In his original programme, Berlioz called the last two movements a dream – or nightmare. Despairing of the chances of a production of his opera Les Francs-juges, he took from it a ferocious ‘Marche des gardes’ – soldiers who, in the opera, are obedient to a tyrant. The orchestra is enlarged by additional brass (trombones and ophicleide or tuba) and percussion. The main theme is presented in many guises, with much harmonic and instrumental originality. To fit the March into the symphony Berlioz added a recollection of the idée fixe at the end, where the image of the beloved woman is brutally cut off; interrupted as the protagonist dreams of his own execution by guillotine.

All the previous theorems and methods are adapted to percussive problems (problems with sudden changes of velocities). The main idea is to integrate them over the percussion interval. The behavior of unilateral constraints is analyzed in detail. The energy balance in percussive situations is particularly challenging. The formulation of energy dissipation through coefficients of restitution (used extensively in the scientific literature but not always in a consistent way) is discussed with a rigor unusual in many texts on classical mechanics. Multiple-point and rough impacts (impacts with friction) are introduced at the end of the chapter.

Throughout his lifetime, Brahms accompanied dozens of singers in a variety of settings, ranging from huge public halls to his friends’ homes, and conducted many others in choirs. Some of those working relationships were one-offs, arising from the widespread practice of including a set of piano-accompanied songs within most concerts and the expediency and cost-effectiveness of using local talent. Others were deep, enduring partnerships; the timbres and interpretative approaches of those singers are surely ingrained in his vocal music. Overall, Brahms’s singers were generally not part of the international operatic elite associated with Verdi, Bizet and Massenet. Figures like Julius Stockhausen (1826–1906) and Raimund von Zur-Mühlen (1854–1931)were almost exclusively concert singers and, later on, teachers. Most hailed from German-speaking territories, reflecting Brahms’s own concert career.

During the long nineteenth century, the design of most musical instruments changed considerably. While the late nineteenth-century orchestra may be familiar in terms of size, configuration and instrumental design, musicians of Mozart and Haydn’s era would be forgiven for not immediately recognising the descendants of the instruments that they themselves played. The industrial revolution generated new technologies and ways of manufacturing which impacted upon the musical world. Woodwind instruments gained more keys, brass instruments acquired new valve technology, strings would eventually transition from gut to metal strings, and metal-framed pianos allowed for more stable instruments with a larger pitch and dynamic range. Within individual histories of these instruments, it is difficult to pinpoint when changes were accepted and adopted. Communities (e.g. soloists, orchestral musicians, amateurs) and countries varied enormously.