Also known as falso bordone, faux-bourdon is a style of vocal partwriting that was used in liturgical and other religious music in the Renaissance. Instead of providing counterpoints to the cantus, as was usually done in Renaissance vocal music, the other parts of faux-bourdon proceed in step. The upper voice sings the cantus, the middle voice is a fourth below, and the lower voice is a third below the middle voice, or a sixth below the upper voice. The qualities of these intervals may change as the parts move through whatever mode the piece is in. For example, in the dorian mode, if the upper voice is singing a D, the middle voice is a perfect fourth below, and the lower voice a major sixth below. To our modern ears, this would sound like a d minor triad in first inversion, but Renaissance composers thought in terms of intervals, not triads.
Composers didn't always stick to strict faux-bourdon structure throughout a given piece; for one thing, they had to avoid all tritones, either vertically or horizontally, and the style of the period required certain notes to move in certain directions. A good example of faux-bourdon mixed with more polyphonic writing is Miserere Mei, by Gregorio Allegri.