4.1. Reduction of Music to an Harmonic Outline.
When discussing a chord progression analysis the central issue that has to be addressed is that the analysis ultimately depends on decisions about what vertical note combinations in the music are significant in the analysis and what should be excluded. Given this necessary selection process, the suspicion always has to be that in finding patterns in chord progressions, the theorist has in some way selected chords and progressions that fit the theory and has rejected note combinations that do not fit the 'desired' results. (In fact, this is one of the principle objection to Schenker's fundamental structure - that the means of its derivation are not independent of the results themselves. 1) To overcome this suspicion, any means of selection of note combinations as significant in the analysis must satisfy the following conditions:
- It must be clear and unambiguous
- It must be based on concrete research data
- It must be applied in a consistent manner
- It must be applied in such a way that is independent of the end result
In other words, the reduction of music to an harmonic outline that lends itself to adequate root analysis, must be carried out by a system of rules that is as objective as possible.
In this current work, the answer to this problem has been arrived at empirically by applying candidate exclusion rules in a variety of combinations until a set of rules was found that produced consistent results across all the musical compositions currently analysed. (This currently includes mostly western art music of the classical and romantic periods. I am currently extending the analysis to include 20th century classical and popular music.) The results have thus been derived at as a result of practical analysis rather than from any a priori theoretical or philosophical standpoint.
The main process I have used to do this analysis is what I call a comparative root progression technique. By carrying out comparative statistical analyses of chord progressions with different assumptions about what exclusion rules might be relevant, it is possible to conclude what vertical note combinations should be included and what should not. The assumption is, as in any theory, that the simplest set of rules that explains what is observed is the best (This principal is sometimes referred to as Occam's Razor).
Consequently, rules will be deployed to derive the harmonic outline, for this piece, in the following way:
Step 1.
I will take as my starting point that any vertical combination of notes with a reasonable duration, that can be arranged into a triadic form (with or without additions such as a 7th, 9th etc.) should be taken as a chord which is significant in the root progression.
I will include all chords as candidates, whether:
- diatonic or chromatic,
- major, minor, diminished or augmented
- with or without added 7ths, 9ths, 11ths or 13ths
- with or without essential 3rds or 5ths.
However, I will exclude note combinations that arise out of:
- passing notes,
- auxiliary notes
- suspensions, appoggiaturas
For some readers, some of the following discussion will be fairly obvious but I've included it fully so as to be clear about how I have carried out the analysis process.
Please note, the process used to reduce the music to an harmonic outline, is different from a Schenkerian foreground reduction process and produces different results and conclusions. This is because the reduction process is used only to uncover the underlying root progressions. A Schenkerian analysis takes the reduction process beyond this point. There are several other differences too. I plan to add a section to this website which explains the differences in more detail, when time permits.
For example, the passing notes B and D in bar 5 will be ignored as foreground detail:
The F in bar 3 will be ignored on the basis that it is an auxiliary note:
Similarly, the appoggiaturas at the start of bars 6 and 7 will be ignored in the analysis.
In a similar way, the resolution from suspension (5 4 chord) to resolution (5 3 chord) in bar 13 does not constitute a root progression in its own right (this may seem to be a fairly obvious position to take but was not the position always taken by Rameau). For the purpose of this reduction process, I will not make any distinction between a suspension and an appoggiatura.
None of these details are indicated as requiring separate roots in the analysis in Appendix B.
Step 2.
I will now extend this idea one step further. I will exclude all passing chords, auxiliary chords and appoggiatura chords. I will explain what I mean by these in the paragraphs that follow:
It is possible for simple passing, auxiliary and appoggiatura elaboration to extend in time in such a way that what is produced appears, on a cursory examination, to be chords in their own right. In the following example:
The C secondary dominant chord at bar 12 could be consider the result of two passing notes: G filling in between the A of the A minor chord and the F of the F major chord; the B-flat filling in between the C of the A minor chord and the A of the F chord and the remaining notes (C and E) being retained from the proceeding chord. This could thus be interpreted as a passing chord rather than a structural chord in its own right. (This type of chord has its origins in second species counterpoint)
Passing chords are shown in curved brackets in the analysis in Appendix B. Arising out of the Comparative Root Analysis Technique mentioned above these chords are excluded from the root progression analysis. In this way, the progression in Example 4. is taken as a falling third progression (A minor to F major).
In the following example, the same idea is extended one step further:
At bar 73, the passing notes extend through several chords as follows: the A of the A minor chord descends through G, F#, F-natural to E. In parallel with this movement is the descending linear movement: C, B, A, G#. The D# results from an auxiliary note to the shared E note. The E note is retained throughout except for the auxiliary motion. Thus all the notes between the A minor chord and E major chord are retained from the starting chord or are the direct result of linear movement (with the exception of the chromatic auxiliary motion already mentioned). This justifies the treatment of the three intervening harmonies as multiple passing chords. The chord progression is thus interpreted as a tonic to dominant (A minor to E major) progression elaborated by linear filling in chords. Due to the multi-step linear nature of the movement, I will refer to this as a linear progression. For those readers familiar with Schenkerian analysis, I should mention that this concept is similar to the concept of linear progressions in Schenkerian analysis with the exception that I will only use the term to apply to the limited situation just described. I will not extend it to other situations which according to my own analyses are quite different. At bar 88 to 93 there is a similar, but longer, linear progression where the harmonies are similarly formed by linear filling in between the starting A minor chord in bar 88 and the ending E minor chord on the third beat of bar 93.
These are indicated in the full analysis by a sloping line or are annotated directly with the words linear progression.
In a similar way, we can eliminate chords which arise out of the auxiliary motion of parts. In the following example:
The G# diminished 7th chord at bar 10 could be considered to arise out of the auxiliary motion of individual parts: the A falling to G# and returning to A and so on. This interpretation is supported by the retention of the A pedal note in the bass which emphasises the transient nature of the movement above. This approach also removes the difficulty of deciding objectively what the correct root is for the inherently ambiguous diminished seventh chord. Note that auxiliary chords are often triadic and there are many such examples in the analysis. This process also eliminates the C E auxiliary movement over the pedal B at bar 59. (This type of chord has its origins in a contrapuntal technique called third species counterpoint)
All auxiliary chords are indicated in the music in square brackets. They are ignored for the purpose of counting root progressions but are nevertheless annotated so that they can be taken account of, later in the analysis.
We can also look for examples of appoggiaturas which appear to produce chords in their own right. These produce a familiar type of chord: the cadential 6 4 chord. In the following example:
At bar 8 we could consider the A minor chord to arise out of two parallel appoggiaturas rather than out of a root progression. The C in the A minor chord falls to the B of the E chord and the A falls to the G# of the E chord. This analysis is supported by the retention of the E in the bass.
These appoggiatura chords are shown in the analysis in appendix B by a horizontal line beneath the piano staves. The progression here is thus taken as a rising 5th progression ( b° to E). (appoggiatura chords have their origins in a contrapuntal technique called fourth species counterpoint, where they normally occur as suspensions.)
Step 3.
As a final stage in the elimination of chords as nonessential to the root progression, I will, for the time being, eliminate progressions that pair up.
For instance, in eliminating the G# diminished 7th chord at bar 10, it could be argued that we eliminated two opposing progressions that paired up to produce the auxiliary motion. We could now extend this process of elimination to other paired up progressions. We find frequent pairing between rising 4th and falling 4th progressions and a few other pairings. For instance, in the following example the rising 4th progression is paired up with the falling 4th progression resulting in no net root movement:
This pairing is widespread to the extent that every occurrence of a rising 4th progression, in this movement, is paired up with a preceding or following falling 4th progression.
These pairings must be significant in their own right but, for the time being, these auxiliary chords will be ignored for the purpose of counting root progressions. They are, however, shown in the analysis in square brackets as they will be examined later in the analysis. Where there is a succession of such auxiliary chords, these are indicated by a horizontal square bracket, below the stave.
One interesting situation that can arise when carrying out this pairing is the situation at bars 72 to 84. The pairing starts on an E chord and the chords: a, F+6 (augmented 6th), up to bar 79 are auxiliary. It is tempting to continue, taking the E as the main chord, through to bar 83. But it is quite clear from the context that at bar 80 (start of the Recapitulation) A minor takes over as the main chord and the E becomes the auxiliary chord.
Appendix D contains a full commentary on the root progression analysis .
1 Narmour, "Beyond Schenkerism"
Ver. 2.3.