4.2. Analysis of Root Progressions

4.2.1 Polarisation

If we tabulate the occurrence of root progression types in this analysis we get the following results:

 

It turns out that the polarisation is so strong that three of the possible progressions are selected to the almost total exclusion of the other three possibilities. Within these progressions there is a clear order of preference, as follows:

 

This can be represented graphically, as follows:

 

I will, in future refer to these progressions as alpha (α) beta (β) and gamma (γ) progressions in order to highlight their relative frequency of use. It is in this way that the progressions are annotated in appendix B. The opposite progressions I will refer to as α', β' and γ' respectively to highlight their frequent function which is to pair with the corresponding strong progressions. (It is worth noting that the single occurrence of the falling 2nd progression in this example (γ') occurs at a point of quick modulation (bar 56 to 58). This progression is frequently associated with modulation in this way. Please refer to Chapter 7 in the book section for further discussion of this progression and of modulation in general. Analysis of a larger sample of pieces in the classical and romantic periods shows that a similar distribution is found in all of the pieces analysed, although the proportion of β progressions in this example is slightly higher than the average. In some pieces the ratios deviate from the average to the extent that the proportion of γ progressions is slightly greater than the proportion of β progressions. The slightly higher than average proportion of β progressions in this piece is due to the cluster of β progressions at bars 42 - 43; bars 71 - 72 and at bars 123 - 124.

Also worth noting is the way I've treated the pedal notes at bars 58 to 69. For the sake of simplicity, I've ignored the pedal notes. Analysis of a larger sample of musical examples suggests that chord progressions over pedal notes tend to behave more like chord successions than chord progressions. However, this discussion goes beyond the scope of this limited analysis and does not make a material difference to the outcome. See some further notes on this in the full commentary on the analysis.

4.2.2 Clustering

Now let's examine the paired progressions that we eliminated from the analysis. Further analysis of these paired progressions reveals a further surprising fact as follows: Most of these pairings are clustered, as follows:

This is a sufficiently high number to be statistically significant. At bars 16 to 22 there is a cluster of 9 adjacent paired progressions. The probability of this happening by chance is very low indeed.

If we remove the three paired progressions over the pedal notes discussed above, the clustering is even greater (90%). See note above and the commentary.

Analyses of a larger sample of music shows that these results are the norm. Clustering is therefore of great importance.In a similar way, we can measure the clustering of the non-paired progressions. This is as follows:

Percentage clustering is thus over 99 %. A very high number.

As a foot note, the single non-adjacent non-paired progression (an alpha progression) is the perfect cadence which leads back to the recapitulation and it would be reasonable to assume that this may have something to do with the fact that it is not part of a cluster!

What is clear from this analysis (and this is confirmed by many other similar analyses) is that (once passing chords , auxiliary chords and appoggiatura chords are removed) the following are true (expressed in purely logical terms, firstly):

1. Root progressions are of two types: paired progressions and non-paired progressions.

2. These two types show very strong clustering.

3. Non-paired progressions are polarised i.e. are almost exclusively selected form three of the possible set of six diatonic root progressions.

Expressed in musical terms this means that:

1. Chord progressions exist in two forms:

alternation of chords (could be referred to as static harmony as the roots do not progress). These are based on either the tonic chord alternating with other chords (tonic prolongation) or the dominant chord alternating with other chords (dominant prolongation).

• a succession of strong root progressions (rising 4th, falling 3rd and rising 2nd) (could be referred to as dynamic harmony as the root progressions show movement)

2. These states exist in distinct episodes and music constantly alternates between these two states.

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