First Species of Counterpoint rule involving the next to last bar

Question

In Alfred Mann's The Study of Counterpoint, there is a passage that explains a rule basically restricting the interval of the last bar to a minor third if the cantus firmus is in a lower register, or a major sixth if the cantus firmus is in an upper register. Can somebody explain why this rule exists?

Answer 1

Why a third and a sixth?

The restriction on the penultimate bar is a consequence of other rules:

1. The exercise must end on an octave or unison; and
2. Voices must move stepwise.
3. Parallel octaves and unisons are forbidden.

Also, not a rule, but the general expectation is that voices remain within an octave of each other.

Given these restrictions...

Suppose we end on a unison

1. Both voices arrive stepwise from above: means a parallel unison.
2. Both voices arrive stepwise from below: means a parallel unison.
3. One voice arrives from below and one from above: means the two voices form a third in the second-to-last measure.

Suppose we end on an octave

1. Both voices arrive stepwise from above or below: means a parallel octave.
2. One voice arrives form below and one from above: means the two voices form a sixth in the second-to-last measure. (It could also mean a tenth, but this is presumed not to be an option based on the idea of keeping the voices within an octave of each other.)

But why major sixth and minor third specifically?

Aloysius says:

Finally it should be noticed that in the next to last bar there must be a major sixth if the cantus firmus is in the lower part; and a minor third, if it is in the upper part. (p. 28)

Which Mann explains in a footnote:

Thus, the seventh degree has to be raised in the Dorian (D), Mixolydian (M), and Aeolian (A) modes. (The second degree of the mode always occurs as the next to last tone in the cantus firmus, the seventh degree always as the next to last tone in the counterpoint.) (p. 28–29n2)

In other modes, the seventh is already a half-step below the octave.