Unfortunately, the “skipping a fifth” example as an explanation of the WOTG often seems to generate more questions that it answers.
I have my own thought process on this topic.
The Lydian scale is a finite, 7-note entity. There is a natural limiting factor that does not allow it to go beyond the 7th note. I will give my theory on that below.
The order of the last five notes [9 tone order through 12 tone order] is really what is in question.
In fact, it is so much in question that Russell himself changed the order of these last five notes sometime between his earlier books and the later book.
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First of all, why is the Lydian scale only 7 tones? If we are using the ladder of fifths to create the Lydian scale, why do we stop at 7 tones and not keep going up to 8, 9, 10, etc?
My thoughts on this are based on the following:
1) imagining that this ladder is not just a straight ladder, but is on the cycle of fifths
2) the idea of interval tonics
Let's use a ladder of fifths built on C (purposefully shown here with the C#, which is outside of the 7 note C Lydian scale).
C#
F#
B
E
A
D
G
C
I'm sure we agree that the tonic of the interval C-G is C, but what if we wanted to test this idea using the ladder of fifths?
To test if G is the tonic of the C-G interval, we can ask if C is in a ladder of fifths built on G.
Here's what that would look like:
C <-
F
Bb
Eb
Ab
Db
F#
B
E
A
D
G <-
C is 11 fifths up the ladder, or around the cycle, from G. C is very distantly related to G in G's ladder of fifths.
To test if C is the tonic of the C-G interval, we can ask if G is in the chain of fifths built on C.
Here's what that looks like:
E
A
D
G <-
C <-
G is one fifth around the cycle from C. G is very closely related to C in the context of C's ladder of fifths.
Based on the number of fifths, we find G is more closely tied to a root of C than C is to a root of G, and therefore, the conclusion that we draw here is that C is the tonic of the C-G interval.
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Let's try C-E.
E is four fifths around the cycle from C.
E <-
A
D
G
C <-
C is eight fifths around the cycle from E.
C <-
F
Bb
Eb
Ab
Db
F#
B
E <-
Therefore, C is the tonic of the C-E interval.
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What about C-F#.
They are exactly opposite on the cycle, so either one could be the tonic.
F# <-
B
E
A
D
G
C <-
C <-
F
Bb
Eb
Ab
Db
F# <-
===
Here's the turning point: the interval C-C#.
C# <-
F#
B
E
A
D
G
C <-
C# is 7 fifths above C.
However, C is only 5 fifths up from C#.
C <-
F
A#
D#
G#
C# <-
C# has more pull on C than C does on C#.
C# is the tonic of the C#-C interval.
C no longer "owns" the notes beyond six fifths away because, beyond this, the gravity of the intervals is inverted.
Ab owns the Ab-C interval, Eb owns the Eb-C interval, Bb owns the Bb-C interval, and F owns the F-C interval.
This limiting factor creates a ladder of fifths of exactly 7 notes because C no longer "owns" ANY of the notes above F#. Therefore, F# is the end of the C ladder of fifths, and the C Lydian scale is defined as 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.”
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===
Let’s build on this idea of interval tonics and the 7-note ladder as a “self-organized GRAVITY FIELD” to imagine the function of the last 5 notes of the chromatic scale.
Here is our 7-note ladder of fifths on C
F# <-
B <-
E <-
A <-
D <-
G <-
C <-
This is an organized set of notes with a very clear hierarchy.
Now let’s introduce the first of the chromatic notes, Ab (G#)
C#
F# <-
B <-
E <-
A <-
D <-
G <-
C <-
F
Bb
Eb
Ab <=!
In the context of C Lydian, the note Ab is a challenge to the authority of the C. Why? Because Ab is the tonic of the Ab-C interval, and Ab also owns two other intervals as well: Ab-G, and Ab-D.
Ab is disruptive in the context of C Lydian - it challenges the authority of C - and is therefore felt to be dissonant.
Now let's look at Eb:
C#
F# <-
B <-
E <-
A <-
D <-
G <-
C <-
F
Bb
Eb <=!!
Ab
The note Eb is even more of a challenge to the authority of C. After all, Eb is the tonic of the Eb-C interval, and three other intervals as well: Eb-G, Eb-D and Eb-A.
Apply the same concept to Bb, which is even more disruptive:
C#
F# <-
B <-
E <-
A <-
D <-
G <-
C <-
F
Bb <=!!!
Eb
Ab
Bb owns the Bb-C interval, and four other intervals as well: Bb-G, Bb-D, Bb-A, Bb-E. Bb really asserts its own power in the context of C Lydian.
You get the idea.
Up until now, we haven't really produced a full, unbroken, competing 7-note ladder of fifths. Ab, Eb, and Bb were kind of stand-alone disruptors (or magnets, or competing gravity sources) entering into the field of gravity that is C Lydian.
But with F, we introduce a competing, unbroken, 7-note ladder of fifths:
F#
B <=
E
A
D
G
C <-
F <=!!!!
This is almost as dissonant as it can get, with a fully formed ladder of fifths on based on a competing tonic. This situation represent two keys being sounded at once (F and C).
However, notice that this competing ladder on F still supports the note C, as C is in the ladder of fifths built on F. F doesn’t banish C to outsider status, it just usurps the authority of C.
In fact, notice that all of the competing tonics: Ab, Eb, Bb, and F, while presenting a challenge to C, also still *support* C by forming consonant intervals with C. (C is within the imaginary ladder of fifths built upon any of these notes).
A ladder of fifths built on any of these notes is in a *flat-lying* direction from C.
However, enter C#, the worst kind of villain.
The addition of C# forms another full competing ladder of fifths, built on G.
C# <=!!!!!
F#
B
E
A
D
G <=!!!!!
C XX
The G ladder of fifths forms a competing gravity field (G lydian) that is *sharp-lying* from C Lydian, and in fact excludes the note C. If the listener (or reader) accepts G as the new Lydian Tonic, C has very little place in this new order.
Remember how dissonant the F is, in the context of C Lydian?
Well, the C, our Lydian Tonic, is equally as dissonant in the new context of G Lydian!
The note G could never have done this to C on its own! It needed C# to yank the cycle into a new order.
Contemplate this quote from p.15 of the modern LCC book, and apply it to the two examples above: "As the strongest of the Lydian Chromatic Scale's vertical tones, the raised fourth degree has a neutralizing effect upon the strongest of its horizontal tones, the fourth degree." According to this statement of Russell's, in C Lydian, F is the strongest horizontal tone. Meanwhile, F# is the strongest vertical tone, and it has the capability to have a neutralizing effect upon F. Apply this logic then, to the G ladder of fifths. Does the C# have a similar effect upon the C (in the key of G) as the F# does upon the F (in the key of C)? Therefore, does the b9 (C#) have a neutralizing effect upon the Lydian Tonic (C) in the key of C?
So, in this fairy tale, C# is the rift in the C Lydian universe. C# is the note that excludes C the most, by completing a ladder of fifths on G, a sharp lying key. C# forms a gateway into an alternate universe! (Much as if you played C# over A minor).
Hope you enjoyed the story!
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Anyway, feet back on the ground now, the "Western Order of Tonal Gravity" is the title of the modified ladder of fifths, and causes the b9 to be skipped, as mentioned. The "Western Order" part of the title tells us that we are taking history into account in the organization of these notes. Why do we skip the b9 (the C#, in C Lydian)? Because we accept the sound of minor (A minor/dorian) as another facet of tonality.
Here is the start of page 231, in summary:
Let's make intervals - combinations of two notes.
We will combine C (the Lyd Tonic) with G, then with D, and then A.
Each note includes, by default, a harmonic series. Take the first few harmonics in the overtone series, for example [C C G C E] and simplify it to a triad on each note [C E G].
Combine the C Triad, with the next note in the ladder of fifths, G, and its triad G B D.
The notes C E G and G B D are all very consonant with C Lydian.
Next, let's combine a C triad and a D Triad:
C E G | D F# A. All very consonant with C Lydian.
In these three triads, we already have all the notes of C Lydian.
Next. What about a C triad and an A triad. C E G | A C# E. This gives us a bitonal sound. However, our Western history (This is where The Western Order title comes into play) allows us to bend the A triad to be minor, which is a stable sound in itself. The Minor is another aspect, or facet, of its "relative Lydian mode". This is the first instance where we reject a note - in favor of staying in the original mode of C.
So, on the one hand we have an unbending ladder of fifths. But on the other hand, we have a 7-note mode that has multiple facets, or various forms (Major, Minor, etc.)
Something has to give to accommodate both of these organizations.
So the ladder of fifths/cycle of fifths is still used as a organizing structure within tonality. But the ladder can't be maintained in its objective form of perfectly regular chromaticism (in P5ths) and still represent all the facets of each individual key without bending.
