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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?

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    Just to clarify it though, the tritones you mention are diminished 5ths. On page 35, Aloysius only mentions the augmented 4th. Don't forget that it is a historic book and you have to adhere to the practices of the time. It was written before the equal temperament tuning become standard and these ecclesiastical modes where not meant to be transposed to distant keys. So there might have been a difference in the sound between a diminished 5th and an augmented 4th. – Chris Mar 16 '15 at 23:26
  • Very astute! The counter examples I found all have to do with the diminished 5th. I wonder if this could resolve the issue? – Limited Atonement Mar 17 '15 at 13:25
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    @LimitedAtonement Yes, that's the resolution. The original meaning of "tritone"--as used by Fux/Mann here--is three tones, or three whole steps. Nowadays, we call that an augmented fourth. A diminished fifth is not technically a "tritone," since it is 2 tones and 2 semitones away, and it has different tendencies and functions in the context of basic counterpoint. – Pat Muchmore Apr 15 '15 at 21:23
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"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.

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  • Excellent. This makes sense, and coordinates with my reading. It would be nice to find a reference for this, too! – Limited Atonement Apr 16 '15 at 18:05
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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.

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  • Please let me know if you have more questions about this, or if you don't understand what I wrote. :) – Mark Mar 16 '15 at 17:57
  • In [G, C, A, G, F#, G], what is being outlined if not a tritone? – Limited Atonement Mar 16 '15 at 18:20
  • Great question. When I say outlined, I mean the first and last notes. So G - G is the outlined, or "outside" interval. Can you please edit your question by adding the octave numbers for these notes? Such as G3 C2, etc. – Mark Mar 16 '15 at 18:32
  • I have modified my question to use Scientific Pitch Notation. – Limited Atonement Mar 16 '15 at 19:35
  • In what context does the note sequence [F,G,A,B] violate the rule (except by itself)? Is [F,G,A,B,G] okay? What about [G,F,G,A,B,G]? – Limited Atonement Mar 16 '15 at 19:38
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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.

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  • "don't focus so much on the little specifics here and there" For a beginner just beginning to learn about music and the theory of music, this makes sense. I'm currently writing a program to compose music (currently, I want it to compose in the style of species counterpoint), so I really care about the little specifics ;-) – Limited Atonement Apr 16 '15 at 18:07
  • That sounds very interesting. Do you have a resource online where I can learn more about it? Or is it a personal private project? – lobi Apr 16 '15 at 20:58
  • The project is hosted at sourceforge.net/projects/partwriter. Feel free to post a "Discussion" on that page if you want to know more, etc. – Limited Atonement Apr 17 '15 at 14:35

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