Method of golden section in musical works

"Golden Section" is a concept, rather, mathematical and its study is the task of science. This division of a certain quantity into two parts in this respect, when the greater part will so treat the lesser as the whole to the larger. This ratio turns out to be equal to the transcendental number Ф = 1,6180339 ... with surprising properties.

Method Of gold Section is the search for the values of a function on a given interval. This method is based on the principle of dividing the segment in the so-called golden proportion. The most widespread he got for finding extreme values in solving problems related to optimization. In addition to mathematics, the golden section method is used in a variety of areas, from architecture, art and to astronomy. So, for example, the famous Soviet director Sergei Eisenstein used it in his film "Battleship Potemkin", and Leonardo da Vinci - when he wrote the famous "Gioconda".

The golden section method is also used in music. It turned out that in the works of music , this golden proportion is very often met. At the beginning of the 20th century, at a meeting of the Moscow Musical Circle, a message was made containing information on how the golden Cross-section at Music. The members of the music circle listened to the message with great interest. They were composers S. Rachmaninov, S. Taneyev, R. Glier and others. Report musicologist Rozenov E.K. "The law of the golden section In music and poetry "initiated the research of mathematical regularities connected with the golden proportion in music. He analyzed the musical works of Mozart, Bach, Beethoven, Wagner, Chopin, Glinka and other composers and showed that in their works this "divine proportion" is present.

The culmination of many musical works is not located in the center, and is slightly shifted towards the end of the work in the ratio 62:38 - this is the point of the golden proportion. Doctor of Arts, Professor L. Mazel noted, studying the eight-stroke melodies of Chopin, Beethoven, Scriabin, that in many creations of these composers the culmination, as a rule, falls on the weak part of the fifth, that is, the point of the golden section - 5/8. L. Mazel believed that almost every composer who is a follower of a harmonious style can find such a musical structure: five clock cycles and three clock cycles of descent. This suggests that the method of the golden section was actively used by composers, consciously or unconsciously. Probably, this structural arrangement of the climaxes gives the music a harmonious sound and emotional coloring.

A serious study of musical works for the manifestation of the golden proportion in them was made by the composer and musicologist L. Sabaneyev. He studied about two thousand works of different composers and came to the conclusion that in about 75% of cases the golden section was present in the music at least once. The greatest number of works in which the golden proportion occurs is noted by such composers as Arensky (95%), Beethoven (97%), Haydn (97%), Mozart (91%), Scriabin (90%), Chopin 92%), Schubert (91%). He most closely studied the studies of Chopin and came to the conclusion that the golden section was determined in 24 sketches out of 27. Only in the three sketches of Chopin the golden proportion was not found. Sometimes the structure of a musical work included both symmetry and a golden section. For example, in Beethoven many works are divided into symmetrical parts, and in each of them a golden section is shown.

So, we can say that the presence of the golden section in a musical work is one of the criteria for the harmony of the musical composition.