The story goes something like this. Gunther Schuller, who died yesterday aged 89, offered Ornette Coleman, who died two weeks ago aged 85, a course of lessons aimed at ironing out Coleman’s autodidactic view of musical nuts-and-bolts. The saxophonist had already recorded his breakthrough 1959 album The Shape of Jazz To Come, but Schuller concluded, quite reasonably, that what Coleman really needed if he wanted to expand on his discoveries about form and harmony was a consistent grounding in music theory.

And Coleman came to his apartment every week for months but Schuller utterly failed to unravel his pet theories about how music operated. Until one day when at last something he said shifted a block in Ornette’s brain – who was promptly violently sick and covered his ears in case more of Schuller’s ostensible ‘good sense’ upset a personal belief system about music that was central to his being.

The obituaries are already tumbling out and there seems little point in repeating them, but this story about Schuller and Coleman says something fundamental about both men, and neatly delineates the musical boundaries, or lack of them, that defined Schuller’s life.

Schuller was the very definition of a complete musician. Beginning life as a French horn player, in 1950 he was appointed principal horn with the Metropolitan Opera Orchestra – and played with the short-lived Miles Davis Nonet, taking part in the final recording session that would later be issued as The Birth of the Cool. He was a champion of Milton Babbitt’s 12-tone music; and of Scott Joplin’s ragtime. He wrote two scholarly books about jazz history, Early Jazz and The Swing Era; and a primer about conducting, The Compleat Conductor. He reconstructed and recorded Charles Mingus’s two-hour orchestral work Epitaph; and music by Bruno Maderna, even a Beethoven Fifth.

Coleman was single-minded in his pursuit of a personal goal; Schuller’s own compositions could at times cower under the weight of his broad stylistic sympathies and knowledge. In 1957, he coined the term ‘Third Stream music’ to describe an evolving idea of a new music that might pitch up somewhere between jazz and modern composition. Schuller was clear what Third Stream music wasn’t; it wasn’t Bach with a backbeat, nor Broadway standards underscored by strings, nor classical musicians playing Duke Ellington. The noble idea was to generate pieces, and a surrounding culture, where improvisation was integrated into compositional structures at such a deep formal level that no one would notice the join.

Milton Babbitt dedicated his Third Stream piece All Set to Schuller; and Schuller himself, in collaboration with the guitarist Jim Hall, came up with the rather structurally creaky Variants on a theme of Thelonious Monk, a work only taken to another level by the improvised solos of Ornette Coleman and Eric Dolphy – a paradox that some have suggested exposed a basic flaw at the core of Third Stream thinking.

Elsewhere, too, as in his orchestral Seven Studies on Themes of Paul Klee, Schuller’s music could feel more like a juxtaposition of different styles than a bespoke fusion. Today the notion of burdening music that chooses to bridge stylistic allegiance with an ideological label feels clumsy and unnecessarily bureaucratic. But his insistence on pressing the point – and hard – that this cross-disciplinarian exploration might yield something useful set an important example; today every composer owes something to Schuller opening up the terrain.

Schuller’s own music was at its best when he wasn’t trying to point-score, when the grandiose ambition was slightly reined in – as with this artfully crafted, beautifully imagined Tuba Concerto, composed in 2008.