Explain that in the Middle Ages there were no key centers; there were modes. Each mode began on a different pitch. The pitch on which each mode began was known as its final, because any tune in this mode would end on this pitch. The mode with the final of E was known as Phrygian.
Play an all white key scale on the piano from E to E and back down again to demonstrate the sound of the Phrygian mode.
Explain that the Phrygian mode went only from E to E, and that it cadenced on E (the final).
Explain that modes came in pairs with the same final. The first of each pair was known as authentic, and the second was known as plagal.
Explain that each mode beginning and ending on its final is known as authentic.
Explain that each mode beginning on its fifth scale degree and having the same final as the authentic mode is known as plagal. This is the most evident in a chant or tune. Every mode will only span an octave, but if the tune cadences in the middle (and not the bottom) of its scale, the tune is using a plagal mode.
Play the Phrygian scale once more from E to E on all white keys.
Play an all white key ascending scale from B to B.
Play an all white key ascending scale from B to B, then arpeggiate down from B to G to E, ending on E. This is a good example of how Hypophrygian uses an ambitus (range) from B to B, but that it will cadence on E.
Explain that Locrian is the authentic mode with the scale of B to B, and its final is B.
Explain that Hypophrygian is a scale from B to B, but its final is E. These modes are very similar and they have the same pitches, but the pitch around which Locrian centers is B, while the most important pitch in Hypophrygian is E.
Conclude that Hypophrygian is the same scale as Locrian, but that any tune or chant in Hypophrygian will end on E, while any tune or chant in Locrian will end on B. Explain that when in doubt, listen for the end. Chants and tunes always end on the final.