On Rosh Hashana and Yom Kipur, sometimes the prayer leader needs to cough or swallow or clear his throat or catch his breath or the like. The congregation then sings a little tune, I guess to avoid an awkward silence. In most synagogues I've been in, it goes (if I'm not mistaken) E͜EDEFEDCB͜BCBA͜A (rough representation). Does anyone know the provenance of this tune, or of the use of this tune for this purpose? (I'm not seeking a source for singing (anything) when the leader pauses. I'm wondering about this tune specifically.)
Are you sure about this? It sounds like the (Ashkenazic) tune used between paragraphs in the third blessing, and it's quite common. Even Virtual Cantor does it, and I've heard it in recordings that are much older. That's not a tune for the shatz to catch his or her breath; it's just to break up the reading. It just happens to be a good moment for the shatz to pause. You'll notice that it's roughly in the mode of the nusach -- it's really just part of the nusach. There are other little niggunim used the same way in Avot, Misod, V'hakohanim, and even Geshem. Abraham Baer's Ba-al T'fillah even has a few, and that's from Germany in the 1870's. If you have one handy (there's one here), check out number 1165, the Avot from Rosh Hashanah Musaf, where there are lots of repeated niggunim, and if the shatz can convince the kahal to sing one, he or she can take a break for a few seconds. The niggun you quoted is actually listed in Zamru Lo: The Next Generation, Vol. II, p. 81, as "Traditional" under High Holy Days Musaf Interludes (that one's not available for free online, unfortunately). So it's probably fairly old, maybe of Chasidic or Eastern European origin (since Baer didn't have it), and it just spread throughout Ashkenazic communities. But again, the niggun isn't the "shatz needs to cough" niggun; it's part of the nusach. Your leader probably just had an itchy throat!