I think you're sticking to rather old fashioned books if you think you need 1st 3rd and 5th to make a chord.Makes a triad all right, but that's only one kind of chord.Sing a C E and G together. Drop the top two notes by a step. You've got C D and F. Nobody's going to a shout "Hey they were singing a chord a minute ago and now they are'nt"