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Many thanks to the great Ethan Iverson (owner of the Jazz Journalist Association's "Blog of the Year") for sharing this blog with his readers! 

"Transition"

"In 1987, Horace Silver told me [Dave Douglas] and Vincent Herring, ‘You young musicians all got the wrong musical lessons from John Coltrane.’… Horace’s point was that people who listen to Trane play all that stuff over a D minor seventh chord. They want to know how to superimpose and put the most complex substitution in. In Horace’s musical world, what was hip wasn’t what you played over one chord, but how you got from one chord to the next—voice-leading. … If you look closer at what Coltrane played, the reason all those superimpositions worked is because they were so brilliantly rhythmically voice-led.”  

Horace Silver, RIP. 

...

"Transition" is one of John Coltrane's greatest recorded performances. A harmonic reduction of his improvisation at 12:27 to 13:32 of the recording shows the smoothness and parsimony of Coltrane's voice-leading as he ascends through two octaves to a local climax. (Here I have relied on Miles Okazaki's transcription.)

The climactic altissimo A/B-flat is reached after a steady stepwise ascent in the upper voice of Coltrane's polyphonic melody, which composes out a tonic D minor triad over the course of seven (mostly) eight-bar phrases. (Emphasis on C-sharp at the start of the sixth phrase temporarily upsets the rhythmic contour of the composed-out triad.) 

In addition to D minor, tonicizations of chromatic-mediant related E-flat minor and C minor are prevalent, usually facilitated by way of the diminished triad formed by lowering the fifth degree of the tonic triad. Occasionally other sonorities containing the D/A-flat tritone are substituted for the D diminished chord; for example: A-flat diminished, B-flat7b13, and D half-diminished seventh. There are five tonicizations of E-flat minor and two of C minor in this section. 

The climactic melodic gesture consists of two transpositions of a diminished collection. The second more complete statement highlights the diminished tetrad (D F A-flat B), the "vagrant chord" which helps tonicize the remote key areas of vii and flat-ii as described above.

The D diminished triad is also central to the melodies of "Vigil" and "Suite" on the same album, suggesting the kind of compositional interrelation found in A Love Supreme, First Meditations, and Sun Ship, all from roughly the same time period. 

Following the climax, Coltrane reverses course and follows a quasi-symmetrical descent to 14:13 on the recording, again mostly by stepwise voice-leading. This section includes three tonicizations of E-flat minor and four of C minor, including a concluding "cadence" on C minor that returns him to the lower tessitura of the saxophone. 

Why the prevalence of E-flat minor and C minor tonicizations over a background D minor harmony? These harmonies originate with the theme to "Transition," which emphasizes the Phrygian scale degrees of the lowered second and seventh. The C gravitates toward the dominant and the E-flat towards the tonic, melodically embellishing that structural axis. It is quite logically beautiful, therefore, that Coltrane's improvisation harmonically integrates at a climactic juncture this notable feature of the thematic melody.

Geometric Representation of Whole Tone and Diminished Scales

Miles Okazaki shares a geometric representation of the two whole tone scales and three diminished scales in pitch-class space. 

Okazaki's hexagon with inscribed triangles is certainly more elegant than a traditional Tonnetz diagram of the same scales. 

Okazaki's diagram relates pitch classes to their proximate neighbors by step, allowing for the easy recognition of scales. The Tonnetz is intended to show tonal relations and is thus based on triads rather than steps. As a result, connecting pitch-classes in order to form scales is somewhat ungainly. 

On the other hand, it does allow for the visualization of the chords---augmented triads and diminished seventh chords---that form the whole tone and diminished scales. It also provides an easy way to visualize how the two whole tone scales and three diminished scales are related by transposition and how the diminished scales are related by inversion: by geometric "translation" and "reflection," respectively. 

Quite a bit more on this general topic here, and more about Miles.