I have studied the above volume and found it an unrewarding task but I also fall into the category enthusiastic amateurs. The first thing is that he uses his own terminology so you have learn that.

There was a site I found via Google which gave are reasonable explanation but I can't be bothered to look for it again. I found  nothing in the wlkipedia.

OK I will try and explain it as I understand.

I assume you understand modes.

The objective is to provide a 'scales' of graded dissonance which can be used to improvise over different chord types. e.g minor 7th, minor 7b5. major 7th. (including compound intervals e.g. dom 7 9 11 13).

This is achieved by the use the 'Lydian concept' which provides according to Russell is a more authentic way of looking at scales than conventional western harmony which is based on the Ionian mode (the intervals of the major scale). The minor scale being  A  (6th) to A' (6th) Aeolian mode. 

The Lydian concept is based on the Lydian mode which the major scale with the  4th shapened (the intervals of the white notes of the piano played F to F'). Russell wishes emphasise the importance of the tetrachord (b5 interval). This occurs naturally in a dom 7 chord (3rd to the 7th).

So in C you get C, D E. F#  G A B this bifurcates the scale into two tetrachords which Russell thinks is more useful as it creates non-rest. This is connected to his idea of tonal gravity. in which a note always tries to resolve to a note a 5th below. There are degrees of tonal gravity.  But I can't remember exactly how this is meant to work. 

However the fundamental scale is the Lydian chromatic, which is to all intents and purposes the chromatic scale. All other scales are abstractions from this and depending on the intervals and there starting points can be selected as basis for improvisation.

There is chart that goes with the whole thing which allows you to look up a chord type and decide which of the 'scales' you might choose to use.

 

I'm tryingto work with GR's ideas as a composer, but not purely in jazz. The value in his thinking, which would be
tough for anyone to explain, is that they link developments in jazz and c20th classical harmony. I'd say its on a par
with advanced serialism as a sympathetic aid to harmonic organisation. I think Gunther Schuller ,as a guy with real
clout in these matters-taught M.A.Turnage of course- thinks so too.

I'm flattered that your'e interested Calum ,thanks.   Friend of mine will be doing a piano piece soon and
I had a meeting yesterday afternoon that bodes well for a recorded performance around August. I'll be glad to
keep you posted.