Two of the great cornerstones of 19th-century chamber music were written by Brahms.

Each, in a very different way, is a colossus, and each is a total masterpiece: of structure, craftsmanship and invention, as well as melodic, harmonic and textural integrity. Each of them is a quintet and neither of them is ever far from chamber concert performance schedules. They are the Piano Quintet in F minor for piano and string quartet and the late, sublime Clarinet Quintet for clarinet and strings.

Needless to say, then, when a new recording of both masterpieces arrived in the post, I pounced, ready to devour them at a single sitting. Looking with curiosity at the presentation, something was missing. In fact, two things were missing: in the Piano Quintet there was no piano; and in the Clarinet Quintet there was no clarinet. What's this? Music Minus One for grown-ups and sophisticates?

The CD cover gives a clue, describing both pieces as String Quintets, while the clever title of the CD gets closer still to unravelling the conundrum: Brahms By Arrangement, Volume One. Both works are arrangements for string quintets with, in the Piano Quintet, two cellos beefing out the texture and, in the Clarinet Quintet, a second viola replacing the clarinet.

There are two very different stories here. That of the Clarinet Quintet arrangement is simpler and more straightfoward. Brahms made this arrangement himself, immediately after the composition of the Clarinet Quintet in 1891. It was a commonplace practice; Brahms did it also with his Clarinet Trio and two Clarinet Sonatas in 1894, replacing the woodwind instrument with a viola in alternative instrumentations that have become standard repertoire for solo violists.

The Clarinet Quintet arrangement for five strings is particularly intriguing as it is rarely played. (I've never heard it in concert performance.) It is – incredibly – enormously effective and very beautiful, suffused with that sense of melancholy that Brahms did better than anyone.

The case of the other quintet is more intricate and complex. It's not an arrangement at all, by Brahms or anyone else. It's an imaginative reconstruction. Brahms actually did write a two-cello string quintet in 1862. He sent it to one of his closest friends and confidantes, Clara Schumann, one of the century's greatest pianists and the widow of composer Robert Schumann. Brahms wanted her observations. Clara forwarded the score to Joseph Joachim, the violinist of the day and a champion of Brahms' music.

Joachim said it was too difficult and not effective enough on strings alone. He suggested Brahms rework it. So the next year Brahms recast it and it came out as a Sonata for Two Pianos, in which form it still exists. Clara then tried to persuade Brahms to rethink it as an orchestral work. So off he went and reworked it again. And this time it came out, not as an orchestral work, but as the great Piano Quintet in F minor, which, to this day, is a central plank in Romantic chamber music repertoire.

But it's what Brahms did next that was critical. Having got the music into a form and format with which he was presumably satisfied, he then returned to the original two-cello String Quintet manuscript and destroyed it. And since that day, musicians have wondered what the original was like. A version described as a "conjectural restoration" was made in the 1940s. Six years ago, Finnish cellist Anssi Karttunen made his own reconstruction; and here it is, played by Karttunen's Zebra String Trio – Ernst Kovacic (violin) Steven Dann (viola) and Kartunnen, joined by guests Krysia Osostowicz (second violin), Richard Lester (cello 2) and James Boyd (viola 2).

It's a brilliant piece of work, played with consummate authority and credibility. I sat mesmerised, listening and wondering, about both quintets: where are the gaps with the removal of piano and clarinet? I can't find them; the music's all there. The sonority is obviously different, but there are no holes; such an exciting discovery. You'll find it on the ever-enterprising Martin Anderson's Toccata Classics label. Check the website toccataclassics.com.