II. Diatonic Polyphony and Functional Harmony

# 22.1 Introduction

Music, like all art, is communicative. Most composers, performers, and listeners agree that music conveys something—an idea, a story, an emotion—to the listener. A comparison with spoken or written language, then, will be instructive, particularly since a number of music theory terms are borrowed from linguistics. In this chapter, we will consider some of the similarities between listening to tonal Western art music and listening to speech. We will begin to discuss how a composition is structured over time and how certain musical features aid the listener in parsing their experience into manageable, meaningful parts.

As we will soon see, these features are closely related to relationships heard between successive members of a harmonic progression. We have already discussed how a single chord may convey different musical meaning in different contexts. An A-minor chord, for example, will be heard as a vi chord in C major and as a iv chord in E minor. (See Chapter 8 and Chapter 9.) But even within a single key, one chord may play different roles depending on the more immediate context. Furthermore, these roles—or harmonic functions as we will call them—may be played by different chords with different musical effects. This chapter will introduce the two most prominent harmonic functions: the tonic function and the dominant function.

# 22.2 Tonic (T) and dominant (D) functions

Consider the following example:

The music in this excerpt does not flow unceasingly from beginning to end. The melody and harmonic progression are punctuated in three places, corresponding with punctuation marks in the text: m. 4, m. 8, and m. 16. We find rests at the end of each of these measures, but if the rests were omitted we would likely still hear these moments as conveying some sense of melodic and harmonic closure. We refer to these momentary gestures of closure as cadences. They divide this excerpt into three smaller passages. Borrowing a term from linguistics, we refer to each of these smaller passages as a phrase. In tonal Western art music, then, a phrase is a span of music that ends with a cadence.

In linguistics, a phrase functions as a complete unit. It conveys an idea. Sometimes this idea is complete and sometimes it links to other ideas coming before or after. It divides a communication into logical, manageable segments and in doing so allows us to make logical sense of the communication. A musical phrase does much the same thing—though, admittedly, musical ideas are usually much more abstract than spoken or written ideas! Most tonal Western art music proceeds in a manner similar to what we see in Example 22–1. The flow of music is punctuated by cadences that demarcate the boundaries of different sections in a piece.

Note: More often than not, phrases in tonal Western art music tend to be four bars long. In some cases this may be halved or doubled, depending on the tempo of the piece or passage. It not a requirement, however, that a phrase be two, four, or eight bars. You will frequently encounter phrases of all different lengths.

For the most part, the length of a musical phrase is similar to the length of a linguistic phrase. If a phrase is too short, a listener might not register it as a complete thought. If it is too long, it risks becoming unmanageable—or even incomprehensible—to the listener who is trying to make sense of it.

Note, too, that not all passages ending with cadences are phrases. For example, a lengthy transitional passage connecting two phrases may end with a cadence, but it should not be considered a phrase itself. We will discuss these distinctions in greater length in Chapter 35.

Activity 22-1

Activity 22–1

In this exercise, you will identify the the number and locations of all the cadences in an excerpt.

### Question

How many cadences are in the following excerpt? Where are they?

Hint

Listen for resting points or moments of melodic and harmonic closure.

There are four cadences in this excerpt: m. 4, m. 8, m. 12, and m. 16

### Question

How many cadences are in the following excerpt? Where are they?

Hint

Listen for resting points or moments of melodic and harmonic closure.

There are four cadences in this excerpt: m. 3, m. 6, and m. 14.

### Question

How many cadences are in the following excerpt? Where are they?

Hint

Listen for resting points or moments of melodic and harmonic closure.

There are two cadences in this excerpt: m. 5 and m. 10.

Listen to Example 22–1 again, and pay close attention to the way each of the three phrases (mm. 1–4, 5–8, and 9–16) begins and ends. The first phrase starts with a I chord. When it ends with another I chord in m. 4, the listener has a sense of closure. The end of the second phrase, on the other hand, feels less conclusive. It begins with the same tonic harmony, but ends with an unresolved V chord. Ending on V leads the listener to expect more music. The third phrase fulfills this expectation. It begins once again with a I chord. This phrase is longer—equal in length to both of the phrases that came before it—and it ends with the most conclusive gesture of all three cadences bringing a sense of closure to the whole excerpt.

The following example reproduces just the first phrase from the excerpt above:

Harmonically, this phrase is very simple. It consists of only tonic and dominant chords. There is a tonic chord in every measure, but despite this apparent redundancy we hear these chords in different ways. The I chord in m. 1, for example, introduces the phrase. Like the others that follow, it is consonant and stable, but this initial tonic performs the important task of orienting the listener in Eb major. Compare this to the I chords in mm. 2–3. These harmonies echo the initial tonic and reaffirm the key. The I chord in m. 4 follows suit. At this point, the listener is firmly grounded in the key and so the final return to I feels like an ending, a return home.

Now consider the dominant harmonies. Dominant sevenths appear in both m. 1 and m. 4, but again these chords play slightly different roles. The V7 in m. 4 feels more substantial. When it moves to I we hear it as an integral part of the cadence that ends the phrase. The V7 in m. 1 feels somewhat inconsequential by comparison. It supports the opening tonic and even though it too proceeds to a I chord, we do not get the sense that it is participating in a cadence.

The role a chord plays within a phrase is its harmonic function. When we label a chord with a Roman numeral, we are cataloging the content and structure of an individual sonority. When we analyze a piece or passage and consider how a sonority relates to its context and the effect it has on us as listeners, we are identifying its function. The two most important harmonic functions in tonal Western art music are named after their most common representatives: tonic function and dominant function. (We will add a third harmonic function to this list in Chapter 24.)

The tonic function is usually performed by a tonic triad. The pitches in a tonic triad ( $\hat1$, $\hat3$, and $\hat5$) are stable and consonant. Correspondingly, the tonic function conveys a sense of stability. At the beginning of a phrase it establishes a tonal center, in the middle of a phrase it reaffirms the tonality, and in a cadence it provides a sense of closure or finality.

It is possible to conceive of a composition that consists entirely of tonic harmony. For many listeners, though, this would not be very exciting. In tonal Western art music, composers invariably strive for some sense of harmonic contrast. The dominant function—usually performed by a V or V7 chord—achieves exactly this. It is a foil to the tonic. As we saw in Chapter 14, certain scale degrees have a tendency to move to areas of greater melodic or harmonic stability. Two of the three pitches in a V chord are tendency tones: $\hat7$ and $\hat2$. When we hear them, we expect them to resolve, making the dominant function less stable than the tonic. Furthermore, a dominant chord like V7which adds one more tendency tone: $\hat4$is inherently dissonant and contributes to a sense of urgency in the harmonic progression. The dominant function, then, provides contrast to the sense of groundedness conveyed by the tonic.

The resolutions of tendency tones are shown with arrows in the following example:

The tonic and dominant functions together are essential in completing a listener’s sense of key. Between a I chord and a V7, we hear all but one of the scale degrees—only $\hat6$ is missing. And when we hear the key-defining tritone of the dominant seventh ( $\hat7$ and $\hat4$) resolve to a tonic triad, we have a very clear sense of the tonality, told through a harmonic narrative of stability leading to contrast leading to resolution.

Chords performing tonic and dominant functions may appear at different locations within a phrase, but their role is clearest when they form part of a cadence. The remaining sections of this chapter will discuss several of the most common types of cadences.

As we have seen, not all cadences are equal in terms of the sense of resolution they convey to a listener. Some cadences feel very strong and conclusive while others feel weak and open-ended. In Example 22–1, most listeners will hear the third cadence as the strongest and the second cadence as the weakest. The first cadence, then, lies somewhere in between with regards to its strength or conclusiveness. The relative strength of a cadence depends on a number of factors. Rhythm, metrical placement, dynamics and other musical dimensions all play a role in determining how conclusive a cadence sounds, but the most important factors are melodic and harmonic.

A cadence that consists of a dominant-function chord (usually V or V7) resolving directly to a phrase-ending tonic-function chord (usually I) is known as an authentic cadence. Authentic cadences are considered the most conclusive sounding cadences in tonal Western art music. Both the first and third phrases in the Example 22–1 end with authentic cadences:

Both of these cadences consist of a two-part gesture: the dissonance and tendency tones of a V7 resolve to a I at the end of the phrase. They are equivalent in terms of their harmonic content and both convey a sense of closure. The second one, however, feels more conclusive than the first. Again, there are numerous factors contributing to our hearing of these two endings, but the most prominent has to do with the melodic content of the outer voices.

In Example 22–4b, the highest melodic line (the voice part) and the lowest melodic line (the left hand of the piano part) both end on scale degree $\hat1$, the most stable and resolved pitch class in the key. The two most prominent voices, in other words, have both arrived at a strong point of resolution. Compare that to Example 22–4a. Here we see the same bass motion ( $\hat5$ to $\hat1$), but the voice part here ends with $\hat4$ stepping down to $\hat3$. Scale degree $\hat3$, being part of the key-defining tonic triad, is a stable tone, but does not sound quite as resolved as the root of that same harmony. We feel a sense of closure, but not quite so much as an arrival on $\hat1$.

Activity 22-2

Activity 22–2

### Question

The excerpt below has three authentic cadences. Where are they?

Hint

Listen for resting points or moments of melodic and harmonic closure that feel relatively conclusive.

There are authentic cadences in m. 8, m. 16, and m. 18. (The cadences in m. 4 and m. 12 are not authentic cadences.)

We categorize authentic cadences by the type of motion found in the outer voices. If an authentic cadence has $\hat5$ $\hat1$ in the bass and either $\hat7$ $\hat1$ or $\hat2$ $\hat1$ in the soprano, it is said to be a perfect authentic cadence (PAC). Put another way, a perfect authentic cadence consists of a root-position V(7) chord moving to a root-position I chord with stepwise motion to $\hat1$ in the highest voice.

The following example shows another PAC, this time in D major:

As in Example 22–4b, the highest voice steps down to the tonic from scale degree $\hat2$. Here, though, instead of leaping from $\hat5$ down to $\hat1$, the bass leaps up from $\hat5$ to $\hat1$. This has no bearing on the status of the cadence as a PAC.

Of course, PACs can occur in minor keys as well. The following excerpt concludes with the same cadence as the one heard in Example 22–5, only this time in D minor:

Notice that this passage is comprised of just two melodic lines. Despite the absence of inner voices, however, the harmonic progression at the end is quite clear. Scale degrees $\hat5$ and $\hat2$ (the root and fifth of the cadential V chord) each move to $\hat1$ (the root of the concluding i chord) and the cadence has the same conclusive effect as the PACs shown above. Tracking the implied harmonic progression in a two-voice texture can be difficult. Listening for authentic cadences can be very helpful in this task since they are highly recognizable and offer moments of harmonic clarity in ambiguous settings.

The following example also has just two melodic lines:

In this case, the harmonic progression is somewhat easier to analyze. In the second half of m. 66, the right hand outlines an A-major triad (V) which leads to a pair of Ds on the downbeat of m. 67 (I). Even without such an arpeggiation, the combination of stepwise motion to $\hat1$ in an upper voice with a leap from $\hat5$ to $\hat1$ in a lower voice with nothing else is enough to sound like a PAC. In this PAC, the upper voice steps up to the tonic from $\hat7$, although at this rapid tempo we may be more attuned to the metrically accented notes in which case the voice-leading resolves as in the examples above ( $\hat2$ $\hat1$).

As we have seen, the upper voice in a PAC may resolve $\hat2$ $\hat1$ or $\hat7$ $\hat1$. You will also very frequently encounter a combination of the two, similar to what we heard in Example 22–7. The PAC in the example below does exactly this:

PACs are frequently decorated with nonharmonic tones. (See Chapter 15 to review the different types of nonharmonic tones.) In the following example, the concluding motion to $\hat1$ in each of the two melodic lines does not happen simultaneously:

Below the surface, this PAC is no different from those shown above: the upper voice steps down from $\hat2$ to $\hat1$ while the bass leaps up from $\hat5$ to $\hat1$. In this case, however, the resolution to $\hat1$ in the upper voice is delayed. The G—which, as the diagonal line indicates, belongs to the V chord—is suspended into m. 16, temporarily creating a dissonant major ninth before resolving to an octave on beat two.

In the following example, the upper voice resolution is rhythmically altered in the opposite direction:

Here, instead of delaying the resolution to the tonic with a suspension, the motion to $\hat1$ in the upper voice is sped up with an anticipation. The D in the upper voice appears just before the arrival on $\hat1$ in the bass, temporarily forming a nonharmonic fourth with the bass before resolving to the octave at the end of the passage.

The following example has two PACs with a different kind of nonharmonic tone:

In both of the PACs shown above, the resolution from $\hat2$ to $\hat1$ in the upper voice is decorated with a type of incomplete upper neighbor tone—an échappée or escape tone. This is still considered a PAC, even though the upper voice does not resolve directly to $\hat1$ with stepwise motion.

Despite the voice-leading restrictions that define a PAC, there is quite a bit of variety in how conclusive these cadences sound. The finality of a cadence is largely determined by the melodic motion in the outer voices, but as we have already suggested, there are many other factors as well. Consider the two PACs in the following example:

The PAC in m. 16 might be said to sound slightly more conclusive than the one in m. 8. Many listeners feel that descending melodic resolutions feel more restful than their ascending counterparts. To such listeners, the second PAC in Example 22–12 feels more conclusive than the second because both voices move to $\hat1$ in a downward direction. Keep in mind that variations between cadences of the same type are often very subtle and that their effect may be influenced by subjective listening experiences. Different listeners, in other words, hear different cadences in different ways.

Any dominant-to-tonic cadence that is not a PAC is said to be an imperfect authentic cadence (IAC). The cadence seen earlier in Example 22–4a is an IAC because, even though the V and the following I are both in root position, the melody does not resolve by step to $\hat1$. Instead we hear the comparatively less conclusive motion from $\hat4$ to $\hat3$.

The following example shows another IAC, this time in D major:

As the V7 moves to I in mm. 3–4, we hear $\hat4$ (the seventh of the dominant chord) stepping down to $\hat3$ (the third of the tonic). Again, the effect is less conclusive than if the outer voices had both come to rest on octave $\hat1$s.

In the following example, the authentic cadence is considered imperfect because of melodic events in both voices:

Again, we see the upper voice resolving to the third of the tonic from scale degree $\hat4$ (this time as a passing seventh from the root of the dominant). Supporting this in the bass we find a dominant seventh in second inversion resolving to a root-position tonic. With a step from $\hat2$ to $\hat1$ in the bass instead of a leap from $\hat5$ to $\hat1$, the cadence sounds much less conclusive than a PAC.

The first three phrases in the following example all end with IACs:

Each of the first three cadences in this excerpt is imperfect because the dominant chord (V6/5) appears in inversion. Each of these cadences is also imperfect because of the melody in the highest voice: $\hat2$ $\hat3$ in the first and third cadence and $\hat4$ $\hat3$ in the second. Finally, in mm. 15-16, we hear root-position harmonies and stepwise motion to $\hat1$ in the highest voice: a PAC that brings a sense of closure to the passage.

So far we have seen only cadences with V or V7 as the functional dominant. Now consider the IAC at the end of the following phrase:

The penultimate chord in this phrase is a diminished leading-tone triad: viio6. All three chord members—F#, A, and C ( $\hat7$, $\hat2$, and $\hat4$, respectively)—are tendency tones that resolve in the same manner as when they appear in a V7 chord. Notice the similarities between the following three chords:

Every V chord includes scale degrees $\hat7$ and $\hat2$, strong tendency tones that usually resolve to $\hat1$. A V7 chord adds $\hat4$, another tendency tone that pulls toward a stable chord member in a tonic harmony ( $\hat3$). A viio chord has all of the same tendency tones but without the root of a typical dominant-function chord. In Example 22–16 we see viio6 performing a dominant function, resolving to I in a phrase-ending cadence. We may think of viio6, then, as a weak dominant substitute.

The following example shows a similar progression:

This IAC is even weaker than the one in Example 22–16. The melody resolves conclusively from $\hat2$ to $\hat1$, but the dominant substitute viio6, resolves to a tonic in first inversion. With $\hat3$ in the bass at its conclusion, the phrase feels very unsettled.

Activity 22-3

Activity 22–3

### Question

The following excerpt has four cadences, three of which are authentic cadences. Which of the three authentic cadences are perfect and which are imperfect? Which is not an authentic cadence?

Hint

A PAC will have $\hat5$ $\hat1$ in the bass and either $\hat2$ $\hat1$ or $\hat7$ $\hat1$ in the highest voice.

The authentic cadences in m. 4 and m. 12 are IACs. The authentic cadence in m. 16 is a PAC. The cadence in m. 8 is not an authentic cadence.

As we have seen, an authentic cadence consists of a two-part harmonic gesture at the end of a phrase: an unstable dominant-function chord resolves to the tonic and conveys a sense of conclusiveness. In the cadences at the end of the following examples, we hear only the first half of this gesture:

Both of the phrases above end with a dominant harmony. We do not hear the expected resolutions to the tonic. Such a phrase ending is known as a half cadence, since it consists of only the first half of an authentic cadence. (The half cadence is labeled “HC” in the example above.) The effect of a half cadence is remarkably different from what we have heard so far. It sounds very unresolved. Rather than moving to the expected points of stability, the tendency tones in the dominant are left hanging.

A half cadence brings a sense of closure to a phrase, but the listener is left wanting more. This expectation for harmonic resolution is often fulfilled by the following phrase, as in the following two examples:

In each of the cases above, the first phrase ends with an inconclusive half cadence. A second, similar phrase follows and ends with an authentic cadence providing the expected resolution of the dominant. Note that in each of the examples above, the first chord after the half cadence is a tonic harmony. This should not be confused with an authentic cadence. In an authentic cadence, the tonic is heard at the end of a phrase. Here, the tonic in m. 5 of each example appears at the beginning of a new and separate phrase and so is not considered a part of the cadence.

Because a half cadence so strongly implies a need for further music, it can be sometimes be difficult to distinguish them from their context. In the following example, a half cadence may be heard at the end of m. 4, followed by a PAC four bars later:

In Example 22–23, a stream of sixteenth notes flows steadily until the second beat of m. 8. It is possible, then, to hear this as a single, eight-bar phrase. There are, however, several factors that break the passage into two four-bar phrases: the contour of the sixteenth-note figure is altered and switches to the left hand in m. 5, the bass repeats a note (B) for the first time, and the dotted-quarter-note melody in mm. 5–7 (B–C–B–A–B) recalls the melody heard in mm. 1–3. Listen again to Example 22–21 and Example 22–22 and compare them to Example 22–23.

Note: Phrases often work together. In each of the three examples above, we heard a pair of similar phrases, the first ending inconclusively with a half cadence and the second ending conclusively with an authentic cadence. Such a pair of phrases is known as a period and will be discussed in greater detail in Chapter 35.

Activity 22-4

Activity 22–4

### Question

Identify each of the cadences in the following excerpt:

Hint

Consider how conclusive each cadence sounds, and keep in mind the various parameters used to categorize different cadences.

The cadence in m. 4 is an IAC, the cadence in m. 8 is a HC, the cadence in m. 12 is an IAC, and the cadence in m. 16 is a PAC..

In the following example, the first phrase ends with an unexpected harmony:

The V7 at the end of m. 35 resolves to a vi chord. This is known as a deceptive cadence. The cadential dominant sets up an expectation for a resolution to the tonic, but moves instead to the submediant. The only difference between a deceptive cadence and the corresponding authentic cadence is that the bass moves to $\hat6$ instead of $\hat1$. Compare this deceptive cadence to the PAC at the end of the second phrase, where the bass leaps up as expected to Eb.

Note: Deceptive cadences are much less common than authentic and half cadences. It is rare to encounter a piece of tonal Western art music that does not include authentic and half cadences. Deceptive cadences, on the other hand, appear only occasionally and a majority of compositions do not include them at all.

The effect of a deceptive cadence is somewhat surprising, though the vi chord does not seem entirely out of place. Consider the similarities between a I chord and a vi chord:

The vi chord shares two out of the three scale degrees that make up a tonic triad: $\hat1$ (the root of the I chord) and $\hat3$ (the quality determining third of the I chord). Only the fifth of the I chord ( $\hat5$), the least essential member, is missing. Like how the viio chords in Example 22–17 and Example 22–18 have the potential to perform a dominant function due to their similarity with V, a vi chord may be seen as a weak tonic substitute.

The following examples both include deceptive cadences:

In each of the examples above, the deceptive cadence is soon followed by a conclusive PAC. The deceptive cadence sets up an expectation, the fulfillment of which is delayed until the PAC provides the implied tonic harmony. The effect is a provocative stretching out of the gesture which for many listeners makes the eventual conclusion all the more satisfying.

Activity 22-5

Activity 22–5

In this exercise, you will resolve dominant chords deceptively.

### Question

Resolve the following dominant chord with a deceptive cadence:

Hint

Aside from the bass, all of the other voices may move as though the dominant were resolving to a tonic triad. (Beware of parallel octaves with the bass!)

### Question

Resolve the following dominant chord with a deceptive cadence:

Hint

Aside from the bass, all of the other voices may move as though the dominant were resolving to a tonic triad.

### Question

Resolve the following dominant chord with a deceptive cadence:

Hint

Aside from the bass, all of the other voices may move as though the dominant were resolving to a tonic triad. (Beware of parallel octaves with the bass.)

### Question

Resolve the following dominant chord with a deceptive cadence:

Hint

Aside from the bass, all of the other voices may move as though the dominant were resolving to a tonic triad.

# 22.8 Summary

Chords can have different musical meanings depending on their context in a piece or passage. We refer to this meaning as a chord’s harmonic function. The two most important functions in tonal Western art music are the tonic function and the dominant function. The tonic function—usually performed by a tonic triad—conveys a sense of restfulness to a listener. It is characteristically consonant and stable. At the beginning of a musical expression, it establishes the key and gives the listener a sense of tonal groundedness. At the end of a musical expression, it conveys a sense of repose and, after contrasting harmonic material, a sense of closure. The dominant function—usually performed by a V or V7 chord—acts as a foil to the tonic. It contains tendency tones which convey a sense of urgency to resolve to points of greater stability.

A phrase is a passage of music, typically four bars long, that conveys a musical idea. The harmonic gesture that signals the end of a phrase is known as a cadence. The relationship between tonic and dominant is most apparent at cadences, which are ranked and labeled according to how conclusive they sound to a listener.

An authentic cadence is one in which a dominant-function chord resolves to a tonic chord. A perfect authentic cadence (PAC) is the most stable and resolved type of cadence. It consists of a root position dominant chord (V or V7) moving to a root position tonic (I or i) with stepwise motion to $\hat1$ in the highest voice. Any authentic cadence that does not fulfill these stipulations is referred to as an imperfect authentic cadence. A cadence may be imperfect for one or more of several reasons: either the dominant or tonic is in an inverted position, the highest voice ends on some scale degree other than $\hat1$, or the V chord has been replaced with a substitute dominant-function chord such as viio.

A half cadence ends a phrase with only the first half of an authentic cadence: the dominant. Compared to an authentic cadence, a half cadence feels very unresolved and is characteristically followed by a second phrase ending more conclusively. A deceptive cadence swaps out the tonic triad with a weak tonic substitute: the submediant—which, like the viio chord, has several scale degrees in common with the chord it replaces. Authentic and half cadences are extremely common, deceptive cadences much less so.