1st Species Counterpoint Tritone Reached Melodically by Step

Question

I'm reading Alfred Mann's translation of Fux's Gradus (original available here). On page 35, footnote 9 (presumably Alfred Mann, certainly not the original author) says "The tritone is to be avoided even when reached stepwise [examples] if the line is not continued stepwise and in the same direction." (He goes on to talk about tritones arrived at "by the progression of two voices".) The examples he shows are on the treble clef: F4, G4, A4, B4, and another: F4, G4, B4.

However, on page 36, Figure 14, the bass part is: F3, E3, F3, A3, Bb3, G3. This violates the rule given in the footnote by moving from E3 to Bb3 without continuing stepwise in the same direction. I say to myself, "No problem. Nobody's perfect" and continue only to find the same situation in the next example. Page 37, Figure 15, soprano line, notes nine to the end are: G4, C5, A4, G4, F#4, G4. Again, from notes 10 to 13, we have a descending tritone without continuing stepwise in the same direction! Okay, two "flukes" is a bit much, but warily, I continue. The next example, on Page 39 -- Figure 21 -- tenor (bottom) part, notes 8 - 12 are B3, C4, A3, F#3, G3. Again, a descending tritone not followed by step in the opposite direction. Three examples in a row?

It seems that Fux and Mann simply disagree here, but which is correct? Did Fux not care about this "rule"? Did others care about this "rule"? Why would Mann write something like that just before 3 (!) counterexamples?

Answer 1

"Tritone" did not originally refer to both augmented fourths and diminished fifths. In fact, thinking of augmented fourths and diminished fifths as enharmonic respellings of the "same" interval didn't exist either. "Tritone" meant, literally, three tones or whole steps: e.g. F-G-A-B is a tritone, and is what we would now call an augmented fourth. A diminished fifth, such as B-C-D-E-F, is reached via two tones and two semitones and thus can't technically be called a tritone.

More importantly, these two intervals came up in different situations in medieval and renaissance music and had different rules. It's important to remember that—although augmented fourths and diminished fifths are the same size, 600 cents, in equal-tempered music—they are different intervals in just intonation and in the various meantone and well temperaments that predate the far more recent practice of equal temperament. In fact, there are many different possible sizes for these intervals. One fairly common instantiation of the augmented fourth would be about 570 cents, and its partner diminished fifth would have been about 630 cents (thus adding up to a perfect octave, 1200 cents).

Bottom line, the "tritone" had a far more specific meaning in Fux's day, and had rules distinct from the diminished fifth.

Answer 2

When Mann wrote:

 The tritone is to be avoided even when reached stepwise [examples] if the line is not continued stepwise and in the same direction.

...what he was saying is that the tritone must not be outlined in stepwise motion. With both melodic snippets [F, G, A, B] and [F, G, B], the F and B, which are the outside notes that outline these steps, create the undesired tritone. This can be fixed, though, if we follow Mann's direction to

(continue) stepwise and in the same direction

So if we took the first [F, G, A, B] and added C at the end, it would outline a perfect fifth, and this would be okay. This is what makes [G, C, A, G, F#, G] and [F, E, F, A, Bb, G] acceptable. Granted, if there was [F, E, Bb, G], that would not work, because there is a direct tritone between the E and Bb (of course that is not stepwise like the other examples). But [F, E, F, A, Bb, G] solves the outlined tritone by adding F and G to the outside.

Answer 3

I should point out that I've read both this book and (more recently) a book on counterpoint in the style of Palestrina. I noticed, very consistently, that the Palestrina book contradicted itself on many many occasions throughout the entire book.

I've learned to realize that a) mistakes are made b) exceptions are allowed and c) don't focus so much on the little specifics here and there (although it is very frustrating to not have resolution of a contradiction before moving on) and instead focus on the bigger picture. With that said, I was doing self-study, not in a class, so that might be different (but then again, you can simply bring your questions to the professor...).

Hope this helps.