The church modes weren't characterized by beginning on any note, only by their finalis on that note, and by an ambitus, which, however, could be exceeded slightly. Melodies of very narrow or broad range, like those with gapped scales, could be modally ambiguous. Since these were formal classifications, not every melody fit precisely. But the classification of hexatonic folk-melodies according as dorian, phrygian, and so forth, strikes me as perfectly appropriate. If Jesu Nostra Redemptio belongs to mode 8 (sometimes called hypomixolydian) on G, "The New Irish Girl" is just as reasonably assigned to mode 7 (mixolydian) on D (a transposed mode). Nor is the validity of this assignment of the melody necessarily destroyed by adding harmony or polyphony.