The "Golden Ratio" is a mathematical concept rather, and its study is the task of science. This is a division of a certain value into two parts in such a way that the larger part will relate to the smaller as the whole to the larger. This ratio is equal to the transcendental number = 1,6180339 ... with amazing properties.
Method of gold sections is a search for function values on a given segment. This method is based on the principle of dividing a segment in the so-called golden ratio. He received the greatest distribution for the search for extreme values in solving problems related to optimization. In addition to mathematics, the golden ratio method is used in a variety of fields, ranging from architecture, art, and ending with astronomy. So, for example, the famous Soviet director Sergei Eisenstein used it in his film “The Battleship Potemkin”, and Leonardo da Vinci - when he wrote the famous “Gioconda”.
The golden ratio method is also used in music. It turned out that in musical works this golden proportion is very common. At the beginning of the 20th century, at a meeting of the Moscow Musical Circle, a message was made containing information about what kind of use the golden section in to music. The composers S. Rachmaninov, S. Taneyev, R. Glier and others listened to the message with great interest. Report by musicologist E. Rozenov "The law of the golden ratio in music and poetry ”initiated the study of mathematical laws related to the golden ratio in music. He analyzed the musical works of Mozart, Bach, Beethoven, Wagner, Chopin, Glinka and other composers and showed that this “divine proportion” is present in their works.
The culmination of many musical works is not located in the center, but is slightly biased towards the end of the work in a ratio of 62:38 - this is the point of the golden ratio. Doctor of Art Criticism, Professor L. Mazel noted, studying the eight-cycle melodies of Chopin, Beethoven, Scriabin, that in many of the works of these composers, the climax, as a rule, falls on the fifth, that is, the point of the golden ratio - 5/8. L. Mazel believed that almost every composer - a follower of harmonic style, can find a similar musical structure: five measures of ascent and three measures of descent. This suggests that the golden ratio method was actively used by composers consciously or unconsciously. Probably, such a structural arrangement of climax moments gives the piece of music a harmonious sound and emotional coloring.
A serious study of musical works for the manifestation of a golden proportion in them was undertaken by the composer and musicologist L. Sabaneev. He studied about two thousand works of various composers and came to the conclusion that in about 75% of cases, the golden ratio was present in a piece of music at least once. The largest number of works in which the golden proportion is found, he noted among such composers as Arensky (95%), Beethoven (97%), Haydn (97%), Mozart (91%), Scriabin (90%), Chopin ( 92%), Schubert (91%). He studied Chopin's studies most closely and came to the conclusion that the golden ratio was determined in 24 sketches of 27. Only in three sketches of Chopin a golden proportion was not found. Sometimes the structure of a musical work included both symmetry and the golden ratio. For example, at Beethoven many works are divided into symmetrical parts, and in each of them a golden ratio appears.
So, we can say that the presence of the golden ratio in a musical work is one of the criteria for the harmony of a musical composition.