I wrote a lot of four-part harmony exercises in freshman music theory classes at the U. of W. School of Music. If I got myself into a voice-leading bind and tried to get out of it by cobbling together a chord with the 3rd missing, the paper invariably came back with a big red circle around the chord and a note, also in red, saying "Missing 3rd!" or "Incomplete chord!" or some other indication that I wasn't going to get an "A" on that paper. Double the root, leave out the 5th if it's absolutely necessary, never double the 3rd, but never leave the 3rd out. It was drilled into my head that a chord requires a root, a 3rd, and a 5th. If the 5th was missing but the tripled root and 3rd sufficiently maintain the identity of the implied chord within the context of what went before and came after, that -- on occasion -- was permissible.
This was freshman theory, and as Prof. John Verrall said, "Here, we are learning the strict rules of harmony and voice leading. Later, when you know the rules thoroughly, we'll learn some ways of breaking them to good effect."
Hand goes up in the back row: "If these rules can be broken, what's the point of spending all this time learning them in the first place?"
Prof. Verrall responds: "If you start breaking the rules now, without knowing them thoroughly, the results will be haphazard and unpredictable. Following the rules helps guarantee that what you write is correct and will sound harmonious. But -- if you always follow the rules strictly, what you write may not be very interesting and certainly not innovative. Once you know the rules, when you break them, you will know exactly what you are doing and why you are doing it!"
There are three songs I sing, Bonnie Dundee, The Earl of Moray, and MacPherson's Lament, in which the accompaniments I play "break the rules" in exactly the manner that started this discussion. To try to evoke the drone sound of a bagpipe, I play open 5ths in the bass -- with no 3rd in sight (but I don't call them "chords," I call them "open 5ths"). I don't do it all the way through; I move into standard chords in the verses and I play a chunk of the melody as a intro and between verses, but the "vamp 'til ready" is open 5ths, to keep the effect going. I'm a bass, not a tenor, and I can't sing the songs in the same keys that he does, so the arrangements are my own, but I snitched the idea from Richard Dyer-Bennet.
It may sound kind of hokey the way I described it here, but it creates the effect I am after and I works for me. I met an old Scotsman once who hadn't been to the old country in years. He liked the Pacific Northwest, but he was, as he told me, "homesick for the heather." He asked me if I knew any Scottish songs, so I played and sang The Earl of Moray for him. At the end of the song, he brushed away a tear and asked me to sing it again.
As I say, it works for me.
P. S. Try going to google.com and type in chord. It's cuh-RAZY out there! Here are a couple of good ones:
here (you can back-track on this into a wealth of information that I sure a lot of Mudcatters have already found. I've spent a lot of time milling around this website.) and here for a lifetime supply of chords.