The most of this book is dedicated to the Roman numerals analysis. It originates from Austrian music theorist and composer Simon Sechter who viewed chord progressions as chord root movements. In Baroque, chord progressions were viewed as bass note movements and the figured bass was used to annotate how chords can be created above their bass notes. Today, Roman numerals are used to annotate chord roots and the figured bass is used to annotate inversions of chords. 

However, another theory exists that is called functional harmony. It examines roles which harmonies play in chord progressions and is described in the following chapters. It originates from German music theorists and composer Hugo Riemman.

Harmonic functions

A harmonic function is the role that particular harmony plays in the chord progression in which it occurs. A harmonic function can be tonic, dominant and pre-dominant. 

• The role of the harmony that has the tonic function is to be the destination of the chord progression. That is, the chord progression ends with the tonic function. 

• The role of the harmony that has the dominant function is to lead to the tonic function. That is, it leads to the harmony that has the tonic function.

• The role of the harmony that has the pre-dominant function is to lead to the dominant function. That is, it leads to the harmony that has the dominant function. 

This is illustrated in the next figure. The letters T, D and P are used as acronyms for the tonic, dominant and pre-dominant functions. 

P > D > T 

Figure: The harmonic functions

The important thing here is that the same harmony can play different roles depending on the chord progression and the position in which it occurs. However, in the compositions from the common practice period, for many harmonies we can find a role that the harmony plays most often. For instance, we can say that the harmony IV in the major key (that is, the diatonic triad chord on the fourth scale degree of the major key) usually has the pre-dominant function. However, this is not always true.

When we have talked about the diagram that shows typical chord progressions (in chapter Typical chord progressions), we identified two groups of diatonic triad chords. One of them contains the diatonic triad chords on the fifth and seventh scale degree and leads to tonic triad chord. Thus, we can say that this group contains the harmonies that typically have the dominant function. The other group contains the diatonic triad chords on the fourth and second scale degree and leads to the first mentioned group. Thus, we can say that this group contains the harmonies that typically have the pre-dominant function. 

Therefore, the diatonic triad chord on the first scale degree typically has the tonic function, the diatonic triad chords on the fourth and second scale degrees typically have the pre-dominant function and the diatonic triad chords on the fifth and seventh scale degrees typically have the dominant function. We know that the diatonic seventh chord on the fifth scale degree strongly resolves to the tonic chord. Thus, it also typically has the dominant function. We know that V/V, the Neapolitan chord and augmented sixth chords most often lead to a chord on the fifth scale degree. Thus, they typically have the pre-dominant function. 

Idealized chord progressions

The next figure shows a so-called idealized chord progression. The idealized chord progression is the sequence of the following chords: the chord that has the tonic function, the chord that has the pre-dominant function, the chord that has the dominant function and the chord that has the tonic function. 

T > P > D > T 

Figure: The idealized chord progression

As its name suggests, such a chord progression is an idealization. That is, while many chord progressions have this form or they can be reduced to this form, many chord progressions are just different. Still, the idealization is very useful. Phrases often adhere to this form. For instance, I – ii – V – I or I – IV – viio – I, in the major scale, are very frequent chord progressions that have the form of the idealized chord progression.

The idealized chord progressions are often used to confirm the key. This is the most important at the beginning of a composition, after a modulation and at the end of a composition. Such a chord progression gives a good opportunity to the listener to perceive the key. For instance, the next figure shows the chord progression at the beginning of Bach’s Prelude No. 15 in G Major from Well-Tempered Clavier, Book I. This chord progression has the form of the idealized  chord progression and it occurs right at the beginning of the composition. It sounds like: “Hello from the G major key!”

Figure: The chord progression at the beginning of Bach’s Prelude No. 15 in G major from Well-Tempered Clavier, Book I

The next figure shows the chord progression at the end of the same composition. Since both the diatonic triad chord on the fourth scale degree and the diatonic triad chord on the second scale degree have the pre-dominant function, this chord progression can be reduced to match the idealized chord progression. The perfect authentic cadence at the end of the chord progression gives a strong sense of finality. The chord progression sounds like: “Good buy from the G major key!” 

Figure: The chord progression at the end of Bach's Prelude No. 15 in G major from Well-Tempered Clavier, Book I

Idealized chord progressions often appear as a chain of idealized chord progressions. In this case, the ending T of each previous idealized chord progression is also the beginning T of the next idealized chord progression. For instance: T > P > D > T > P > D > T 

Prolongations of harmonic function

Many chord progressions that look different from the idealized chord progression can be reduced to it by using the prolongation. We explained how a single harmony can be prolongated. In chapter Prolongations of harmony, we explained how a single harmony can be prolongated. However, there exists another type of prolongation: the prolongation of harmonic function which prolongs a single harmonic function. For instance, the previous example contains the prolongation of the pre-dominant function. In this example, the diatonic triad chord on the second scale degree prolongs the pre-dominant function that was triggered by the previous chord (the diatonic triad chord on the fourth scale degree). This is possible because both of these chords typically have the pre-dominant function. 

The diagrams that show typical chord progressions (explained in chapter Typical chord progressions), contain two possible prolongations of the harmonic functions: the chord bigram from the fourth to the second scale degree which, like in the previous example, can be used as the prolongation of the pre-dominant function and the chord bigram from the seventh to the fifth scale degree which can be used as the prolongation of the dominant function. For instance, the chord progression I – IV – viio – V – I, in the major key, uses the later chord bigram to achieve the prolongation of the dominant function.