copyright © 2002 by Larry J Solomon
Abstract: The following demonstrates a possible systematic approach to Schenkerian style analysis by use of an algorithm. It uses Bach's Prelude in C Major, WTC I, as an example. A different version of this essay was published in Musicus 2/i, UK, 1990. Some additional new ideas are presented here. As such, it is an original contribution that attempts to develop a consistent and logical series of steps for carrying out such an analysis. Following these steps objectifies the analysis and yields a clear and consistent result.
See the table of symbols link
The ideas in this primer are developed from those originated by Heinrich Schenker1. Symbols, methods, and ideas are consolidations of those used by various scholars in the field, and some are newly invented in the interest of consistency. There is no attempt here to petrify Schenker's methods. His own writings show them to be in constant evolution. No doubt, if he had lived longer he would have continued to improve upon and change his methods and representations. The term "Schenkerian" no longer refers to a petrifaction of one phase of Schenker's development, but instead to serve as a springboard for further evolution. Nevertheless, the author believes that the following representation adheres to Schenker's general spirit and goals of analysis. The purpose here is to reveal the structure of music, not to dogmatize a theory that was clearly in a state of flux.
A melody is more than an arbitrary string of notes and rhythmic values. Perception plays a part in determining intelligible melodies. For this reason it is necessary to examine how melodic lines are perceived. Recent psychological studies have shown that the mind is continually and involuntarily trying to make sense of the sensory data that it receives from our ears and eyes. This is done by making connections, or relationships, between the incoming data and information that is already stored in memory. The mind tends to favor the easiest connections which are usually also the shortest routes. In a polyphonic texture, the ear and mind tend to form melodic lines from notes that are close together in pitch, i.e., those that repeat or move stepwise. Leaping notes are more difficult to connect melodically. The larger and more frequent the leaps, the more difficult or unintelligible a melody becomes. So, as a rule, step motion predominates.
Ex 1. Tchaikovsky, Symphony No. 6, movement IV, violins
a. actual score b.
play v1 play v2 play v1 + v2
Ex1a displays the notes that begin the last movement of Tchaikovsky's Symphony No. 6. The violins have leaping lines with no step motion. But when this is played, especially by two instruments of the same timbre, what is heard is the illusion of a stepwise descending line shown as Ex1b. This surprising result shows that the mind prefers smooth motion, or shortest routes, over leaps for purposes of intelligibility. These are not only easier to hear and understand but are also easier to sing and play. Here are some other examples:
Ex 2. Illusion from Rachmaninov's Sonata No. 2 for two pianos
a. actual b.
play v1 play v2 play v1+v2
Ex 3. Scale illusion by Diana Deutsch
play v1 play v2 play v1+v2
Although these phenomena were not well known in the nineteenth century, Schenker (1868-1935) hinted at them in his discussions of scale steps (Stufe) and linear progressions (Zug), and his analyses show a persistent recognition of the primacy of step motion. This inevitably and logically leads to the concept of step progression. Voice leading is here defined as the motion of a single voice requiring step progression. Moreover, step progression is necessary for maintaining voice identity. Repeating and sustained notes continue voice identity but are not essential to voice leading.
From the above evidence we can say that the ear and mind tend to connect melodic notes by listening for step motion even when step motion is not being played or is not apparent in the notation. The principle of step progression is paramount for melodic perception. Many scholars overlook this essential point. A series of notes that moves by step, as in Ex 4a, represents a single voice line. A series that leaps, even by intervals of a third, represents divided voices, with each note separated by a leap representing a different voice. In Ex 4b two voices are moving in parallel thirds. The v1 moves stepwise as E-D-C-B, and v2 moves as C-B-A-G. No leaps are involved in the actual perception of these voices. Thus, the voice leading of these two voices may be reduced to Ex 4c.
Ex 4. Examples of linear progressions and divided voices
The same prinicple holds in Ex 4d with voices moving in parallel sixths, simplified as Ex 4e. In Exs 4f-4h arpeggios of four notes represent four different voices.
Some common factors are known about tonal music. First, nearly all such music begins and ends in the same key, which we call the tonic. Second, nearly all tonal music then diverges away from the tonic in the middle of the piece, often to the dominant key or to some other related key such as the relative major. Most often it moves to the dominant. In fact, this is the plan in binary and sonata form. From this we can induce a common plan for most tonal music, namely I V I. Now this itself, in the final analysis further reduces to simply tonic. Therefore, all such tonal music finally reduces to just tonic, because it starts there and ends there. That is why we can state that a particular piece of music is "in the key of" whatever it may be. This is enormously simplifying. It means that all such music is really just an elaboration of the tonic chord, and the most common divergence is to the dominant. So the basic harmonic scheme is I V I, with the roots of these chords in the bass. This is the Bass Arpeggiation, which we can illustrate as follows:
Ex 5. Bass Arpeggiation
The bass voice often has a harmonic rather than a melodic function. It has a tendency to outline essential harmonies, especially tonic and dominant. Therefore, the bass voice typically leaps by fourths and fifths, but it may also have step motion that fills in these leaps.
Another Schenker discovery is that there are three essential structural soprano lines, which can be called Fundamental Lines (Urlinie). All of these lines descend. This is due to the commonly perceived sense of resolution at the end. Therefore, there must be a higher energy level preceeding that resolution. The downward line may be regarded as the graphic representation of the dissipation of this energy level throughout the course of the music.
Due to the normal establishment of the tonic key at the outset, each Fundamental Line starts with a note of the tonic chord, and each can only have one of three possible starting notes: 3, 5, or 8. Additionally, tonal music normally ends with structural 1 (tonic) in the soprano. Thus, the three schemas are:
Ex 6. Fundamental Lines (Urlinie)
When a Fundamental Line is combined with the Bass Arpeggiation, the result is the Primordial Structure, (Ursatz). The Primordial Structure is an abstraction and is not the same as the Background. There are three possible Primordial Structures:
Ex 7. Primordial Structures (Ursatzen)
Schenker believed these to be the fundamental structures of tonal composition. Through analysis, it has been shown that a large number of compositions share these basic structures.
Although voice leading is defined by step progression, there are many examples, especially in instrumental music, where the principle of step motion seems to be violated. The following well-known example is from Bach's Prelude in C major in his Well-Tempered Clavier (WTC), Book I.
Ex. 8. First four measures of WTC I, Prelude
The notes seem to persistently leap as arpeggiated chords, the first measure being tonic and the second being a ii7, the third a V7, etc. However, a verticalization of the chords of these four measures discloses the essential voice leading:
Ex. 9. First four measures of WTC I, Prelude
Ex 9 distills the essential voice leading. Five voices are revealed: 2 sopranos, 1 alto, 1 tenor, and 1 bass. Thus, each is moving stepwise as smoothly as possible. In Ex 9.1 each measure is compressed to a single chord and contains all the notes of the original version. Ex 9.2 displays the basic structure of the voice leading, called the foreground. It views the first four measures as simply tonic with nonharmonic step motion in and out of the notes of the tonic. All the notes are still represented.
Some symbols are introduced to indicate aspects of the voice leading. Notice that white and black notes are used not as rhythmic notations but as hierarchical tonal symbols; i.e., white notes are structurally more important than the black notes. The black notes are nonharmonic tones that show the step connections between the essential white notes, which in this case are those of the tonic C major domain chord. All of the black notes are neighbor notes in our example.
Solid beams indicate voice leading that is changing, whereas the broken beam denotes stationary voices (repeating or sustained notes). The key to this type of analysis is the recognition of step motion moving through the harmonies. Not only does this reveal important aspects of the voice leading, but it is also invaluable for developing sight-reading skill by simplifying the music and engaging a broader scan.
On a more "distant" level, or middleground, Ex 9.3
simplifies the first four measures as simply a prolongation of tonic. This
is accomplished by deleting notes of lesser importance (black). Finally,
Ex 9.4 shows these four measures to be essentially a tonic which is part
of a larger directed motion that continues to the end of the work, i.e.,
A reduction of the harmonies of the complete prelude is as follows:
Ex 10. Preliminary reduction
This analysis should be studied carefully. The entire prelude can be played from this graph by arpeggiating the chords. Compare it to the score of the Prelude. Barlines are shown here only every four measures. Each chord represents one measure. Every note of the prelude is accounted for. The white notes are the harmonies, and the few black notes are nonharmonic; the one in m24 is passing, in m27 and 30 they are suspensions, and the black bass notes are pedals.
There are several secondary dominants and leading-tone sevenths. The arrows among the chord symbols show the directed harmonies. The arrows on the staff indicate chromatic notes that have a strong sense of direction toward resolution. Sharp notes are usually secondary leading tones, resolving up, while flat notes are sevenths of chords, which resolve down. These strong chromatic motions signal important changes in the music.
Notice at the bottom of Ex 10 there is a line labeled domains. The first four measures have already been shown to be basically an elaboration of tonic, which is why these measures are shown as a tonic "domain". The harmonic domains show that not all the chords are equally important, but some dominate specific segments of the music. These domains are always diatonic chords and normally fall into one of four types: tonic, dominant, subdominant, or supertonic, representing the essential functions. The second domain is dominant, which extends from measures 7-11. Notice that the F# s are chromatic notes that "point to" this domain.
The next domain is supertonic in measures 12-13. Once more, the chromatic notes in m12 point to this domain. Measures 14-19 return to the tonic domain; the strongest indicators are the return to B natural (leading tone) and the Ab, which is the seventh of the vii°7 chord. Measures 20-21 emphasize the subdominant due to the preceding secondary dominant. Measures 22-31 center on the dominant, which is emphasized as a pedal in the bass. The F# in m22 functions as the leading tone of G and the Ab in m23 functions as a downward leaning note to G.
Measures 32-33 focus again on the subdominant. Measure 34 has its own domain, the dominant, as does the last chord, tonic. It is common, although not universal, in this kind of analysis to regard the final cadential chords as important domain chords.
If one knows the rhythmic figuration that runs through this prelude, one can play the entire piece from this diagrammatic analysis. Therefore, by grasping this structure, the prelude is readily comprehended and played. This makes it quite powerful for sight reading as well as for understanding the tonal structure.
The reduction in Ex 10 of the C major Prelude can be regarded as the first stage of a preliminary foreground. The next step is shown in Ex 11.
Ex 11. Preliminary Foreground
In this intermediary step, ties are used to show note repetitions, and some of the previously white notes are filled in to show their lesser significance on a larger scale. These black notes will be deleted in the foreground analysis. Notice that in most of the repetitions the notes following the intial one are blackened, but in a few the intial note is blackened. This choice is based upon the domain harmony. Specifically, the initial note is blackened only if it precedes (anticipates) a chord of the domain harmony. Otherwise, only the immediate repetitions are blackened.
The following is the Foreground analysis. In this step, the black notes of Ex 11 are deleted. Additionally, any notes that are not part of the domain harmonies are now blackened, which shows their lesser significance on a larger scale.
Ex 12. Foreground
Beams now connect notes of domain chords. Broken beams indicate repetitions. Black notes are also beamed to show the step progressions between domain notes. Slurs are used to show step progressions between one domain and the next. Occasionally, they are also used for step progressions within a domain when lines split or diverge; e.g., the F# and Ab in the bass of m23-24 converging on G in m24.
Octave displaced step progressions are shown as lines and interval numbers between the connected notes in Ex 12. All of these occur near the end of the prelude. "Octave displacements" include leaps of sevenths and ninths, which are really displaced step progressions.
White notes in the bass voice are examined for fourth and fifth leaps within a single domain. These are not beamed together but show a harmonic rather than a melodic significance. These are normally authentic cadences; e.g., m10-11 and m18-19. Each of these pair of notes will have a principal and an auxilary note. In upward motions of a fourth, the initial note is auxiliary. Otherwise, the second one is auxiliary. The auxiliary note is shown as a white eighth-note connected to the principal with a slur.
Notice that all notes of the original score are preserved in the foreground except for repetitions. Therefore, it represents the essential structure on an immediate scale. Again, all of music can be played from this chart if the original arpeggiations are added.
Between this foreground analysis and the background, a middleground analysis shows the link between the two, revealing more of the basic structure and less of the details than the foreground. The same steps of note elimination are used after reducing the foreground domains to simply tonic-dominant-tonic.
[To see the intermediary analytic steps between the foreground and middleground, click on this link.]
Ex 13. Middleground
Note the bass line descending stepwise an octave to m20. Then it moves to the dominant via an incomplete neighbor (F). Finally, in m35 it returns to tonic. The soprano line starts on 3 (E) and descends stepwise to C in m16, moves up to D in 24 and returns to C at the end. Thus, it is fundamentally 3-2-1. Inner voices are also shown on this graph, all of which move stepwise, mostly descending. From 24 to 34 the soprano rises through an arpeggiated dominant, thus transferring the register from D4 to D5. In the background this will be reflected as a stationary D. Thus, we can begin to see the outline of the background in the middleground graph. By eliminating the black notes of the Middleground and duplicate lines we get:
Ex 14. Background
[follow this link to see an intermediate step between the Middleground and Background]
Although a detailed study of Schenkerian analysis is beyond the scope of this exposition, this serves as an introduction to this type of analysis.
A problem that is debated about Schenkerian analysis concerns its subjectivity. The opposing concept of objectivity is brazenly attacked today, especially by the school of late postmodernists and deconstructionists. It is oftened maintained, even among some analysts who are not of these persuasions, that subjectivity is not only inevitable but desirable2. Although a small degree of subjectivity is inescapable in any type of analysis, the problem with a liberal subjectivity is that it means that anything goes and that any analysis is as good as any other. This leads to an unprincipled type of "analysis" (which is actually not an analysis but a personal interpretation) that reveals more about the analyst than it does about the music.That is exactly how subjectivity is defined. It is very doubtful that Schenker had subjectivity in mind. His strict, single-minded dogmatism certainly shows his belief in objectivity. An objective analyst believes that structure is inherent in the music itself, that it can be found, and that an objective analysis at least attempts to show the structure of the music itself, rather than analyzing what the analyst feels about the music.
Schenkerian analysis is mostly done unsystematically today. Discarding subjectivity, this is probably because there is little agreement among scholars about a system, which has led to many pitfalls and flaws that have had no resolution. One analysis may be radically different from another, and both are equally justified or unjustified. This author believes a subjective approach is of very little value to anyone other than the particular analyst doing it. It is hoped that a systematic approach will help this kind of analysis reveal more about the music and gain a wider acceptance.
A Comparison with Schenker's Analysis of the C Major Prelude
To distinguish Schenker's analysis from the above analysis, the latter will be refered to as the "algorithm" or as "algorithmic".
Ex 15. Schenker, H. Five Graphic Analyses, Dover, 1969
In Schenker's own analysis of the C major Prelude, Ex 15, the Foreground (Vdg.) (bottom level), is directly comparable to Ex 7and Ex 12 of the algorithm. His reduction proceeds more rapidly from one level to the next (moving upwards in Ex 15), and some of the symbols are different, but the essential meanings of the graphs are comparable and similar. The initial step of verticalization is immediately apparent in Schenker's first (lowest) graph. However, there are a few details in his graph that are not shown in this model. For instances, the large scale parallel-tenth (Oberdezimen) movement between soprano and bass from the beginning to measure 19, and the large scale fourth and fifth motions in the bass (Quartzug and Quintzug). Schenker also shows the 3-2-1Urlinie in his Foreground graph. This is reserved for the final stage the algorithm here, Ex 14, the Background. Generally, Schenker incorporates some of the large scale analysis on his lowest level, whereas the algorithmic method has built in controls that allow these large scale structures only in the last stages of the analysis.
On the other hand, there are many aspects of voice-leading and tonal hierarchy shown in the algorithmic graphs that are not in Schenker's. This includes most of what is in the Foreground and Middleground graphs; e.g., the contextual beams show many lines that are completely absent from Schenker's graphs. The hierarchical harmonic-domains are also absent. The algorithmic graphs show more detail.
In some cases, Schenker's observations are shown with a different symbol by the algorithm; e.g., many of the contextual slurs he used in the Foreground are found with beams in Ex 12 and Ex 13. The Arabic numbers below the staff in Schenker's graphs, which outline the voice-leading, are shown graphically with beams or slurs in the algorithmic Foreground.
There are some basic disagreements, too. First, some of Schenker's notes are in error. The alto E in measure 15 should be a C, and the tenor A in measure 30 should be a G. More serious, however, is the deletion of the leading-tone, B, in measure 23. From a published letter concerning this measure, we know that this is not an oversight, but Schenker's only explanation for its obliteration is that Bach merely "feigns" the outline of a third. This is the only note that Schenker omitted from the score; that is, until measures 33-34, where the omissions are understood as the beginning of note eliminations in this cadenza-like passage. It is difficult to justify his intent. The leading tone is not superfluous to the structure, and surely Bach did not put it there merely as any note to fill in the finger work. Schenker's omission of the leading-tone is unique in measure 23, and leads him into an erroneous interpretation of its context. He labels the chord there as a II6 (it certainly is a II if the B is ignored). What is more surprising is that Schenker actually deletes the leading-tone and keeps the result as a significant harmony in his middleground. However, with the B present, the chord is a leading-tone seventh, not a II6 (curiously, he seems to show its position differently in his Background). The B cannot be ignored, and its presence considerably weakens the credibility of II as a unique middleground domain. This chord is simply a continuation of the dominant-preparation begun by the IV preceding it.
Schenker gives a great deal of attention to the chromatic motion in measures 22-23, and even includes an exclamatory note on the bass there. If the leading-tone were not deleted in his analysis, it seems certain that the result would have been different, and the II would then have to be removed as a significant middleground harmony.
Another significant difference in Schenker's analysis is the position of the final tonic (domain). He considers the tonic to be prolonged from measures 32-35, presumably because of the bass pedal in these measures (curiously, he shows its position differently in his Background). The aspect of a prolonged bass is considered in the algorithm, but aside from the bass C, there is no other corroboration for the tonic until measure 35. In fact, a close examination shows measures 32-33 to be a IV64, a common cadential six-four, until measure 34 where the dominant occurs, immediately followed by the final tonic.
In a level-by-level comparison, Schenker's Background is comparable in meaning, if not in appearance, to the results achieved by the algorithm. His Foreground and Middleground are much less detailed but what is there is contained in the middleground graph of Ex 13, notwithstanding the differences just mentioned. The Foreground graph, Ex 12, has no direct counterpart in Schenker's analysis, but instead, the Preliminary Foreground, Ex 7, is a parallel to Schenker's Foreground, with the exception of Schenker's errors in copy, mentioned above. Additionally, his contextual slurs are mixtures of the separate contexts defined for slurs and beams incorporated in the foreground graph of Ex 12.
In conclusion, the algorithm designed here gives results comparable, but not identical, to those obtained by Schenker, yet yields more structural detail. More important is the application of a consistent procedure that may be followed for other analyses. This procedure has been designed to illuminate the structural aspects of the harmony and voice-leading, but also contains aspects of the formal and rhythmic structure.
1. Heinrich Schenker (1868-1935), Austrian theorist, was the originator of this type of analysis.
2. It is mistakenly claimed that since music is a subjective art, that analysis of it should also be subjective. This is the same as saying that because human beings have emotions, thoughts, ideas, dreams, etc, that we should not attempt to logically or scientifically analyze them. This would dismiss all of psychology, philosophy, medicine, and other human sciences, as well as the logical analysis of any art.