Accidentals in First Species Counterpoint?

Question

On page 39 of Alfred Mann's *The Study of Counterpoint* there is a passage concerning justification of using accidentals in first species counterpoint.Here is the passage:>Aloys: [...]but why did you put a sharp in the eleventh bar? This is generally not used in the diatonic system.>Joseph: I wanted to write a sixth here. But when I studied singing, I learned that *fa* leads down and *mi* leads up. Since the progression moves upward from the sixth into a third I have used a sharp in order to emphasize the tendency to ascend. Besides, the "F" in bar eleven would result in a harsh relation with the "F#" in bar thirteen.----------What does Joseph mean by "*fa* leads down while *mi* leads up"? What is his reasoning for changing the "F" to a "F#"?Also, what does Joseph mean when she says that "the progression moves upward from the sixth into a third"?

Answer 1

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Here's a brief summary of what I explain in the video. Let me know if you need clarification!

The ideas in Johann Fux's 1725 book Steps to Parnassus: The Study of Counterpoint differ slightly from what we're familiar with today.

1. Hexachords, the six-note scales present in music theory back then, have only 1 semitone (half-step), between fa and mi. Joseph is saying that these tones have tendencies to resolve into one another.
2. His reasoning to change "F" (we would call it F-natural) into F-sharp is that the F in measure 11 would clash with the F-sharp two measures later. Accidentals are used sparingly in strict counterpoint rules, and you typically only see them as the leading tone right before the final measure of music. This is why Joseph had to explain himself: his usage of the sharp in measure 11 was not at the end.
3. "the progression moves upward from the sixth into a third" means that the harmonic progression of the passage (measures 11 &12) moves from a minor sixth (D-F#) into a major third (B-G). The F# resolves upward into the G.

Adam