Comparative Root Progression Analysis Technique
4.1.1. Introduction
The purpose of this section is to demonstrate how the Comparative Root Progression Analysis Technique works. This technique has been used to derive the theory presented on this website. You can read this section on its own or in conjunction with the interactive graphical tool which produces graphs of root progressions for pieces currently on the progression database.
If the graph is not already open, press here: GRAPH to use the interactive graph for your self. You may wish to narrow this window in order to be able to read this window and view the graph at the same time.
Press here: HELP for help on how to use the graph then press Back to return to this page.
By viewing graphs based on different selection criteria you will be able to see that different pieces of tonal music show a similar distribution (or polarisation) of root progressions but only when passing chords, appoggiatura chords and auxiliary chords are eliminated from the analysis. This suggests that these types of chords are surface detail rather than structural chords in their own right. It can be shown that these kinds of chords arise out of contrapuntal movement in one or more voices.
This removal of surface detail is different from the foreground reduction process in a Schenkerian analysis and produces different results. I am currently preparing a section on this topic which I will add to the site as soon as possible.
You can also use the interactive graph tool to examine how historically this change came about. To do this, see the section Historical Process.
The term polarisation is used in this discussion to describe (after non-structural (or non-functional) chords are eliminated) the degree to which three of the possible 6 diatonic root progressions are used in preference to the other three progressions. Total polarisation involves the use of three progressions: the rising fourth (α); falling third (β) and rising second (γ) progressions to the total exclusion of the remaining reverse progressions: falling fourth (α'); rising third (β') and falling second (γ') progressions.
Most pieces of tonal music display close to total polarisation. The purpose of the Comparative Root Progression Analysis Technique and the Interactive Graphical Tool are to demonstrate how this works for the pieces currently on the progression database.
Polarisation is not the only important thing to note about the patterns in root progressions. Other points of interest are the clustering of progressions and the segmentation of chord progressions into static harmony and dynamic harmony. These are discussed later in this thesis.
Following, is an explanation of how the technique works, taking one example from the database - The analysis of the Mozart Piano Sonata in A minor KV310 (300d). You can either use the Interactive Graphical Tool for yourself or just follow the screen shots included below throughout the explanation. You can also use the interactive graphical tool to examine the graphs for other pieces on the database.
If the graph is not already open, press here: GRAPH to open the graph page and try this for yourself on this or other pieces.
Press here: HELP for help on how to use the graph then press Back to get back to this page.
There is a more detailed description of the Mozart analysis in the Commentary Section of the Thesis. There is also an brief explanation of the chord progression analysis of each piece on the database in the section Description of Pieces on the Database.
Before continuing with this section, you should have already read the sections in the Thesis up to 4.1. Reduction of Music to Harmonic Outline
4.1.2.1. All Chords Included
In an attempt to discover the patterns in root progressions in tonal music we will start with a first hypothesis that includes all chords in a piece of music. I will include all combinations of notes that can reasonably be considered to be chords and ignore only those combinations that obviously arise out of the short term melodic movement of voices such as: passing notes, auxiliary notes, appoggiaturas etc. For more information about how chords are determined then please refer to Thesis section 3. Method of Working
This means that just about anything that could be considered to be a chord will be included. This hypothesis is just a "first cut" root progression analysis. One issue at this stage is that of the difficulty of identifying correct roots for some chromatic chords such as the diminished 7th chord at bar 10. The fact that this is over a pedal note is an additional complication. We cannot be certain that we should consider this to be a structural chord in its own right with a valid root. It is difficult to identify correct roots in these cases for two reasons:
- Firstly, because these chromatic chords are inherently ambiguous.
At this stage, we will rely on the composer's (or the editor's) spelling
of these chords to be able to assign a root.
- Secondly, because these chords do not contain a perfect fourth or fifth that enables us to determine the root of the chord. There are acoustical reasons why these fourths and fifths enable us to determine roots but this goes beyond the scope of this discussion.
This problem applies to chords such as the augmented 5th chord and augmented 6th chords. It is possible to conclude that these should not be assigned roots but should be treated as non- functional chords. We have to include these chords, initially, in order to determine if they show any root progression patterns, even if this is only in order to eliminate them.
It is, of course, much easier to assign roots to triadic and seventh chords where the presence of the interval of the fourth or fifth helps us to determine the root both analytically and audibly. But this does not automatically mean that we will observe any interesting patterns, even for these chords. Only by testing different assumptions can we come to a conclusion.
By including all chords, there are also difficulties interpreting progressions such as C to C# and C# to C as diatonic progressions. Because of the diatonic nature of the root progression analysis, the graphical tool does not count these as progressions. However, C# to D is treated as a rising second progression.
I have, at this stage, ignored the potential relevance of pedal notes on root progressions.
The detailed results of this analysis are documented in Appendix B. Analysis of Root Progressions which shows chord symbols under the music. (a = a minor, E = E major etc.). The progressions are summarised in the table in Appendix C. List of Root Progressions.
The analysis of the relative occurrence of progressions for this basic hypothesis is as follows:
1. ALL CHORDS INCLUDED
R4 = rising fourth (or alpha) progression
F3 = falling third (or beta) progression
R2 = rising second (or gamma) progression
Etc.
If the graph is not already open, press here: GRAPH to open the graph and try this for yourself on this or other pieces.
Press here: HELP for help on how to use the graph then press Back to get back to this page.
Apart from demonstrating a preference for the rising and falling 4th progressions and a relative lack of falling 3rd progressions, there is no other clear pattern to be concluded from this analysis. This pattern is not similar to that predicted by Rameau's, Schoenberg's or Schenkerian theory. For example, Schenker would predict a larger number of rising 3rd progressions. See Thesis section: 2. Introduction.
4.1.2.2. Passing Chords Excluded
In the "first cut" analysis, we removed only obvious surface detail such as passing notes, auxiliary notes, appoggiaturas etc. We can now extend this idea one step further. It is possible for these types of elaboration to extend in time in such a way that they produce what appear, on a cursory examination, to be chords in their own right. For instance, the C dominant 7th chord at bar 12 could be considered the result merely of two passing notes, in line with the rules of second species counterpoint. (G filling in between A and F and the B-flat filling in between C and A) This can then be classified as a passing chord rather than a structural (or functional) chord in its own right. The important point about the elimination of chords in this way is that it should be done consistently across all pieces analysed.
At some places in the score there are multiple passing chords due to the linear movement in one or more parts see the detailed commentary for more information.
If we eliminate passing chords from the analysis in this way we get the following result. Note that under 'exclusion rules', I've ticked 'Passing Chords' and pressed 'UPDATE GRAPH'.
2. PASSING CHORDS EXCLUDED
This makes only a small difference to the graph. Two of the rising 3rd progressions are eliminated because these arose as the result of passing chords. Apart from this the graphs are similar. Note: the exclusion of a chord results in the removal of two progressions and the creation one new progression. This means that counts can increase as well as increase.
4.1.2.3. Passing Chords and Appoggiatura Chords Excluded
Now let's also eliminate appoggiatura
chords. These are chords that arise from appoggiatura note(s) but
to the extent that they appear to create a chord it its own right. The
most common form of appoggiatura chord in this piece is the cadential
6 4 chord. The result this time is as follows:
Note that under 'exclusion rules', I've ticked 'Passing Chords' and 'Appoggiatura Chords' and pressed 'UPDATE GRAPH'. (You could try just excluding appoggiatura chords)
3. PASSING & APPOGGIATURA CHORDS EXCLUDED
The main changes are that the number of falling second progressions is reduced from 15 to 9 and the number of falling 4th progressions is reduced from 50 to 40. This is because many of these progressions arise out of cadential 6 4 chords, for example, as follows:
This also partially accounts for the small increase in the number of rising 4th progressions.
This exclusion from the count does not mean that these chords are ignored in the analysis. It just means that they are explained in terms of their contrapuntal relationship with adjacent chords rather than their root progression relationship and indicates that they have a less structural significance in the musical phrase.
4.1.2.4. All Non-functional Chords Excluded
Now let's see what happens when we eliminate passing chords, appoggiatura chords and auxiliary chords.
Note that under 'Exclusion Rules', I've ticked 'Passing Chords', 'Appoggiatura Chords' and 'Auxiliary Chords' and pressed 'UPDATE GRAPH'.
4. ALL NON-FUNCTIONAL CHORDS EXCLUDED
This makes a substantial change to the distribution as follows: The rising 3rd and falling fourth progressions now score zero and the falling second progression is substantially reduced from 17 to just 1. The reason for this dramatic change is that these progressions are almost always used in pairs with their opposite progressions in order to create auxiliary chord formations, for example, as follows:
What this demonstrates is that when passing chords, appoggiatura chords and auxiliary chords are eliminated from the analysis, the remaining structural (or functional) chords are polarised to the extent that three of the possible diatonic progressions are used to the almost total exclusion of the remaining three progressions. It also shows that there is an order of priorities within these selected progressions which is similar across tonal pieces.
Examination of the graphs for the remaining pieces on the database (and analyses to be added in the future) will show that this pattern is common for all the tonal pieces analysed and acts as a kind of signature pattern for tonal music. Examination of the patterns for other combinations of exclusion rules does not show the same similarity across all pieces. This suggests that the three types of chord excluded acts as surface detail which elaborates the underlying root progression patterns.
As will be shown later, the root progression analysis is important as an indicator of structure rather than a means within itself.
I will in future refer to these progression as alpha (α) beta (β) and gamma (γ) progressions in order to highlight their relative importance. The opposite progressions I will refer to as α', β' and γ', respectively, to highlight their normal function which is to pair with their opposite 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). As will be seen from the detailed descriptions of the pieces on the database, the use of weak progressions normally occurs systematically rather than randomly and is usually associated with a quick modulation or a chord sequence where the composer appears to be overriding the normal subconscious selection of strong progression by consciously learned processes. See section Chapter 7 - Modulation and the Description of Pieces on the Database for more details on this topic.
Analysis of a larger sample of pieces of tonal music shows that a similar distribution is found in all pieces of tonal music, 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. These are stylistic differences between pieces rather than differences of tonal syntax. You can check this by selecting each piece on the database, selecting all the exclusion rules and then comparing the results.