“Very traditional, but fun,” ran the verdict of one fellow-traveller as we waited for a bus outside the Royal Albert Hall on Saturday night. No one can gainsay the infectious fun that the Budapest Festival Orchestra bring to every gig. For all its musical accomplishment, Iván Fischer’s all-singing (yes, they did) if not quite all-dancing (yet) outfit never forget that they belong to a, rather elevated, branch of show business.

You’d feel short-changed if an orchestra like the Budapest Festival Orchestra came to the Edinburgh Festival and didn’t play some Hungarian music, so why not put together a whole concert of the stuff?

It’s an everyday story of festival folk. The festival’s Queen’s Hall concert on Wednesday morning was meant to be a song recital from Günther Groissböck, but he cancelled at (I’m told very) short notice due to illness and the festival team had to scrabble around to find a replacement pronto.

And we’re off! This concert marked the beginning not just of the 2023 Edinburgh International Festival but, perhaps more importantly, of Nicola Benedetti’s tenure in charge as the EIF’s Director. She came onstage for a chat before a note of music was played. Part of her mission as director appears to be to make the arts more accessible, and if her introductory chat wasn’t much more than a gentle hello then it still did the job. Any aim to demystify classical music has to be welcomed.

Programming works from the same decade – in this case the 1940s – can reveal fascinating contrasts: what an impressive gulf, for instance, between two masterpieces by Hindemith and Strauss in this first half, and what sensitivity to very different styles from the NYOGB under Carlos Miguel Prieto. Be careful what you choose as the big symphony, though. I’d always had my doubts about Copland’s Third, and though it couldn’t have been more compellingly lit and shaped, it paled by comparison.

Yuja Wang and Klaus Mäkelä, two of the classical world’s biggest hitters, have recently united to make that even more powerful item, the “power couple”. But much as they are both photogenic and charismatic, their reputations are also based on musical excellence, as was on display at last night’s sizzling Prom.

It stunned me to discover that last night was only the sixth time Carmina Burana had been heard at the Proms. It seems tailor-made for the festival: large-scale and bombastic in a way that fits the proportions of the Albert Hall, familiar to occasional concert-goers but with much more to it than the "famous bit". And in this performance the CBSO and an array of choirs went at it with gusto, raising the audience to its feet at the end.

Perhaps music and politics should always stay at a decent arm’s length; in the modern world, they seldom can. The Hallé’s annual visit to the Proms presented an all-Russian bill and closed with Shostakovich’s Fifth Symphony: his much-disputed “Soviet artist’s response to just criticism” and a classic instance of the collision between art and power as, in 1937, the composer struggled to survive Stalin’s potentially fatal disapproval.

One of the undoubted highlights of Prom 14 was unprogrammed – following his commanding performance of Beethoven’s third piano concerto, Jan Lisiecki returned to the stage to give an encore of Chopin’s Nocturne in E Flat, Opus 9 No 2.

The Three Choirs is (are?) off again, for the 295th time, but with a very different look, even from the festivals of my youth, never mind 1715, or whenever the first one was held (there seems to be some doubt about it).