Day Nine: Modulation

Key relations are among the most powerful means of expression in music.

We've spoken a lot about stability in music: how the ear hears stability in a single musical tone; how one tone can be more stable than another even if it's heard later in time; how chords can be organized by scales with degree-relations that contribute to or take away from the stability of the tonic; how cadences can establish a stable chord or interval through the use of dissonances and resolving harmonic tension. 

But frankly, stability 

can be so 

... boring. 

Modulation is the art of harmonic mutiny!

It is the challenging of the stability of the tonic and eventually allowing the victory of a new tone to emerge as the stable tone! It is about leaving the key and moving on a clear path from one stable tonal center to another. At its core essence, it is the transference of stability of one sound into a different one. 

Variety is the spice of life and in tonal music we find the same principle at work. Modulation is musical revolution. Recall the experience that musical tension itself is felt from contrast, the concrete juxtaposition of two sounds. We hear how they are different, and how they are the same. If life is to emerge from a work of art, then we have to engage in this conflict that is made possible by the emergence of musical tension. In a tonal work of art, the tonality must be placed in danger of losing its sovereignty, and the struggle of the other tones for dominance must eventually yield to those victories. The total ruler of the piece is then allowed to "take pleasure in ruling live subjects that will attack and plunder."

In this lesson we are pivoting toward learning how to destabilize a tone and move away from it to somewhere else. This struggle is central to tonality, the system of musical organization in the West based on relative stability and key relations. Renaissance musicians in the 16th and 17th century composed using a different system that we call modal that relied more on scales and intervalic relations. Our tonal system was codified by Johann Sebastian Bach (1685-1750) whose music expressed all that was harmonically and contrapuntally possible with the 12 major and minor keys. Eventually, composers would find ways to express new key relations, and establish one of the most powerful means of human expression in music. 

Tonality took centuries to develop. After Bach, tonal music has been enormously expanded upon by Haydn, Beethoven, Brahms, Bruckner, and so many others who have used tonality to build musical forms. After decades of development, the fundamentals themselves were eventually treated not as the most stable note but as a leading tone, yielding to newer forms of expression through use of chromatic harmony. Eventually history would birth composers who have treated the 12 tones in a very different way. These are the composers of the Second Viennese School. Despite having modulated vastly their tonal origins, their dodecaphonic works clearly show a highly thought out musical means that are similar to the techniques of harmonic and contrapuntal expression in music by J. S. Bach. And in our contemporary understanding, these relations all have their roots in the relations that the tones have among themselves, organized by the structures of Nature and appreciated by the human body. 

The Tonal System: What defines a stable tonality?

A tonic chord is defined by its dominant (the chord built on top of the upper fifth of the tonic) and its subdominant (the chord built on top of the lower fifth of the tonic). Tonality as a system is a way of thinking about the musical tones as major and minor chords, and these chords each have their respective scales and keys, determined by the accidentals that are in the descending scale. The richness of the harmonic universe could be freely understood by the following. Each of the 12 tones has a chord (commonly either major or minor) and a key associated with it. So the key relations themselves can be understood and experienced by the intervallic relation between the two tonics.

Oversimplified: Tones = Chords = Keys

What defines a stable tonality?

The stable tonic is defined and strengthened by its dominant (upper fifth) and subdominant (lower fifth). Essentially, a cadence is the expression of a tonality.

Figure 1. The reign of the tonic as the 'central' tonality is established by cadence harmonies. Because the system of tonality allows all of the 12 tones with their own respective major and minor keys, any of the 12 tones can be established as stable provided we hear its upper and lower fifth functioning as dominant and subdominant harmony.

Part of the goal and charm of a modulation comes from how the transferral of those subdominant and dominant qualities are given to notes outside of the original key. To understand how to easily do that, it's best to develop a deep appreciation for fifth relations.

Fifth Relations

When a note is actually sounding, the perfect fifth above the fundamental is naturally produced. If we arrange these fifths on the staff then the relations can be expressed as:

Figure 2. This figure makes clear why harmonic direction is perceived in the way it is. Each note has two possible directions, up and down, and so the interval of the fifth from the tonic can be expressed in two ways. Each of those two notes, the upper fifth and the lower, has its own upper fifth present in the sounding musical tone that we hear as its second overtone. The innate quality of sound gives rise to its own upper fifth sounding as the third harmonic.

From the A's point of view, the upper fifth present in the subdominant (D) yields back to itself (A). The upper fifth of the dominant (E) yields to a new note: B.

This generative property of the dominant, of the upper fifth, can be heard in the extroversion of the ascending fifth. The self-yielding property of the subdominant, the lower fifth, is heard in the introversion of the descending fifth. Part of the reason why we study the intervals is because it can contain all that is in music. 

The major and minor scales can be formed starting on any of the 12 tones, each forming its own unique signature sound world. A key is like a sound world. When you start using notes that are not in the scale of the key, you begin to travel away from one world and you can navigate to somewhere else. When you're done, and if you've done it in a way where the most stable note from the beginning is no longer stable by the end, then the tonic has been usurped by different note and you can say that a modulation has taken place.

Our goal for writing a modulation is to write the smoothest and most direct chord progression from one stable tonality to another. 

Figure 3. From one cadence... to another.

"Art is creativity. Art is anything done for its own sake." - Naval

Key relations are among the most powerful means of expression in music. 

When modulating, it's helpful to have the map of all possible key relations on hand. This map is called the Circle of Fifths. The circle maps fifth relations in the 12 notes, and clearly show the tonic-dominant and tonic-subdominant relation of every key. Any note on the circle can be the tonal center defined by its two adjacent fifths.

Figure 4. The circle moves to upper fifths in the clockwise direction and lower fifths in the counterclockwise direction. Every key has its dominant and subdominant right next to it. For example, pick E major. The upper fifth is B, and the lower fifth is A, the two keys next to it on the circle. It also works with the minor keys too; d minor has g and a as the subdominant and dominant respectively. In a chord progression, you'll always use the major dominant, since that's the chord that functions as dominant harmony; the minor v chord functions more like predominant harmony. So in the case for D minor, you'll use A major as dominant. Again, it's useful to develop a free exchange between the tones, major and minor chords, and keys—that was one of Beethoven's contribution to the tonal system.

A tonal center is like a center of gravity for the ear. You can think of a modulation as a rocket launch OUT of the center of gravity to a new region of space. Harmonic energy needs to be used to guide the ear to a new harmonic region. Suppose you want to modulate from F major to C# minor. You're moving up 5 fifths, adding 5 sharps to the scale, and that will require much more harmonic energy than if you were to move down 2 fifths (or add 2 flats to the scale). Similarly, if you want to move up 1 fifth it requires more harmonic energy than if you want to move down 1 fifth. Why do you think that is? 

Keeping the circle of fifths handy, let's modulate from C major to F major. 

How to write a modulation from C to F

Figure 5. The goal. As it stands now, the ear will hear the F chord as relating to the first chord, not as a stable center. 

C major, no sharps, no flats. Modulating from there to F major one fifth down, adding 1 flat to the scale. (The first flat is Bb for that reason). 

You'll need two cadences. One in C major and another in F major.

The modulation is a change from one tonality to another. Cadences will express the tonalities in each key. Use an opening cadence in C, and a final cadence in F. (Here's a link to the cadences page.)

Figure 6. The modulation is all about bridging the gap between these two regions. We know that we can't just play a series of chords in C major and then in F major, because the ear won't ever have the chance to change from one tonality to the next. Therefore, we must have a third sound, a third key that functions as a neutral field of harmony that can unite the two keys. 

As you can hear, the ear jumps when it hears the Bb because it's not in the key of C. So we have to try and find a way to introduce the Bb and prepare the ear for it, and that's what the neutral field can do and what the harmony between the two cadences is for. 

The pivot chord is the revolution.

The chord that bridges the two keys is known as a pivot chord because the ear hears that it can have two functions, one connecting to the original tonal center, and another connecting to the final tonal center. The best way to find a pivot chord to just memorize the chords in each key. In the meantime, you can write out the diatonic chords and look for any chords that the two have in common. 

Figure 7 makes clear that D minor and A minor are common in both F major and C major. So that makes them good candidates to guide the ear from one key to another! When writing a modulation, you must always pick a neutral field that will function as the actual modulatory part of the progression. It's as important as knowing your starting and ending key. 

How to pick a neutral field.

Going back to the circle of fifths, note how D minor is a strategic choice in our traveling from C major to F major. D minor brings us closer to F in terms of fifth relations. If you're modulating to a key several fifth above or below, then choosing a neutral field between them will help smooth the progression. If you’re modulating to a major key, use a minor key as the neutral field and vice versa.

Figure 8. The pivot harmony has two functions. In C, it's the supertonic, F, the submediant. That ambiguity is perfect for the ear to hear it differently as it moves through the chord progression in time.

Now at this point it sounds pretty smooth. But, does it function? The ultimate test is to hear the progression as it is and ask if F major really feels stable. To my ears, it kind of does and kind of doesn't. This doesn't make for a compelling modulation. I would say that as it stands, this modulation does not function. If a piece of music had this progression in it, it would be weaker than it could be. The F major isn't really putting up that much of a fight for tonal dominance, and the ear still grants the C a victory over all the other tones even with the Bb in the bass. 

Concretely, the Bb pulls us out of the modulation by surprising the ear. D minor brings us closer to the Bb because Bb is in the D minor scale, but the Bb comes off as too strong of a contrast in the overall progression. We want that Bb to be in the cadence harmony, not as the first moment of harmonic contrast. 

Another way of looking at the problem is, have we really left C major? Is there another chord that can come before the pivot chord that pulls us strongly out of C major and propels us to a D minor chord that can function clearly in an F major cadence?

Think again to the circle of fifths. How can we smoothly prepare D minor? What chord will anticipate D minor and help establish it away from C so that it can progress to F?

Let go for the dominant of D, or A major. To make it as smooth as possible, we'll add a seventh to the A major chord so that it's clearly a dominant preparation of D minor. This way, D minor can live more on its own, freed from C major, and the harmonic contrast from the tonic C major chord to A major helps propel the ear away from the tonic so that it can definitively land in F major. 

Figure 9. Placing A major before D minor is a clear contrast from the beginning and allows the D to be heard as resolving down to C, the new fifth of the tonic.

A modulation has 3 harmonic moments. The opening tonality, the neutral field, and the final tonality. The neutral field functions to pull us away from the stable tonic and opens the harmonic possibility to establish one somewhere else. 

Modulation Directions

This section has both uses of the word direction. First, I want you to hear how the d minor changes between coming from C major and pointing at F major in the same note. Listen to the modulation from the beginning and turn off the logical categorizing portion of the mind. It’s easier to use the string version for this. Where/when do you hear the change?

The ii7 chord of F is used as the subdominant chord. If you analyze it, you can see the subdominant fifth present in the chord, between Bb and F. The subdominant of F, Bb major, is strengthened by the G which resolves down to the tonic, pointing very strongly to F. The reason that the Bb-F fifth is the stronger fifth in the overall progression is why I usually think of the ii65 chord as a IV chord with an added 6th (Bb-D-F-G). So it is really the presence of this subdominant fifth that turns D minor from having a direction pointing forward from the beginning to pointing backward from the end. The orientation changes on D minor from beginning to ending, and the ending function isn't heard until the subdominant chord (the next chord) is heard. In this sense of changing directionality on D minor makes D minor the climax of the modulation. This isn't to attribute musical intensity to the climax. It indicates the one point in the piece that makes it possible for us to live the elements of the piece as a singular, whole, unity. 

When writing modulations—

1. Avoid cadence harmonies in the neutral field because you want to save them when they're strongest: at the beginning and end. 

2. Avoid fully diminished chords in the neutral field because they could lead anywhere. Or, use them sparingly. 

3. Think more in terms of fifth relations. We want the modulation to be economical. It is not a composition. Concentrate and use as little as possible. Hear them to feel the central pulse of harmony. 

4.  Tip: when modulating to a major key, use a minor chord as the neutral field and vise versa.

Learn to hear modulations in terms of harmonic directionality: is something beginning or ending? Feel the charm of the progression, the satisfaction of the completed challenge, and the eventual expression of yourself. That's how we will come to building music with key relations. 

Try one: The Modulation Exercise

A modulation exercise is a short study that focuses on how to do it. It’s a bit like the 12 tone exercise. A concentrated effort with huge musical consequences. Try writing one from C major to F major via A minor. You can email it in at jordan@jordanali.com if you want some feedback.

This post is part of 14 Days of Harmony, a free course for musicians who want to deepen their understanding of harmony, and learn how to develop their connection with sound as a result. You can view the entire course here.

Next day (and the ❤️ of the course): Modulation Practice

Previous day: The Bach Chorale