Despite the real successes in creating monophonic melodies with the genre features, the students have not yet mastered the creation of a developed polyphonic fabric. Therefore, S. Filatov-Beckman and E. Sokolovski again organized a master class on creating the polyrhythmic polyphony and introducing young musicians to various polyphonic techniques.
Mathematical music as a new means of developing compositional skills
The Third Essay
How is the rhythmic diversity realized in the polyphonic fabric? One of the developments preserved in the author's archive contains musical examples of the emergence of several variants of the third stage from the polyphony obtained at the first stage. These variants are very different from each other. Let's follow this process.
The scientist and composer chose a polyphonic fragment contained in the book of S. Filatov-Beckman (page 145, example 9 {62TT9}). It was created on the basis of the line No. 1, placed there. This example was obtained at the first stage of the work, it is completely monorhythmic (this example was already shown in the first Essay under No. 10, but in this case it was not taken completely for further work).
To get a two-voiced example, you must first select two monophonic lines from the original monorhythmic polyphony. For the first two-voice example, the upper voice, the soprano, was singled out. Thereupon it was rhythmically modified. Further, it was decided to isolate and modify the alto, but in a different rhythm than the soprano, and at the same time lower it an octave down. When these two voices were combined, the rhythmic complementarity was observed. The second option is significantly different from the first. Alto and bass are combined here, the number of the identical sounds of the bass is slightly increased. In the third version, tenor and bass are combined. The number of the bass sounds is reduced.
These small two-voiced examples differ from each other in character and genre features. This was precisely the task of composing the exercises and became the result of work on the melodicization of the fragments of the sound line (see Essay No. 2). Thus, the first of these examples resembles a lyrical duet of voices; the second, thanks to the organ point in the bass, possesses a somewhat disturbing character and is closer to the scherzo; the third is associated with some kind of narration with a slight bourdon accompaniment.
The exercises for creating three-voice examples required the same technique as the two-voice ones. The same example of the monorhythmic polyphony from which the previous examples were derived was used as a basis. We present these exercises.
When comparing two-voice and three-voice exercises, the greater richness of the three-voice from the point of view of the genre appearance becomes noticeable.
In the first and second examples for three-voice, the features of danceability appear quite clearly, especially since the second is a variant of the first, where a permutation of the voices is used. The third example is very different from the previous two, because it possesses the features of Lamento or elegy.
Turning to the four-voice, we again return to the monorhythmic polyphony of the example No. 19. Carefully studying this source, one can notice in it a descending scale-like course, starting in the alto voice of four parts and continuing in the tenor. If the three voices of the example No. 19 are rhythmically modified, and this descending scale move is recorded with the same duration, for example, the eighth notes, then the following variant can be obtained, in which this descending scale move will be especially noticeable. We give the example No. 20.
If, however, this scale-like move is specially selected rhythmically (for example, carried out in the «Hungarian» rhythm), and the rest of the voices are slowed down, their rhythm is changed and an accompanying function is left to them, then in this example some features of the homophony will appear, which are not present in the original polyphony. The homophony brings completely new features to the fabric. (example No. 21).
While working on the four-voice examples, the students began to look not only for the polyrhythmic and textural possibilities of the fabric, but also began to search for the timbre solutions (all examples, No. 20-25, sound in different timbre colors). One can rhythmically transform the example No. 21 even more and strengthen its homophonic features by introducing the «Hungarian» rhythm into other voices. The emerges a polyrhythmic polyphony, which possesses the genre features of a sad aria with the accompaniment and counterpoints. The students found an instrumental color for this example (example No. 22): the melody is performed by a trumpet accompanied by a piano.
In the following example (No. 23), the task was to use the principle of the rhythmic imitation polyphony. The imitations occur between the top fabric layer and the bass. Thanks to the well-known rhythmic figure, this example acquired a distinctly audible character and some genre features of the old sarabande dance. The example is performed in the piano timbre.
Among the numerous study examples preserved in the archive of S. Filatov-Beckman, we selected two more examples (No. 24 and No. 25). One of them (No. 24) is presented in the sonority of a string quartet. Thanks to the counterpoints in the middle voices, it is presented in a homophonic-polyphonic storage.
The following example (No. 25) is a variant of the example No. 24. It uses the technique of the vertical movement of voices with a number of the rhythmic changes: the former alto voice becomes soprano, the bass voice moves to the tenor register, etc. The piano timbre emphasizes these changes.
Thus, throughout the second and third essays, a process is presented that shows how, according to the author's method of S. Filatov-Beckman, genre-specific features are increasingly manifested in examples of mathematical music. This happens due to metro-rhythmic and pitch modifications of fragments of the sound line. They are transformed from “strict” into “free”, timbre colors are introduced into them and a polyphonic texture is created. In this essay, examples are limited to four voices, but, of course, the number of voices in the fabric can be more or less – it depends on the intentions of the young musicians.
Thanks to the complex and painstaking work on obtaining such different small examples of an educational nature, S. Filatov-Beckman 's students acquired the ability to embody not only various genre-specific features in the selected fragments of the sound line. They learned to find in these fragments the opportunities for interesting texture-timbre sonorities. And this happened on the basis of one source, from which the maximum of its capabilities was derived.
The ability to work pursuant to the strict rules of handling the sound line and at the same time to achieve good results in solving the texture-timbre tasks and genre-specific features has borne fruit. Some students became seriously interested in the composition, the professionalism of the young musicians has distinctly increased.
It is important to note the following circumstance. All the presented small musical examples of the second stage (No. 22-25) were created, as mentioned above, on the basis of the same polyphonic example (No. 19) of the first stage, and this polyphony, in turn, arose on the basis of one and the same computer line. If we combine these six examples (No. 20-25), we can imagine them as a "cycle of variations", where the theme is the polyphonic example of the first stage.