II. Diatonic Polyphony and Functional Harmony

17.1 Introduction

As discussed in Chapter 13, it is possible to build a triad of any quality on any given note. We saw that it is likewise possible to build triads on each scale degree of a major key using only diatonic notes—that is, using only the pitch classes belonging to that particular key. Since there are seven scale degrees, there are seven distinct triads. Together, these sonorities make up the harmonic palette of the key to which they belong.

But of these seven diatonic triads, one sounds unique. The triad built on the leading tone has a quality that puts it in stark contrast to those built on the other six scale degrees.

Activity 17-1

Activity 17–1

Question

What makes the viio chord unique?

1. It is built entirely of scale steps from the corresponding scale.
2. It is dissonant.
3. It is a minor triad.

Answer B is correct. It is dissonant: the viio chord constitutes the only dissonant member of the set of diatonic triads. All of the other triads are consonant; they are either major or minor. (Answer A: While the viio chord is made entirely of diatonic pitches, it is not alone. All of the diatonic triads are built using only scale steps. Answer C: Although the symbol for the viio chord is written with lowercase letters, it also has a small circle indicating that it is not minor, but rather diminished.)

With the exception of the viio chord, all of the diatonic triads in a major key are either major or minor. The triads built on scale degrees $\hat1$, $\hat4$, and $\hat5$ are major while those built on $\hat2$, $\hat3$, and $\hat6$ are minor:

Triads I through vi are framed by a perfect fifth. They are all heard as being consonant. The triad built on scale degree $\hat7$, however, is dissonant. The viio chord is dissonant because it is framed instead by a diminished fifth:

In our discussions of basic interval progressions (Chapter 12 and Chapter 14), we adhered to the practice of using only consonant intervals. When dissonances appeared, they were shown to be a byproduct of consonant progressions between other pairs of voices happening at the same time. We called them resultant intervals. The diminished fifth of the viio chord, however, is inherently dissonant. We must therefore make an exception to the constraints described earlier when we use this chord in three- and four-part textures.

When one of the tritone forming notes—the root or fifth—appears in the bass, composers generally feel that the dissonant interval sounds too harsh. In order to avoid accentuating the dissonance in this manner, they invert the viio chord. The third of the chord—scale degree $\hat2$appears in the bass and the inversion is indicated with the bass figure 6 appearing next to the Roman numeral, as shown here:

Putting the triad in first inversion creates consonant intervals between the bass and all of the upper voices. In this case, the bass forms a sixth with the tenor (hence the bass figure 6), a third with the alto, and an octave with the soprano. The diminished fifth is hidden in the inner voices between the tenor and alto: it is a resultant interval formed by the avoidance of dissonance with the bass.

Activity 17-2

Activity 17–2

This exercise will reinforce your understanding of the various intervals found in a viio6 in SATB setting.

Question

Identify the two pitches forming the tritone (augmented fourth/diminished fifth) in the following viio6 chord:

C# and G

Follow-up question

What intervals do these pitches form with the bass?

C# forms a M6 (major sixth) with the bass and G forms a m3 (minor third) with the bass.

Follow-up question

Is the tritone in this chord a d5 (diminished fifth) or A4 (augmented fourth)?

d5 (diminished fifth)

Question

Identify the two pitches forming the tritone (augmented fourth/diminished fifth) in the following viio6 chord:

A and D#

Follow-up question

What intervals do these pitches form with the bass?

A forms a m3 (minor third) with the bass and D# forms a M6 (major sixth) with the bass.

Follow-up question

Is the tritone in this chord a d5 (diminished fifth) or A4 (augmented fourth)?

A4 (augmented fourth)

Question

Identify the two pitches forming the tritone (augmented fourth/diminished fifth) in the following viio6 chord:

G and Db

Follow-up question

What intervals do these pitches form with the bass?

G forms a M6 (major sixth) with the bass and Db forms a m3 (minor third) with the bass.

Follow-up question

Is the tritone in this chord a d5 (diminished fifth) or A4 (augmented fourth)?

d5 (diminished fifth)

Question

Identify the two pitches forming the tritone (augmented fourth/diminished fifth) in the following viio6 chord:

C# and G

Follow-up question

What intervals do these pitches form with the bass?

C# forms a M6 (major sixth) with the bass and G forms a m3 (minor third) with the bass.

Follow-up question

Is the tritone in this chord a d5 (diminished fifth) or A4 (augmented fourth)?

d5 (diminished fifth)

17.2 Treatment of the viio chord

Dissonances are unstable and therefore require motion towards stability (resolution). In order to allow tritones to be used under these circumstances—as they routinely are in music—we must expand our catalog of interval progressions. There are four distinct interval progressions associated with the tritone: two with contrary motion and two with similar motion. The choice of interval progression depends on the movement of the bass, as we shall see. If the tritone appears as an augmented fourth, its voices may move by step in contrary motion, expanding to either a minor sixth in major keys or a major sixth in minor keys:

Contrary motion, such as in the above example, is the preferred type of intervallic movement involving a tritone. Alternatively, the voices may move in similar motion to a perfect fourth (the same is true for major or minor keys):

If the tritone appears as a diminished fifth, the voices usually contract to a third (a major third in major keys or a minor third in minor keys):

Although less common, they may also move in similar motion to a perfect fifth:

At first glance, Example 17–7 appears to be a case of parallel fifths. Recall, however, that parallel perfect fifths are avoided in this style of music because they undermine the independence of the voices. Voices a perfect fifth apart blend together so well that they almost sound as though they are singing the same pitch. In the progression from a diminished fifth to a perfect fifth, the voices sing two very different intervals. Each voice is clearly heard as the dissonance of the tritone resolves to the stable consonance.

Note: The resolution of a tritone by similar motion is less common than the resolution by contrary motion. Because it resembles parallel fifths, some teachers and texts advise students to avoid it altogether.

Although this resolution of the tritone is less common than those shown in Example 17–4 through Example 17–6, it does appear in the tonal repertoire. Example 17–8 and Example 17–9 show two excerpts from Bach chorales where diminished fifths resolve to perfect fifths (in both cases, the tritone occurs between the alto and soprano):

Similar-motion resolutions of the tritone (A4 to P4 and d5 to P5) may seem counterintuitive. Their validity is explained by the interval progressions formed with the bass. In both cases, one of the voices moves in parallel thirds with the bass while the other creates parallel sixths with the bass.

Look again at Example 17–8. The alto and soprano voices form a tritone on beat three which resolves to a perfect fifth on beat four. Now consider the intervals formed with the bass; the alto moves in parallel sixths with the bass while the soprano moves in parallel thirds with the bass. The same is true for Example 17–9. The interval progressions formed with the bass validate the similar motion in the upper voices. With permissible interval progressions occurring between the bass and each of the upper voices, the tritone in each case may be thought of as a resultant interval, and the similar-motion progression as a byproduct.

It is also worth mentioning that the same interval progression in reverse—movement from a P5 to a d5—is perfectly acceptable and happens regularly in common practice.

Activity 17-3

Activity 17–3

In each of the following examples you are presented with a tritone (consisting of the leading tone and scale degree $\hat4$) and one of the notes in the resolution. Complete each tritone resolution by providing the pitch for the second voice.

Question

Hint

Remember, both voices must move by step when resolving a tritone. You cannot have disjunct motion in the lower voice.

Stepping up to E would be a valid answer. If the lower voice ascends to E, the tritone resolves from an A4 (augmented fourth) to a P4 (perfect fourth). Stepping down to C# would be another valid answer. If the lower voice descends to C, the tritone resolves from an A4 (augmented fourth) to a m6 (minor sixth). (Note: Sustaining D would not a valid answer as oblique motion is not permitted when resolving a tritone.)

Question

Hint

Remember, both voices must move by step when resolving a tritone. You cannot have disjunct motion in the lower voice.

Stepping up to A is the only valid answer. If the lower voice ascends to A, the tritone resolves from a d5 (diminished fifth) to a m3 (minor third). (Note: Sustaining G# would not be a valid answer as oblique motion is not permitted when resolving a tritone. Stepping down to F would not be a valid answer either as the tritone may only resolve from a d5 (diminished fifth) to a P5 (perfect fifth) if both voices are ascending by step. In this example, the lower voice may not descend because it is the leading tone and must resolve upwards to the tonic.)

Question

Hint

Remember, both voices must move by step when resolving a tritone. You cannot have disjunct motion in the upper voice.

Stepping up to F would be a valid answer. If the upper voice ascends to F, the tritone resolves from an d5 (diminished fifth) to a P5 (perfect fifth). Stepping down to D would also be a valid answer. If the upper voice descends to D, the tritone resolves from an d5 (diminished fifth) to a M3 (major third). (Note: Sustaining Eb would not be a valid answer as oblique motion is not permitted when resolving a tritone.)

Question

Hint

Remember, both voices must move by step when resolving a tritone. You cannot have disjunct motion in the upper voice.

Stepping up to F is the only valid answer. If the upper voice ascends to F, the tritone resolves from an A4 (augmented fourth) to a M6 (major sixth). (Note: Stepping down to Db would not be a valid answer. The tritone may only resolve from a A4 (augmented fourth) to a P4 (perfect fourth) if both voices are ascending by step. In this example, the upper voice may not descend because it is the leading tone and must resolve upwards to the tonic. Sustaining E would also not be a valid answer as oblique motion is not permitted when resolving a tritone.)

Activity 17-4

Activity 17–4

[Activity 17–4:

In each of the following exercises, you will be presented with a tritone (consisting of the leading tone and scale degree $\hat4$). Resolve the tritone according to the rules given in this chapter by first providing a pitch for the upper voice, then one for the lower voice.

Question

Hint

Remember, both voices must move by step when resolving a tritone. You cannot have disjunct motion in either voice.

Upper voice: Stepping up to E is the only valid answer. In this exercise, the upper voice has the leading tone and must resolve to the tonic. (Note: Sustaining D# would not be a valid answer as oblique motion is not permitted when resolving a tritone.)

Lower voice: Stepping up to B would be a valid answer. If the lower voice ascends to B, the tritone resolves from an A4 (augmented fourth) to a P4 (perfect fourth). Stepping down to G would also be a valid answer. If the lower voice descends to G, the tritone resolves from an A4 (augmented fourth) to a M6 (major sixth). (Note: Sustaining A would not be a valid answer as oblique motion is not permitted when resolving a tritone.)

Question

Hint

Remember, both voices must move by step when resolving a tritone. You cannot have disjunct motion in either voice.

Upper voice: Stepping up to C is the only valid answer. In this exercise, the upper voice has the leading tone and must resolve to the tonic. (Note: Sustaining B would not be a valid answer as oblique motion is not permitted when resolving a tritone.)

Lower voice: Stepping up to G would be a valid answer. If the lower voice ascends to G, the tritone resolves from an A4 (augmented fourth) to a P4 (perfect fourth). Stepping down to E would also be a valid answer. If the lower voice descends to E, the tritone resolves from an A4 (augmented fourth) to a M6 (major sixth). (Note: Sustaining F would not be a valid answer as oblique motion is not permitted when resolving a tritone.)

Question

Hint

Remember, both voices must move by step when resolving a tritone. You cannot have disjunct motion in either voice.

Upper voice: The upper voice may move by step to either B or G#. (Note: Sustaining A would not be a valid answer as oblique motion is not permitted when resolving a tritone.)

Lower voice (with upper voice moving to B): Stepping up to E is the only valid answer. If the upper voice ascends out of a d5 (diminished fifth), the lower voice must also ascend. The tritone then resolves from a d5 to a P5 (perfect fifth). Furthermore, D# is the leading tone and must resolve to the tonic. (Note: Stepping down to C# would not be a valid answer. If the lower voice descends to C#, the resulting interval will be a m7 (minor seventh). This is also a dissonance and cannot be used as a resolution from a tritone. Sustaining D# would also not be a valid answer as oblique motion is not permitted when resolving a tritone.)

Lower voice (with upper voice moving to G#): Stepping up to E is the only valid answer. If the lower voice ascends to E, the tritone resolves from a d5 (diminished fifth) to a M3 (major third). Furthermore, D# is the leading tone and must resolve to the tonic. (Note: Stepping down to C# would not be a valid answer. The tritone may only resolve from a d5 (diminished fifth) to a P5 (perfect fifth) if both voices are ascending by step. In this example, the lower voice may not descend because it is the leading tone and must resolve upwards to the tonic. Sustaining D# would also not be a valid answer as oblique motion is not permitted when resolving a tritone.)

Question

Hint

Remember, both voices must move by step when resolving a tritone. You cannot have disjunct motion in either voice.

Upper voice: The upper voice may move by step to either D or Bb. (Note: Sustaining C would not be a valid answer as oblique motion is not permitted when resolving a tritone.)

Lower voice (with upper voice moving to D): Stepping up to G is the only valid answer. If the upper voice ascends out of a d5 (diminished fifth), the lower voice must also ascend. The tritone then resolves from a d5 to a P5 (perfect fifth). Furthermore, F# is the leading tone and must resolve to the tonic. (Note: Stepping down to Eb would not be a valid answer. If the lower voice descends to Eb, the resulting interval will be a m7 (minor seventh). This is also a dissonance and cannot be used as a resolution from a tritone. Sustaining F# would also not be a valid answer as oblique motion is not permitted when resolving a tritone.)

Lower voice (with upper voice moving to Bb): Stepping up to G is the only valid answer. If the lower voice ascends to G, the tritone resolves from a d5 (diminished fifth) to a M3 (major third). Furthermore, F# is the leading tone and must resolve to the tonic. (Note: Stepping down to Eb would not be a valid answer. The tritone may only resolve from a d5 (diminished fifth) to a P5 (perfect fifth) if both voices are ascending by step. In this example, the lower voice may not descend because it is the leading tone and must resolve upwards to the tonic. Sustaining F# would also not be a valid answer as oblique motion is not permitted when resolving a tritone.)

We now turn to a progression involving the viio chord. In the following three-voice example, a viio6 chord resolves to I6:

In this case the tritone of the viio6 chord appears between the soprano and the middle voice. In the following example, the tritone is resolved using a d5–P5 progression:

More importantly, note that the bass forms familiar progressions with the soprano (3–3) and the middle voice (6–6). Because of these consonant interval progressions, we can think of the tritone as a resultant interval formed by pitches that are consonant with the bass. At this point, we can also add a fourth voice for a full SATB texture:

Activity 17-5

Activity 17–5

[Activity 17–5:

In each of the following exercises, you will be presented with a short progression consisting of a viio chord resolving to a I chord. In each case, you will be asked whether or not the voice leading is correct in every voice.

Question

Is the voice leading correct in this resolution of a viio chord? If not, which voice needs to be adjusted and to which pitch should it resolve?

Hint

Look at the alto voice.

No. The D# in the alto should step up to E. D# is the leading tone and must resolve to the tonic.

Question

Is the voice leading correct in this resolution of a viio chord? If not, which voice needs to be adjusted and to which pitch should it resolve?

Yes, all of the voices resolve properly. There is no need to adjust any of the voices.

Question

Is the voice leading correct in this resolution of a viio chord? If not, which voice needs to be adjusted and to which pitch should it resolve?

Yes, all of the voices resolve properly. There is no need to adjust any of the voices.

Question

Is the voice leading correct in this resolution of a viio chord? If not, which voice needs to be adjusted and to which pitch should it resolve?

Hint

Look at the soprano voice.

No. The D in the soprano should step up to Eb. D is the leading tone and must resolve to the tonic (Eb).

For the sake of clarity, we have been looking at only one example of a diminished triad so far: viio in a major key. Of course, this is not the only diminished triad you will encounter. Consider, for example, the triads of the minor scale. (Note that the V and viio chords in the following example include B§ instead of Bb. As discussed in Chapter 16, these altered chords are products of the harmonic minor and are far more common than their fully diatonic counterparts, v and VII.)

As you can see, there are two diminished triads in minor, one on scale degree $\hat2$ and one on scale degree $\hat7$. (In minor, viio is the result of the raised leading tone creating a diminished triad from a naturally major one.) In both cases, the tritone must be handled carefully according to the same interval progressions outlined above.

17.4 Summary

As the above examples show, one must adjust the constraints on using dissonances in order to accommodate the viio chord. Tritones are permitted when they are part of this particular triad. (Later, we will explore other permissible uses of the tritone. See, for example, Chapter 19.) However, because of this interval’s dissonant quality, it must be treated delicately. First, the viio chord appears in first inversion in order to avoid a dissonance with the bass. By inverting the triad, we hide the tritone within the inner voices, presenting it as a resultant interval. All of the upper voices are then consonant with the bass. Second, the tritone must also be resolved properly. This is to be done with both voices moving by step in either contrary or similar motion. Other tritone-containing chords such as the iio in minor are treated in the exact same fashion.