The Glyndebourne Festival Opera has staged two new productions of operatic comedies this year: Franz Lehár’s The Merry Widow, mounted for the first time by Cal McCrystal and played and sung in English, and Georges Bizet’s Carmen from Diane Paulus. The first is a textbook example of Viennese operetta, whose English incarnations were masterminded by Gilbert & Sullivan. Carmen, as a French opéra-comique, is tonally and generically a complex affair: operetta for three acts before turning distinctly dark.

Both are cooked up from the same ingredients — sung numbers intercut with spoken dialogue, both comic and melodramatic in character; the Festival this year is also reviving Renaud Doucet and André Barbe’s production of Mozart’s Die Zauberflöte, which, as a Singspiel, provides the mould from which operetta would be later cast.

Operetta has multiple roots and many registers: it draws on pantomime, the stock horseplay of the commedia dell’arte, vaudevillian comedy, melodrama, as well, in the case of Bizet, drawing in the symphonic sweep of 19th-century orchestral writing, whose use of leitmotifs in its storytelling borrows from Wagner.

Its many sources give it a remarkable expressive and tonal variety, as well as a flexibility in its realisation and dramaturgical freedom — exploited by directors for better or worse. Carmen herself might be the best avatar for its playfulness, irreverence, adaptability, and melodiousness (she sings one of music’s great earworms when she is told she cannot speak). In Paulus’ modern production, she is more trickster than seductress, cavorting about in jeans and Converse trainers and tussling with the soldiers.

Carmen says she will die free. In Paulus’ staging, she is strangled by Don José with his tie in an unsparing final scene – killed off by bourgeois respectability, to which the countercultural rock club aesthetic of Lilas Pastias’ bar is an earlier counterpoint. Freedom is an important subject in post-18th-century opera, which, especially after the French Revolution, shifts its focus from (as in the world of Rameau or Handel) questions around enlightened absolutist monarchy and the correct conduct of its leaders to ideas of individual and communal liberty.

In Mozart and Da Ponte’s Don Giovanni (1788) the eponymous antihero proclaims ‘Viva la libertà’ — championing a life unconstrained by moral stricture or social mores; the freedom to rape, kill, lie, and seduce – the abyssal freedom imagined by the Marquis de Sade, whose libertinage was seen by some as the ultimate completion of the revolutionary project. Its obverse is to be found in Beethoven’s Fidelio, which sees heroine Leonora cross-dressing to free her husband Florestan from political imprisonment; its second act is an extended essay on the nature of human freedom and dignity. From the depths of the dungeons, Florestan and the other prisoners sing their dreams of emancipation, anticipating the great Verdi choruses of the 19th century that allegorised the aspiring freedom of the Italian people.

Though singspiel and its antecedent operetta are, on the surface, less serious forms, narratives of liberation are part of their DNA: rescues from capture are at the heart of both Mozart’s Der Zauberflöte and Die Entführung aus dem Serail. The premise of Johann Strauss’ Die Fledermaus — the apogee of Viennese light opera — also sees its male lead trying to evade a night in the cells; one of its distinctive elements is the stand-up skit given to the laconic gaoler at the top of act three.

In Carmen the heroine manages to dodge prison by ensnaring Don José. Freedom is conceived on personal and social axes, intersecting in Carmen: sexual liberty, also deployed by the heroine as a means of navigating her rough world of bullfighters, soldiers and petty criminals, and the apparent freedom of life on the road as a Romany smuggler, away from the constraints of duty and routine – what looks these days like a rather starry-eyed, borderline racist set of clichés. In Fidelio it is the offstage trumpet fanfare that announces Florestan’s liberation – in Carmen it is what calls Don José back to the barracks and ultimately to his home and his obligations towards his mother, away from whose safety Carmen is the apparent lure.

He does end up in jail. ‘Vous pouvez m’arrêter / C’est moi qui l’ai tuée!’ (You can arrest me, I’m the one who killed her!), he tells the young girl who stumbles across him at the scene of the murder; we first met her at the beginning of the show, and drops the flower Carmen finds by chance and uses to seal the seductive deal with José. Is her future the same as Carmen’s?

Paulus’ production upends any straightforward notions of freedom in the piece. Act two closes with a jaunty, vaudevillian sextet in which Carmen, Frasquita, Mercédès, and the smugglers Le Dancaïre and Le Remendado appeal to Don José to join them for their next job. José, now in so deep that he has no choice, ambivalently assents. As initiation he slits the throat of his army boss Zuniga. Out in the wilds, José is told, ‘pour loi, sa volonté’ – the law is whatever you want. Zuniga’s lifeless body suggests that the freedom they champion might be little more than the state of nature.

When we get to the hills life is less than bucolic. In Paulus’ staging the smugglers traffic not contraband booze nor cigarettes, but people – migrants, refugees, and others without papers who set up an encampment around an electricity pylon-cum-watchtower in what could be any grim and ambiguously administered borderlands. ‘Listen friends’, they sing in the subdued march that opens act three, ‘fortune lies over there – but be careful on the way’. This is the ‘freedom’ of the world outside the law, ungoverned and vulnerable, whose gifts – anonymity, mobility, escape from one life into another – are ultimately controlled by guys with AK-47s.

It captures the unease stitched into Bizet’s score, whose stylistic and tonal freedom allows it to pivot dizzyingly from operetta to tragedy. The music of act three is a case in point. An exquisite entr’acte – an airy duet of flute and harp – depicts the promised paradise, tranquil, easy, even edenic, that might lie just over the next ridge. It is succeeded by a rather more muted march that again opens with the flute, but in a gloomier register. The dark-hued male voices intone their warnings about the way ahead in drab, unadorned octaves, with a suddenly pained falling chromatic figure intruding on the march’s progress.

There’s not much freedom in a world subject to the whims of soldiers and gangsters – we see the former picking on the poor as they pass through the square at the opening of the opera. Who will they decide to harass today? Entire fates hang on the unpredictable caprices of men with guns at checkpoints. This terrifying changeability is summed up in the card trio of act three, which sees the heroine and her two friends try to read the future as they cut and shuffle the deck. Musically, it pivots between a uneasily skirling woodwind pattern of semiquavers – Theodor Adorno likened their whirl to that of the roulette wheel – intercut with playful entrances from Mercédès and Frasquita, with a jolly little music hall refrain as the cards fall. The former discovers that she’ll marry a rich old duffer not long for this world; the latter that she will meet the love of her life.

When Carmen’s card comes up the music makes a dramatic turn into a portentous and dark-hued andante with shadowy strings: she has drawn Death. As the jaunty earlier music is reprised, Carmen intones ‘death’ underneath, though the music pretends not to notice and acts as if nothing consequential has happened; on- and offstage, men kill women every day.

Many of the trafficked will end up working somewhere like the cigarette factory of act one, whose exterior canteen is enclosed by a wire mesh cage, which is both protection from the leering soldiers outside and a symbolic and literal prison of its own. Paulus’ production is a bitterly effective commentary on the promises made of a new life in a world where you have your labour and body to sell, and the unfreedom entailed in entering that marketplace, by choice or otherwise. As Kris Kristofferson had it, ‘freedom’s just another word for nothing left to lose’.

In The Merry Widow, directed by Cal McCrystal, no one faces imprisonment. Instead the narrative traces the self-emancipation of Count Danilo (Germán Olvera) and Hannah Glawari (Danielle de Niese) from their emotional chilliness and world-weariness about love, born of both grief and an unhappy youthful fling. Her mooted remarriage is of special interest to Baron Zeta and the lead diplomats of ‘Pontevedro’, whose national bank is on the brink of collapse.

McCrystal has previously made fine use of the dramaturgical freedoms afforded by operetta, where performing traditions welcome spicy contemporary updates or even wholesale reworkings of text and music. His productions of Iolanthe and HMS Pinafore for English National Opera drew on the madcap slapstick he conceived for One Man, Two Guvnors in the West End as well as for the Paddington movies.

The updated scripts for both were stuffed with blue jokes and zippy satirical interventions – in Pinafore Boris Johnson was dunked in the sea after falling from a zipwire, a bit of business the ENO artistic director tried to persuade him to cut – that draws into operetta the traditions of revue, music hall, and pantomime. In The Merry Widow it makes for a distinctly English filling in a crisp Viennese pastry shell, with its waltzes, czárdás, and polkas.

Like his Pinafore, McCrystal’s Merry Widow begins with a stand-up skit from Tom Edden, who plays the non-singing role of Njegus, whose high-camp delivery and naughty jokes about shenanigans in John Lewis fuses Lehár’s rather wistful Viennese romantic comedy with old-school British variety humour. There are some smart touches, with Thomas Allen’s Zeta arch and witty, especially when mooting the creation of a new Ponteverdrian state opera to be called ‘P.N.O.’; hopefully it will prompt the Glyndebourne audience to put their hands in their pockets for the beleaguered English National Opera up the road in London. Creative licence is exploited to the max, and it results in a rather odd coupling where extensive dialogue scenes – the show came down 45 minutes late on opening night – are fitfully interrupted by ravishing orchestral playing, all Schlagobers strings, from John Wilson and the London Philharmonic Orchestra.

Operetta might be a Frankenstein form, but in McCrystal’s Widow it feels rather like there are two tenuously related plays running concurrently, with the antics of one crowding out the emotional movement of the other. Hannah Glawari’s gentle softening towards Danilo, who finally finds the emotional courage to reveal his long-smouldering feelings, feels rather incidental to the pratfalls and shenanigans. Some of it is very funny indeed – a slapstick sequence where the drunken Danilo tumbles down the stairs and is dragged around and propped by Njegus as if from some Buster Keaton classic, is superbly executed. Some of it is not – adulterous Camille (Michael McDermott) cavorting with an inflatable sex doll in Act two’s summerhouse gets effortful very quickly.

But the production has plenty of other things to say. Gary McCann’s designs set the piece in the era of the great Hollywood Weepies, with its three-framed set like a cinematic title card as we swoop down into the Pontevedrian Embassy. It’s a setting that reminds us of the way operetta nourished the media that superseded it: the film music of the 1940s and 50s Hollywood – silvery strings with a sensuously overexcited vibrato – and the great open-hearted musicals of Rodgers and Hammerstein and Lionel Bart. It is helped enormously by John Wilson’s fabulous conducting, a specialist in what is sometimes uncharitably termed ‘light’ music, which in fact requires the skill and precision of an upmarket jeweller or patissier. Flawed, ultimately, but still beautiful, and a fine reminder of operetta’s inventive resilience.

Carmen returns to the Glyndebourne Festival Opera on 1 August and plays at the BBC Proms on 29 August.