III. Modulation and Chromatic Harmony

# 30.1 Introduction

In Chapter 16, we discussed how minor-key compositions frequently borrow certain scale degrees from the parallel major: scale degree $\hat7$ in the case of the harmonic minor composite and scale degrees $\hat6$ and $\hat7$ in the melodic minor composite. As we saw in Chapter 29, however, the practice of sharing scale degrees between parallel keys goes far beyond these variants of the minor scale. Many tonal compositions—particularly those from the Romantic era—include passages where all three modal scale degrees ( $\hat3$ in addition to $\hat6$ and $\hat7$) move fluidly between their major and minor forms. By swapping certain notes for their parallel-key counterparts, composers are able to alter chord qualities and create some surprisingly dramatic harmonies. In this chapter we will look at other types of chromatic alterations. In these cases, though, the dramatic changes in chord quality cannot be explained as resulting from the interplay between parallel keys.

In describing these increasingly chromatic harmonies, we will push our chord-labeling system toward its limit. Roman numerals are a useful, shorthand way of naming chords, identifying their function, and showing their connection to a tonic. Although their primary purpose is to identify diatonic relationships, conventional Roman numerals may be adapted to indicate chromaticism when harmonic practice becomes more complex. Being able to recognize and use such adjustments is essential for analyzing a wide spectrum of music. This chapter will show how we expand Roman numeral conventions to deal with chromatic elements in cases of advanced mixture.

# 30.2 Roman numerals and simple mixture

Because Roman numerals indicate the quality of a chord, they are especially useful for analyzing instances of mixture, where triads change quality by incorporating scale degrees from the parallel key. Consider the triads built on $\hat5$ and $\hat7$ in a minor key. As discussed in Chapter 7 and Chapter 29, these two chords borrow the leading tone from the parallel major. The resulting difference in quality is reflected in the Roman numerals: v becomes V and VII becomes viio.

The following example shows several other examples of how Roman numerals may be used to indicate mixture:

In a major key, the triad built on scale degree $\hat1$ is naturally major (I). When mixture is applied to the chord—lowering the third to Eb, in this case—the Roman numeral is correspondingly changed to lowercase i to indicate the minor quality. Example 30–1 also demonstrates how, through mixture, ii (minor) becomes iio (diminished) and IV (major) becomes iv (minor).

The fourth case in Example 30–1 shows simple mixture applied to a vi chord in a major key. Here, the root of the chord is affected (A is lowered to Ab). When the root of a chord is altered, we put an accidental before the Roman numeral. The chord is labeled bVI: the b indicates that the root—scale degree $\hat6$, as indicated by VI—has been lowered by one semitone. The case of the Roman numeral also tells us that the chord is major, and requires Eb instead of E (with E the chord would have been augmented). The chord built on a lowered scale degree $\hat7$ (bVII) is labeled the same way.

Note: In Example 30–1, the altered submediant chord was labeled bVI in C major. This same alteration in E major would require a natural sign instead of a flat in front of the note representing degree $\hat6$:

Notice, however, that the lowered submediant in E major is still labeled bVI. This brings up an important idiosyncrasy about using accidentals with Roman numerals. A lowered root is always indicated by b and a raised root is always indicated by #, even in cases like Example 30–2 where it may not correspond with the accidentals appearing in the staff notation. (The chord built on # $\hat7$ in minor is simply labeled viio instead of #viio because it is so commonly used.)

You should become familiar enough with the qualities of diatonic chords that you can immediately recognize Roman numerals that indicate chromatic alterations. In the context of a major key, for instance, chords such as i and iio should instantly alert you to chromatic alterations.

# 30.3 Secondary mixture

All of the examples of mixture presented so far have been relatively straightforward. In each case, one or two tones are borrowed from the parallel key to emphasize or intensify certain harmonies. Occasionally, however, you will encounter instances of chromatic alterations that cannot be explained in such simple terms. The following excerpt includes two chromatically-altered harmonies, one in mm. 2–3 and another in m. 7:

The Ab-major chord in m. 2 is a familiar case of simple mixture. Scale degrees $\hat3$ and $\hat6$ are borrowed from the parallel key (Eb and Ab, respectively), changing the submediant triad from minor to major by lowering the root and fifth by a semitone each. The Roman numeral bVI indicates both the chord quality and the lowered root.

The sonority in m. 7, on the other hand, cannot be explained in these same terms. The major III chord is not a case of simple mixture since G# (# $\hat5$) is not drawn from the parallel minor key). In other words, an E-major triad uses pitches that do appear in C major or in C minor. This type of alteration, where a chord’s third is modified by an accidental that is not borrowed from the parallel key, is sometimes referred to as secondary mixture. Like regular mixture, the quality of the chord in question is changed, but not through borrowing of tones from the parallel key. In Example 30–3, scale degree $\hat5$ soon reverts back to G§ (m. 8).

The following example shows another, similar case of secondary mixture:

The chord in mm. 57–58 cannot be explained as a case of simple mixture. As with the G# in Example 30–3, the quality-altering chromatic pitch D# (# $\hat5$) is not drawn from the parallel minor. This is another case of secondary mixture. The chord is again labeled with an uppercase Roman numeral III to reflect the major quality.

Despite the chromaticism, we still use Roman numerals to identify the chords described above. The following example illustrates:

In instances of secondary mixture the quality of the chord in question is changed, but not through borrowing of tones from the parallel key. The minor quality of the diatonic ii chord becomes major (II) through secondary mixture. Likewise, iii and vi become III and VI, respectively. In each instance, the alteration is indicated by a change in the case (uppercase or lowercase) of the Roman numeral.

The following example shows both simple mixture (bVII) and secondary mixture (II):

The F-major harmony heard in mm. 93-94 is a secondary dominant tonicizing Bb. When it resolves in m. 95, however, we hear a Bb-major chord instead of the expected Bb-minor. This chord cannot be explained in terms of simple mixture since the parallel key (Ab-minor) does not include D§. This is a case of secondary mixture: the quality of the diatonic ii chord is changed simply by raising the chordal third to the chromatic scale degree # $\hat4$. The Gb-major chord in m. 97 (bVII), on the other hand, is a case of simple mixture—all three pitches (Gb, Bb, and Db) appear in the parallel key.

Secondary mixture appears in minor keys as well:

As shown by Example 30–7, III, VI, and VII in a minor key may be made minor through secondary mixture. In such cases, they are labeled iii, vi, and vii, respectively.

The following example includes a minor vii chord in a minor key:

Example 30–8 shows the concluding phrase from a chorale. On the third beat of m. 12, we find a case of secondary mixture. The third of the chord has been lowered to the chromatic Bb, changing what would diatonically be a major VII chord to a minor vii chord.

Activity 30-1

Activity 30–1

Identify each of the following chords in various keys with a Roman numeral. (Note: Each chord is in root position.)

### Question

Identify the following chord in Ab major with a Roman numeral:

Hint

Remember to use accidentals to indicate altered roots. The case of the Roman numeral should correspond with the quality of the triad.

III

### Follow-up question

Is this a case of simple or secondary mixture?

secondary

### Question

Identify the following chord in A major with a Roman numeral:

Hint

Remember to use accidentals to indicate altered roots. The case of the Roman numeral should correspond with the quality of the triad.

bVI

### Follow-up question

Is this a case of simple or secondary mixture?

simple

### Question

Identify the following chord in F minor with a Roman numeral:

Hint

Remember to use accidentals to indicate altered roots. The case of the Roman numeral should correspond with the quality of the triad.

i

### Follow-up question

Is this a case of simple or secondary mixture?

simple

### Question

Identify the following chord in C# minor with a Roman numeral:

Hint

Remember to use accidentals to indicate altered roots. The case of the Roman numeral should correspond with the quality of the triad.

vi

### Follow-up question

Is this a case of simple or secondary mixture?

secondary

# 30.4 Double mixture

A third type of mixture may be found in the following example, which begins in the key of C minor:

At the beginning of m. 5 we find a clear-cut case of mixture: the E§s in both the voice and the piano change the quality of the tonic triad from minor to major, and C major is then tonicized with an authentic cadence in m. 6. However, in the second half of m. 5, we find an A major triad (#VI). With respect to C minor, all three members of that triad have been raised. This is not a case of simple mixture because C major, the parallel to C minor, does not include C#.

This is a case of double mixture. Double mixture involves changing the quality of a chord derived from simple mixture. In the key of C minor, simple mixture allows for an A-minor triad (#vi). When we encounter an A-major triad (#VI), we may think of it as being a case in which secondary mixture has been applied to a simple mixture chord, hence double mixture.

Once again, we may use Roman numerals to represent these chords. Consider the following cases of double mixture in a major key:

In a major key, bIII and bVI are both examples of simple mixture because they are derived through borrowing tones from the parallel minor. When their chord quality is changed by lowering their thirds, they become instances of double mixture. Again the Roman numeral analysis indicates the root and quality of the triad. The chord labeled biii is a minor triad built on b $\hat3$ while bvi is a minor triad built on b $\hat6$.

Example 30–11 shows two examples of Roman numeral labeling for double mixture in a minor key:

Note again that despite the natural signs altering the roots of these chords, the Roman numerals are preceded by # symbols.

Now consider the following example:

In this excerpt we find a pair of surprisingly chromatic sonorities in m. 2 and m. 4. (We hear the same chords again in m. 6 and m. 8, after the singer joins in.) Let us look at the chord in m. 4 first. If we hear the E-major opening as tonic, the chord in m. 4 may be heard as a chromatic mediant. If this was a case of simple mixture we would see and hear a G§-major triad—G§ (b $\hat3$) and D§ (b $\hat7$) being borrowed from the parallel minor. Here, however, we encounter a G§minor triad. It is as though secondary mixture was applied to a simple mixture chord and so we may think of this as an instance of double mixture.

The sonority in m. 2 (and m. 6) is complicated by the presence of pitch-class B. If the B is heard as a pedal tone—sustained through the framing E-major chords in m. 1 and m. 3—we would consider this another case of double mixture: a simple-mixture bVII with a secondary minor inflection. It is also possible to hear this as a neighboring sonority prolonging the tonic: the D§s are neighbors to the Es, the F§s are neighbors to the Es, and the As are neighbors to the Bs. (Example 30–12b demonstrates this hearing by placing the neighbor tones in parentheses.)

Note: As you can see from the explanation above, discussions involving secondary and double mixture can quickly become quite complicated. Furthermore, faced with such abundant chromaticism, you may have even found it challenging to hear E major as the tonic in Example 30–12. Given the prevalence of chromaticism in such music, one must wonder if the harmonic relationships encountered here are fundamentally different from those found in the tonal music of the high Classical era. Many listeners hear enough of a difference that they consider such music to be, in a way, post-tonal. Indeed, upon hearing the music, one quickly becomes aware that the tonic-dominant relationship forming the foundation of tonal harmony plays a much more peripheral role here.

It follows, then, that one should use caution when employing analytical strategies that were designed for describing a different kind of music. In recent years, music theorists have begun hearing this music on its on terms. According to this new perspective, the relationship heard between, for example, I and biii may be as distinctive and foundational as any authentic cadence. It all depends on the context.

Activity 30-2

Activity 30–2

The following triads exhibit various types of mixture: simple (borrows tones from the parallel key), secondary (alters the quality with pitches not from the parallel key), or double (secondary mixture applied to chords derived through simple mixture). For each exercise, label the triad with the appropriate Roman numeral and identify the type of mixture present:

### Question

Provide a Roman numeral for the following triad in G major:

Hint

Remember to match the case of the Roman numerals with the quality of the triad and indicate altered roots with accidentals.

VI

### Follow-up question

What type of mixture is exhibited by this triad?

secondary

### Question

Provide a Roman numeral for the following triad in D minor:

Hint

Remember to match the case of the Roman numerals with the quality of the triad and indicate altered roots with accidentals.

IV

### Follow-up question

What type of mixture is exhibited by this triad?

simple

### Question

Provide a Roman numeral for the following triad in B minor:

Hint

Remember to match the case of the Roman numerals with the quality of the triad and indicate altered roots with accidentals.

#VI

### Follow-up question

What type of mixture is exhibited by this triad?

double

### Question

Provide a Roman numeral for the following triad in Bb major:

Hint

Remember to match the case of the Roman numerals with the quality of the triad and indicate altered roots with accidentals.

bVI

### Follow-up question

What type of mixture is exhibited by this triad?

simple

### Question

Provide a Roman numeral for the following triad in A major:

Hint

Remember to match the case of the Roman numerals with the quality of the triad and indicate altered roots with accidentals.

biii

### Follow-up question

What type of mixture is exhibited by this triad?

double

### Question

Provide a Roman numeral for the following triad Eb major:

Hint

Remember to match the case of the Roman numerals with the quality of the triad and indicate altered roots with accidentals.

III

### Follow-up question

What type of mixture is exhibited by this triad?

secondary

# 30.5 Other chromatic chords

A different type of chromatic chord can be seen in the following example:

Example 30–13 begins with a pair of root-position auxiliary V chords prolonging the minor tonic. In the second half of m. 3, however, the bass leaps down to an F (scale degree $\hat4$), harmonized in the upper voices with Db (b $\hat2$) and A ( $\hat6$). This chord leads to a V6–5/4–3 in the following measure before resolving back to the tonic. (The entire progression then repeats.) Obviously, this is not a typical case of a borrowed chord since Db appears in neither C major nor C minor. Furthermore, there is no such thing as a major diatonic II chord.

This particular sonority, commonly refereed to as a Neapolitan chord, nonetheless occurs quite frequently. It typically appears with scale degree $\hat4$ in the bass and performs a pre-dominant function. It is a derived chord that can arise in different ways, so we label it N6 to avoid the confusion of Roman numerals. Though not a case of mixture in the purest sense of the word, Neapolitan chords are often grouped with borrowed chords for this similarity in behavior.

Another Neapolitan chord appears in the following example:

Following a prolonged tonic chord (with passing octave Bs in the bass), the harmony moves to VI and then to the Neapolitan—spelled F#, A, D§ ( $\hat4$, $\hat6$, b $\hat2$)—before progressing to the cadential dominant.

A more detailed discussion of the Neapolitan chord can be found in Chapter 31.

# 30.6 Summary

Simple mixture consists of borrowing one or more scale degrees from the parallel key to alter the quality of a chord. Other types of mixture also involve changing the quality of a chord, but with chromatic tones that are foreign to both the key at hand as well as its parallel. With secondary mixture, a composer may change the quality of a triad by raising or lowering its root with such a chromatic tone. In cases of double mixture, a composer changes the quality of a simple mixture chord by applying secondary mixture.

Because Roman numerals are able to express both the root and quality of a chord, they are also useful for indicating the three types of mixture. As always, the case of the Roman numerals indicates chord quality. You should become familiar enough with the qualities of diatonic triads that by simply looking at a Roman numeral you are immediately able to recognize the presence of chromatic alterations. Accidentals preceding a Roman numeral signal an altered root: a prefix b indicates a lowered root while # indicates a raised root.