I rely on Miles Okazaki's transcription for my analysis. In the musical examples below, the enharmonic spelling of pitches has occasionally been adjusted from that in Okazaki's transcription in order to illustrate my interpretation. I have also omitted consecutively repeated pitches.
The first excerpt comes from 0:31 in the recording and lies on the third staff of the transcription. It follows directly from the melody and launches Coltrane's first improvisatory statement. Okazaki has labeled this bit "Aug.," though it contains an extra pitch not found in the hexatonic or "augmented" scale. Rather, these eight pitches represent a fragment of Messiaen's third mode of limited transposition, which can be conceptualized in Coltrane's melodic language as scale-steps 7-1-2-3 in an ascending major-third cycle.
The music on the fourth staff of the transcription (0:38--0:41 in the recording) provides another instance of harmonic ambiguity. The phrase can be thought of as primarily "in" D-sharp minor, with some chromatic alterations; e.g., in the order that they appear: lowered-2, lowered-4 (or major-3), major-6 and -7, lowered-5, etc. But it is also clear that the phrase mostly uses an octatonic collection, interspersed with some four-note subsets of the two whole-tone collections.
As with the first example, however, we can simultaneously hear this phrase as moving through two transpositions of the major-third cycle. The notation below has been adjusted to illustrate this.
A similar example comes from the beginning of the sixth staff of the transcription, or 0:49--0:51 in the recording. This phrase contains a segment of a whole-tone collection elided with a full octatonic collection. Yet it also outlines subsets of the diatonic collection transposed through major thirds.
A final example, from the second half of the eighth staff of the transcription (1:02--1:06 in the recording), shows a similar multiplicity of possible interpretations.
Like the second example, there is a sense in which this phrase is "in" D-sharp minor, with various chromatic alterations (more or less the same ones we hear in the second example, above). Also like the previous examples, this phrase is simultaneously operating both with octatonic collections and a major-third cycle.
While each of the three different harmonic interpretations (minor tonality; synthetic-scale collections; and diatonic major-third transpositions) can be justified by the musical surface as well as by what we know about Coltrane's musical practice, this music's abstraction from predetermined harmony and the fact that Coltrane's is a monophonic instrument make it impossible to provide a precise and unambiguous harmonic analysis.
The listener will probably notice the melodic prominence of D-sharp against E in the music cited here, dramatizing scale degrees 2 and 3 in the tonic C-sharp minor. Higher-level voice-leading by step predominates, and it is apparent that much of Coltrane's improvisation on "Ogunde" prolongs V and especially V/V in C-sharp minor. For these reasons (which deserve elaboration elsewhere) as well as the arguments above, I must dissent from Jeff Pressing's view that in "Ogunde" Coltrane "juxtapose[s] lyrical directness with eruptions of broken quasi-atonality." To the contrary, Coltrane's playing, even to the last, remains grounded in blues tonality. Where Coltrane applies the tools of early 20th-century atonality, like octatonic and whole-tone sets, his melodic sensibility is nonetheless guided by blues-based minor-key tonality, harmonic prolongation, and the nearly ubiquitous diatonic major-third cycle.