Chris - you have many questions in a row, so I thought it easier to submit an answer rather than post several comments. To moderators, I apologize if this is cluttersome.
Here we go:
1.) You are correct, you did not in fact make that statement - I did not realize that you were quoting Cuthbert. I will expand upon this answer momentarily.
2.) The term "diatonic" refers to what naturally occurs in the scale. In major keys, each degree of the scale corresponds to a different mode - even the root pitch. The exercises are in modes that are for the most part diatonic. The exception includes use of Musica Ficta.
During this time period, modal counterpoint dominates the contrapuntal landscape - if not the entire musical landscape. Much music of this period is referred to as "motor music" because of the constant rhythms and motion of independent voices.
3.) Ascension cannot be on downward motion. Your confusion stems from a typographical error on my part. The sentence should read "You are incorrect in saying that the F# is temporarily "Fa" because the voice-motion indicates ascension and not downward motion."
That said, I would like to redact that sentence as it is incorrect.
Regarding your confusion concerning "Mi" and "Fa", I would like to direct your attention to page 35. Read the footnotes carefully, particularly #8, and refer to the bottom of page 31 for further clarification.
According to the text, the Mi / Fa relationship in this instance is not as Cuthbart describes in his answer, but regards relationships within different hexachords.
As as aside to Cuthbert - the popular phrase mentioned in your answer appears in Fux's text, which was published in 1725, the early part of the 18th-century. It may be understood that this phrase does not actually hold a religious connotation as some may ascribe, but rather that the interval is a "devil" to work with because it offers no readily available diatonic solutions.
4.) Joseph's comment about "harsh relation with the F# in the thirteenth bar" is an apt one. Here Joseph is pointing out tendencies of aural memory. Since it is toward the end of the exercise, listeners would have perceived that non-diatonic semitone relationship between the "F" in bar 11 and the "F#" in bar 13 as jarring dissonance and therefore unsatisfying. Composers during this time period were extremely cautious (as you will read in further species of counterpoint) about approaching and resolving dissonances. Aural palettes were simultaneously refined and immature.
In order for the relationship not to see too jarring, Joseph would have needed more time in between each occurrence. Due to the brevity of remaining measures after bar 11, Joseph astutely concludes that he does not have the space necessary to facilitate appropriate pacing to leave the "F" as natural.
5.) Also: How does the progression move upward from a sixth to a third? What does that mean?
If you look on pg. 39 of the book, you will see that the line for counterpoint contains a treble clef with an "8" below it. This indicates that all written pitches sound 1 octave lower. By comparison, the "D" in bar 11 on the top staff would sound as the fourth line from the bottom on the lower staff in the same measure. The numbers above the lower staff indicate the interval between the two pitches, but do not qualify the type of interval (such as major or minor.) In this case, it is a minor-sixth.
In the following measure, the voices move by contrary motion to another consonant interval: a major third. Since the exercise involves counterpoint below the cantus firmus, the progression may be aurally perceived as moving upwards.
6.) By the way: c and c# are used in a previous example so does Johann Fux contradict himself for using a good example (as his teacher
Aloy. calls it) with a c and c# and then avoiding the use of f and f#
in the invention that I wrote out above?
My understanding is that you are referring to the exercise on page 31. Here the context is different which is why it may seem contradictory and therefore incorrect. Before I answer, you should know that Fux was a master of counterpoint. He was very, very famous during his lifetime and regarded as one of if not the foremost counterpoint pedagogue. The Fux text has not remained the authority on counterpoint for almost 300 years because it is full of mistakes. Any mistakes in the text are more likely the result of translation, copying, or editing. That said, Mann provides a wonderful translation.
As the book states on page 32:
"...put a minor third in the next to the last bar since the cantus firmus is in the upper voice..."
From then on, the exercise resolves to a unison. If the counterpoint was left as "C" then it would have been incorrect. Moreover, if the "C" in bar 8 were written with as a "C#" instead, two worse issues would have occurred: the interval between the two voices would have been a tritone, which is bad news for everyone, and the counterpoint motion from bar 7-8 would have been the interval of an augmented fifth - another big no-no. In this instance, any jarring dissonance experienced by a listener between the "C" and "C#" would pale in comparison to the dissonance of a tritone or voice movement of a augmented fifth. Sometimes you need to break small rules to avoid breaking larger ones.
