The conflict between Brahms and his posse, and Wagner and his, resulted in a “manifesto” written by Brahms and published in the Berliner Musik-Zeitung Echo in 1860:

The undersigned have long followed with regret the proceedings of a certain party whose organ is Brendel’s Zeitschrift für Musik. The said Zeitschrift unceasingly promulgates the theory that the most prominent striving musicians are in accord with the aims represented in its pages, that they recognise in the compositions of the leaders of the new school works of artistic value, and that the contention for and against the so-called Music of the Future has been finally fought out, especially in North Germany, and decided in its favour. The undersigned regard it as their duty to protest against such a distortion of fact, and declare, at least for their own part, that they do not acknowledge the principles avowed by the Zeitschrift, and that they can only lament and condemn the productions of the leaders and pupils of the so-called New-German school, which on the one hand apply those principles practically, and on the other necessitate the constant setting up of new and unheard-of theories which are contrary to the very nature of music.

Wagner was outraged by this screed, and branded its authors “Jews,” one of his favorite epithets.

A few days later, an answer appeared in the Neue Zeitschrift:

Dread Mr. Editor,

All is out!——I learn that a political coup has been carried out, the entire new world rooted out stump and branch, and Weimar and Leipzig, especially, struck out of the musical map of the world. To compass this end, a widely outreaching letter was thought out and sent out to the chosen-out faithful of all lands, in which strongly outspoken protest was made against the increasing epidemic of the Music of the Future. Amongst the select of the out-worthies [paragons] are to be reckoned several outsiders whose names, however, the modern historian of art has not been able to find out. Nevertheless, should the avalanche of signatures widen out sufficiently, the storm will break out suddenly. Although the strictest secrecy has been enjoined upon the chosen-out by the hatchers-out of this musico-tragic out-and-outer, I have succeeded in obtaining sight of the original, and I am glad, dread Mr. Editor, to be able to communicate to you, in what follows, the contents of this aptly conceived state paper—I remain, yours most truly,

Crossing-Sweeper.

Office of the Music of the Future [Zukunftsmusik]

In spite of this mocking response, the anxiety over who would inherit the mantle of Beethoven caused real anxiety among culture-minded Germans; David Thatcher goes so far as to call the Absolute vs. Program Music dispute a “civil war”; indeed, it’s even been called “The War of the Romantics.” Brahms and Wagner were each competing, as it were, to wear the mantle of Beethoven, and to carry the genius of Germanic music into a new era. As conductor Russell Ger puts it:

From our perspective it seems fantastically overblown. Could people really get that worked up about something like this? Well, unfortunately we have a tragic parallel in the two great rap artists Biggie and 2Pac. The feud between these musicians resulted in two gang-related homicides, with both men being cut down in their prime. This is equally beyond comprehension. 

In the 19th Century, the conflict was somewhat more restrained, with only occasional outbursts of physical violence at concerts. The war was predominantly restricted to vociferous condemnations in print and vocal denunciations at performances.

Brahms despised Liszt’s music, and was widely believed to hold the same low opinion of Wagner’s. Liszt was a great supporter of Wagner; his daughter Cosima became Wagner’s second wife. Wagner, in turn, hated Brahms and everything he believed Brahms stood for. As Wagner’s defender, the philosopher Friedrich Nietzsche, put it:

What does Johannes Brahms matter now? His good fortune was a German misunderstanding: he was taken for Wagner’s antagonist — an antagonist was needed. That does not make for necessary music, that makes, above all, for too much music.-If one is not rich one should have pride enough for poverty. The sympathy Brahms inspires undeniably at certain points . . . long seemed enigmatic to me — until finally I discovered, almost by accident, that he affects a certain type of man. His is the melancholy of impotence; he does not create out of an abundance, he languishes for abundance. If we discount what he imitates [e.g., Beethoven], what he borrows from great old or exotic-modern styles — he is a master of imitation — what remains as specifically his is yearning. This is felt by all who are full of yearning and dissatisfaction of any kind. He is too little a person . . .  This is understood by those . . . on the  periphery [of socity] and they love him for that. In particular, he is the  musician for a certain type of dissatisfied women . . . . Brahms is  touching as long as he is secretly enraptured or mourns for himself — in this he is “modern”; he becomes cold and of no further concern  to us as soon as he becomes the heir of the classical composers. People like to call Brahms the heir of Beethoven: I know no more cautious euphemism.

In brief, Nietzsche says that Brahms is impotent, small-minded, with no original ideas, and appeals only to people with frustrated lives. Do you agree?

Whatever the case, Brahms quite clearly paid homage to Wagner in the second movement of his Symphony no. 1 in C minor, op. 68.

The symphony’s second movement contains several obvious allusions to Wagner’s groundbreaking “Tristan chord” (movement 2 starts at 12:52):

The Tristan chord occurs first in the prelude of Wagner’s 1865 opera Tristan und Isolde, and consists of F-B-D#-G#: an augmented fourth, sixth, and ninth. Any chord that contained these intervallic relationships became known as a Tristan chord.

More on the Tristan chord:

Brahms was a collector of manuscript scores, and had an autograph score of a scene from Wagner’s opera Tannhäuser. When Wagner found out, he demanded that Brahms return it to him. They exchanged frosty letters, which you can read here, and Brahms eventually did return the score. Wagner relented by sending him a first-edition of Das Rheingold.