II. Diatonic Polyphony and Functional Harmony

# 20.1 Introduction

Consider the particularly dissonant chord on the downbeat of m. 2 in the following excerpt:

This sonority, a fully-diminished seventh chord built on the leading tone of G minor, is comprised of two interlocking tritones, both of which resolve in the manner described in Chapter 16. The tritone between F# and C resolves inward to a third and the tritone between the A and Eb does the same. The chord reappears in mm. 3–4, though here the members have been rearranged. Each chord member resolves in the same way, but now the tritone between F# and C has been inverted and resolves outward to a sixth. The effect of these two chords is striking, but the voice-leading smooth and consistent with that seen in most tonal Western art music.

In this chapter, we will discuss different kinds of seventh chords built on scale degree $\hat7$, the leading tone. These sonorities have a unique effect that composers exploit in a number of ways. After examining the construction of these chords, we will go on to discuss the ways in which they typically resolve and the roles they typically play in a musical context. Leading-tone seventh chords may be half-diminished or fully-diminished, but we will here pay particular attention to the latter. As we have already seen, fully diminished seventh chords consist of two tritones and therefore require careful treatment due to their strong voice-leading tendencies.

# 20.2 Construction: viio7 in minor and viiø7 in major

In Chapter 16 we discussed the diminished leading-tone triad: viio. There, we examined why the tritone between the root and fifth of the chord requires special attention. The chord usually appears in first inversion precisely to avoid the dissonant interval sounding against the bass when viio is in root position.

As Example 20–2 demonstrates, placing the chord in first inversion ensures that the upper voices are consonant with the bass. The diminished fifth is between the alto and soprano, concealed within the upper voices. It is therefore best understood as a resultant interval formed by avoiding dissonances that involve the bass.

Leading-tone seventh chords (viio7 in minor and viiø7 in major) are even more dissonant than the triads on which they are based. The following example shows leading-tone seventh chords in both C minor and C major:

Both of these chords add a dissonant seventh to an already dissonant triad. In a minor key, the chordal seventh above a leading-tone triad forms a diminished seventh above the root. (See Example 18–16a.) This is a fully-diminished seventh chord. The prefix “fully-” refers to the fact that the chord is constructed of a diminished triad and a diminished seventh.

The dissonant sound of a fully-diminished seventh chord is striking. The combination of a diminished triad with a diminished seventh above the root yields two interlocking tritones. One between $\hat7$ and $\hat4$, the other between $\hat2$ and $\hat6$:

As Example 20–4 demonstrates, every member of the chord forms a tritone with some other member. Regardless of the chord’s position, then, one of the tritones inevitably involves the bass:

Unlike the viio6 chord, where the single tritone can be hidden between upper voices, one of the two tritones in a fully-diminished chord will be accentuated by the bass and, as we will discuss momentarily, must be treated with care.

Activity 20-1

Activity 20–1

Identify the diminished fifths and augmented fourths in the following inverted fully-diminished seventh chords.

### Question

Which two pitches form the tritone between the root and fifth of the following fully-diminished seventh chord? Which two voices form the tritone between the third and seventh? (Keep in mind that a tritone may appear as a diminished fifth or in inversion as an augmented fourth.)

Tritone between root and fifth: G# (tenor) and D (alto)

Tritone between third and seventh: B (bass) and F (soprano)

### Question

Which two pitches form the tritone between the root and fifth of the following fully-diminished seventh chord? Which two voices form the tritone between the third and seventh? (Keep in mind that a tritone may appear as a diminished fifth or in inversion as an augmented fourth.)

Tritone between root and fifth: A# (soprano) and E (bass)

Tritone between third and seventh: C# (alto) and G (tenor)

### Question

Which two pitches form the tritone between the root and fifth of the following fully-diminished seventh chord? Which two voices form the tritone between the third and seventh? (Keep in mind that a tritone may appear as a diminished fifth or in inversion as an augmented fourth.)

Tritone between root and fifth: B§ (tenor) and F (bass)

Tritone between third and seventh: D (alto) and Ab (soprano)

### Question

Which two pitches form the tritone between the root and fifth of the following fully-diminished seventh chord? Which two voices form the tritone between the third and seventh? (Keep in mind that a tritone may appear as a diminished fifth or in inversion as an augmented fourth.)

Tritone between root and fifth: C# (tenor) and G (alto)

Tritone between third and seventh: E (bass) and Bb (soprano)

In a major key, on the other hand, the diatonic seventh of a leading-tone chord does not form a diminished seventh. (Recall Example 18–16b.) Scale degree $\hat6$ instead forms a minor seventh above the root. This sonority is therefore referred to as a half-diminished seventh chord. Half-diminished seventh chords occur somewhat less frequently than their fully-diminished counterparts, and only rarely in minor keys.

# 20.3 Resolving a viio7 chord

Resolutions of leading-tone seventh chords follow many of the same voice-leading conventions as the viio triad. In all chords built on the leading tone, scale degree $\hat7$ forms a dissonant tritone with scale degree $\hat4$, a dissonance that must be resolved properly. In Chapter 16 we expanded the list of basic interval progressions to accommodate chords that include a tritone. There, we discussed several possible resolutions: augmented fourths resolve outward to sixths or in similar motion up to perfect fourths while diminished fifths resolve inward to thirds or in similar motion up to a perfect fifth. Example 20–6 summarizes:

The tritone formed by the leading tone and scale degree $\hat4$ usually resolves by contrary motion. If the tritone appears as a diminished fifth, both voices typically resolve inwards by step to form a third. If, on the other hand, the tritone is in the form of an augmented fourth, the voices will expand outwards by step to form a sixth. In either case, scale degrees $\hat7$ and $\hat4$ usually fulfill their tendencies to resolve to $\hat1$ and $\hat3$, respectively.

As discussed above, however, the fully-diminished viio7 chord contains an additional tritone (between scale degrees $\hat2$ and $\hat6$). The same rules for resolution apply to this interval. Typically, this has scale degree $\hat2$ stepping to either $\hat1$ or $\hat3$ while scale degree $\hat6$ steps down to $\hat5$.

The following example shows a typical resolution of a viio7 chord to i in C minor:

In Example 20–7, the bass (B§) forms a diminished fifth with the tenor (F). As viio7 resolves to i, we can see this tritone contracting to a minor third (C and Eb). Likewise, the augmented fourth between the alto and soprano (Ab and D) expands to form a minor sixth (G and Eb). Note the resulting doubled third in the tonic, which is common after fully-diminished leading-tone chords. Typically, contrary motion of this sort is the favored method of resolving tritones. Composers will occasionally resolve a tritone using similar motion, but will frequently restrict such a progression to the upper voices.

Most of the tones in a fully-diminished leading-tone chord have a strong tendency to resolve to the pitches of a tonic triad. The leading tone, for example, is pulled towards scale degree $\hat1$. The seventh of the chord (b $\hat6$) resolves like any other seventh: down by step (in this case to $\hat5$). (Refer to Chapter 18 for more information on seventh chords.) And finally, $\hat4$ is drawn downward to $\hat3$. For these reasons, it may be helpful to think about the resolution of a fully-diminished seventh chord in terms of its tendency tones.

Look again at Example 20–7. Each of the tendency tones resolves as expected: the leading tone steps up to the tonic in the bass while $\hat6$ steps down to $\hat5$ in the alto and $\hat4$ steps down to $\hat3$ in the tenor. The remaining voice, scale degree $\hat2$, can move to either $\hat3$as it does in Example 20–7—or to the tonic, as in the following example:

Note that in Example 20–8 the augmented fourth formed by the alto and soprano now resolves with similar motion to a perfect fourth.

Fully-diminished seventh chords resolve similarly in any position:

Third-inversion fully-diminished seventh chords are less common than the other positions. With scale degree $\hat6$ in the bass (b $\hat6$ in major), viio4/2 tends to resolve to a chord built using the pitches of a tonic triad but with scale degree $\hat5$ in the bass. (This sonority is commonly known as a “cadential 6/4 chord” and is discussed in greater detail in Chapter 23.) Note that in the resolution of viio4/2 in Example 20–9 scale degree $\hat4$ in the alto voice steps up to $\hat5$, resolving the augmented fourth in similar motion to a perfect fourth.

Now consider the following example (Example 20–10b provides a reduction of mm. 5–6):

In m. 5 we encounter a fully-diminished seventh chord in second inversion. Looking at the left-hand part, we see that all of the voices resolve as expected. Both of the tritones appear as augmented fourths and expand outward by contrary motion to sixths: F and B§ move in contrary motion to Eb and C while Ab and D do the same, to G and Eb.

A very similar resolution may be found in the following excerpt:

The voice leading in this example is nearly identical to what we saw in Example 20–10. The only difference is that here the tritone between scale degrees $\hat2$ and $\hat6$ (G and Db) is written as a diminished fifth and resolves inward to a third (Ab and C). The other tritone, the augmented fourth between scale degrees $\hat4$ and $\hat7$ (Bb and E§), resolves in the manner described above: by contrary motion outward to a sixth (Ab and F).

Activity 20-2

Activity 20–2

Resolve the following fully-diminished seventh chords according to the voice leading procedures outlined above.

### Question

To what chord would the following fully-diminished seventh in first inversion normally resolve to?

Hint

Which chord member is in the bass? To where does this pitch normally resolve?

i6

### Follow-up question

Resolve the fully-diminished seventh chord:

Hint

Remember to resolve all tendency tones in the usual manner.

### Question

To what chord would the following fully-diminished seventh in second inversion normally resolve to?

Hint

Which chord member is in the bass? To where does this pitch normally resolve?

i6

### Follow-up question

Resolve the fully-diminished seventh chord:

Hint

Remember to resolve all tendency tones in the usual manner.

(Alternatively, C# in the alto may resolve to B).

### Question

To what chord would the following fully-diminished seventh in third inversion normally resolve to?

Hint

Which chord member is in the bass? To where does this pitch normally resolve?

cadential 6/4 or auxiliary 6/4

### Follow-up question

Resolve the fully-diminished seventh chord:

Hint

Remember to resolve all tendency tones in the usual manner.

### Question

To what chord would the following fully-diminished seventh in root position normally resolve to?

Hint

Which chord member is in the bass? To where does this pitch normally resolve?

I

### Follow-up question

Resolve the fully-diminished seventh chord:

Hint

Remember to resolve all tendency tones in the usual manner.

# 20.4 Resolving a viiø7 chord

The resolution of a half-diminished seventh chord is similar to that of a fully-diminished seventh chord. The tritone formed by the leading tone and scale degree $\hat4$ should again resolve according to the interval progressions outlined above. Unlike fully-diminished seventh chords, however, half-diminished seventh chords contain only one tritone since the interval between scale degrees $\hat2$ and $\hat6$ is a perfect fifth instead of a diminished fifth. The resolution of a half-diminished seventh chord therefore runs a greater risk of parallel fifths:

In Example 20–12, the alto ( $\hat2$) and soprano ( $\hat6$) both resolve down by step, resulting in parallel fifths. This can be avoided by having $\hat2$ resolve upwards to $\hat3$, similar to what was seen in Example 20–7. The following resolution avoids the parallel fifths by doing just this:

Note again that resolving $\hat2$ up to $\hat3$ instead of down to $\hat1$ will result in a doubled third in the chord of resolution.

Activity 20-3

Activity 20–3

In this activity, you will be presented with a series of fully-diminished and half-diminished seventh chords. For each exercise you will be asked to identify the tritones and then resolve the chord to the tonic triad.

### Question

Identify the tritones in the viio7 chord:

Hint

Remember, in a viio7 chord, scale degrees $\hat7$ and $\hat4$ form one tritone and $\hat2$ and $\hat6$ form the other.

G# and D form a diminished fifth, while B and F also form a diminished fifth.

### Follow-up question

Now resolve the viio7 to i.

### Question

Identify the tritones in the viio7 chord:

Hint

Remember, in a viio7 chord, scale degrees $\hat7$ and $\hat4$ form one tritone and $\hat2$ and $\hat6$ form the other.

C# and G form a diminished fifth, while Bb and E form augmented fourth.

### Follow-up question

Now resolve the viio7 to i.

### Question

Identify the tritone in the viiø7 chord:

Hint

Remember, in a viiø7 chord, scale degrees $\hat7$ and $\hat4$ form a tritone.

C# and G form a diminished fifth.

### Follow-up question

Now resolve the viiø7 to I.

### Question

Identify the tritone in the viiø7 chord:

Hint

Remember, in a viiø7 chord, scale degrees $\hat7$ and $\hat4$ form a tritone.

D and Ab form a diminished fifth.

### Follow-up question

Now resolve the viiø7 to I.

# 20.5 viio7 chords in major keys

The sound of a fully-diminished leading-tone chord is both striking and immediately recognizable, a characteristic that composers will often exploit even when the chord does not belong to the key at hand. Unlike viiø7 chords, which typically appear only in major-key contexts, viio7 chords are commonly encountered in both major and minor. The following example shows the construction of the chord in C major and in C minor:

As Example 20–14 demonstrates, fully-diminished seventh chords cannot be constructed from diatonic notes alone. In other words, a tone borrowed from the parallel key is always required. The viio7 in C major borrows Ab (b $\hat6$) from C minor and the viio7 chord in C minor borrows B§ (the leading tone) from C major.

The following example is in Ab major and includes several prominent fully-diminished leading tone sevenths:

The leading-tone chords on the downbeats of the opening bars in this excerpt all include Fb instead of F§. Combined with the suspended Cs in the right hand, these borrowed tones enhance the peculiar, dissonant character of the passage.

Activity 20-4

Activity 20–4

Build fully-diminished seventh chords on the leading tone of each of the following keys. (Remember to use b $\hat6$ in major keys and to raise the leading tone in minor.)

### Question

Build a root position fully-diminished leading tone seventh chord in D minor.

Hint

Remember to raise the leading tone in minor.

(Answers may vary, provided C# is in the bass with E, G, Bb in the upper voices.)

### Question

Build a root position fully-diminished leading tone seventh chord in F major.

Hint

Remember that viio7 uses b $\hat6$ in major keys.

(Answers may vary, provided E is in the bass with G, Bb, Db in the upper voices.)

### Question

Build a root position fully-diminished leading tone seventh chord in E minor.

Hint

Remember to raise the leading tone in minor.

(Answers may vary, provided D# is in the bass with F#, A, C in the upper voices.)

### Question

Build a root position fully-diminished leading tone seventh chord in G major.

Hint

Remember that viio7 uses b $\hat6$ in major keys.

(Answers may vary, provided F# is in the bass with A, C, Eb in the upper voices.)

# 20.6 Diminished-seventh chords as dominant substitutes

As its name indicates, the leading-tone seventh chord includes the leading tone as its root. As discussed in Chapter 19, the leading tone is an important member of the dominant seventh chord. The similarity between these two chords, however, does not end here. The following example compares dominant seventh chords with fully-diminished leading-tone chords in both C major and C minor:

As you can see from Example 20–16, fully-diminished seventh chords built on the leading tone have three pitches in common with dominant sevenths (scale degrees $\hat7$, $\hat2$, and $\hat4$). The only difference is that they include b $\hat6$ (diatonic $\hat6$ in minor) instead of scale degree $\hat5$. (Recall from earlier in this chapter that fully-diminished seventh chords cannot be built using only diatonic notes and must include a note borrowed from the parallel key.) Because they share three chord members, fully-diminished leading-tone seventh chords typically function as substitutes for dominant seventh chords.

Notice the similarity between the V7 and viio4/3 chords in m. 8 of the following excerpt:

In the first half of m. 8 we see a dominant seventh chord with pitch classes E, G#, B, and D. In the second half of the same measure, after beat three, the D is transferred down an octave while the G# and B are held in place. The root of the V7 chord (E) steps up to form the seventh of the viio4/3 (F).

# 20.7 Summary

Leading-tone seventh chords are constructed by adding a seventh to a diminished triad whose root is scale degree $\hat7$. In minor keys, they are fully-diminished in quality. In major keys they are half-diminished, but will occasionally include a chromatic b $\hat6$ to make them fully-diminished. Both fully- and half-diminished seventh chords include a tritone between scale degrees $\hat7$ and $\hat4$ which typically resolves in the manner described in Chapter 16. Typically, composers resolve the tritones by contrary motion: augmented fourths expanding outward to sixths, diminished fifths contracting inward to thirds. Fully-diminished chords include a second tritone between $\hat2$ and $\hat6$. With all four chord members forming either a diminished fifth or augmented fourth with some other voice, one of the tritones will inevitably be emphasized by the bass. Occasionally, one of the tritones will resolve in similar motion to a perfect fourth or fifth, but that voice-leading is usually restricted to upper voices.

These chords, when built on the leading tone, include three strong tendency tones leading to pitches of the tonic triad. Scale degree $\hat7$ is pulled upward to $\hat1$ while $\hat4$ and $\hat6$ are pulled down to $\hat3$ and $\hat5$, respectively. $\hat2$ resolves either to $\hat1$ or $\hat3$ depending on context. While the chord can appear in any position, third-inversion fully-diminished chords are rare. With b $\hat6$ in the bass, they tend to resolve to a cadential 6/4 chord.