Harmonic Analysis as a Method to Build a Musical Composition Model

Zakharov Yurii ORCID: 0000-0002-3873-3934 Doctor of Art History Professor; Department of History and Theory of Music; Victor Popov Academy of Choral Arts 125565, Russia, Moscow, Festivalnaya str., 2 n-station@yandex.ru Other publications by this author

There are a large number of methods for analyzing harmony, theories that allow us to establish the laws according to which a musical composition is built.

Among them are the identification of the fret or frets and how the fret unfolds and is realized in the work, tonal and functional analysis, analysis in terms of central and derived elements, analysis using the method of G. Schenker, analysis using neo—Riemannian theory, etc.

Each of these methods is based on its own theory, has its own tools and leads to its own results. This can be a graphic figure (for example, a wave or a bundle of lines), a combination of functional paints (shown in different colors on the diagram), or even just the unfolding of the qualities of a chosen interval — a fifth, a quarter, a tritone (see more details in ^[1]).

The results of a musical theoretical analysis are largely determined by its language, that is, the way these results are presented. Theorists try to avoid simply describing what happens in each bar. They use tonal and functional signs, form diagrams, special analytical musical examples, etc. Musical works of the twentieth and twenty-first centuries require the invention of more and more new methods of analysis and new ways of presenting results.

Any analysis involves identifying the logical structure of the work, separating the more significant from the less significant. Identifying and reflecting the semantic structure means building a model of the work. Such models will include fret polychords represented by notes, a sequence of functional symbols, shape schemes, serial disposition, and many other ways of graphically representing musical compositions.

The concept of "model" is increasingly used by musicologists in order to clarify both the process of a composer's work on a composition and the process of analyzing music. Back in 1975, M. G. Aranovsky spoke about a heuristic model that should "become a piece of music, transform into it, generate it" ^{[2, p. 138]}. If we talk about modeling as a method of analysis, then it is impossible not to mention the analytical methods of B. M. Gasparov, who at the time of his work at the University of Tartu (1968-1980) published a number of articles devoted, in particular, to the structural analysis of the harmonic system in some works by L. Beethoven. Gasparov puts forward an algorithm for classifying chords based on their contextual compatibility with other chords, which allows him to experimentally create a functional system that is close to the traditional one, but differs from it in some details ^{[3; 4]}. L. O. Hakobyan, talking about Gasparov's experiments, writes that "the model obtained at the output of the analysis is a set of rules for compatibility and mutual dependence of segments belonging to different functional classes" ^{[5, p. 121]}. Similarly, Gasparov gets a new system of kinship of keys.

In recent years, musicologists have increasingly resorted to various modeling methods, including graphic ones aimed at visual perception, in order to obtain a one–time image of the analyzed work or to penetrate the mystery of the compositional process.

A. N. Pleskonosov (2021) emphasizes the importance of a "simultaneous image" for the perception of musical form ^{[6, pp. 63-64]}. A. A. Amrakhova (2022) speaks of "mental models that determine the features of shaping in the music of composers of the late XX — early XXI centuries" ^{[7, p. 15]}. Referring to the American linguist J. According to Lakoff, she names 4 main types of mental models — proposition, image-schema, metaphor and metonymy ^{[7, p. 16]}.

In his article "The principle of modeling in the light of text theory" S. S. Goncharenko (2016) talks about "the study of compositional models that have crystallized in the music of the European professional tradition" ^{[8, p. 19]}.

As if picking up Goncharenko's idea and developing it on concrete examples, M. S. Vysotskaya (2021) writes about a compositional model that "can be understood as a draft, preliminary plan, foundation and architectonic framework of the future "building". <...> An artistic canon or an author's text acts as a compositional model; it is set from the outside or built according to an individually developed algorithm; the method of its representation can be a verbalized system (description) of rules, a graphic scheme or an objectified sample in sound form" ^{[9, p. 152]}. Vysotskaya cites analytical examples where such a model is a rhetorical disposition or a four-part scheme of a symphonic cycle; the method of serial work by A. Webern turns out to be a model for F. Karadzhev when he composes Concerto grosso in memory of Anton Webern for chamber Orchestra.

All this is a search for a way to either find an algorithm for the high–pitched unfolding of a work in time, or to present the work as a graphic image.

This article is devoted to the search for new ways of analytical representation of a musical composition in a simultaneous form. These methods are based on the theory of G. Schenker, outlined in his work "Free Writing" ^[10], and are developing, firstly, in the direction of a more detailed identification of the middle plan of the work and the sound-pitch lines operating in it, and secondly, in the direction of attempts to free the method of reduction from the strict dogmas of the Schenkerian methodology and "inculcate" it includes methods for analyzing modal harmonic systems, as well as ways to identify melodic functions.

To present a work in a simultaneous form, which has become the result of analytical procedures, means precisely to build a model of the work. This model may be more or less adequate to the work itself. The degree of adequacy is partly determined by the method: most likely, models that use musical notation, but at the same time represent the work in a generalized and abbreviated form, will be more adequate.

From this point of view, the activity of a music theorist can be characterized as follows. Based on his knowledge of the principles of the design of high-pitch systems at this stage of the evolution of musical language, he assumes with the help of which of the theories known to him (and the methods implied by it) he can identify the laws that the high-pitch structure of a particular work follows, and, accordingly, find the logic according to which its pitch is built and deployed in time.

The starting point here will be the identification of the fret organization in the broadest sense of the word — the fret as a system of connections between high-pitch elements (pre-designed and individually implemented in this particular work), as a balance of foundations and weaknesses at different levels, as the interaction of horizontal and vertical aspects of the harmonic system.

After defining the basic fret categories, a choice of theories is possible, with the help of which further analysis will be carried out. This will be a different set of theories for different centuries of music evolution.

An obvious set for 19th century music and 20th century tonal music:

1) theory of major-minor tonality and tonal functions;

2) theory of extended tonality, extended functional system;

3) Theory of (neo)modality, diatonic and chromatic modes (if applicable);

4) reduction, revealing the harmonic backbone, including by the method of G. Schenker;

5) reduction leading to a scheme of pitch lines outside the context of Schenker's theory;

6) melody analysis in terms of identifying the main and secondary melodic foundations (which may be different from tonal and harmonic functions); fret polychords on which the melody is based;

7) analysis of chord verticals from the point of view of neo-Riemannian theory (if applicable).

On the way to the model, the musicologist takes several steps from empirical reality to abstraction. The question is when to stop.

To clarify this point, let us turn to the Schenkerian methods of analyzing music through a series of successive reductions. The musical text is taken as it is given by the composer. First, the diminutives are removed, i.e. melisms, chord figuration, and scale-like movements. Then the texture is reduced to a strict four-tone. Delays and rhythmic difficulties, cases of voice exchange and other subtleties of voice management are removed. The linear moves of the foreground are highlighted. Next, the transition to the middle plan is carried out — the main (supporting) linear moves and the sequence of steps in the bass are revealed. L.O. Hakobyan writes about the first structure, which turns out to be the final step of reduction: "The idea that the I–V–I structure serves as a kind of primary core for tonal music, and any tonal piece can be interpreted as its prolongation, is quite consistent with our understanding of tonal music and, most likely, will not cause protest among theorists of most trends, including those that have nothing to do with Schenkerianism." ^{[5, pp. 109-110]}.

Using the method of step-by-step reductions, we get more and more compressed images or models of the work. Firstly, these models are visual, as they graphically reflect the high-pitch structure. Secondly, they correspond to the nature of music, because they reflect precisely the pitch in its temporal formation. Thirdly, they reveal the harmonic framework of the work, allowing you to distinguish the more significant from the less significant. Fourthly, the analyst can stop at the very moment when the individuality of the work begins to fade.

The intermediate schemes that arise in the course of Schenkerian analysis reveal three aspects of the musical fabric — linear, stepwise (i.e., in a sense vertical), as well as the work as an unfolding of the tonic triad. On the one hand, we see the work as a complex of high-pitched lines that play the role of branches on which the musical fabric "grows". On the other hand, we see the steps of the middle and background placed under these lines (and, in fact, controlling them).

However, the reduction of musical fabric can be carried out outside of the Schenkerian methodology. In this case, it does not lead to a primary structure, but to a pattern of pitch lines, which emphasizes the horizontal aspect of the musical fabric. At the same time, the basic tones of the sounding chords and the logic of their sequence are not always taken into account or are not taken into account at all. Interestingly, this reduction can also be used in atonal music, including dodecaphonic music. Linear analysis makes visible the hidden sound-pitch lines traced behind the visible and audible melodic pattern.

For example, a similar analysis, applied by us to the melody of A. Webern's song Das Dunkle Herz, led to such a scheme ^{[11, p. 36]}.

Figure 1. — A. Webern. Linear analysis of the melody of the song Op. 23 No. 1 (vol. 1-11)

Another method aimed at melody research is melody analysis in terms of identifying the main and secondary melodic foundations (melodic functions), as well as the fret polychords on which it is based.

This method has not yet become generally accepted and elaborated in detail. Yu. N. Kholopov wrote about the functions of the melodic tonic and melodic dominant in the article "Melody" from the Musical Encyclopedia ^[12]; these ideas go back to B. Yavorsky ^[13] and L. Kulakovsky ^[14]. The latter wrote: "The melodic center is the normal sound level for a given melodic curve. <Analyzing folk melodies, it is not difficult to notice that along with the frequent coincidence of the melodic center in them with the main tone of the tonic, its relation to the fret elements is also different, i.e., that the melodic center of a folk melody is much more free compared to classical music. Its coincidence with the fifth tone of the tonic (Tv — the classic “dominant”) is especially common, which is understandable due to the stability and brightness of this sound. Sometimes there is another position of the melodic center, which coincides even with unstable elements, most often with the least striking restlessness — So.s." ^{[14, No. 6, p. 19]} ("the inversely conjugated tone of the subdominant" is the II step in a major and the IV step in a minor).

And in the Musical Encyclopedia we read: "The unity and definiteness of melody are determined by the attraction of the sound stream to a firmly fixed reference point — the abutment ("melodic tonic", according to B. V. Asafyev), around which the gravitational field of adjacent sounds is formed. Based on the acoustic affinity felt by the ear, a second support arises (most often a quarter or fifth higher than the final abutment). Due to the quart-fifth coordination, the mobile tones that fill the space between the abutments eventually align in the order of the diatonic scale" ^{[12, stb. 514]}.

In our opinion, to analyze the tonal melody, methods developed for monody can be used, implying the identification of the main, intermediate and secondary foundations and various types of weaknesses. These methods are necessarily related to the identification of fret polychords, the extreme sounds of which turn out to be foundations.

Is it possible to synthesize these methods and present them on a single analytical musical example?

First, let's turn to the rather simple harmonic structure of the play "Winter II" by R. Schumann from the Album for Youth (composed in 1848).

It is not quite ordinary in form: after the repeated period c-moll → g-moll comes the interlayer period c-moll → Es-dur, followed by the middle section in the modulating simple two-part form g-moll – d-moll– g-moll – c-moll. The abridged reprise is interrupted by a C-dur insertion with grossfather themes, after which the resumption of the first theme is no longer heard as a reprise, but as a coda. Although the play is small, the contours of a complex three-part form are visible.

If we analyze "Winter" using the Schenker method, we will identify the third stage as the head tone, which quickly descends to the second; the second stands for almost the entire second part and at the end of it goes to the first. This is where the play could have ended, as the first structure "did its job." And indeed, the reprise seems to be "attached", not so necessary from the point of view of the general harmonic construction. But in order to include this reprise in the overall design, let's imagine the previous descent 3-2-1 as a mid-plan line, and in the reprise we will see a real first line.

Everything else, from Schenker's point of view, is definitely a code, and the first line is set to "1".

Figure 2. Analysis of R. Schumann's play "Winter II"

In addition, Schenker's analysis easily reveals the logic of the tonal plan of the play. This is the third row of C–Es–G–G–B–D, followed by a quick descent through the fifths of D–G–C (the third notation).

This, in fact, is the compressed model of the work.

Compared to Schenker, I pay more attention to some foreground phenomena. In this case, the melodic unfolding of the leading, most noticeable intervals.

Such an interval in the "Winter" turned out to be a minor sexta, i.e. a progressive upward movement from the fifth to the third, and sometimes from the first to the sixth minor scale. The movement from 5 to 3 is called the tonic unfolding of the sextet, and from 1 to 6 it is called subdominant (denoted respectively by T(c) and S(g), where "c" and "g" are tonalities).

Figure 3. — The unfolding of the sexta in the play "Winter II"

In the initial period, this movement is seen by the upper sounds of the triads, from which the melody is born. The upward movement is replaced by a downward one, which leads to the tonicalization of the fifth step.

The construction adjacent to this period leads to Es-dur and is based on the reversal of the minor sexte, i.e., the major third, which underlies the melody.

In the initial period of the second theme (with the tonal plan g–B–d), the subdominant unfolding of the minor sexta is directly present in the melody (from the first to the sixth step of G minor). In the second sentence, at this point, there is a movement in the octave range b-c-d-e-f-g-a-b, leading from B-dur to d-moll. And in the reprise sentence (c-moll)— subdominant deployment is again used.

In the major "intermezzo" there is no movement within the minor sexta at first, but in the last bars it (in the descending version) unexpectedly appears as 6-1 F-dur, where f is replaced at the last moment by fis (a reduced third quarter chord).

After that, the first theme returns at the dominant organ point, and with it the tonic unfolding of the minor sexta, which, a few bars before the end, leads not to es, but to e, as if finally resolving the tension that had been building up throughout the piece.

The proposed analysis reveals, among other things, the plot of the play, but this plot is absolutely intramusical and is based on the life of the melodic unfolding of the minor sexta, which: a) it is given as the ascending and descending backbone of the melody (while the ascending movement occurs within c-moll, and the descending modulates in g-moll, and the sexte turns out to be subdominant); b) turns into its own reversal — a large third; c) It is represented directly in the melody and grows to an octave; d) occurs within the framework of F major, where f suddenly changes to fis (leaving the sextet small); e) overcomes its minor character by replacing es with e.

Now let's turn to the analysis of the play by K. Debussy. Debussy's music is more difficult to analyze, as the composer tends to build individual harmonic systems, and tonal principles are often combined with modal ones. And Schenker's laws can only partially apply in it. The object of consideration will be the play "Ruins of a temple in the light of the moon" from the second notebook of "Images" (L. Kokoreva suggests a more accurate translation of the title: "And the moon descends on the once former temple" ^{[15, p. 123]}).

The form of the play allows for various interpretations; I would characterize it as a second rondo (according to A.B. Marx).

The first theme consists of two contrasting elements: 1) a melody set out in chords with a 5.2 structure; 2) parallel triads (as well as one large minor seventh chord) against the background of the tonic fifth of e-moll.

Figure 4. — K. Debussy. Ruins of the temple by moonlight, vol. 1-9

Although the melody follows the sounds of harmonic E minor (sometimes capturing f), the fifth step still turns out to be the central tone.

The second theme (the middle section of the play) begins with a modal episode in Dorian c. It seems that the si tone is the main pillar of this section. Its shape is shown in Table 1.

Table 1. The shape of the middle section

The "C" represents the counterpoint of two modal melodies, summing up the h-fis fifth as the main mouthpiece. In the lower layer, symmetrical figures 2.3 down and 2.3 up from h attract attention.

Figure 5. — K. Debussy. Ruins of the temple by moonlight, vol. 13-15

The "d" material, as an unstable construction (thematically close to the first theme), reveals the side foundations of g and c, stopping at chord 4.2 (however, now h claims the role of its main tone).

The "E" material is a sequence on basses A and G (VII and VI in H-dur), which is why it is marked as unstable.

Next, "C" is a combination of a modal theme in the upper register (h Mixolydian) and tonal chords in H-dur (T-⁰ D-T-⁰ D-M).

Figure 6. — K. Debussy. Ruins of the temple by moonlight, vol. 29-30

From the 29th measure, the most unstable section of the play begins, which can be considered as a return move (to the reprise). First, the modal-tonal complex h Mixolydian reappears on the theme of "C" + cis-moll in the chord layer. Next, gis claims to be a tonic with very unstable harmonies and a sequence of m3 downwards. Finally, in verses 35-38, the material D on b2 above is repeated, after which the pre-reprise begins. However, this predicate is not dominant, but on the tritonant (which alternates with the parallel of the minor dominant and with M).

An abbreviated reprise of the code combines the material of the first and second topics ("A"). But in the first theme, the chord 5.2 and 6.1 in the sixth step takes the dominant place, and the second one is still in Dorian h. In the last cadence, the tritonant and tonic alternate.

Consider Figure 7.

Figure 7. Diagram of the shape and pitch structure of Debussy's play

What does this musical example show us?

1) Form: Part I — Part II — return stroke and predicate — Part III (reprise code).

2) Tonal plan, functions, steps.

3) The modal layer in "C" and in the reprise.

4) Some chord verticals.

5) Traces of Schenkerian primordial structure. However, the head tone (5) does not move. At the same time, the bass, as it should, makes the I–V–I move. At the beginning of the prelude, before the reprise, V is shifted to a lowered V. Schenker did not know about the tritonant, but Debussy uses it, making the original reverse arpeggiation of B–Gis–E.

In the following figure, the same piece is presented in an abbreviated form, with bar numbers; the first structure is more clearly visible on it.

Figure 8. — Abbreviated scheme

Figure 9 shows the melodic layer, as well as a combination of tonal and modal layers. It shows the polychords on which the melody is based, and the supporting tones (foundations).

Figure 9. Melodic polychords, basic harmonic verticals and basses

In this play, attention is drawn to the alternation of tonal episodes with modal ones and their subsequent overlap. In most cases, the presence of chords in a large octave is a marker of tonal episodes. However, this does not apply to the first five-step. It is written in e-moll (the central tone in the melody is h) and is based on alternating tonics with a quart (in semitones — 5.2) and a specific minor dominant with a lowered fifth (4.2). The sounds a—g-f-d (below the V step) feature a chord.

The second line (verses 13-15) shows a different picture. First it's the pentatonic, and then the Dorian h, and then the main sound is h, and the secondary sound is fis (melodic tonic and melodic dominant). In addition, in the lower layer there are so-called intermediate abutments per quart down and up from h. The upper and lower sounds of the tetrachord usually act as such foundations (see this in more detail in ^{[16, p. 395]}).

For example, in bars 25 and 29, a two-layered structure appears: at the bottom is the tonal layer (H-dur and cis—moll), and at the top is the modal layer. I believe that these layers can coexist in simultaneity.

Therefore, it is advisable to combine the usual tonal and functional analysis (if there is a tonal system) and the fret analysis of the melody in terms of identifying the main and secondary melodic foundations, which may be different from the tonal and harmonic functions.

Finally, this musical example reveals the special role of the "trichord in the quarta" — the motif fis-a-h (3-2) and its inversion h-cis-e (2-3). These motifs are prominent in verses 13-15 and at the end of the piece. However, they permeate the entire piece, starting from the 4th bar. Motif 2-3 (in semitones) is shown in black, and motif 3-2 is shown in green.

The following table shows the ratio of modal and tonal layers in the piece.

Table 2. Tonal and modal layers in the play "Ruins of a Temple by Moonlight"

As you can see, if there is a modal layer, in most cases its mouth is a fifth higher than the tonal one; this also happens in the last bars. It turns out that the piece as a whole is written in e-moll, but its main melodic foundation is h.

Thus, combining a vertical and horizontal slice of a harmonic structure on one musical example creates a kind of three-dimensional model, allowing you to see their constant interaction with each other.

In the music of the twentieth century, including the music of Debussy and the tonal composers of subsequent decades, Schenker's original structure can manifest itself in an unconventional way. In particular, her tones can "stand still" and not move anywhere.

Let us turn to the analysis of Myaskovsky's novel "I go out alone on the road", composed in 1936.

By many indications, the head tone in romance is the third stage. But it doesn't go anywhere into the second and the first. And the bass diligently avoids the fifth step and stops at it only once — before the reprise (the romance is written in a simple three-part form).

Figure 10. — N.Ya. Myaskovsky. "I go out alone on the road." Diagram of the shape and sound-height structure

Obviously, the harmonic structure of romance is based on completely different factors. In the middle, it is a persistent rise in the sounds of the chromatic scale (from c to fis). In the extreme parts, there is a 3-4-3 move (shown in green), where the "4" is deliberately harmonized by a major subdominant (Schenker denied the IV step an independent function, but in some places indicated it in his examples as if unrelated to the tonic). If we look at the idea of the work as a whole, we will see that, having shifted to "4" in the upper layer, the composer first grows the mentioned chromatic line from it (reaching almost to the upper tonic, but still could not reach it), and then - in the last bars of the romance — brings the "4" up to "5", which is where everything ends (but it's too late for these five to claim the role of the head tone, it's the covering tone).

The middle is most interesting. An analytical musical example (Figure 10) reveals parallel fifths between the bass and the upper voice. However, these are not real fifths (they are avoided in the real texture), but the backbone of minor triads: each subsequent step reached during the ascent is supported precisely by the minor triad (as a local tonic). In addition, we see that the rise is gradually accelerating.: first, the bass moves in "threes" (es-e-f, f-fis-g), and then in "twos". After reaching the height of f 2, the composer temporarily retreats from further ascent, but after some chromatic wanderings, he finally reaches fis 2, which is supported by a completely unobvious chord — a large nonaccord from Gis without a third. In fact, this is the upper gate. On the eve of the reprise, the bass goes G–As–G–As–Cis–D. Special auditory attention is attracted by the tritonant, realized by a small fifth with a reduced seventh chord (apparently, this chord symbolizes the grave).

We should also note one more detail that testifies to the thoughtfulness and integrity of the harmonic design: in the reprise, after three phrases, the composer resumes the chromatic ascent that was in the middle, but brings it only to the first step; in the upper voice, the "4", not resolved into "3", goes to "5" (the composer as it pushes her up, contrary to tonal gravity).

In the extreme parts, Figure 11 shows that the composer avoids pure tonic triad, replacing it with a tonic with a second, then a tonic with a sexta.

Figure 11. Diagram of the initial period of N. Myaskovsky's romance

This musical example also shows the main melodic turns of the first movement (in green). The key intonation of the romance is es—g-a, or 4.2. It emphasizes the expressiveness of the leitharmony of the romance (small with a reduced fifth of a seventh chord).

Let's try to summarize all our observations in one sentence.

The harmonic idea of the romance is based on a double rise: "3" turns into "4" first as an auxiliary tone (with a return), then "4" generates a rise in the chromatic scale to "7#"; the second wave is weakened (again from "4", but only reaches "5", before the tonic). This is a deeper interpretation of the form than the simple three-part one.

Conclusions

By describing and analyzing a piece of music, a theorist inevitably creates a model of it. Even if he confines himself to a verbal description, the model will be the system of concepts that he has built up in relation to the work, and the schematized plot that he may have found in this work.

The model will also be a concrete implementation of the functional system, which is embodied in the work. In this case, the functions can be tonal, modal, and "melodic" (the ratio of foundations of different weights).

And, of course, any analytical musical example (according to Schenker or not according to Schenker), where the work is presented in a concise form, will be a model. Such an example reveals its "supporting structure" and allows you to look at the entire work with a single glance.

This article is devoted to the development of methods for constructing such analytical musical examples. Their construction, although based on well-known techniques and techniques, requires consideration of a huge number of subtleties and small details and, in fact, is a creative process, the final result of which cannot be predicted. The more such models can be built for more works, the more the methods of their construction will be honed.

In our opinion, such models have a purely heuristic, cognitive value. Having learned the inner structure and organics of a piece of music, embracing it with a single glance, feeling the balances of foundations and weaknesses, the attraction of some tones or chords to others, we certainly understand the work better and gain the ability to more accurately embody the composer's intention. In addition, the built-up models often allow you to see the musical form from a different angle and, thus, get a three-dimensional theoretical picture. These methods can also be used in the course of analysis of musical forms at the university.

Let's also pay attention to the fact that, having built a high-pitch model of the work using various reduction methods, we are not limited to just examining it. We can immerse the drawing in a verbal and conceptual environment and find more precise formulations describing what we see.

Thus, in relation to Schumann's play, we begin to see an intramusical plot that is based on the life of the melodic unfolding of the minor sexte. In the last bars, this sexta finally gives way to a major (with a major tonic third), which dispels the oppressive atmosphere of slow, gloomy ascents, finally aggravated by the dominant organ point.

There is no plot in Debussy's play, but we see a picture of an increasingly intense interaction between tonality and modality, and to describe many of its episodes we can use the following metaphor: modal fluctuations arise on the surface of the tonal sea (or ocean) — currents, wave movements, and we hear both principles simultaneously.

Finally, in Myaskovsky's romance, as the basis of a harmonic design, we consider an attempt to rise (from the "3" minor), which for the first time reaches the seventh elevated (but not the eighth), and in the second, in greater impotence or pacification— the fifth step.