CHAPTER 4
LINEAR PROGRESSIONS
Those readers who are familiar with Schenker's theory should note that the use of the term 'linear progression' in this book is similar but not exactly the same as that used by Schenker. Here the term is used in a more limited sense to describe a phenomenon that is observed when analysing root progressions.
Where a chord progression arises out of a step by step movement in one or more voices rather than by root progression, I will refer to these progressions as 'linear progressions'. The term is thus limited to this situation and is not used in the more general sense as used by some Schenkerian analysts.
In the last chapter, all the examples of passing chords were created by one or more passing notes resulting in a single chord filling in between two functional chords. An important extension of the single passing chord is a series of passing chords created by a step by step movement in one or more voices. I will refer to this as a linear progression. The succession of filling-in chords is neither recognisable as static harmony nor as dynamic harmony. This does not mean, however, that the linear progression is a third type of movement. This is because linear progressions are always incorporated within static or dynamic patterns. The chord succession results out of the linear movement of one or more voices rather than a root progression. A linear progression may be either diatonic or chromatic or a mixture of both and may involve more than one voice in parallel motion or in contrary motion. Where there are concurrent linear progressions these may move at different rates.
The stepwise movement is always contiguous. If there is a break in the step wise movement then the progression is not a linear progression and the chords formed are not passing chords. In general, the same rules as for passing chords are apparent: The passing chords in the linear progression are made up mainly (if not completely) of notes from the starting chord and linear moving notes. Occasionally, chromatic auxiliary notes are also deployed.
Linear Progressions in Static Harmony
Linear progressions in static harmony can link successive tonic chords or link the tonic chord to an auxiliary chord involved in the static harmony, in which case the auxiliary chord behaves as a functional chord in its own right.
The following example contains just about the simplest kind of linear progression possible. It is totally diatonic, it is created by a linear movement between two successive tonic chords and is not extended by any further elaboration.
It may look at first sight that this is a I - V - IV - I progression but on further examination it is clear that the succession is created by a linear descending bass movement: D - C# - B - A, linking the D in the tonic chord in the upbeat bar to the A in the tonic chord in bar 2. As well as this linear movement there is a parallel downward stepwise movement F# - E - D and also A - G - F#. Note that the common properties normally observed in linear progressions are present here, even in this very simple example: The chords involved in the linear progression are made up of the notes of the starting chord and notes which are moving (i.e. taking part in the linear progression). Hence, the A (first beat, bar 1) and the D (first bar 2) are retained as starting chord harmonies against the other notes that are involved in the movement. The static harmony continues for several bars. What is going on here is a contrapuntal or melodic movement rather than a harmonic chord progression.
The following slightly more elaborated example involves a simple two note diatonic linear progression filling in the gap in the bass between the root of chord I and the root of chord V:
The linear progression shows clearly in the descending bass movement linking the C in the tonic chord via B and A to the G in the dominant chord. The example also includes a chromatic ascending passing note connecting the C via C# to D. The progression is further elaborated by appoggiaturas to the dominant chord. Note again the normal property of linear progressions is apparent: The chords involved in the linear progression are made up of the notes of the starting chord and notes which are moving. Hence, the E and the G are retained throughout the linear progression and the intervening chords are made up of these notes plus the notes which are moving: B and A in the descending linear progression and the C# in the ascending linear progression. Readers familiar with strict counterpoint will note that this process arises out of the rules of 3rd species counterpoint.
After bar 3 the base continues its stepwise movement, downward. However, this movement continues as I - V- I static harmony.
Chromatic linear progressions are more common than diatonic linear progressions as there are more possibilities. Following is an example of a chromatic linear progression within static harmony. The bass voice descends by 5 semitone steps from the key note to the dominant. A second voice follows the linear progression in parallel minor thirds, whilst a further voice decorates with a chromatic auxiliary note. The minor third (or its inversion the major 6th) is the most common interval for chromatic linear progressions moving in parallel motion.
Note: This piece is in D minor (hence the D minor key signature) but modulates at this point to A minor.
At bar 26, in the same composition, Mozart employs a combination of diatonic and chromatic linear progressions in contrary motion. There are three linear progressions in all: two rising in parallel thirds and one descending. In spite of this complexity the harmonies produced make perfect sense. The progression is again within static harmony and starts and ends on the tonic chord.
Linear Progressions in Dynamic Harmony
The following example includes a linear progression moving chromatically from the dominant chord to the tonic. The underlying harmony is dynamic as the dominant is preceded by chord II and the tonic followed by chord IV. The purpose of the linear progression is to temporarily delay and extend the movement of the dynamic harmony. The linear progression involves: two diminished 7th chords (one supertonic leading; one dominant leading) and one triad (the supertonic). The normal rules about linear progressions are followed. The chords in bars 13 and 14 retain the D form the starting chord although this is temporarily displaced in bar 12 by the C# auxiliary note. The G from the starting chord is retained in bar 12 but then forms a passing note F which descends to the E in the ending chord.
The full book will contain further examples of linear progression in dynamic harmony.
Linear Progressions Elaborating Dominant Harmony
The third area where linear progressions are important is in the prolongation of the dominant chord. This is similar to the use of linear progressions in the opening section static harmony where the linear progressing prolongs the tonic chord. However, as we have not yet examined the prolongation of the dominant harmony, I will leave this subject until chapter 5.
Ver. 2.3.