What is the Linear Harmonic Transform actually?
It is a transformation of the notes of a sequence of chords to yield a similar sequence of chords that are all related with respect to the over all harmonic implications of those chords.
It is accomplished by sequencing the notes in harmonic order and then rotating them.
C Am gives C E G A C E
E G A C E C
From which you have the chords E G A = Am7 and C E C = Cmaj. Notice that, at least in this specific case, we simply exchanged the chord order and got a 7th. See how the implication of the key is retained? It's usefulness lines in that the sense of key tends to be retained even when the chords become quite awkward(but they work well with pedal tones).
The thing to realize is that it's simply a tool to vary the harmonic framework. It can take rather uninteresting chords and create some complex sounding harmonies.
You can continue the rotation to get more alterations of the root harmony. e.g., G A C E C E. Again we get Am7 and Cmaj.
Other chord combination's and non-diatonic relations can be used.