hypothesis - the only harmonic ratio that is necessary for LCC to be derived (or the circle of fifths for that matter) is 3:2, the perfect fifth (russell merely calls it a "fifth")

my view on harmonic ratios is:

first, that it is easily demonstrable that the first non-octave interval produced by the overtone series is what classical theory calls the perfect fifth. guitarists may play the twelfth fret harmonic followed by the seventh fret harmonic to hear this. overtones on wind or brass also...

secondly, twelve-tone equal-temperament alters this ratio slightly (2 cents flat) so that the circle of fifths may produce octaves and unisons of exactly twelve pitch classes.

further, if you play a fifth of a fifth of a fifth, it has been suggested that a ratio of 27:8 (3/2x3/2x3/2) results and that the human ear may perceive this but what i hear is an octave above a major sixth (ideal frequency ratio 5:3) and if you tune a pitch to exactly 27:16 (an octave below a fifth of a fifth of a fifth) all i am going to hear is a slightly out of tune sixth.

you see, i am simply not going to buy into the suggestion that i am going to be able to perceive the difference between something like 27:16 and 28:16 or worse still, 243:128...

what i am trying to say is that 3:2 is a fifth and the circle of slightly adjusted fifths gives us the familiar pitch classes that APPROXIMATE other harmonic ratios and that no more than the simplest ratios need be dealt with:

2:1 the perfect octave or 12.00 half steps

3:2 the perfect fifth or 7.02 half steps

4:3 the perfect fourth or 4.98 half steps

5:4 the major third or 3.86 half steps

6:5 the minor third or 3.16 half steps

major sixths are inverted minor thirds - 6:5 becomes 5:6 or 5:3 (8.84 half-steps)

minor sixths are inverted major thirds - 5:4 becomes 4:5 or 8:5 (8.14 half-steps)***

these are the most commonly accepted ratios of musical intervals. multiple major seconds may be produced but 9:8 is the most widely cited although 10:9 shows up as well as 8:7 and others. the half-step is usually defined as 16:15.

you see, whomever has created, discovered, or otherwise invented equal-temperament has saved us from a whole lot of trouble. i am not suggesting that players of professional groups aren't making microtonal adjustments in order to blend their harmonies to maximum consonance. of course they are; however, what i am suggesting is that for me to perceive 243:128 (a pythagorean major seventh) as being unique from something close like 15:8 (the most commonly accepted ratio for the major seventh) other than being slightly out of tune...

the term "absurd" comes to mind. see also "the science of musical sound" by john r pierce...

LCC is based on the fifth. george russell told me so himself. the ratio 3:2 is all that is necessary to generate not only the circle of fifths but the entire LCC.

another two cents

***ps - all of the math for converting intervals in terms of harmonic ratios to half-steps can be derived from:

R = 2 raised to the power of (H/12) or R = 2^(H/12)

where R is the harmonic ratio and H = number of half-steps in twelve tone equal temperament.

therefore H = 12 log R / log 2