"Music always wins" may be an unexpected statement to come from a Nobel prizewinner for literature. But those who knew Samuel Beckett also knew that his was a life embedded in music, both making and listening to it, usually in the company of friends. The writer uttered his three-word resignation when composing his radio play Words and Music, itself probably triggered by an occasion at the piano with his Romanian composer friend Marcel Mihalovici. Both were labouring over Mihalovici's operatic version of Beckett's play Krapp's Last Tape, for which Beckett had agreed to write the libretto. Re-engaging with his own words at the behest of music was a struggle. But it led him to the creation of two highly innovative radio plays where music itself became a central character: as well as Words and Music there was Cascando; his composer cousin John Beckett writing the original music for the former, and Mihalovici for the latter.

Music was always going to win out for a schoolboy sent to lessons with two German spinsters in a place called Stillorgan where the young Beckett grafted at the piano, and word had it that his style of playing was "intense". His cousin Morris Sinclair recalled evenings accompanying Beckett on the violin and remembered "well with what conviction and elan he would play the last movement of Beethoven's Pathétique. The intensity of his absorption was almost ferocious." In the late 1960s, when his sight was beginning to fail, Beckett wrote a humorous description of himself: "... bought a little German piano (a Schimmel) in the country and take it out on Haydn and Schubert ... my nose so close to the score that the keyboard feels behind my back. Get it by heart in the end and lean back."

For many Irish families music making was akin to drawing breath: it was in the blood. Beckett's grandmother was musical and his uncle Gerald and aunt Cissie were talented pianists. Gerald (John Beckett's father) would play duets with Sam in four-hand arrangements of the Haydn symphonies and quartets, and Mozart and Beethoven symphonies. Beckett's first published work of juvenilia was actually a music review, "To a Toy Symphony (Haydn)", written while attending Portora Royal School in Enniskillen in the early 1920s and which he chose to sign with the pen name John Peel. As fate would have it, Beckett met his future wife, Suzanne Deschevaux-Dumesnil, herself an accomplished pianist, while she was studying the piano.

The Beckett musical inheritance would continue with Edward Beckett, Sam's nephew and son of his elder brother, Frank. Edward studied the flute at the Paris Conservatoire, living with his uncle Sam and Suzanne during his Paris years, and later worked in London as a professional flautist with several London symphony orchestras. Edward Beckett (now at the helm of the Beckett Estate) makes a rare sortie from semi-retirement to Enniskillen to contribute just one of the many musical events – Ravel's Les Chansons Madécasses – that form the Happy Days Enniskillen Beckett festival that opened on 31 July. Audiences will also be able to hear Words and Music performed by the Crash Ensemble; Krapp's Last Tape, with Klaus Maria Brandauer; and Heiner Goebbels's music-theatre piece I went to the house but did not enter.

Samuel Beckett was an avid concert‑goer, whether residing in Dublin, London, Berlin or Paris. His closest friend in Paris, the painter Avigdor Arikha, said "Listening to music was essential to him ... we used to listen to music (mainly Beethoven chamber music, Schubert) during the day in my studio ... later with Anne [Arikha's wife], always after dinner, it was a ritual. Concerning pianists his favourites were Yves Nat, [Alfred] Cortot, [Artur] Schnabel, Solomon [Cutner], [Rudolf] Serkin."

When the chance afforded him, Beckett would gladly get to know musicians, forming a friendship with the concert pianist Andor Földes, for example, after hearing him play the Beethoven piano concertos over two evenings in Paris. Beckett would also write critically about the concerts he heard. When, under Wilhelm Fürtwangler, the Berlin Philharmonic Orchestra gave a disappointing performance of Beckett's favourite Beethoven symphony, No 7 in A major, he reportedly said of the conductor: "what can one expect from a recent convert to Nazism but an absence of mystery and a disintegration of formal structures?" There's also a poetic one-liner on the symphony that could only have been written by Beckett: "tonal surfaces eaten by great black pauses".

Although contemporary composers – Poulenc, Debussy, Ravel and Bartók, whose Microkosmos he would play – were in his firmament, it was the classical and Romantic composers that he truly loved; Haydn, Beethoven and Schubert being his holy trinity. He didn't much like Bach ("the divine sewing machine"), nor Brahms: "That old piddler! Pizzicatoeing himself off in the best of all possible worlds." That said, playing the Brahms Intermezzi for piano was one of his most relaxing experiences.

He also disliked Mahler, but it was Wagner who received Beckett's ultimate dismissal, being, according to Arikha, "actually antithetical to his sense of less is more". The character of Molloy mocks Siegfried's Forest Murmurs with his bicycle horn, and Belacqua (the hero Beckett stole from Dante) slags off Wagner as "a roaring Meg" and "a demi‑tarif"! Yet to the actor Jessica Tandy, when rehearsing Happy Days, Beckett was heard to say: "Work on the nerves of the audience, not the intellect" – a Wagnerian call to action if ever there was one. Both men were "total" control-oriented artists, revolutionaries in their art forms. As well as seeing himself as both composer and poet, Wagner engaged with technology and set out the best performance practice for his works. Beckett shared a similar desire and expertise, as his production notebooks attest, as well as a profound engagement with technology as a means for expression. Where they part is in exteriority and ego: Wagner deals with expansion and limitlessness, while Beckett drills deeper into the idea of limitation, nowhere to go. But both shared a love of Schopenhauer who wrote "the effect of music is so very much more powerful and penetrating than is that of the other arts, for these others speak only of the shadow, but music of the essence''. So perhaps there may yet be a rib of Wagner in Beckett.

With his musical fervour and knowledge, it is hardly surprising to find Beckett multiple referencing from the music storehouse. Whoroscope, his early award-winning poem, has a go at Galileo's "consecutive thirds". Belacqua describes the starfield of the universe as "an abstract density of music" and, when uncharacteristically happy, listens to Schubert's Lied An die Musik (Beckett's favourite Lied). Ravel's Pavanne pops up in "Love and Lethe", and another short story, "Yellow", in the More Pricks than Kicks volume, has the colour of anaemia as Mozartian. Schubert's Death and the Maiden quartet wafts eerily through the cracks of the house in his radio play All That Fall (Matthias Claudius's "Death and the Maiden" poem remained a spectral favourite ) and Chopin's Waltz No 5 in A flat major gets an outing in another radio play, Embers (though here he also refers to Chopin as "WinkelMusick" [toilet sounds] and as "tubercular"). Winnie in Happy Days hum-sings the waltz from Lehar's The Merry Widow and Krapp croaks the hymn "Now the day is over'". It was at Portora Royal School where Beckett was inured to hymns with trips into town for Sunday service at St Macartin's Cathedral. The school today proudly displays three blue plaques commemorating former pupils: Beckett, Oscar Wilde and Henry Francis Lyte, hymnist and composer of "Abide With Me".

Many writers have a background of music training, but few went as far as Beckett to integrate its spirit and its formalism. Where Beckett's lifelong mentor, Dante, may have failed in his imagining of Paradiso, Beckett found his in music which he considered the "pure spirit" and which he sought to place in the very soul of his own work. I feel he achieved this in the late short prose of Ill Seen Ill Said, Stirrings Still and Worstward Ho, works that have a pureness of concision, beauty and sonority that is as consummate as that of Lieder. Perhaps they are Beckett's Lieder.

Beckett said that stage directors "don't seem to be sensitive to form in motion. The kind of form you find in music, for example, where themes are reprised." But that is the intersection where my own passion for Beckett has been aroused. Reading Not I, I found the fixating presence of a musical score: snatches of staccato words, spacings with as many dots as pauses, are a sonic notation that guides tempo, rhythm and even a sense of pitch. There is a musical structure underlying Beckett's work with its use of da capo, theme and variation, reprise and counterpoint. When directing his own plays, he was routinely heard to use the musical terms piano and fortissimo, andante and allegro. Some would think it too much to place a metronome in front of an actor and leave it there ticking in order to get the rhythm he desired, but Beckett did just that to the poor Brenda Bruce in Happy Days.

A lifetime of looking to music for inspiration is sublimated in three of his late works from the 1970s and 80s, made for a new medium: television. In The Ghost Trio Beckett reconfigures Beethoven's piano trio The Ghost (played by Pinchas Zukerman, Jacqueline du Pré and Daniel Barenboim in the original BBC rendition). In Nacht und Träume the actor first hums and then sings the last seven bars of Schubert's late Lied to the words "Sweet dreams, come again!"("The music whispers away into nothingness," says Dietrich Fischer-Dieskau, Beckett's most favoured Lieder singer). And his last stage-and-television play, What Where, digs deep into the cultural memory of his single favourite piece of music, Schubert's Die Winterreise. "To find a form that accommodates the mess, that is the task of the artist now," he wrote. Words might fail him. But music never did. As V says in Ghost Trio: "Kindly tune accordingly (pause)."

• Happy Days Enniskillen Beckett festival takes place until 10 August. Details: happy-days-enniskillen.com.