Grey-haired, immaculately groomed, peering down through rimless glasses at his instrument and music, Jordi Savall’s stage persona is that of a benevolent but rather patrician professor. Until, that is, you look at his left hand. Towards the end of one of the Renaissance dances that he so loves, as the music accelerates to a frenzy, you can’t actually see his fingers: they have transformed into a blur of semiquavers or even demisemiquavers. The music that streams out cannot fail to thrill, both from its melodic richness and the excitement of its ever-changing rhythmic pulse.

Savall is more than a master musician: he is a master curator of programmes. Last night’s concert was proof of this, as Hespèrion XXI – the collective name for Savall, harpist Andrew Lawrence-King and guitarist/theorbist Xaver Díaz-Latorre – dazzled the St John's Smith Square audience as they took us on a tour of Europe, tracing the path of the Portuguese “Folia” dance. (It’s one of the earliest and best remembered Renaissance melodies/ chord sequences, an especially famous later variant being Handel’s Sarabande used in Stanley Kubrick’s Baroque epic Barry Lyndon). Savall treats his themes with a light touch: rather than subjecting us to two hours of the La Folia motif, he simply ensured that it kept cropping up in our travels, but always in a different guise.

The variety of the evening came from two sources. Firstly, we saw various combinations of solo/duet/trio, with each musician playing a choice of two instruments (treble and bass viols, Baroque guitar and theorbo, Spanish or Italian harps). But what enthralled most was the variety and complexity of rhythms. The concert opened with Diego Ortiz’ three time sarabande rhythm, the Savall time machine allowing us to soak in the atmosphere of a medieval Spanish court. Soon, we moved into prototype flamenco from Gaspar Sanz, a chance for Díaz-Latorre to display exceptional virtuosity as strums, runs and rasgueados gushed forth from his guitar with rhythmic solidity that never faltered in the changes, the high and low strings perfectly balanced to achieve clarity of melody conjoined with underlying pulse.

We moved on to England’s John Hume and the chance for Savall to display solo virtuosity with an immense variety of playing techniques: plucking his viol, bowing it, thumping the strings with the horsehair of the bow in complex patterns, sometimes mimicking military drums and trumpets, projecting his evident relish at the humour in the music. We heard classic English melody in the shape of Greensleeves played over a Folia-like ground bass, and to close the first half, we moved swiftly across the Atlantic for a Mexican fiesta with variations on a “Tixtla” dance.

The second half took us to Germany and France, giving us hints of music yet to be composed – the music of Carl Friedrich Abel showing us that Bach was not the only composer for solo stringed instrument who could put together complex chord progressions, the music of Marin Marais showing hints of motifs that would become Mozart’s Symphony no. 40 in G minor or Paganini’s A minor Caprice, Savall astounding us with the ability to create birdsong with harmonics played almost at the bridge of the viol.

If Savall appears professorial when you first see him, you come to realise that his real persona is the Doctor Who of music, whisking you in space and time around places and centuries, an unpredictable but ultimately reliable guide who somehow makes aural sense of the links between the 15th century of Ortiz and the 18th of Bach, the medieval courts of Spain, the great Baroque halls of England and all points around the Grand Tour of Europe. In the introduction to the second half of the concert, Lawrence-King expressed his “special pleasure to be celebrating musical Europe here in the heart of Westminster” – the particular goal of this year's London Festival of Baroque Music. Amen to that.