Who can tire of the sight of two world-class musicians, friends since childhood in Buenos Aires, parading round the Royal Albert Hall stage in their own victory lap, he holding up her hand as if leading her to the tango floor, protective, encouraging, she looking bemused but safe and gracious. This (Prom 34) was the year’s hottest ticket: two septuagenarians who each made their Proms debuts, in separate concerts, in 1966. The pianist Martha Argerich and the conductor Daniel Barenboim, also pianist, peace ambassador and musical polymath, were taking bows after a wild, almost feral performance of Tchaikovsky’s Piano Concerto No 1 in B flat minor. The orchestra was the West-Eastern Divan, which Barenboim founded, with the author Edward Said, two decades ago.

Argerich has acquired near mythical status, chiefly for her phenomenal playing – still brilliant and technically assured – in part because she is rarely seen here. Her last Prom, three years ago, was with the same forces. (She and Barenboim ended up playing a Schubert piano duet, an encore not repeated last week alas.) Argerich has a long association with the Tchaikovsky, one of the most popular concertos in the romantic warhorse repertoire. With ferocious opening chords, she led the tempo as if daring the orchestra to stay with her, which they did, mostly.

Barenboim knows her playing so well he barely needed to look at her, adjusting his beat accordingly, ushering the orchestra into orderly action out of the miasma and fantasy of her solo passages. It must have been hair-raising for the players, and there were moments of slippage, but the work held us gripped. Together with a slightly unsettled account of Schubert’s “Unfinished” Symphony No 8, the other major piece was Witold Lutosławski’s Concerto for Orchestra (1950-54), a richly coloured masterpiece by a composer who, a quarter of a century after his death, deserves fresh attention. With its rip-roaring brass chorales, thunderous timpani and flickering string writing, it showed the Divan at its best.

The other highlight of the week – of the season, too – was a Prom (32) on a Sunday morning by musicians in their teens: the 16- to 19-year-olds of the National Youth Orchestra of the USA, conducted by Antonio Pappano, with the mezzo-soprano Joyce DiDonato as soloist. (They had also performed together at Edinburgh’s Usher Hall, two days earlier.) DiDonato could have stolen the show in her gloriously shaded, sensuous performance of Berlioz’s Les nuits d’été but was determined not to, giving no encore and generously air-punching and applauding her talented young compatriots. A boisterous, minimalist-inspired curtain-raiser, Occidentalis, by Benjamin Beckman, showed that even at 19 you can have a strong compositional voice.

The main work was Richard Strauss’s An Alpine Symphony, which depicts a mountain climb of epic extravagance requiring massive forces (some 120 players on stage). Pappano responded to the NYO-USA’s youthful exuberance – reinforced with brass from the National Youth Orchestra of Great Britain in the gallery – by urging them up to the summit in record speed, with no loss of detail or nuance. This symphony, with its endless peaks, can prompt the “are we there yet?” query. Here, instead, it was more an enthusiastic exclamation: “There so soon!”

Their encore was “Nimrod”, heard again in context two days later in Elgar’s Enigma Variations, part of the BBC Scottish Symphony Orchestra’s Prom 35. The conductor was Martyn Brabbins, who has served contemporary music – not only, but especially, British – with conviction and imagination all his career. For his 60th birthday last Tuesday, 14 composers had contributed to a new set of variations using Elgar’s example as the framework. Pictured Within: Birthday Variations for M.C.B. worked unexpectedly well as an entity. It also captures, in one work, some of the contrasting voices, from romantic to witty to downright noisy, currently writing for symphonic forces. Each composer – including Sally Beamish, Judith Weir, Harrison Birtwistle, Colin Matthews, Brett Dean and David Sawer – honoured Brabbins with carefully crafted, distinctive offerings. His chorus from English National Opera, where he is music director, joined the BBC Singers for Brahms’s Song of Destiny and Vaughan Williams’s Serenade to Music, each work melancholy and consoling in their different ways.

No war is good, even when waged between teenagers brandishing lacrosse sticks on one side, hockey sticks on the other, lethal weapons both. Robert Carsen’s production of Handel’s Rinaldo, new at Glyndebourne in 2011, sidesteps the thorny First Crusade setting and has the action occur in a dream-cum-school history lesson. It’s cleverly designed (by Gideon Davey), with witty routines in the playground, on old black bicycles or in many-bedded dormitories. Braided blazers, or gymslips and plaits, are standard, unless you happen to be the stiletto-heeled sorceress Armida (Kristina Mkhitaryan), in which case shrink-wrapped black latex is obligatory.

This revival, by Bruno Ravella, is slickly done. The show is played for laughs, especially in the near-farcical last act. You can take or leave the staging, a concept with a capital K, but it’s one way to negotiate the heroes/villains, Christians/Saracens plot, which nevertheless turns out happily. High musical standards make this a revival to relish. The title role was sung by the rising-star Polish countertenor Jakub Józef Orliński, stepping in at short notice for Elizabeth DeShong. In his short career to date, Orliński has sung Rinaldo at Frankfurt, but this was his Glyndebourne debut.

The New Yorker magazine recently called him a millennial countertenor with pop-star appeal who “brings a swooning sultriness – and a bunch of break-dancing moves – to the baroque music revival”. Many may indeed have swooned at his sultriness as the Crusader hero, in love with the sweet-natured, sweet-voiced Giulia Semenzato as Almirena. The rest of us valued his conviction, control and lack of showiness. Some of his physical moves alluded discreetly to his dance skills, but wisely these were held in check. His voice has a rounded, mellow beauty, none too common in this voice type, with easy flexibility on trills, runs and ornaments. In a work with four countertenors – among them Tim Mead as a noble Goffredo – the bass-baritone Brandon Cedel had the stage to himself at the bottom end of the register. He was robust and vigorous as the Saracen Argante.

The Orchestra of the Age of Enlightenment skipped through Handel’s obstacle course of a score, rapid string passages immaculate, the continuo – harpsichord, cello, double bass and theorbo – imaginative and often dazzling, especially in all those virtuosic keyboard flurries. This was an auspicious debut for the young Russian conductor Maxim Emelyanychev, a baroque specialist but also an all-rounder. Next month he will succeed Glyndebourne’s music director, Robin Ticciati, as principal conductor of the Scottish Chamber Orchestra. Early days, but we’ll be watching keenly.

Star ratings (out of five)
Prom 32 ★★★★
Prom 34 ★★★★
Prom 35 ★★★★
Rinaldo ★★★★

• Watch Prom 32 on BBC Four iPlayer. Prom 35 will be televised on BBC Four on 8 September. All Proms can be heard on BBC Sounds. The Proms continue until 14 September

• Rinaldo is in rep at Glyndebourne, East Sussex, until 25 August