Modes are scale patterns, but they're patterns associated with particular tonic notes. Consider just the white keys on a piano. If you play any eight contiguous white keys, you'll have played a scale in one of the ecclesiastical modes. (The exception is B-- for some reason there's no mode associated with B.) For example, if you choose C, you'd play C-D-E-F-G-A-B-C. The interval pattern of that series of notes is W-W-H-W-W-W-H. That's the scale of the Ionian mode, and it's familiar to us as a major scale. If you choose A, you'd play A-B-C-D-E-F-G-A. That interval pattern is W-H-W-W-H-W-W. It's the Aeolian mode, and familiar to us as a natural minor scale.
The other modes each have their own interval pattern that you can discover by starting on a different white key as the tonic. Those other modes haven't survived in popular music, but are all over the place in medieval music.
|Aeolian||W H W W H W W||A|
|Ionian||W W H W W W H||C|
|Dorian||W H W W W H W||D|
|Phyrgian||H W W W H W W||E|
|Lydian||W W W H W W H||F|
|Mixolydian||W W H W W H W||G|
The upshot: when Aloysius says to Josephus, "find a counterpoint to the cantus firmus which I am writing down for you in E," he doesn't mean the key of E. Instead, he means the Phrygian mode. In the Phrygian mode, if you choose E as the tonic, you don't need to use any accidentals-- you can confine yourself to the white keys of the piano.