In the first Essay we have already mentioned the technique of modifying the fragments of the computer sound line, which S. Filatov-Beckman proposes to implement. Let's get back to the computer line. How are the sound line fragments modified, what methods should be used for the original fragments to acquire a new quality? The scientist has already prepared the theoretical basis for answering these questions. But for the practical acquaintance of students with a variety of techniques, Sergei Anatolyevich decided to organize a master class, in which the professional composer Elena Sokolovski, who was well acquainted with the methodology of S. Filatov-Beckman, took part. Several such meetings took place at S. Filatov-Beckman's home, and different groups of students mastered this technique over the course of a number of years. The method of modifying a sound computer line became the second stage of work on mathematical music.
Mathematical Music as a New Means of Developing Compositional Skills
The Second Essay
Working together, the scientist and the composer developed an algorithm for the sound line modifications. First of all, especially great importance was attached to the metro-rhythmic modifications of the monophonic fragments. This was the first step of the second stage. As a result of these changes, the metro-rhythmic structure of the original monorhythmic line was unrecognizably transformed. A variety of rhythmic figures appeared - syncopations, triplets, dotted rhythm, and so on. Any musical beats were used, up to a variable meter.
What specific methods were used? One of the rhythmic modifications of the original fragments is the merging of a number of identical fast sounds into one long one (multiple repetitions of the sixteenth notes are constantly found in the lines). Here is an example of such a merger:
In the modifications of the original sound line, various, sometimes even bizarre rhythmic figures are allowed. Here are the examples of the original monorhythmic fragment and it's metro-rhythmic modifications:
The pitch modifications are also possible (second step of the second stаge), for example, the octave shifting of the individual sounds and even of the entire fragments up or down. Here is the original fragment and its metro-rhythmic and pitch modifications:
By introducing multiple modifications into the monorhythmic fragments selected from the sound line, the students received a large number of examples. S. Filatov-Beckman and his friend, a professional composer, set the students a task: each sound of the transformed monophonic fragment must strictly correspond to the sounds of the original monorhythmic fragment. At the same time, it is necessary to ensure that some well-known characteristic features of this or that musical genre are observed — march, waltz, song, scherzo. Such characteristic genre features should have been known to the students studying the history of music, the analysis of the musical form, etc.
And gradually there appeared the examples containing certain genre features. Some monophonic students’ works were especially interesting, especially if they were created on the basis of some original unmodified fragment selected from the sound line, since from one fragment undergoing various modifications, you can get a number of the melodies that differ in the genre characteristics.
Here passed the border between the sound line converted from the geophysical data by S. Filatov - Beckman and the transformation of its fragments into a melodic line with genre characteristics. And it was this potential contained in the sound line that became the reason for the creation of the mathematical music by the scientist.
The examples below really combine the full adherence to the original line and at the same time contain some genre features. Thus, the first modified fragment (see the exercises for the «strict» monophonic fragments) may resemble the mazurka genre: time signature ¾, characteristic rhythm and structure often found in the dance music. In the future it would be possible to grow a rather interesting melody from this fragment, and later, when the accompaniment and the counterpoints were connected to it, a small piece would arise. The same goes for other examples. Let's get acquainted with the most interesting melodies.
Exercises on the «strict» monophonic fragments.
S. Filatov-Beckman and composer called the above examples strict, since each sound of the modified monophonic fragment exactly corresponded to the original line. The order of the sounds remained unchanged. However, after some time, obeying the algorithm of the methodology, the students began to enjoy certain liberties.
Now it was possible to introduce the repetitions of sounds and phrases in the modified fragments, varying them according to the metric-rhythmic and pitch parameters. Such repetitions strengthened the genre features of the created fragments. One could already speak of the melodies that urgently demanded further musical development. These requirements led later to the creation of the well-formed polyphonic plays, which we will meet a little later. And now we present the most interesting monophonic modified fragments turned into the melodies (please note that to create a melody based on sound line №4, the line itself is given in the form of a diagram).
Exercises for the «free» monophonic fragments.
Many educational examples of the melodies created by the students have been preserved in the scientist's archive. Of course, they are not all equal in character, but the rules were persistently observed in them: a strict sequence of the order of sounds in the monophonic fragments selected from the computer line, and their metro-rhythmic and octave-register modifications.
Here are two examples of the melodies created by the students. S. Filatov-Beckman considered these examples to be quite interesting and recorded them in the archive. They are built from the monophonic fragments, their modifications and further horizontal connection, forming melodic phrases.
The first melody possesses a lyrical character and largely corresponds to the genres of romance or aria. The melody contains quite interesting features. Its deep-minor harmonic structure (juxtaposition of the keys in D minor, B flat minor and D minor again) and the intonations of detentions, «the sighs», give the melody the features of an elegy. One should also note the wide ascending jumps to the septima and nona, which sound expressively and brightly. We cite the fragments from the computer line No. 2, selected by the students (example No. 15), and the melody obtained due to various transformations and horizontal connections (example No. 16). The tempo and shades are put down by the students themselves.
This melody is written in the musical form of the period of the repeated construction of two sentences. It fully complies with the norms of the classical period, found in the slow parts of the classical sonata-symphonic works. It is obvious that the students learned well the course of the musical form analysis and were able to apply this knowledge to their own work.
Having learned to compose melodies, the students became interested in their accompaniment. Each student composed the accompaniment to the melodies independently, improvising it on the piano, guitar, button accordion and not fixing it on paper. The accompaniment to the melodies varied greatly, as the students improvised it without constraining themselves with any restrictions, that is, without using a computer line. S. Filatov-Beckman did not object at all - on the contrary, he believed that this desire for various forms of accompaniment manifests the creative imagination of young musicians, which was the goal of the entire author's methodology. The students picked up «by ear» the chords or arpeggios, typical accompaniment figures or counterpoints - each student tried to express himself most clearly.
The second melody was written by other students. Its basis was the computer line No. 5. The following fragments were chosen by the authors, and their transformations made it possible to create a melody in the genre of a vigorous march. Here are the fragments from the computer line:
The melody possesses the characteristic features of a march: a dotted rhythm, a clear repetition of the melodic figures, and a musical form of a square structure period. The accompaniments to this melody, as in example No. 16, were selected by the students without restriction: basically, they harmonized it differently with chords on the strong beats. Similar themes can sometimes be found in the marching or anniversary marches for a brass band.
Thus, the strict adherence to the computer line was required for composing melodies, whereas the freedom for various solutions was necessary for the accompaniment. The synthesis of rigor and freedom led to an extremely significant result. Beginning composers gradually learned to set themselves a creative task: having determined the nature and genre features of the future melody, to find the corresponding fragments in the computer line and transform them, connecting them horizontally. Thus emerged sufficiently long melodies with a variety of accompaniments.
The most important thing was the ability to conceive exactly the genre features of the melody and determine them for oneself. It is not a mere coincidence that the most interesting melodies have a clear genre appearance and at the same time strongly contrast with each other. In the aforementioned student examples, this contrast arose between the melody of the romance, «vocal» type, and the march of the «instrumental» character.
The students liked this work, they enthusiastically composed extended melodies and, as mentioned above, selected different types of the accompaniment for them. This was the second stage of work on the mathematical music.