‘Human beings are in trouble these days,’ says Herbie Hancock, chatting to us between songs. ‘And do you know who can fix it?’ ‘Herbie!’ comes the instant reply, shouted from somewhere in the stalls.

Hancock might be a jazz legend, but he’s not quite the Saviour. Kicking off this year’s excellent contemporary music programme at the Edinburgh International Festival, he’s a hit from the moment he strolls into view. In his long black frockcoat, Hancock has come tonight as the High Priest of Cool. When he straps on a keytar, he’s a funky gunslinger. When one of his outstanding trio takes a particularly inventive solo, he cracks up with undisguised glee at the sheer showdown-slaying audacity of their playing.

Hancock’s band is sensational, comprising James Genus on electric bass, drummer Justin Tyson and electric guitarist Lionel Loueke, who takes centre stage. Hancock parks himself over to our left, hopping between synthesiser and grand piano, often switching several times during a song.

At 82, he is a living link in the chain of jazz greats. A member of Miles Davis’s legendary Second Great Quintet in the 1960s; creator of a series of brilliant Blue Note albums during the same period; jazz-rock fusioneer in the 1970s; pop star in the 1980s. Still a tireless innovator into his ninth decade, Hancock needs a box set for every era of his career. Two hours on stage can only offer so much.

As such, the opening ‘Overture’ turns out to be a kind of crash course in Herbie, a medley moving – only slightly awkwardly – from spacey atmospherics to acoustic piano improvisations to Loueke’s impressively scatted guitar and vocal version of Hancock’s jazz/funk/rap/pop crossover ‘Rockit’, which you will remember as the old Top of the Popstheme in the 1980s.

That entrée buffet out of the way, it’s time for the more substantive main course. A new arrangement of Wayne Shorter’s ‘Footprints’ is mellow, airy and expansive. ‘Actual Proof’, plucked from Hancock’s funky fusion era, is by contrast busy and glitchy. An urgent morse code piano figure leads into an extended improvisation, solos shared around the band. ‘Come Running To Me’, a track from his 1978 album Sunlight, is eerie and strangely beautiful. Manipulating his unremarkable singing voice with a Vocoder, Hancock multiplies before our ears, the song ending in a chorale of treated harmonies.

This disquietingly smooth future soul from the past feels fresh yet weirdly disembodied from any current reference point. At other moments, the music has more contemporary resonance. The clicking rhythm of ‘Footprints’ finds a fruitful connection between Miles and Massive Attack. Later, Genus spends several minutes alone on his bass, improvising a series of loops and waves into an extraordinary wash of sound which any post-rock band would covet. Around the edges of ‘Secret Sauce’, Loueke and Hancock lock into a beautifully off-kilter guitar and piano refrain which recalls the Radiohead of ‘Pyramid Song’.

Loueke takes the lead melody on ‘Cantaloupe Island’, which is closer to the classic 1964 original than the later reggae-tinged reinterpretation. Hancock holds down the lovely loping blues groove before spinning off into a highwire solo. As a crowd-pleasing encore he performs ‘Chameleon’, the epically kinetic opening track from the groundbreaking Head Hunters album. He prowls the apron of the stage stabbing at his keytar, exhorting the crowd to get up and dance. Hancock is no slouch as a showman but for all that, it’s clear that pushing the boundaries of his music still comes first.

A week later, at Leith Theatre, British jazz quartet Sons of Kemet go considerably further in bringing the new stuff to the EIF. Led by saxophonist Shabaka Hutchings, the group comprise tuba player Theon Cross and two drummers, Tom Skinner and Eddie Hick (who do the work of four). It’s not a standard line-up and they are not a standard group. The sound they make is highly percussive with a thick low end: a soupy, intense kind of modern jazz with strong flavours of funk, Afrobeat, rock and reggae.

While the trio lays down energetic, complex rhythms, Hutchings adds a series of dazzling top lines. He is an adventurer. Sometimes – as in the balmy lament which ends the set – his style is soft and lyrical; during a solo interlude, he weaves a gentle spell on the quena, the Andean wooden flute. More often, it comprises a wayfaring repertoire of peals, skronks and squalls.

For 90 minutes there is no chat and barely a pause. By the end, Hutchings and Cross are slugging it out like two prize fighters. Sons of Kemet are a fiercely good band and this was a thrilling show.