Functional Harmony in contrapuntal Music

Hi,

is there any guide that describes how to imply functional harmony progressions in contrapuntal music? Most articles on counterpoint seem to be mostly concerned with which intervals are allowed and how dissonances have to be resolved but none explains properly how to combine this approach to composition with functional harmony.

All I can do is recommend a book by Kendall Briggs - Counterpoint. He does a nice job linking harmony with counterpoint by discussing the harmonic implications of scale degrees in common practice music and then takes it from there.

Syntactic Structures in Music by Tom Sutcliffe

http://www.harmony.o...tonal_music.htm

In the first paragraph of the Introduction, he makes the following claim:

Conventional theories of structure in tonal music concentrate either totally on root progression patterns (Rameau. Schoenberg, etc) or totally on voice leading. (Schenker etc) This book is the first to explain how root progression patterns and voice leading work together.

Although the site refers to a forthcoming book, I'm not sure it has actually been published (yet). Nevertheless, I've found the contents of the website to be extremely helpful. Especially his distinction between static and dynamic harmony, his labeling of chord progressions as alpha, beta, and gamma progressions, and his formulation of musical phrases in terms of linguistic phrase structure trees.

Granted, it's more focused on the structure of musical phrases, rather than the intimate details of species counterpoint, but in focusing on this structure, he demonstrates the interactions between contrapuntal voice leading and functional harmony. Appendix A does contain an overview of voice leading principles in terms of species counterpoint, though.

I still didnt find a precise rule for determining which notes belong to the currently implied harmony and which dont.

Look at this: http://www.harmony.org.uk/book/examples/ex_7p6.gif

in the third bar: why are only c and G considered in the harmonic analysis?many different chords appear afterwards. By which principle do you identify that they are nonharmonic (and are they really?)

Well, you need to understand non-harmonic tones, and what inversions are suggested. In the analysis presented you would understand it better (and I fault the analysis too) if you see the first three eighth note beats in bar 3 as two first inversion (6-3 chords) chords going to G , eg Eflat-C, B natural-G, G-G . Bach's figuration avoids many contrapuntal problems with the figuration in the left hand. Yet as the G-G octave falls on a weaker beat of the bar and the quality of the chord hinted at - G major chord, (to serve as a V of C minor) fails to do something which Bach's statement of the fugal subject in the dominant is pushing to do - move the harmony into the dominant to create a contrasting section and expand the form of the piece. So he has to figure a way to reinforce the B flat to get to G minor and convincing chords to lead to G minor. IN this instance the I in Cminor becomes a iv in G minor leading to a i6- viidim- i. The diminished chord supports the fugal subject and reinforces the move to the root position G minor chord.

Another way to understand it is to reduce it to the chords suggested - starting on bar 3 - Eflat-G-C to D-G-Bnatural to C-Eflat-G to Bflat-D-G to A natural-c-F# to G-Bflat -D. The implied roots to the subject then are E flat- D-C-Bflat- A natural- G, the first six degrees of the G minor scale!

In fact, here is one dirty little secret, you can with practice improvise in a Bach style by taking fugues reduce it to the chordal skeleton and then add figuration of your own! A rather advanced technique of improv organists learn and employ to gussy a hymn when they are up to the 7th stanza.

In addition to the text I gave and the other one listed, Aldwell and Schachter do a nice job of linking Harmony and elements of counterpoint in their Harmony and Voice leading book. They offer the best exercises - realization of figured bass, harmonization of melodies and basses. They are hard but you really learn harmony and key concepts of counterpoint. You won't learn counterpoint thoroughly per se - that requires specialized study. But such texts will help you very much when you do get to that topic.

Another way to understand it is to reduce it to the chords suggested - starting on bar 3 - Eflat-G-C to D-G-Bnatural to C-Eflat-G to Bflat-D-G to A natural-c-F# to G-Bflat -D. The implied roots to the subject then are E flat- D-C-Bflat- A natural- G, the first six degrees of the G minor scale!

My question was actually HOW you determine, which tones/chords belong to the chordal skeleton.

For example in bar3: why aren't the last two chords D-G and C-A considered in as "harmonic"? They don't even resolve to the supposedly implied harmony c-minor.

Or in bar4: why isnt A-G considered as harmonic?. It's metrical position isn't even any weaker than that of the G major chord in bar 3.

Of course i already know harmonic functions and all that stuff , i don't really see how it relates to my question?

By which principle did you identify the A as an auxillary tone, whereas the lower B in the same metrical position in bar 3 isn't auxillary?

When you're analysing the above, why don't you analyse the whole thing as just one big C minor chord? Or why don't you just analyse every single note as a chord (that'd be fun to do for ornaments)? Why don't you analyse the first beat of bar 3 as an A dim 7 with an appogiatura? Why wouldn't you analyze the first half of the first bar as a C major chord?

I don't think any of the stuff on its own has any sort of function, it's only in the context and in how it's used that it does.

Looking at the first beat of bar 4, I can say to myself, 'Bb and D... Well, G has a B, so it could be a VII without the fifth'; I look at the next half and go, 'Oh right... G and A... What the hell is that? Oh, maybe the G's an apppogiatura and it's a diminished chord' and I could reason the same way for the next beat.

On the other hand, I could look and say to myself, 'D jumping to G in the top with a Bb on the first beat; that's a v... But what about that A on the next beat? Oh look, he's going back to a Bb, so I'll stick with this G business'.

Or (and I don't know if this is a thing), I could analyse the whole first half of the bar as revolving around Gm (by the second method above) and still take into account that actual spellings of chords occur because of non-chord tones; however, I would be wary and would rely on my ear if I wanted to give these 'auxiliary' chords functions, since they happen so quickly/incidentally when Gm is so prominent.

If I look at how the whole thing moves, I could notice that the first half of the bar is very G minorish, and the second half emphasizes a diminished 7th on an F#, which would be the vii7 chord in the key of G minor and analyse it: i - vii7 in the G minor context or i/v - vii7/v in the context of the key of the C minor key of the piece. Of course, there's nothing incorrect about analysing it iii/#III/i/i - I/v/I/i, or something; but why the hell would you? The tonic chord, for instance, could be analysed as I/I/I/I/I...; but people don't think about that when they write their V/V's &c., just like they don't think about the infinite amount of zeros afters an integer's decimal point, though they consider things like that when it comes to the irrationality of a number like pi.

It's all about which is simplest and about how it's all used, I think. There are a million ways to analyse it that could be hella complicated, but make just as much sense (i.e., be equivalent); for example, analysing F#-A-C as a dim chord on the 7th degree or as a V without the root: what's the difference and who cares? They're the same thing, as far as function is concerned.

Basically, I think most analyses just try to strip it down to be as simple as possible: if you can analyse 180 bars coherently in the context of one single chord, go for it; if, as above in the third bar, you can't make sense of even half a bar with a single chord, that's when you change. It's just like in physics when some people will tell you that the earth revolves around the sun and some will tell you that the sun revolves around the earth: both and neither are true; it's just a combination of perspective and simplicity; and that could relate exactly to whether the fourth bar above contains a change to a dominant function on the second half-beat or not.

In short: whatevs.

Also: beats and speed and stuff.

You could analyse the A as an auxillary tone if you suppose that g is still the implied harmony. But how do you determine that it really is? Maybe the harmony changed to f#-dim (with the nonharmonic g in the upper voice resolving to the chord-tone f#)

How do you know one tree is taller than the other? How do you know what time it is? How do you know any of the chords' functions at all, if they could have others? How do you know the piece isn't in F# major with a different key signature and accidentals?

Just 'cause.

If you want to think about it differently, think about it as analysing what was going through the composer's head at the time, and not analysing the music in the one true objective way to do it.

Ok, so the answer is actually that there aren't any mathematically precise rules for determining chord progressions in contrapuntal music, rather one judges by ear or intuition which chords are the most prominent/important?

Exactly. I used to have the same notion you did (which comes from lack of practise/usage and/or being physically attracted to logic, I think); but it's incorrect. Some people might disagree, but I imagine those people have been putting into practise what they learn and have absorbed more from that than they think and/or let on and/or they're liars who want to show everyone how smart they aren't.

If you wanted to program a computer to do it, I think you'd have to supply it with certain information about certain music (the commonness of I and V, for example), if you wanted it to be accurate/useful, 'cause it couldn't derive everything from the rules alone.

I think the way to do it is just apply the rules in as clear and simple a context as possible; i.e., "as simple as possible but no simpler"..

Well, DAI, there is a mathematical relation there. BUT it arises from conventions developed over a hundred + years.

As for your statement - why don't you assume it goes to F# diminished shows that a) you have not learned deeply Baroque practice b) you have to look at the subject.

First off, look at the subject in the first measure, C-B natural-C then a leap to G, in reality Bach develops the piece from that entire figure. The move to A flat is somewhat of a red herring at first until you realize it is outlining in an inverted and slower rhythmic values the c-Bnatural-C motive. So the C-Bnatural-C figure transposed to G would be G-F#-G, so the inverted form would be G-Aflat-G. (see pick up note to beat 3 in bar 1 to the A flat on beat 3 - G-Aflat, then on beat 1 measure two it goes to G). BTW, you can easily see how serial procedures stem in part from Bach's contrapuntal practices.

Now, recall I said the transposed version of that little figure C-Bnatural-C is G-F#-G? Well Bach makes it clear he is connecting this tiny figure to the harmony by 1) having the diminished harmony ALWAYS on the weaker beats of the measure, it NEVER falls on beat 1, the strongest beat of 4/4. Second in bar three the motion of G-Bnatural to F#-Anatural to G-G in sixteenths shows the A natural as passing as in the next two beats C minor chord is strongly emphasized by having all the chords of triad heard. The only foreign notes are the F# and A natural. F # is acting as a neighbor as it returns to the G and the A natural is what is called an "incomplete neighbor" as it leaps down to D instead of going back to G. We soon see though in measure 4 beat 3 and 4 that the harmony is a diminished chord by the lower voice and the move to E flat. In Baroque music the ONLY chord it could be is a diminished chord.

I go into detail for you to understand that by understanding the change in compositional practices and how rhythm and voice leading (what are the harmonic tones versus non harmonic and how they are treated) all lead to an understanding of the analysis.

BUT, one suggestion to show that you will veer out of the Baroque convention - as that is what this boils down to - treat F# as the new tonic, this would make G minor a Flat ii. My question is, could you rewrite measures 4 onward to show the two main tonal points of the piece are now C-F#? That would be an excellent composition/theory exercise for you. And who knows you may create some cool music.

Finally, you will find in some late 17th century music that it is very difficult to determine the harmonic function - especially some French keyboard works. So, your questions are quite valid but it is being applied not quite to the correct era and composer.

Something I don't see discussed much (composerorganist does touch on it), and which takes out some of the arbitrariness, is the rhythmic hierarchy. With the exception of appoggiaturas and sustained notes, nonchordal notes are much more likely to be unaccented rather than accented. In 4/4, the 1st and 3rd beats are strongest, followed by the 2nd and 4th beats, followed by 8th notes that fall in between beats, followed by 16th notes that fall in between those 8th notes.

It's not so much a matter of "right or wrong", so much as a matter of scale -- how detailed do you want to look? You could analyze every 16th note in a run as a separate chord, but that gets tedious real fast. Nonchord tones are basically just a lazy excuse to not have to harmonically analyze every single note. For the most part, the accented notes tend to be much more important. A list of different types of nonchord tones (passing tones, neighbor notes, appoggiaturas, etc) is also really helpful here, because you can look at it and find similar patterns. It's basically a catalog of melodic patterns that musicians over the years have found helpful to ignore.

Let's look at the second half of bar three at various levels:

Technically, since this is only two parts, there's no actual chords at all, just a series of intervals. In early contrapuntal music this is largely the way would have been thought of. So in the second half of bar three you have: 10th, 3rd, 2nd, 4th, 6th. You *could* try to put one or more chords with each of these, but first...

Let's "zoom out" one level, away from the 16th notes. We see that there is an unaccented 16th note: an F# between two G's. This note is unaccented and creates a dissonance (aug. 2nd) with the E-flat, so we can safely call it a nonchord tone. Looking at a chart of nonharmonic tones, we see that this is a classic example of something called a "neighbor note". So let's remove the F#.

Now we have a series of 8th notes C-E-flat, E-flat-G, D-G, C-A. You could analyze each of those as being a chord. I'll assume you're OK with assigning the first two to both be a C-minor chord (instead of, say, an A-flat followed by an E-flat). If you're not OK with that, then think of it as an application of Occam's Razor - it's the simplest chord (tonic) that fits the most notes (all of them). In this analysis, beat four would have a G minor chord followed by perhaps an A dim or an F# dim chord.

But let's zoom out above the level 8th notes. The C-minor is clearly accented here, since it comes on the stronger beat, and lasts twice as long as the other two "chords". Those other two each last half as long, and are relatively unaccented (and we can't even tell what the root for one of them is!), making them seem more transitory. Now, if we look from a purely melodic perspective, the alto line goes C E-flat D C, which outlines an interval of a third (ignoring the fact that the C is in a different octave). In this context, the D looks just like a "passing tone" to get back to C. The soprano line goes E-flat G G A (then leaps downward). This looks just like an "escape tone" (aka an incomplete neighbor, as composerorganist calls it).

Since, at this level, these last two chords are short, unaccented, and created by nonchord tones, they aren't as important. So at this higher level, we see that this entire half of the measure is just a C minor chord with decorations. Hope that helps!

TLDR: Zoom out using rhythmic hierachy, and Occam's Razor.

So when you write contrapuntal music yourself, how do you go about realizing harmonic progressions? do you just make sure that the essential tones are accented?