Shopen's etudes in lighting of the Golden Section

The music is the kind of art, which reflects a reality and affects on the person by means of interpreted and especially organized sound sequences. Saving some similarity to real sounds the musical soundings in essence differ from the latter by stringent high-altitude and temporary (rhythmic) organization ("musical harmony"). Since the antique period the search of the "musical harmony laws" is one of the relevant scientific directions.

It is assigned to Pythagor the discovery of two fundamental harmonic musical laws: (1) if the ratio of oscillation frequencies of two tones is described by small numbers they give harmonic sounding; (2) to get harmonic triad it is necessary to add the third tone to the chord from two consonance tones and the oscillation frequency of third tone should be in harmonic proportional connection with the two latter's. A significance of Pythagor's work on scientific explanation of the musical harmony is difficult for overestimation. It was the first scientifically well-grounded theory of musical harmony. By learning a verity and beauty of his musical theory, Pythagor attempted to distribute it to the cosmology; according to his ideas the planets of the Solar System are placed pursuant to the musical octave ("harmony of spheres").

Any musical piece has a temporary duration and is divided into separate parts by some "stakes" ("aesthetic stakes"), which ðày our attention and easy our perception of the piece. The dynamic and intonational culmination points of the musical piece can be by these "stakes". Whether there are some regularity in originating the "aesthetic stakes" in the musical piece? The attempt to answer this question was made by the Russian composer Sabaneev. In the article "Shopen's etudes in the lighting of the golden section " (1925) he shows, that the separate time intervals of the musical pieces connected by the "culmination event", as a rule, are in the ratio of the golden section. Sabaneev writes:

"All such events are timed by author's instinct to such points of the whole length that they divide temporary durations into separate parts being in the ratio of the "golden section". As the observation shows, the timing of the similar "aesthetic stakes" to the division points of the general or partial duration in the "golden" ratio is realized quite often with a huge accuracy, that it is surprising because especially some knowledge about similar things is absent for many composers and all this is extremely by consequence of the internal orderliness feeling".

The analysis of a huge number of the musical pieces made by Sabaneev allowed to him to make a conclusion that the musical piece organization is constructed so, that its cardinal parts divided by the "stakes" form the golden section series. Such organization of the musical piece corresponds to most economical its perception and consequently gives the impression of the best "ordering" of the piece form. In Sabaneev's opinion, the quantity and the frequency of the golden section usage in musical composition depends on the "composer rank". The musical pieces of the ingenious composers distinguish by the highest percent of the golden section usage, that is "the intuition of the form and ordering, as it is necessary to expect, is strongest for the first class geniuses".

On Sabaneev's observations, in the musical pieces of different composers not only golden section, conjugated with some "aesthetic event", but a number of similar golden sections are observed. Each such golden section reflects some musical event, quantitative jump in development of the musical theme. In 1770 musical pieces of 42 composers studied by him the golden section met in 3275 times; even one golden section met in 1338 musical pieces. The greatest number of musical pieces based on the golden section meets in musical works of Arensky (95 %), Beethoven (97%), Gaidn (97%), Mozart (91%), Skrjabin (90 %), Shopen (92%), Schubert (91%).

All 27 Shopen's etudes were studied by Sabaneev in detail. In them 154 golden sections are revealed; only in three etudes the golden section was missed. In some cases the constitution of the musical piece combined in itself symmetry and golden section simultaneously; in these cases the musical piece was divided on some symmetrical parts, in each of which there was the golden section. Beethoven compositions are divided into two symmetrical parts, and inside of each part the golden section is observed.

The Russian art critic Rosenov paid big attention to research of the musical harmony laws. He asserted that in the musical pieces and poetry there are stringent proportional relations and that the golden section should play an outstanding role in music as the means for reduction of homogeneous phenomena to conformity created by the nature.

Rosenov selects for analysis a number of typical musical works of the outstanding composers: Bach, Beethoven, Shopen, Vagner. For example, studying Beethoven's Chromatic fantasy and fugue he used the quarter duration as the measure unit. 330 such measure units are in this musical work. The 204th quarter from the beginning corresponds to the golden section of this musical work. This moment of the golden section coincides precisely with the fermata (in the musical notation the fermata sign augments the note or pause duration usually in 1,5-2 times), which separates the first part of the musical work (prelude) from the second one. The fugue following after the fantasy demonstrates also an astonishing harmony of parts. At the look on the scheme of the harmonic fugue's analysis "one comes in sacred thrill by the genius of the composer who embodied by effect of art keenness to such accuracy extend the innermost laws of natural creativity".

Rosenov analyzed the ending of the sonata of cis-moll by Beethoven, the Fantasia-Impromptu by Shopen, the introduction to the "Tristin and Isolda" by Vagner. In all these musical works the golden section meets very often. He paid special attention to Shopen's fantasy, which was created "by impromptu" and was not subjected to any edition; it means that there was no consciousness application of the golden section, which is present at this musical piece down to small-sized music parts.

So, it is possible to recognize, that the golden proportion is the harmony criteria of music piece composition.