Double Concerto (Brahms)

The Double Concerto in A minor, Op. 102, by Johannes Brahms is a concerto for violin, cello and orchestra. The orchestra consists of 2 flutes, 2 oboes, 2 clarinets, 2 bassoons, 4 horns, 2 trumpets, timpani and strings.
Origin of the work.
The Double Concerto was Brahms' final work for orchestra. It was composed in the summer of 1887, and first performed on 18 October of that year in the in Cologne, Germany. Brahms approached the project with anxiety over writing for instruments that were not his own. He wrote it for the cellist Robert Hausmann, a frequent chamber music collaborator, and his old but estranged friend, the violinist Joseph Joachim. The concerto was, in part, a gesture of reconciliation towards Joachim, after their long friendship had ruptured following Joachim's divorce from his wife Amalie. (Brahms had sided with Amalie in the dispute.)
The concerto makes use of the musical motif A–E–F, a permutation of F–A–E, which stood for a personal motto of Joachim, "Frei aber einsam" ("free but lonely"). Thirty-four years earlier, Brahms had been involved in a collaborative work using the F-A-E motif in tribute to Joachim: the F-A-E Sonata of 1853.
Structure.
The composition consists of three movements in the fast–slow–fast pattern typical of classical instrumental concerti:
Performance and reception.
Joachim and Hausmann performed the concerto, with Brahms at the podium, several times in its initial 1887–88 season, and Brahms gave the manuscript to Joachim, with the inscription "To him for whom it was written." Clara Schumann reacted unfavourably to the concerto, considering the work "not brilliant for the instruments". Richard Specht also thought critically of the concerto, describing it as "one of Brahms' most inapproachable and joyless compositions". Brahms had sketched a second concerto for violin and cello but destroyed his notes in the wake of its cold reception. Later critics have warmed to it: Donald Tovey wrote of the concerto as having "vast and sweeping humour". Its performance requires two brilliant and equally matched soloists.
Scholarly discussion.
Richard Cohn has included the first movement of this concerto in his discussions of triadic progressions from a Neo-Riemannian perspective. Cohn has also analysed such progressions mathematically. Cohn notes several progressions that divide the octave equally into three parts, and which can be analyzed using the triadic transformations proposed by Hugo Riemann.