It’s easy to get springtime wrong. The history of classical music is rich with composers who seem to have never actually experienced the month of May, let alone April—composers for whom the end of winter is summed up by pretty melodies, soaring themes, bucolic strings. Basically: all lambs, no mud.

But spring is more than that. It’s ice-crack and tulip bud, wet basements and birdsong, mulch and meltwater. Lambs don’t just gambol: They’re born too, messily. Allergies ramp up, rain clouds open, troubles surface. With the exception of Stravinsky’s glorious Rite of Spring, teeming with primal appetites, the best-known vernal works render the season at its most banal: Schumann’s exhaustingly upbeat Spring Symphony; Mendelssohn’s dementedly Arcadian, Looney Tunes–like “Sp...