Pronounced as if it is an Italian word, ("kwin-deh-chee-leh"), it refers to 15 degree increments of aspects in an astrological chart between two planets, corollary points, or cusps. This is based around the twenty-fourth harmonic, and 360° of a circle divided by 24 equals 15. Since this has to do with increments, the quindecile expression refers to relationships of 15, 30, 45, 60, 75, 90, 105, 130, 135, 150, 165, and 180 degrees.
According to Noel Tyl, one or more quindecile aspects appear in as many as 85% of horoscopes. They can suggest an important focal point or extreme concern, mania, upheaval, etc.