J. S. Bach’s Well-Tempered Clavier as a Semantic Whole:
Peculiarities of Dramaturgic Structure
Polina Samotina and Oksana Verba
"Both parts [of the Well Tempered Clavier] represent a single authentic artistic value…"
J. N. Forkel
The preludes and fugues in Bach’s Well-Tempered Clavier (henceforth WTC) have been always regarded as masterpieces commensurate with his genius, which emanate most clearly the individuality of the composer’s musical thinking. Bach worked on this composition for almost half of his life. Book I was completed in 1722 in Coethen, Book II 22 years later, in 1744, in Leipzig. Mikhail Druskin remarks that the idea of this musical work is unusual: "Bach wanted to convey his image-bearing concepts which were connected in his mind with definite keys."
N.A. Goryukhina in her article "Small and Large Cycles in Bach’s Well-Tempered Clavier" (based on the original idea of T. N. Livanova) explores the specific character of the preludes’ and fugues’ relationship in a two-part cycle and concentrates her attention on converting a few preludes and fugues into larger cycles in performance practice. Combinations into larger cycles (sub-cycles) are subjective, but quite possible since Bach’s exploration of musical forms remains an under-studied area. Such cyclic suite-principle recurrences within Bach’s WTC become most evident in a fixed arrangement of several large summarizing fugues, similar to choral or organ fugues, endowed with both a deep figurative concept, as well as with "immediate contrast" (Livanova's term) and, as Goryukhina says, "with the highest form of polyphony - that is, the polyphony of structures and forms".
In our interpretation, Goryukhina's idea about small cycles within a large one reveals itself in a slightly different way. Goryukhina groups the “prelude-fugue” combinations into groups of four, singling out three small cycles of 4 preludes and fugues each (total number 12). From the above-mentioned principle, the Ukranian theorist Il'ya L'vovich Glauberman insists on exploring the small cycles, separating preludes and fugues into two distinct genres and tracing their respective dramaturgic development separately within both parts of the WTC. We will therefore discuss here the WTC's dramaturgic peculiarities from a genre-semantic approach, and with this end in mind, let us explore some of the 48 preludes and the 48 fugues within the WTC.
Before Bach, the prelude as a genre was often considered an improvisatory introduction to another musical piece, mostly to a fugue, and it was often omitted in published music. Alongside the prelude, in Bach's time there were other independent improvisatory genres: toccatas, fantasies, variations etc. It was Bach who transformed improvisation into an independent work of art. The prelude as a genre took stricter contours, becoming a character piece; its improvisatory elements became better identified.
In Bach’s works, the prelude stands as an independent genre mainly in the organ chorale-based preludes and prelude collections including “Nine little preludes from Wilhelm Friedemann Bach’s music book,” “Six little preludes,” and “Five little preludes”. In “Wilhelm Friedemann Bach’s music book,” written for the composer’s son in 1720, the preludes were arranged first on the white keys, then on the black keys, minor alternating with major. Later the themes of eleven of them were used in Book I of the WTC. The fact that the large cycle’s pieces were written at different times, and, as some researchers suggest, sometimes preludes and fugues were written separately from each other, enables us to consider the preludes and fugues as two separate genres.
The WTC’s preludes are filled with a deep meaning and have a characteristic tint of genre. They communicate with each other, as well as with the fugues which they precede on the principle of genre comparison and contrast. After defining each of their genres, we notice that the preludes can be divided into peculiar sub-cycles, consisting of 4 preludes each (see Figure 1).
As it is well known, in Bach’s time there was a specific attitude to numbers, which originated from the Greek and medieval numerology. The idea was popular that numbers and geometric figures expressed philosophical and theological concepts about the world harmony. Besides this, in the medieval consciousness there was not only a horizontal relationship between different events or things (cause effect) but also a vertical one; thus, it was accepted, for example, that every terrestrial thing has its celestial prototype revealing its deeper sense.
Such an interpretation can be valid as to the large cycle’s division into sub-cycles: perhaps it is not accidental that the first 12 preludes are divided into 3 sub-cycles consisting of 4 preludes each, and that each single prelude expresses a definite state of mind that can be called a prototype. What defines such a combination? The first three preludes of each sub-cycle have clear genre signs (they belong to a single genre, normally), convey a definite character and proceed at a fast or lively tempo. The fourth (closing) preludes serve to sum up the sub-cycles and to provide points of culmination. They are characterized by a mixture of several genre varieties, both polyphonic and homophonic developmental techniques, as well as by the complexity of the musical language as a whole. These preludes are filled with a deep philosophical sense, they are written in a minor key and proceed at a slow or moderately slow tempo.
Although a Baroque dance suite, consisting of mandatory four pieces (allemande, courante, sarabande, gigue), usually ends up with a lively gigue, it is worth recalling that the lyrical center of the suite was the sarabande. In our groupings of preludes, the fourth prelude is not only the lyrical center (a "sarabande") but also the closing part of the sub-cycle; that is why the last preludes of each sub-cycle acquire such an important summarizing meaning. In the small cycles of preludes, there is also a contrast of images typical of the instrumental Baroque cycles such as suites, concertos, and sonatas.
Toc. – toccata Cor. – courante All. – allemande Sic. – sicilianna Past. – pastorale Gig. - gigue Inv. – invention Ov. –
overture Pass. –
passacaglia Pol. -
polonaise Figure 1: Preludes of WTC books 1 and 2 with Genre Types
Toc. – toccata Cor. – courante All. – allemande Sic. – sicilianna Past. – pastorale Gig. - gigue Inv. – invention
Ov. – overture Pass. – passacaglia Pol. - polonaise
Figure 1: Preludes of WTC books 1 and 2 with Genre Types
The C-major prelude Andante con moto (1) is regarded as an introduction, “a majestic portal to the WTC, Book I”. It provides the beginning for the entire cycle, while its harmonic development provides a kernel to the sequence of keys in the WTC. The resemblance to a chorale prelude should be noted notwithstanding the homophonic figuration of the texture. The plasticity and the importance of five voices point to the polyphonic nature of the prelude; however, an important role also belongs to the harmonic verticality, forming a chord succession I - II4/2-V6/5 - I in the first four bars. This is the kernel of the prelude. (It should be mentioned here that all the preludes in this cycle, regardless of genre or style, have a similar introductory thesis in which the thematic material is shown against the harmonic progression T-S-D-T.)
Example 1: Bach WTC Bk.I, Prelude 1
The c-sharp-minor prelude Andante espressivo (4) combines the features of a siciliana, a sarabande, a passacaglia and an aria lamento. It is the culmination point of the sub-cycle I, just as the e-flat-minor prelude (8) is the same for the sub-cycle II and the f-minor prelude (12) for the sub-cycle III.
Example 2: Bach WTC Bk.I, Prelude 4
Example 3: Bach WTC Bk.I, Prelude 8
Example 4: Bach WTC Bk.I, Prelude 12
Among the following 12 preludes of Book I, the b-flat-minor prelude Adagio lamentoso (22) is distinguished by its tragic and monumental character. The second half of the cycle cannot be grouped into four-prelude sub-cycles as easily as the first half, because each fourth (closing) prelude is of much less importance as the culmination point against such a powerful main climax as the b-flat-minor prelude. It can be associated with a choral fresco and is the culmination point of the first large cycle consisting of the entire preludes of Book I of the WTC:
Example 5: Bach WTC Bk.I, Prelude 22
We can now trace the development of the main dramaturgic line of the WTC, Book I, with the help of the culmination points within the preludes’ combinations. The b-flat-minor prelude (22) being the main culmination point, it is also the only prelude in the whole cycle written in the passacaglia genre – the genre that expresses world-weariness in the best possible way. A clear dramaturgic line appears as the WTC’s preludes from Book I follow one another:
- the first 12 preludes are an "exposition" with three sections;
- the next 9 preludes are a peculiar suite-like succession that carries out a developmental function
and serves to prepare and delay the main culmination point of Book I, the b-flat-minor prelude;
- the last 2 preludes are a closing part, an afterword, a kind of meditation on the previous events.
The internal structure of the cycles of preludes resembles the form of a single prelude with its exposition, development, and the culmination point at the “golden section”. In this connection, what appears quite striking is that the number of bars in the b-flat-minor prelude (24) coincides, accidentally or not, with the total number of preludes in Book I.
The internal structure of the prelude also attracts our attention. Here is what we see: in the b-flat-minor prelude, the exposition section makes up 12 bars, the development makes up 9 bars, the culmination point of the prelude itself (a nine-voice VIIo7) falls on the 22nd bar (!), and the two closing bars are an afterword. Thus the internal structure of this prelude seems to be no less than a key to the dramaturgy of all the preludes in Book I of the WTC !
In Book II, we can also trace four-part sub-cycles within the large cycle. It is remarkable that the climax preludes (4, c-sharp-minor, and 12, f-minor) are similar as to their tone and genre (see Figure 1): prelude 4 is a siciliana, 12 is a lyric narrative. However, the existence of the second sub-cycle (from the 5th until the 8th prelude) is doubtful however because here the closing prelude (8, d-sharp-minor) never becomes as important as the ones mentioned above.
Example 6: Bach WTC Bk.II, Prelude 4
Example 7: Bach WTC Bk.II, Prelude 12
The culmination points of the cycle's preludes from either book are alike with relation to their tone and genre (see Figure 2). One can make a general conclusion about Bach's treatment of the prelude as a genre in the WTC: in the development sections the prelude is disguised as a toccata, an invention or a pastorale, in the culmination points – as a siciliana, a lyric narrative, a sarabande, an allemande, or a passacaglia. If we put down the preludes’ numbers from both books, one under another, marking the culmination points (see Figure 3) we would see a diagonal cross figure. Prelude (22) from Book I is preceded by a long development, while the climax g-minor prelude (16) from Book II is followed by the a-minor prelude (20) – the culmination point of the entire cycle.
It is noteworthy that the g-minor prelude (16) from Book II is written as a large French overture, which appears only once within the 48 preludes. The a-minor prelude (20) is a two-voice prelude. The first bar contains the thematic material: an expressive and skillfully developed chromatic upper voice sounding together with a descending chromatic bass, which stands for Bach’s katabasis, descent into hell. The second bar is a vertical rearrangement of the first. The double contrapuntal device is used within the most part of the prelude and plays a significant role in its development. The form is binary and uncommonly proportional. The second part (m. 17) starts with the inversion of the theme and represents a mirror reflection of the first part. It can be claimed that this prelude, in its balanced proportions, is a type of cipher to the dramaturgic principles of all the 48 preludes of the WTC.
Example 8: Bach WTC Bk.II, Prelude 20
Figure 2: Culmination points of preludes in Bk. 1 and Bk. 2 of Bach’s WTC Figure 3: Correspondence of
Culmination Preludes between Books I and II of Bach’s WTC
Figure 2: Culmination points of preludes in Bk. 1 and Bk. 2 of Bach’s WTC
Figure 3: Correspondence of Culmination Preludes between Books I and II of Bach’s WTC
The following coincidences are also interesting:
- the b-flat-minor and the a-minor preludes’ keys form the first letters of the surname BACH;
- if the b-flat-minor prelude expresses the drama of Book I, the a-minor prelude expresses the
drama of all the 48 preludes.
Now let us consider separately the sequence of the fugues as a cohesive artistic entity within the WTC. As many have noted, in Bach’s heritage the fugue represents the highest imitation-based form, which is constantly re-interpreted by the composer in search of new variants of its structure (Selected Bibliography: 1; 5; 10; 16; 17). Each of Bach's fugues has its own well-marked profile, which is defined by the theme, its character, the ways of its development, the number of voices and their typical interaction, etc.
Bach further endows the fugue’s theme with a certain personality, clear genre signs and style features. It is the theme that determines the fugue’s genre, its structure, the character (or even the absence) of the episodes, the means of polyphonic development, the texture etc. Therefore at the heart of the fugues’ classification used here there is the criterion of the theme’s character, which determines the genre of the fugue.
The WTC fugue themes can be conditionally divided into "serious" and "simple" ones. The "serious" ones are represented by symbol-themes, themes with busy chromaticism, “choral” themes, and freely melodic themes. The "simple" themes are genre themes, theme-motifs and themes with an opening energetic impulse (see Figure 4).
After dividing the fugues into two groups and following the features of their construction, we discover the following similarities: the fugues of the first group ("serious") are written in a two-part form, sometimes with a coda, the principal role in the development process belonging to the theme itself and its transformations (stretti, inversion, augmentation, diminution, etc.). The presence of a through development and the appearance of new counter-subjects create the dynamics of the fugues of this group. These fugues mainly have a four- or a five-part texture, while "simple" ones have a three-part texture.
In the development of the themes of "serious" fugues we see similar features throughout both Book I and II, while Bach’s interpretation of "simple" fugues is somewhat different. In Book I, the themes and the episodes based on them possess almost equal importance. The theme itself is not exposed to considerable transformations owing to its simplicity and the nature of its genre. Such themes exhaust their potential in the exposition. On the other hand, in Book II Bach emphasizes the episode: for example, in the f-minor fugue (12) the episode becomes independent; in the recapitulation it sounds together with the theme resembling a sonata form within the framework of a polyphonic form. Unlike those from Book I, these fugues feature a theme with modifications of all kinds that as a rule is typical for strict polyphonic forms.
One can conclude now that the above-mentioned classification enables us to refer to the first group of fugues as to the polyphonic fugues, owing to their genre vagueness and the stringent polyphonic manners of development, and to the second group of fugues – as to the genre fugues, owing to their genre distinctness and the fragmentation of themes in the episode sections.
Let us explore the dramaturgic role of both groups of fugues within the WTC. The “serious” polyphonic fugues, especially the ones with symbol-themes, come first in the dramaturgy of the whole. These fugues, abounding with tragic events, are significant emotional centers and principal culmination points of the dramaturgic development of the whole. As is often the case with Bach, the themes of these fugues present a symbolic figure of a “cross” i.e. a dissonant melodic interval of a diminished seventh, diminished fourth, or diminished fifth approached and left from within. The genre fugues are different: one can even say that they represent the material world surrounding us, which is secondary (in Bach's mind) to the important philosophic issues raised in the "serious" fugues. They fulfill a developmental function in the dramaturgic line of the large cycle. Among the fugues with symbol-themes the most important role belongs to I, 4 and 22, and to II, 20 (Figure 4). The c-sharp-minor fugue I, 4 plays an important role of a dramaturgic initiation and stimulates the formation of the subsequent symbol-themes of Book I. Its theme projects the most laconic of the “cross” motives:
Example 9: Bach WTC Bk.I, Fugue 4, Subject
The b-flat-minor fugue I, 22, the culmination point of the fugues from Book I, also coincides with the culmination point of the WTC’s preludes from Book I. The fugue becomes so important owing to its five-part texture, the intensity of intonation (a leap downward of a minor II9), the five-voice stretta, ranging wide and sounding powerful.
Example 10: Bach WTC Bk.I, Fugue 22, Subject
The a-minor fugue II, 20 is the culmination point among not only the symbol-themes but also among all the 48 fugues. It is distinguished by the vigorous and lively character of the theme, its developmental intensity due to an important rhythmic contrast of quarters, eighths and 32rd rhythmic values, and also to its dynamic episodes. This fugue’s theme is a relative of the first theme of the double “Kyrie eleison” fugue from Mozart's Requiem.
Example 11: Bach WTC Bk.I, Fugue 20, Subject
Example 12: Mozart Requiem, Kyrie, Fugue Subject
Figure 4: Positioning of “Choral-,” Chromatic-, Symbol-, and Genre-Theme Fugues in Books I and II
of the WTC
Let us explore the dramaturgic role of other fugues with symbol-themes (Figure 4). The g-minor fugue I, 16 is a local culmination point of the "serious" fugues from Book I; it serves to prepare the main climax, that is, the b-flat-minor fugue (22).
Example 13: Bach WTC Bk.I, Fugue 15, Subject
The g-minor fugue II, 16, which is distinguished by its distinct theme’s character and motivic and rhythmic structure, resembles fugue II, 20 and prepares it as the principal climax of the cycle.
Example 14: Bach WTC Bk.II, Fugue 16, Subject
The a-minor fugue II, 20, however, is the main climax of the 48 fugues and it also coincides with the principle climax of the 48 preludes. It is, in a way, a type of encyclopedia of double counterpoint, the fugue containing 13 transformations of the theme – as many as the chorus “Crucifixus” from Bach’s b-minor Mass.
The b-flat-minor fugue II, 22 is of a summarizing and closing character:
Example 15: Bach WTC Bk.II, Fugue 22, Subject
The culmination-point fugues with symbol-themes from Book I disperse, while the climax fugues in Book II concentrate in the second half of the volume and represent a climactic sequence, the closing stage of all the 48 fugues. The culmination-point fugues in Book I do not contrast with their own preludes in character, representing a cohesive semantic line; the prelude’s character receives further development, affirmation and conclusion in the fugue. The prelude-fugue cycles in Book II, on the other hand, are built more on the principle of contrast between the two parts.
We should also pay attention to the voice in which Bach begins a fugue theme (Figure 4). In lively "simple" fugues, the theme first sounds in the soprano voice. According to our classification, this is a common feature of all genre fugues, which, as mentioned above, fulfill an intermediary role in the dramaturgic development of the whole (the b-flat-minor fugue I, 22 excluded). The fugues with expressive and reciting themes ("serious" ones) normally start in the tenor or bass part.
If we follow the starting voices in symbol-theme fugues, we can remark an interesting succession in the dramatically most important fugues:
4 (I) --- 20 (I) --- 22 (II)
The arch-like succession (bass-soprano-bass) allows us to follow the dynamics of the culmination points within the two books, which is evidence of the cycle’s artistic integrity and its strong internal dramaturgic laws. The lower voice (alto) begins less important fugues among the 48, the ones which are the culmination points of local importance (fugues 1, 16, 24 from Book I, fugue 22 from Book II; see Figure 4).
An important dramaturgic role belongs to the choral-theme fugues; they represent a state of prayerful concentration, taking on a certain philosophical undertone. Their location in the cycle is significant. At times the culmination points follow them, at other times they take on a closing character themselves. The “choral” fugues from Book II prepare the entire climactic sequence, while another “choral” theme, the C-major (I,1), prepares at a distance the climatic a-minor fugue (II, 20). The “choral”-theme fugue 23 B-major from Book II is ultimately of a summarizing character.
If we now compare all the 48 fugues of the WTC with the symbolism of a Gothic cathedral, the C-major fugue from Book I would represent the cycle’s portal and the b-flat-minor fugue from Book II would be its altar (similar to the first and last choruses from St. Matthew Passion). The last b-minor fugue of the cycle (II, 24) demonstrates some resignation; it is not accidental that it resembles a lullaby with a doom motif and a “cross“ theme as its basis. The last fugues of both books are connected by a melodic and tonal relationship. Much of the chromaticism in the theme of the b-minor fugue (I, 24) results in its tonal instability; the theme here, so to say, is looking for new horizons and is asking more questions that it can answer. The b-minor fugue (II, 24) echoes it by its very beginning – a basic descending triad – but this time around the music is filled with emotional calm and humility. There is no chromaticism any more at this point; this fugue’s theme is simple and diatonic and can be regarded as a wise philosophical glance on the past events.
The fact that Bach has reconsidered the concept of genre by the end of his life is seen from the f-minor fugue (12) from Book II, which has clear genre characteristics (of a "simple" fugue). At the same time, this fugue's theme contains a cross figure (like the b-flat-minor fugue in Book II), which is typical for symbol-theme ("serious") fugues.
Example 16: Bach WTC Bk.II, Fugue 12, Subject
The f-minor fugue (II, 12) is very important dramaturgically thanks to this combination of features of both fugue types ("serious" and "simple" ones). All in all, in Book II there is the following succession: genre-theme (12), “choral” theme (14), symbol-theme (16), which is the inversion of the succession in Book I: from symbol (4) through “choral” (8) to chromatic theme (12) (Figure 4).
We should take a look now at the last three fugues of each book. As to their genre, they are exact projections of the fugues mentioned above. In Book I, the succession is kept intact: from symbol (4) through “choral” (8) to chromatic theme (12) corresponds to the genre of the fugues 22, 23 and 24. In Book II, there is a genre fugue replacing the chromatic one: the succession genre (12), “choral” (14), symbol (16) thus corresponds to symbol, “choral,” genre (II, 22, 23, 24). This is hardly accidental, considering the inversion principle immanent
to the dramaturgy of Book II. From all this arrangement the a-minor fugue (II, 20) stands out with even more relief and individuality as the culmination point of the entire WTC.
Let us compare now the culmination points of the 48 preludes and those of the 48 fugues. In Book I those are the prelude-fugue combinations number 4 (c-sharp), 8 (e-flat), 12 (f), and 22 (b-flat); in Book II – 12 (f), 16 (g), and 20 (a). Arranging both successions as they follow one after another, we get a series of 7 keys (which is symbolic in itself) with a certain symmetric order from the b-flat-minor fugue (I) to the c-sharp minor fugue (I) and to the a-minor fugue (II):
c-sharp ¬ e-flat ¬ f ¬ b-flat ® f ® g ® a
In the numerology of Bach's time the number 7 had a symbolic meaning and was regarded as a divine number, consisting of 3 and 4, where the number 3 stands for the spiritual world (trinity, etc.) and the number 4 stands for the material world (e.g., four evangelists). Let us recall the division of all the preludes into sub-cycles (4), and the number of sub-cycles in each book (3) (Figures 1-3). The correlation 3+4 forming the number 7 stresses the symbolic identity of the WTC. The above-mentioned series of keys can be compared to a concentric form with the b-flat-minor prelude and fugue (I) in the center. The keys cover mostly whole tones with b-flat at the top and c-sharp at the basis, forming a diminished seventh (c-sharp – b-flat) – the kernel of almost every symbol-theme in Bach.
Based on the above peculiarities of the WTC and the proposed classification of preludes and fugues, one can conclude that the 48 preludes and fugues form a cohesive whole with its own links and relationships at the dramaturgic, compositional and tonal levels. The melodic and semantic ideas of this composition reveal their deeper meaning to us and expand our notions of its artistic conception.
Druskin ends his monograph on Bach with the following words: "The attempts to solve the rebus of Bach’s music will continue forever. We are constantly looking for Bach and are re-discovering him in our ever-changing world. The 'renewed' Bach is our contemporary." To get to the inner sense, to reopen new facets of the great master’s music, to search and finally find in it something that helps us to become more aware of its deeper meaning is the ultimate goal of any theoretical research in music.
Translated from Russian by Marina Lupishko
1. Берков В. (Berkov V.), "«Хорошо темперированный клавир» И.С. Баха" (Bach's WTC), Берков В., Избранные статьи и исследования (Berkov V. Selected writings), Moscow, Muzgiz, 1977: 280-297.
2. Габай Ю. (Gabay Yu.), "О принципе единства в пассионах Баха" (On the principle of unity in Bach's Passions), Вопросы музыкального стиля. Сб ст. (Issues of musical style: A collection of articles), Leningrad: LGITMIK, 1978, 11-25.
3. Горюхина Н. А. (Goryukhina N. A.), "Малые и большие циклы из «Das Wohltemperierte Klavier» И. С. Баха" (Small and large cycles in Bach's WTC), Очерки по вопросам музыкального стиля и формы (Essays on musical form and style), Kiev: Muzychna Ukraina, 1985: 55-80.
4. Друскин М.С. (Druskin M.S.), Игорь Стравинский: Личность. Творчество. Взгляды (Igor Stravinsky: Personality, works, views), 2nd ed., Leningrad: Sovetsky compozitor, 1979.
5. Друскин М.С. (Druskin M.S.), Иоганн Себастьян Бах (Johann Sebastian Bach), Moscow: Muzyka, 1982.
6. Друскин М. С. (Druskin M.S.), Клавирная музыка (Keyboard music), Leningrad, Muzgiz,1960.
7. Друскин Я. С. (Druskin Ya.S.), Про риторичні прийоми в музиці Й.-С. Баха (On the rhetorics of Bach's music), Kiev, Muzychna Ukraina, 1972.
8. Ливанова Т. Н. (Livanova T.N.), Музыкальная драматургия И. С. Баха и ее исторические связи, Ч. 1: Симфонизм (Bach's musical dramaturgy and its historic connections, Part I, Simphonism), Moscow, Leningrad: Muzgiz, 1948.
9. Ливанова Т. Н. (Livanova T.N.), Музыкальная драматургия И. С. Баха и ее исторические связи, Ч. 2: Вокальные формы и проблема большой композиции (Bach's musical dramaturgy and its historic connections, Part II, Vocal forms and the problem of the large-size composition), Moscow: Muzyka, 1980.
10. Мильштейн Я. (Mil'shtein Ya.), Хорошо темперированный клавир И.С. Баха и особенности его исполнения (Bach's WTC and the peculiarity of its interpretation), Moscow: Klassika-XXI, 2004.
11. Протопопов В. В. (Protopopov V. V.), Принципы музыкальной формы И. С. Баха: Очерки (The principles of Bach's musical forms: Essays), Moscow: Muzyka, 1981.
12. Решетняк Л. В. (Reshetnyak L. V.), Восемь очерков о феномене палиндрома в теории и практике музыкального искусства. Монография (Eight essays about the phenomenon of palindrom in the theory and practice of music), Donetsk: Donetsk State Tchaikovsky Conservatory, Vostochnyi Izdatel'skiy Dom, 2002.
13. Узоры симметрии, Ред. М. Сенешаль и Дж. Флек. (The ornaments of symmetry, ed. M. Seneshal and J. Flek), Moscow: Mir, 1980.