I'm not an expert at all on Byzantine Chant but music is my best subject The apechema as described seemingly lets people know exactly what the mode is and how it is sang. How many times do you hear people say, "I don't know my tones." Well here we have an apechema being chanted before the tone begins, thusly letting people know what to sing.
Do I have that right?
I had a great response to this when I was at work, and somehow closed the window before I could post it. Let me try to reconstruct:
You have it right. The reason why the apechema is so critical is less about the pitch and more about the scale - because there is so much variance to the Byzantine scales (as there was in Western scales before the movement for equitemperment) there has to be a way of indicating that one is operating on a new system.
Example (using the piano for reference):
Only third mode can be played properly on a Piano without re-tuning it. It can be played (most simply) on all white keys, with F as the base of the scale. If one wanted to shift the scale up a note, they could (using the D major scale, with G as the base of the scale). However, since it is played from F, it cannot be properly called a major scale (just as A minor is also played on all white keys, but because it is based on A it has acquired a different sound than C major)
Plagal of the 4th is the next easiest, closely corresponding to a major scale (I'll use C major as an example). However, D and B would be too sharp by 1/3 of a half-step. In addition, B would need to be flattened by an additional 1/3 to 2/3 of a half-step when going down from it or from a note higher than it. Finally, in the medium and slow styles of Plagal of the 4th D is sometimes sharpened by 1/3 of a half-step.
*Note* Byzantine music theorists spent a lot of time working on these systems, and determined that 1/3 of a half-step (or 1/6 of a whole step) is the smallest variance in pitch that can still be accurately controlled by the human voice.
Theoretically, Plagal of the 2nd Mode can be rendered fairly well on the piano. If one works on a D-D scale, the notes would go: D-D#-F#-G-A-A#-C#-D. However, even using this system, the notes of F# and C# are still too flat by 1/3 of a half-step.
Now Fourth Mode is a nightmare; If one takes it as a white-key scale, the slow mode is based on G, with Ab and F# frequently needed and each one insufficient on their own (each one is too sharp for the scales by about 1/3 of a half-step). The fast mode, meanwhile, is based on E with an occasional D# (which is too sharp by 1/3 of a half-step) and the standard Ab/G# (which is, again, too sharp). The medium mode only has the G# which is standard for the 4 diatonic tones, but it can switch between its own style and the fast and slow.
Hence, why the apechema is useful - so you know which scale you're working with.