nc&t=1m25s
I'd love to hear some examples of a barbershop n-tet doing it too, I'm sure it's even better. But when I search for 'barbershop overtones' it's all music by groups called The Overtones :)
The principle is actually the same -- she is modifying her singing apparatus to emphasize different upper partials that are already in her voice. Barbershoppers do this as well, though less explicitly (we do vowel matching, which helps emphasize upper partials to produce greater ring).
Here's a video from a perennial favorite quartet (the Gas House Gang): https://www.youtube.com/watch?v=pvYT_yWiLqU The top/tenor note is often the same note as the primary overtone produced by the other 3 parts (adding further emphasis to the overtone), but this effect is what gives barbershop the quality of sounding like more than 4 voices, and produces the "ring" in the sound.
Just had to add (since I'm a giant Barbershop Geek in addition to all of the other ways I'm a geek ;))...
Lest anyone think that barbershop is just for old folks:
https://www.youtube.com/watch?v=XWmFfFx24zs (Vocal Spectrum, 2004 international collegiate quartet champions and 2006 international quartet champions)
I can do it pretty well -- the higher one, not the deep version (I can approximate the growly version, but it'll have me coughing after about 10 seconds, so I'm clearly doing something wrong!) -- after seeing Huun Huur Tu in concert back when I was in school I just tinkered until I could get the overtones reliably.
I also did a bit of barbershop (we called ourselves The Overhead Projectors...) and playing with throat-singing was actually quite useful in that context. I'm not sure how much it affect my actual voice control (probably some), but mostly I just became keenly aware of how much tone could vary just based on little changes in positioning of mouth, throat, tongue, palate, etc..
Overtone singing itself isn't very common in barbershop music, but their harmonies certainly make use of overtones. Here's a performance of a song called Play That Barbershop Chord (a stable in the genre). It's not instructive, but it's got some great harmonies: https://www.youtube.com/watch?v=F5bYnRkwpco
Here's a fairly technical article about three different seventh chords, one (the harmonic seventh) being the so-called barbershop seventh chord. There are embedded YouTube videos with computer-generated audio samples. http://www.garygarrett.me/?p=1575
There are some multi-track barbershop videos on Youtube which are good examples of over-the-top overtones. The fact that it's the same person (with the same timbre, "overtone profile") doing all the voices seems to amplify the effect. Here's one: http://www.youtube.com/watch?v=QpU_X6VKqvA
Here's a great video of the Ambassadors of Harmony (an incredibly good barbershop chorus from St. Louis, MO) doing some exercises to tune overtones, and they're really loud: https://www.youtube.com/watch?v=sCdQVqQXkzc
Listen for when the director sings the overtone note alone, then listen to the chorus as they produce it.
Incidentally, cheesy musical puns are the norm for barbershop and other a capella groups. Aural Fixation, Decibelles, Chiefs of Staff, Vocal Majority, Clef Hangers, and so on.
Is there a solid example of loud overtones being generated by the "barbershop seventh" that you can point to, such as in a recording?
I don't mean to be skeptical in a negative way. It would be great if music has a built-in payoff for singing with just intonation.
I've recently heard some doubt, however, about whether this overtone effect is actually perceptible to listeners, in a way that would pass double-blind tests and stuff. It's not that the alternative is that barbershop singers are delusional or something: the alternative is that barbershop singers -- and some listeners -- have well-trained ears for just intonation, and the perception of the chord they're singing changes inside their brain when the intervals line up.
I don't doubt the power of overtones lining up. This effect is where the entire musical scale comes from, after all. But does the barbershop effect actually physically come from constructive interference (this would limit it to a small number of chords made of undertones, incidentally), or is it just a noticeable auditory illusion that occurs in trained ears?
It doesn't strictly come only from 7ths. Here you can see a chorus practicing producing these overtones, and it's both highly audible and demonstrably there: https://www.youtube.com/watch?v=sCdQVqQXkzc
What should I be listening for in the video? There's the moment at 0:45 when everyone stops and says "wow", but that could come from either explanation.
Should I be expecting to hear a pitch that's higher than all the notes in the chord, lower than all the notes in the chord, or a different phenomenon that's not a distinct pitch?
This effect is sometimes claimed to happen with a 4:5:6:7 dominant 7th chord. In that case, would I hear it at the 28th harmonic? The 30th? The 420th (which is the lowest overtone that all the notes have in common)? The 1st (the undertone they have in common)? Or is 4:5:6:7 actually a bad example?
(I really doubt it's the 420th, of course. That would be a very high pitch around 17 KHz, which would be difficult to hear, and wouldn't be preserved at YouTube quality anyway.)
The pitch is higher than the notes in the chord. At 0:32 I believe the instructor sings the overtone pitch in isolation and says "you should hear that over it," then they sing the chord again and you can hear that same overtone pitch fairly clearly.
As this article points out, it's mathematically impossible to perfectly tune some of these intervals, but depending on the relationships between the notes being sung, you can tune to one singer or the other. It takes a lot of practice and a good ear, but the resulting effect is pretty darned cool.