"Every linear progression shows the eternal shape of life—birth to death. The linear progression begins, lives its own existence in the passing tones, ceases when it has reached its goal—all as organic as life itself." Heinrich Schenker, Free Composition
The higher-level voice-leading of the first fifty-two seconds of John Coltrane's "Number One" features an initial step-wise descent followed by a longer step-wise ascent.
The harmonic content of that initial descent (first staff of the transcription plus the first four notes on the second staff) is ambiguous. The rhythmic emphasis on F and its embellishment with upper and lower neighbors suggests a third-less F (natural) minor at the onset. Hearing F minor is further supported by the subsequent point of rest, albeit fleeting, on C, suggesting the dominant.
It is also possible to hear the first and most repeated pitch, E♭, as a local pitch-center. The whole-tone segment (D♭ E♭ F G) suggests E♭ dominant seventh. A segment from the complementary whole-tone collection, (C D E), intervenes as a "dissonance," followed by a "consonant" return to and completion of the previous whole-tone collection: (E♭ D♭ B A). All but three pitches, plus the last note, then, belong to WT1. If we consider WT1 as a proxy for E♭ seventh, the entire phrase seems to take on the sound of a dominant to tonic resolution in A♭ minor.
Or, perhaps most convincingly, the first four-note whole-tone subset can sound like a dissonant upbeat to a local resolution on C: (D♭ E♭ F G)→(C D E). The next whole-tone segment is similarly heard as resolving to A♭ minor. In both cases, there is a suggestion of the "tritone substitution" on the dominant: one hears a Phrygian 3-2-1 in C and then again in A♭. This T4 transformation or major-third relation is of course significant in Coltrane's music.
In any of these harmonic interpretations of this initial phrase, the step-wise and symmetrical descent of a fifth (or possibly a sixth) arriving on A♭ is clear. (This is indicated by upward-pointing stems on the illustrations above.)
The ascent by step that follows is also clear. Below is an attempt at a voice-leading sketch, beginning with the A♭ on the second line of the above transcription that forms the fulcrum between initial descent and subsequent ascent. Repeated notes have been omitted, and accidentals apply only to the note immediately following.
At this point Coltrane takes hold of B-flat (as the dominant of E-flat) for a centric pitch, and the improvisation continues in a new direction.
This step-wise ascent, similar to the one I identified in "Transition," helps to illustrate that even in a very late performance—"Number One" was recorded on March 7, 1967—Coltrane's improvisation exhibits a remarkable degree of high-level voice-leading coherence. "Free jazz" is often associated with the unstructured, random chaos of frenetic (or equally common in the current era, torpid) emotional expression, timbral experimentation, and critical deconstruction of traditional musical qualities. But Coltrane's free improvisation, in addition to its many other superlative characteristics, effable and otherwise, displays the kind of musical logic that has permeated the great monuments of Western music for centuries.