Harmonic Function--www.martiandances.com

I’ve probably said it before, but now I’m going to say again that there are three types of thinking that have to happen in music theory. The first and most basic is the area with which your textbook, Tonal Harmony, 5^th edition (Kotska-Payne, or K-P) does a great job, and that is putting names with things that happen in music, i.e., nomenclature or taxonomy. If you think of music as a zoo full of animals (intervals, chords, cadences and the like), K-P does a decent job of relating them to each other and giving you a framework to use to identify new musical animals that you might come across.

The second type of thinking in music theory is also covered reasonably well by K-P, and that is analysis. K-P is great about giving you lots of little examples from actual pieces of music from a canon of music so that you know that they are not making this up. There was an age, about two hundred years ago, when the examples weren’t necessary because the only people reading a book like this had only ever heard purely tonal European music. As a result, they were in complete agreement about what sounded right. They could play an author’s original examples and hear for themselves that they were right. K-P sort of drops the ball on analysis, because they hardly ever give you an entire piece of music to analyze, but that’s my job anyway.

The third type of thinking in music theory is where K-P completely misses the boat. We can’t really blame the authors for this, because they have a real challenge for themselves in that they are trying to write a book that will get college students the basics of theory without completely confusing them. So they have pretty much left out what they think is the most advanced information, and in so doing, they have ensured that most students who read their book get completely confused. It is my job to give you this advanced information.

What K-P has declined to include in the book is any kind of system or theory of tonal harmony. That’s the third level of thinking – systematic principles and concepts that apply to many pieces of music (or sometimes just a few) and help to explain why the composer did what the composer did. In science, Newton’s laws of gravitation and Einstein’s theories of relativity are examples of this kind of thinking. In music (which is half-science and half-art), we have something called Harmonic Function.

The version of Harmonic Function that I am going to present to you has its roots in many approaches to music theory, but owes most of its content to the work of Hugo Riemmann, a 19^th-century German theorist and acoustician. Let’s begin:

1. In tonal music, there are three harmonic functions:

Tonic (T)

Dominant (D)

Pre-dominant (PD)

2. These harmonic functions group together to form phrases. (A phrase is a musical idea that ends with a cadence). There are two phrase models:

T-PD-D (a phrase ending with a half-cadence)

T-PD-D-T (a phrase ending with an authentic cadence)

3. In either of these phrase models, PD can be omitted, but only at the risk of being boring if you omit it too frequently.

4. The chords of the major mode are divided between the three functions like this:

And the chords of the minor mode are divided like this:

5. Now – between the two phrase models, and the three functions, we can figure out which chord comes next.

Let’s take the second model (T-PD-D-T) as our example:

T	PD	D	T
I	IV	V	I
I	ii	V	I
I	IV—ii	vii^o	I

As you can see, there are multiple options for some functions, including using more than one chord in an area.

6. What about other chords? We seem to be left with vi and iii. In K-P’s infamous (by now) diagram on p. 107 (the “happy face”), these two chords seem to have the job of being at the beginning of phrases and starting the chain of fifths that leads to I in every phrase. If only that were true! The more you look at music, the more you realize that this simply isn’t what happens. The fact is that vi, while often leading to IV, almost never leads to ii, as page 107 would have you believe. A more common progression is this one, based on the first phrase model (T-PD-D):

I—vi—IV—V⁷

If I were labeling the harmonic function of these chords, I would write the following:

T—Ts—PD—D

(Ts stands for tonic substitute, which is what vi is doing – it’s moving us smoothly between I and IV.

Think of the most basic possible voice-leading in C-major for this progression:

G A A G

E E F F (upper voices, no particular order)

C C C B

C A F G (bass voice)

The only thing that the vi chord is doing is smoothing out the way between I and IV. But it also allows us to have four chords in the progression instead of three. Which is really nifty if the song you’re singing is “Duke of Earl.” I call this the Duke of Earl progression, but composers have used it forever. Think of the opening of the “Moonlight” sonata.

7. Okay… that’s vi, but what about iii? In major keys, iii is extremely rare. If you see iii, take a close look, because whatever is happening is probably interesting. Look at where iii is on the diagram with the three circles – it is part-tonic and part-dominant (it has two notes each from the I and V triads, just like vi has two notes each from I and IV).

Imagine a piece of music as a trip. If it uses the first phrase model (T-PD-D), it’s a one-way trip, and if it uses the second phrase model (T-PD-D-T), you have a round-trip ticket. A single phrase is a trip the mailbox. A Bach minuet might be a trip to the post office. A Beethoven piano sonata is a morning’s worth of errands. A Brahms symphony is a vacation to Disneyworld (although not as commercialized and not nearly as expensive). A Wagner opera is a trip to the moon.

Here’s the problem with iii: it’s like a secret passage in the game Clue or a wormhole on Star Trek. It’s a shortcut that seems to put you both at your destination and at your starting point at the same time. In music, unlike real life, getting there isn’t half the fun: it’s all the fun. So too much iii is like waking up after an alien abduction: you know something just happened, but you’re really freaked out because you don’t know what.

Note: In minor keys, III acts very differently and often is sort of a substitute for V, another goal, but not on the phrase model level.

8. I’m not saying that every piece has a vocabulary of only three or four chords. You know better than that. Composers do two things with the phrase models.

First, they make chains of them:

[T—PD—D]—[T—PD—D—T]

Second, you can use chords from other functions as neighbor chords:

[T—PD—D—T]

[T—PD—T—PD—D—T]

Third, and this is really cool, they nest them:

[T—PD—D—T]… but then, if we look at the first Tonic in the model, we see…

[{T—Ts—PD—D—T}—PD—D—T]… and within the PD…

[{T—Ts—PD—D—T}—{T—D—T}—D—T]

And so on, until the piece is over. Now… in the PD area above, we are talking about tonic and dominant in a different key, either IV or ii, not the original key. This is all stuff we will be going over this quarter. Any function can have multiple functions (phrase models, neighboring movements) nested within it, and the levels can get very deep, especially in long pieces.

We could write the next-to-last progression above in the following way:

I—vi—IV—V⁷–I—ii—V⁷–I …or it could be…

I—vi—ii—V⁷–I—IV—V⁷–I …or even…

I—vi—IV—ii—V⁷–I—ii—vii^o—V⁷–I (using more than one chord for some functions)

This concept of harmonic function starts to allow us to consider bigger, longer and more complex music, but to still relate it to a basic framework. One of the most important music theorists of the 20^th century, Heinrich Schenker, thought that pretty much all Western music from Bach through Brahms could be explained as very long versions of the second phrase model with lots of nested levels. Caveat: this type of generative theory, where a large piece of music is built by expanding and prolonging bits of a smaller piece of music is relatively new. Bach, Beethoven and the rest certainly did not think in these terms, at least not explicitly. The terms are a modern substitute for a rock-solid understanding and feeling for this music on a gut level. A Common-Practice Era composer (at least until 1890 or so) wrote music that sounded right, not to conform to some theory he or she had read about in school.

9. From harmonic function, we can learn about root motions. If you start to look at possible root movements, you will see that there really are three root progressions that stand out and three that are somewhat less common (but still important).

Between two chords, the three most common root motions are:

Falling Fifth (or Rising Fourth) – For example, V-I or I-IV.

Falling Third (or Rising Sixth) – For example, I-vi or IV-ii.

Rising Second (or Falling Seventh) – For example, IV-V or V-vi.

These three root motions cover many, many situations, including half- and authentic-cadences, the deceptive cadence, and the basic moves of the two phrase models. Use these as much as possible.

The opposite motions are less common, and tend to obscure the direction of the functions:

Rising Fifth (or Falling Fourth) – I-V, it’s true, but there should really be a IV or a ii in the middle. Usually appears within some kind of neighboring motion.

Rising Third (or Falling Sixth) – Here is where the ambiguous I-iii progression would happen. Use I-I⁶ instead.

Falling Second (or Rising Seventh) – V-IV doesn’t make sense; neither does ii-I – you’re skipping steps. Again, I-viiº often appears as part of a neighboring motion.

These opposite motions aren’t absent from tonal music – they do appear from time to time, but they are somewhat problematic. Employ them with caution.

You may have heard me say that much rock music lives in a sort of half-tonal world. On one hand, many rock songs were and are written by songwriters with classical training of some kind, for whom it was important to build up a narrative sort of chord progression. Just as common (perhaps more common) are rock songs based on the blues in some way, or built around a repeated “lick.” These songs are often based on pentatonic or expanded pentatonic scales and don’t adhere to the principles of harmonic function. The near-ubiquitous 12-bar blues, for example, uses only dominant-seventh chords, and ends with the progression V-IV-I, using root motions that are not typical of functional harmony. I’m not writing this to argue that one style of music is better than another, just different. Rock and much popular music takes its inspiration from the idea that a song should convey a single, unified musical idea, and achieves through a relatively static harmonic approach—the “changes” that are repeated several times through a song, or through fairly predictable formal structures. In music that is functionally harmonic, the harmonic language is a key component in setting up conflict in the beginning and middle of a piece that is then (usually) resolved at the end.

10. The last component of functional harmony is the sequence. A sequence consists of a short musical pattern (a foot) repeated at various pitch levels. The foot of a sequence may be as short as a beat or half a beat, or as long as eight or sixteen measures (in the development sections of Beethoven’s sonatas, it is not uncommon for groups of phrases to be treated as sequential feet, but Beethoven also incorporates sequences into his thematic material with feet of two sixteenth-notes).

There are three main types of sequences, based on the same intervals as most common chord progression in the functional model:

Falling Fifths Sequence: By far the most common in the music of the Common Practice Era, and nearly ubiquitous in the music of Bach, who seems to write either one of the phrase models or this sequence. This is a very useful sequence in moving from one key to the next—the last chord can easily be made to feel like a new tonic pitch. The sequence follows the Circle of Fifths—I-IV-viiº-iii-vi-ii-V-I—but could start on any of these chords.

Falling Thirds Sequence: Probably best known for its incorporation (in an embellished form) in the Pachelbel Canon in D (an infamous example of lots of interesting compositional techniques). The sequence follows a “Circle of Thirds”—I-vi-IV-ii-viiº-V-iii-I.

Rising Seconds Sequence: Appears quite frequently during the Baroque era—in music by Vivaldi, particularly, often combined with a rising chromatic bass line. Look for the pattern I-ii-iii-IV-V-vi-viiº-I.

As you might guess, each composer has his or her preferred sequential usages, but a very common approach is to build a two measure melodic cell over a D-function chord in the first measure and a T-function chord in the second measure. This figure can then be repeated a step lower to create a falling fifths sequence.

A sequence is a section of music that it outside the phrase model, and it has two main purposes—first, it can be used to extend a musical idea, filling measures either for dramatic purposes (as in a Classical sonata) or practical ones (as in a Baroque dance). In this case, the sequence will be a tonal sequence, and all the notes will remain in the home key; intervals may alternate between major and minor or perfect and augmented/diminished when feet of the sequence are compared. The second use of a sequence is to move toward a new harmonic goal by introducing chromatic pitches that make each foot of the sequence appear to be on a new tonic. This is referred to as a real sequence, and the flavor of intervals will be maintained between feet, with the result that a new key will be implied with each foot, and at the end of a new sequence, the music will be in a different key.

Hopefully, all this hasn’t completely confused you. With any luck, the next time you write a chorale harmonization, you will have a better idea of which chord to pick as you move through each phrase. Ultimately, we will be elaborating on this framework as we mop up the remaining techniques of tonal music during this course and in Theory IV.

Tricks and rules we will be learning this semester:

1. Any major or minor triad can be tonicized, that is, treated as a temporary tonic. As such, the chord that would be V or viiº can be placed in front of it to intensify and clarify this new function.

2. Chords can be borrowed from the parallel major or minor of a key for dramatic effect.

3. The predominant function is a funny place, where strange things happen, like augmented sixth chords and the Neapolitan triad.

4. When we’re using equal temperament, we can occasionally call a horse a horse—if it sounds like something, it just might be that thing.