Here ends one of the finest achievements of recent crime fiction, Ray Celestin’s City Blues Quartet. The first book, The Axeman’s Jazz, introduced us to black Pinkerton detective Ida Davis, hunting a murderer with the help of her friend Louis Armstrong, against the backdrop of the birth of jazz in the New Orleans of 1919; later instalments took the pair to Chicago and New York, and now the tetralogy comes to rest in Los Angeles in the dying days of 1967.

There’s an appropriately wintry vibe to this concluding volume. Ida has lost confidence in herself and retired; Satchmo is mulling over his doctors’ orders to quit touring. Jazz, once unrivalled as an expression of youthful vigour and experiment, is now irredeemably square.

Happily, however, Ida is persuaded to get back in the saddle when a serial killer starts terrorising LA. Celestin, a maximalist when it comes to plotting, also throws into the mix an elderly fixer reluctantly postponing retirement to search for a mob boss’s missing son, and a nurse, disfigured while serving in Vietnam, using her leave to track down her vanished brother. At the murky centre of the tangle of storylines are such real-life outrages as the CIA programme for LSD-based mind-control experiments, and Governor Ronald Reagan’s unhealthily close ties to the mob.

Throughout this series, the counterpoint to Celestin’s stark portrayal of a fundamentally corrupt and gangster-ridden America has been his rare ability to capture in prose something of the glory of the music made by Armstrong and his fellow jazz-men. Here he rises to the final challenge of Armstrong’s last comeback, when he turned the saccharine sentiments of What a Wonderful World into something transcendent. Few artists have combined greatness and lovability to the extent Armstrong did, and these outstanding hard-boiled thrillers double as a worthy tribute to him.

Sunset Swing by Ray Celestin is published by Mantle at £16.99. To order your copy for £14.99, visit Telegraph Books