A sequence of numbers in which each is the sum of the previous two, thus: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34 etc. It is found in nature (in numbers of petals in single flowers, for instance, or in the proportions of snail shells); it has also been used by composers to govern rhythms and forms (seeNumbers and music).
The series was first described by Leonardo da Pisa (Liber abaci, 1202) as the successive population sizes of pairs of rabbits breeding each month from one parent pair. The ratio of successive numbers is an arithmetical expression of Euclid’s geometrical division into extreme and mean ratio (the ‘golden ratio’; seeGolden number), but da Pisa was unaware of this. The first written connection between Fibonacci’s series and Euclid’s ratio appears in a handwritten comment in a copy of Pacioli’s 1509 edition of Euclid’s Elements. The mathematician Johannes Kepler also demonstrated a connection between the two in a letter of 12 May 1608 to Professor Joachim Tanckius.
It has been claimed that composers have used Fibonacci numbers in musical compositions as a deliberate attempt to reproduce the golden ratio. While this is undoubtedly the case in certain 20th-century compositions, it appears to be a historical impossibility for earlier composers. The first attested use of the term ‘golden number’ in a strictly mathematical context was by Martin Ohm in 1835, and it was not until 1843 that the explicit expression for fn in terms of G was published by J.P.M. Binet. Thus any composer using Fibonacci numbers before then would not have done so with the ‘golden number’ in mind. It could be argued that ‘naturally occurring’ Fibonacci sequences appear in compositions written before 1843, but in such cases the musicologist must maintain a clear distinction between an interpretation imposed on the composition and the composer’s conscious intention.
For bibliography seeNumbers and music.
Composer who may be identifiable with Johannes Fedé or Henricus Tik.
(b Odessa, 15 Jan 1896; d Buenos Aires, 9 Sept 1978). Argentine composer, violinist and conductor of Russian origin. A musician's son, he began violin lessons at the age of nine with Stolyarsky and later studied with M.T. Hait. He continued violin studies with Sergei Korguyev and Auer at the St Petersburg Conservatory (1912–17); among his other teachers were Vasily Kalafati, Maximilian Steinberg, Nikolay Tcherepnin and Nikolay Sokolov. In 1919 he won by competition the post of leader of the Petrograd State Academic Theatre orchestra, but he did not take up the appointment. In 1923 he settled in Buenos Aires, where in 1929 he was a founder-member of the Grupo Renovación, devoted to studying and promoting new compositional trends; he was also among the founders of the Argentinian Composers' League (1947). He was appointed in 1956 to teach composition at the University of La Plata, where he eventually became professor, and he has also taught at the National Conservatory in Buenos Aires, the Municipal Conservatory and the Instituto Superior de Arte of the Teatro Colón. He won several prizes, among them the Coolidge Prize for the Second Quartet, and in 1969 he was elected to the National Fine Arts Academy.
Ficher's career stretched over 50 years and his output comprises about 150 works. His work is marked by various influences, among which is his Jewish heritage, apparent in early works such as the Suite for orchestra (1924) as well as in the mature cantata Kadish (1969). Slavonic themes can also be found, especially in the two Chekhov-based operas. In the 1920s there was a flash of French Impressionism, and later he became interested in Hindemith. After settling in Argentina he was stimulated by native gaucho literature (Obertura para Don Segundo Sombra, 1954), popular urban music (Tangos y milongasfor piano, 1948–59), the folk music of the countryside (Tres danzas populares) and historical themes (for example the Seventh Symphony, which commemorates the Argentinian Independence Revolution). Ficher's style oscillates between neo-romanticism and neo-classicism, with polyphonic lines interwoven with harsh polyharmony. Although he explored 12-note technique, serialism, atonality and other fashionable trends, he refused to be tied to a single technique or scheme unless the work's character seemed to require it; thus he maintained a stylistic individualism enriched by his own motifs and themes.
Ops: El oso (chbr op, 1, after A. Chekhov), op.75, 1952; Pedido de mano (chbr op, 1, after Chekhov), op.84, 1955–6
Ballets: Colombina de hoy (1), op.25, 2 pf, 1933; Los invitados (1), op.26, 1933; Melchor (3, C. Tiempo), op.40, chorus, 14 str, 1938–9; Golondrina (3), op.47, 1942
Solo vocal: El organillero (L. Barletta), op.41, B, orch, 1949; 2 Poems of Longfellow, op.70, 1v, pf, 1949; Oda a la libertad (J. Isaacson), op.90, nar, orch, 1957; Mi Aldea, cant. (Rugeles), op.91, S, A, T, chbr orch, 1958