Question Regarding "the Study Of Counterpoint" By Alfred Mann/fux

Hi guys, I just joined this site and I have my first question about counterpoint. Currently, I am studying 3rd Species in Two Parts (Chapter 3) and while I was analyzing one of the figures that Fux completed as an example (Figure 55), I noticed that in some places Fux placed a flat before the B note and in other places he didn't (thus a natural B note). Furthermore, Fux goes on to explain that he used the flats to avoid "harsh relations with Mi against Fa". I just don't understand how it could pertain to Mi against Fa. As previously stated in the book, Mi against Fa was explained as the tritone (Augmented Fourth/Diminished Fifth), but when I was analyzing one of the measures that used the flat on B the note sequence was: D, C, Bb, & A descending against the Cantus Firmus tone D. The following measure was: Bb, C, D, E ascending against the Cantus Firmus tone G. In my amateur eyes, I don't see anything that has to do with the Tritone harmonically. My only remote guess would be that he flatted the B because of the fact that it progresses into A which is a whole tone combination that is a part of the Tritone. Am I right?

Sorry for the incorrect grammar in the title, I tried to edit it but it wouldn't change lol.

This is the figure I'm talking about: 

http://postimg.org/image/uj11xx2jx/

If you take away the Bbs then, from measure 5 to 6, you get B C D E | F' F etc. The scalar ascent from the strong beat in measure 5 to the strong beat of measure 6 would outline a tritone. If you're using the Alfred Mann edition, this is explained in the footnotes on pg. 35.

That figure (55) actually contains a few problems, though: the ascent of the first two measures, beginning on D, culminates on the strong beat of the third on an E, outlining a ninth, and then the cambiata figure of this bar, which begins on this E and creates an octave with the cantus, proceeds to D on the strong beat of the following bar, creating another octave with the cantus; if you go back to pg. 43, he writes that 'it should be said that the skip of a third cannot prevent a succession of either two fifths or two octaves' in relation to second species, but in figures 50 and 53 (and probably other places), he uses the cambiata figure (which is just a quarter-note decoration of a half-note skip of a third—c.f. figures 51 and 52) in a way which creates such octaves—which sound rather horrible.

In terms of the other flats, though, the one in measure 4 is added because the one in measure 5 is necessary, as, if there was a B in 4 and a Bb in 5, it would create quite a startling cross relation. 

In measure 8, if the final quarter-note was B, it would create a tritone relation with the following note of the cantus (again, see Mann's footnote on pg. 35), and so the B is flattened here; it is then flattened on the first beat of this same measure and on the third quarter-note of 7 for similar reasons to 4, above.

In general, if you have the money and are serious about learning counterpoint, I would recommend paying a little for some lessons (even online) because there are plenty of things which aren't dealt with in one single text (Fux's text requires lots of footnotes from Mann, and even these could have things added to them from, for example, Jeppesen, which itself doesn't deal with certain things, etc.). Also, there are so mistakes in publishing, now, that you don't know what to believe. In the Vienna print in the Latin, for example, your A from measure 4 of figure 55 is printed as a C, but with the interval numeral 5 printed; Mann's edition has a little flat above the final quarter-note in measure 6, but the French Pietro Denis edition marks this as a natural; Mann's figure 59 has, in the first two bars, an outlined tritone (which appears in the Latin text) but this is corrected, in the French edition, by adding a flat to the B in the second bar, which was presumably left out of the Latin (especially since there is already a mistake in that figure in this text) and thoughtlessly copied into Mann's edition—in fact, in the antepenultimate and the penultimate bars of this figure in both Mann's and the Latin edition, the same melodic shape occurs, but this time the flat is put in. Teaching yourself—properly—from a textbook would have you jumping from book to book and you still might not get everything.

Yea thanks for the excellent explanation! I did not see the outlining of the tritone from the 5th to 6th measure at all. I guess it was the change of measure that threw me off. With regards to the cross relation, I read about it briefly in another harmony book ("A Guide to the Practical Study of Harmony" by Tchaikovksy) and I think understand what you're saying now. I guess I just didn't have enough practice with the cross relation concept to apply it to this figure.

I have been trying to decide between studying harmony vertically and/or horizontally and I have still failed to come to the conclusion on which topic I should study first. I read in other books that you should study Counterpoint first because you will have better, melodic voices. But yet, I have also heard that colleges teach harmony vertically first and teach Counterpoint second. I haven't been able to figure out this contradiction.

With regards to your advice, I am currently taking piano lessons with an instructor who is also teaching me harmony as well. I plan on going to college soon.

Also, I agree that there are a lot of inconsistencies in a lot of the music books. I have noticed that in both the books I am reading.

In general, it is handy to keep an eye on the relationship between the strong beats of each measure, because although they don't look priveleged on the page, one really hears connections between these (in Figure 55, in fact, if you remove the barlines and play just the upper line with each note equally accented, there is no tritone outlined there when you put the B in, because the A comes before it; but, if the B were to occur on the strong beat and proceed upwards to F on the next strong beat, this outlines a tritone even though the A is present on the weak fourth quarter).

In terms of spotting the cross relations, etc.: it's quite simple. If one were to do the exercise without considering outlined tritones, one would get the following:

When one examines one's work, then, these tritone-relationships should be noticed. The one between measures 5 and 6 is quite bad (marked with a !) while the one between 8 and 9 could be improved, if this is found to be practicable. Thus, in the first case, one needs to either flatten the B or raise the F; since the latter is undoable (for one thing, it creates an augmented octave with the cantus), we choose the former. For measures 8–9, since we can't change the cantus, we can instead flatten the B. These changes produce the following (additions are marked with a *).

When one plays this, the close proximity of the B and Bbs produces a bad effect, which is the cross (or false) relation. Theoretically, what is happening is that the A on the fourth quarter-note of measure 4 is acting as a lower auxiliary note between the two B-named notes of measures 4 and 5 (in the picture of the figure you posted, from the pdf online, you can see that somebody has marked 'aux' at this point); thus, since an auxiliary should happen between two notes which are the same, the change from B to Bb causes a cross relation; this can be amended by simply flattening measure 4's B, also. In measure 8, the use of both B and Bb above the G is a contradiction, since only one of these notes can appear above the cantus' note; since we want the Bb on the fourth quarter to avoid the tritone relation with the following bar, we simply change the B on the first quarter to a Bb also. This produces the following.

This produces one more problem, like the one between measures 4–5 above but with the auxiliary note going the other way ('aux' is marked again, in your link); again, we simply flatten the B causing the problem, producing the final result:

As for studying harmony vertically or horizontally, I suggest getting a good grasp of two-voice counterpoint up to second or third species before moving on to harmony (look up the stories of Brahms' advice to George Henschel and Max Graf, both of which you can find in Pascall's book on Brahms which I think is on Google Books). Lots of colleges teach things in their own way, but you have to remember that, essentially, colleges are businesses, and they do things more often to suit their timetable than to teach their students. In the university here, in fact, several people failed their harmony assignments throughout the year, and it was looking like they were going to flunk out, but it turned out that all they needed was some proper tutoring and they made incredibly surprising strides in a very short amount of time, once they were taught to understand what they were doing.

With regard to the texts, there simply isn't really a good way to teach a craft-based technique like harmony direct from a book. If you look at your Tchaikovsky book, it's little more than chord connection, really, and when you look at newer books, it's pretty much just a tome of rules to follow with no aspiration of understanding on the part of the student. Textbooks, in general, are getting so like this that they're exterminating all sense of musicality from modern music education, which is quite a shame, as students don't even realise it. The way to get a real understanding of the music of the 'Common Practice Era' is through a diet of counterpoint, figured bass, and (provisionally) Formenlehre (which will later be supplemented with an understanding of the implications inherent in the musical material of a piece, which you can't really find in a book and is a perfect example of something really essential that's gradually fading from music education). Learning harmony from a teacher is important for the 'what?'; but these three are important for the 'why?'.