Jeff Brent says:
1. In relation to the three notes G D A, an ‘acoustic root’ of G
explicitly implies the II-V-I progression via extension of the V-I
perfect cadence: D > G.
A > D > G
II > V > I
Taking the ‘acoustic root hypothesis’ out to its logical conclusions:
A > D > G > C
VI > II > V > I
Now C is the acoustic root.
A > D > G > C > F
III > VI > II > V > I
Now F is the acoustic root.
A > D > G > C > F > Bb
VII > III > VI > II > V > I
Now Bb is the acoustic root.
A > D > G > C > F > Bb > Eb
IV# > VII > III > VI > II > V > I
Eb Lydian: Eb F G A Bb C D Eb
Now Eb is the acoustic root.
The ‘acoustic root’ chain bolded above is the veritable backbone of
George Russell’s derivation of the Lydian scale, his main
justification for that scale’s ‘primacy’, and a founding principle of
the entire Lydian Chromatic Concept.
from George Russell’s LCC 1980’s edition (pg 50-iii)
According to Hindemith, the tonic of an interval of a fourth is the
upper note, while the tonic of an interval of a fifth is the lower
note. Consequently, in viewing the major scale as a chord, both
intervals indicate that the tone on the fourth degree is the tonic of
the chord. The C major scale, therefore, is actually an F major chord.
The presence of the interval of a fourth in the lower structure of the
major scale subdivides the tonality of that scale into two tonalities;
the one on its fourth degree, and the one on its root.
The absence of a fourth in the structure of a Lydian Scale enables the
whole scale to emphasize the root tonality.
from GR’s LCCOTO volume 1, 4th edition 2001 (Page 3)
Tonal Gravity, or “tonal magnetism,”within a stack of intervals of
fifths flow in a downward direction […]; the tone F# yields to B as
its tonic – F# and B surrender “tonical” authority to E, and so on
down the ladder of fifths – the entire stack conferring ultimate
tonical authority on its lowermost tone, C. In this way, an order of
six fifths represents a self-organized GRAVITY FIELD.
But, you knew all that already, right?
The fatal flaw here is that there is no inherent limiting factor in
the acoustic root cadential chain (GR: order).
There is no reason to stop at three notes, four notes, five notes, six
notes, seven notes, eight notes, nine notes, … or a hundred notes,
etc. It could keep going on and on forever.
A > D > G > C > F > Bb > Eb > Ab
#I > IV# > VII > III > VI > II > V > I
A > D > G > C > F > Bb > Eb > Ab > Db
#V > #I > IV# > VII > III > VI > II > V > I
A > D > G > C > F > Bb > Eb > Ab > Db > Gb
#II > #V > #I > IV# > VII > III > VI > II > V > I
A > D > G > C > F > Bb > Eb > Ab > Db > Gb > Cb
#VI > #II > #V > #I > IV# > VII > III > VI > II > V > I
A > D > G > C > F > Bb > Eb > Ab > Db > Gb > Cb > Fb
#III > #VI > #II > #V > #I > IV# > VII > III > VI > II > V > I
The acoustic root hypothesis (if I truly understood your meaning of
“works downwards in fifths”) quickly descends down a slippery slope
into a bottomless abyss.
There is no ‘logical conclusion’, because the sequence never actually concludes.