The bird in particular was in a brushy area near the Rio Grande in Albuquerque, cleverly concealed in a small aspen. Its call sounded like “Tea party, tea party, tea party, tea” (though I doubt the bird was announcing its political affiliation.) Sometimes it trilled a bit at the end of the song. I’ve checked the calls of many of the birds found in the area plus a few that aren’t but haven’t come up with anything.
It’s probably one of the towhees. Here in Charleston, SC, we have only the eastern towhee, but in Alberquerque you have the spotted towhee and the canyon towhee. The canyon does not have a trill at the end, according to Sibley, but the spotted does and its song varies geographically: zero to eight identical introductory notes followed by a harsh or buzzy, rapid trill. The towhee was so named because the eastern variety’s song sounds like “Drink your tea,” but usually the introductory “drink your” is not vocalized and all you hear is the drawn-out teeee.
Try this Carolina wren, if it sounds close, there are several more wrens in your locale. Birds have different “dialects” in different regions, so there can be a lot of variation. Wrens in my area sound like “teakettle, teakettle, teakettle” or “jeremy,jeremy,jeremy” I’ll try to find an example of a wren trill. Found it, here’s the trill(complaint)
The Macaulay Library at Cornell is an excellent resource for audio and video of birds, insects, and other animals are being added.
It did have a rhythym similar to the Carolina wren, but somewhat slower. I’ve been wondering if it was a Common Yellowthroat Warbler, which sings in that pattern and is skilled at concealing itself (it’s also found near water.)
The common yellowthroat is usually described as singing “witchitee, witchitee.” My first thought was the Carolina wren, but that’s not found where you live. However, the house wren is, and it’s song is similar to the Carolina wren, only feebler. Not quite as musical.