Harmonic Reduction

Hearing the big within the small. 

The purpose of this lesson is to start putting to use everything that we've learned and apply it to bigger sections of pieces. I think you'll start to see that there are very basic patterns of interconnectedness within any tonal piece of music. 

Just to recap the story so far, we first saw how elemental tones can come together in us to form the smallest level of music: the interval. Then, we built chords and diatonic chord progressions from those 12 tones based on how they all related to one another. We then learned about the expression of a stable tonality with the cadence. Recently, we connected cadences of many different keys through modulation. 

With the exercises, we learned how to build and smoothly connect chords, and discovered that the music of J. S. Bach expresses everything that is harmonically possible in the system of harmony that we still use today: the system of tonality. 

By way of the principle of tonality and the relations of the tones, you can come to understand that the larger structures of music have at its core a much simpler and more basic harmonic skeleton. And the reverse is true too; a simple harmony can form the basis of an entire movement. 

Complex harmonic structures can be "reduced" and still retain the essence of what we experience.

To get a sense of what I mean, let's start with a familiar modulation. Here I've labeled all of the tones based on their interval relation to the chord root: tones with style.

Figure 1. Here's our classic modulation from C major to F major via D minor. The three moments of the modulation are the initial cadence in C major, the neutral field that goes to D minor, and the final cadence in F major. This time I use the unconventional method of labeling the ii65 chord a IV6 chord to mean a IV chord with an added sixth above the bass, not a first inversion IV chord. The benefit here brings attention to the more important fifth in the chord in the context of F major. Labeling as ii puts more importance on G-D, and labeling as IV puts the importance on Bb-F. 

We hear the C major, D minor, and F major as the points of stability in the modulation. It is going by way of D minor from C major that we can live F major as the new tonic, and not hear F as the subdominant of C. Those chords are stabilized with their upper fifths, and the two "tonics" also have their subdominant chords to help define and support their stability. 

Figure 2. A harmonic "summary". You can think of each chord in this measure as a representation of the "backstory" behind the main structural moments in the modulation. This is an example of a harmonic reduction. And, each chord and its backstory in the reduced modulation could itself be expanded upon to form the original modulation or a variation of modulation. The more familiar and in tune you are with the smallest elements of music, the more you can imagine the full universe of musical possibility.

Any element of harmony can be extended in time by other parameters in music: melody, rhythm, timbre. 

Let's look at an example of a mini modulation that appears in one of Beethoven's symphonies, between the first two themes of the Pastorale.

Figure 3. The first triplet eighth note has a very rich "back" story: it represents everything that came before it in the piece—a thoroughly established F major. How is it established in the symphony, starting from bar 1?

We hear the first theme being sung by the first violins (see figure 4). Then, a development of the first theme. We hear a brief expansion where he sits on the dominant chord. After a repetition of the first theme and an increase of tension via the melodic parameter and orchestration, we hear the first theme in its full expression and motivic development with the tutti orchestra. A grand statement in F major. Now there is harmony throughout this F major section that begins the piece, for sure. But the active element that builds the first section is not the melodic parameter. The whole first "minute" of the piece, to reduce it down to the level of mere time, is a sung expression of F major. Naturally, there is harmony within the F major, dominants, mostly, and smaller harmonies, passing tones, mini-harmonies within each bar. But overall, the beginning can be summarized by that first stable F major chord of figure 3. 

Now to go on, Beethoven invokes the harmonic parameter more strongly by starting to leave the key of F. That's what we'll study in this modulation. 

Immediately after hearing the F major chord, we hear a contrast but this time it appears in the rhythmic parameter: triplets are brand new to the ear at this point. But quickly, we hear the beginning motif, and it's more developed. Hearing sounds from the beginning helps stabilize the ear after the contrast.

Figure 4. The opening theme, measures 1-4 of the piece.

Then we introduce a brand new sound—D minor—we haven't heard this chord yet. It's relatively a big harmonic contrast. We then hear the melody again, but this time it feels much different even though it's the same exact collection of notes. The melody has been changed by the relative minor of F, D minor. Then, we hear the C major chord, which is expected in F major. 

Sooner than we would have expected based on what came before it, we hear G major! For the first time, a B natural is in the ear. Beethoven then stays on the G major for six entire bars before resolving to C major, which feels much more stable than it did when it was the dominant of F. The B natural has brought us closer to C. And his stabilizing of C major is done in typical Beethovenian fashion, by repeating "V-I" in C major six and a half times across before arriving at the grand statement of the second theme in C major:

Figure 5. The emerging C major. 

Figure 6. The progression from F major to C major can be reduced to this modulation. Compared to the ones we've built yesterday, this is a pretty weak modulation! 

Extensions of time

Further still, the relation of the beginning and end of the exposition can be expressed as:

Figure 7. The interval of a fifth.

Slow building with repetition. The reason that Beethoven keeps the harmonic contrasts minimal in the exposition is because he has a plan to use harmony in a much more impactful way just around the corner in the development of the piece. It's a use of harmony that hasn't been imagined before in music history. This slow building though constant repetition, little by little. A technique that Bruckner took hold and expanded through the roof. It's remarkable that Beethoven composed the Sixth Symphony simultaneously with his Fifth. Go and listen to the whole symphony! Here's an amazing recording: https://www.youtube.com/watch?v=l02XlWX_6HE

Simple structures lie at the heart of big pieces of music.

Where you are in a piece can be felt by the interval relation that it has with the beginning and the end.

For example, if a piece of music is in D major, and you're hearing a section that plays for a while in Bb major, then you know that in the context of the whole piece this section will feel like the subdominant lower-fifth introverted direction of harmony. Whereas if the piece is in Db major and the section that you're in is Bb major, you have the feeling that in the context of the whole piece, the Bb major will feel brighter, more extroverted. This is just the theory, and you won't know until you're living in the piece yourself. 

At the end of the first theme in a sonata form, very often we've cadenced on the dominant via a modulation, or otherwise left the key somehow. But what if the piece ended there? It can't because we still feel that extroverted fifth present by the time we're in that part of the piece. We have to fall back down a fifth and maybe even approach the tonic from the subdominant to really feel the grounded sense of the stable and ruling tonic. The beauty of composition is exactly that journey from the expansion to the contraction of musical tension created by the musical intervals and tonal relations.

The art of harmonic reduction

Here's the method that I learned from Reza Vali. Take any piece of music and write down all of the notes of a piece in 4 voices to study how they move. Reduce the complexity down to just 4 voices. Write the melodic voice without rhythm and the underlying harmony beneath the melody tones that it supports. 

In the next lesson I'm going to do a harmonic analysis of a solo cello work by J. S. Bach. I've never done one before for that piece, but I can already imagine that I'm going to learn a lot and I can't wait to share it with you in the next post.