I am re-posting this here, having posted it a few days ago at another forum. The post assumes we are operating within the frame of reference of the Ladder of Fifths, and not looking for other organizational principles of tonality. I don't see this as much fact as an excursion in imagination following a certain line of logic.
However, as far as I've heard, the popularly offered answer is usually along the lines of "tradition", or "there is no logic", so this may be an improvement...===
Why isn't the Ladder of Fifths (here built on Ab) eight notes such as:
A > D > G > C > F > Bb > Eb > Ab
#I > IV# > VII > III > VI > II > V > I
I am purposefully including the A, which is outside of the 7 note Ab Lydian scale.
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My thoughts on this are based on the following:
1) imagine that this chain is on the cycle of fifths
2) the idea of interval tonics based on a fifth, and by extension, the chain/ladder of fifths (see the definition of the ladder of fifths and tonal gravity in the LCC)
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First example:
Almost anyone, I think, would agree that the tonic of the interval Ab-Eb is Ab, but what if we wanted to test this idea using the chain/ladder of fifths idea?
To test if Eb is the tonic, we can ask if Ab is in a chain of fifths built on Eb.
Here's what that would look like:
Ab > Db > Gb > B > E > A > D > G > C > F > Bb > Eb
IV > bVII > bIII > #V > #I > IV# > VII > III > VI > II > V > I
Ab is 11 fifths around the cycle from Eb. Ab is very distantly related to Eb in the chain of fifths.
To test if Ab is the tonic of the Ab-Eb interval, we can ask if Eb is in the chain of fifths built on Ab.
Here's what that looks like:
Eb > Ab
V > I
Eb is one fifth around the cycle from Ab. Eb is very closely related to Ab based on the chain of fifths idea.
The conclusion drawn here is that Ab is the tonic of the Ab-Eb P5 interval. Based on the number of fifths, we find Eb is more closely tied to a
root of Ab than Ab is to a root of Eb, and therefore Ab is the tonic of the Ab-Eb interval.
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Let's try Ab-C, to pick a another random interval which is still within the first seven notes of the chain.
C is four fifths around the cycle from Ab.
C > F> Bb > Eb > Ab
III > VI > II > V > I
Ab is eight fifths around the cycle from C.
Ab > Db > Gb > B > E > A > D > G > C
#V > #I > IV# > VII > III > VI > II > V > I
Ab is the tonic of the Ab-C interval.
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What about Ab-D.
They are exactly opposite on the cycle, so either one could be the tonic.
D > G > C > F > Bb > Eb > Ab
IV# > VII > III > VI > II > V > I
G# > C# > F# > B > E > A > D
IV# > VII > III > VI > II > V > I
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Here's the turning point: the interval Ab-A.
A > D > G > C > F > Bb > Eb > Ab
#I > IV# > VII > III > VI > II > V > I
A is 7 fifths above Ab.
However, Ab(G#) is only 5 fifths up from A.
G# > C# > F# > B > E > A
VII > III > VI > II > V > I
A has more pull on Ab(G#) than Ab does on A.
A is the tonic of the A-Ab interval.
Ab no longer "owns" this level of the ladder because it is no longer the tonic of the interval.
Ab no longer "owns" ANY of the notes above D. Therefore, that's the end of the "chain/ladder".
This limiting factor creates a ladder of fifths of exactly 7 notes.
I imagine that this is what G. Russell means by the statement: “an order of six fifths represents a self-organized GRAVITY FIELD.”
