Interval Tonics - Their Effect Upon the LCS

[chespernevins]

I don't suppose I'll convert anyone, but let me try once more. See if this version is less dense to sort through. It's compatible with the Western order.

The question is, if the Western order is based on Tonal Gravity (based on stacked 5ths) and a Ladder of 6 5ths, why do we "skip" a fifth at the b9 and save it for last (and not just because it sounds like crap, to quote Jeff Brent from a while ago)?

The Lydian scale is a ladder of 6 5ths because C is the tonic of the 6 5ths stacked upon it. C is no longer the tonic of the interval it makes with the 7th fifth.

F#
B
E
A
D
G
C <-- the Lydian Tonic "owns" (is the tonic of) 6 5ths

===

The next 5th in the cycle, C#, is the most dissonant because it creates a ladder of fifths built on G, giving G (a sharp lying key) equal strength to the C. As we know, a sharp lying key tends to "cut off" the sound of a flat lying key.

Db <--|
F#
B
E
A
D
G <--|
C x

Notice that G "owns" 6 perfect 5th intervals just like C

===

In the key of C, F is the second most dissonant note because the F creates a ladder of fifths built on F. This gives F as much power as the tonic C, so it clashes with C greatly. It is not quite as dissonant as the ladder of fifths built on G (in the 12 tone order) because F doesn't cut off the sound of the C completely (being a flat lying key).

F#
B <--|
E
A
D
G
C x
F <--|

Notice that F "owns" 6 notes just like C

===

Ab, Eb, and Bb get more dissonant around the cycle of fifths as they progressively get closer to the C (from the flat side
of the cycle) because they claim ownership of more and more perfect 5th intervals in the key of C.

F#
B
E
A
D <--| Ab is co-owner of the notes from here down - only 3 notes.
G <--|
C <--|
-
-
-
Ab <--|

=============

F#
B
E
A <--| Eb is co-owner of the notes from here down - 4 notes
D <--|
G <--|
C <--|
-
-
Eb <--|

=================

F#
B
E <--| Bb is co-owner of the notes from here down - 5 notes.
A <--|
D <--|
G <--|
C <--|
-
Bb <--|

===============

The more notes that are owned by these "non-lydian" tones means the more they will challenge the key center of C.

[Ben Schwendener]

You're correct in talking about how tonal centers relate to each other in terms of fifths - this is what the Circle of Close to Distant Relationships and the Law of Resolving Tendencies is based on. However, the LC Order of TG is not a horizontal matter - rather it represents a contained Vertical organization which places (grades) the 12 chromatic tones in their vertical ingoing to outgoing order relative to one key center, to one Lyd Tonic. Its initial origin derived from how each of the progressively more outgoing tones of this order contributed its unique color to (and is featured by) the 7 vertical chord producing (parenting) Principle scales. Anytime we talk about a relationship between keys, we're addressing the level of HTG. And you are correct in stating that movement in a sharp direction cancels the previous Lydian Tonic. Conversely, movement in a flat direction sounds suspended.

[bobappleton]

Hi Bob,
I appreciate your point, and so renamed this thread...

Good edit strachs. Thanks.

On my way to Istanbul to perform... Back in 10 days. Like ML, the last thing I want is to stop the conversation... I missed his comment and would like to have seen that too...

b

[strachs]

Chesper:

A concise, reasoned argument. Thank you for taking the time.

In the interests of objectivity, I wonder if we do well to consider the differences between a FULL LYDIAN SCALE in the presence of the 'foreign' note, and the actual scales that represent a given tonal order?

Reason being: in the scales themselves, the foreign note actually REPLACES a Lydian note. If we decided to factor that missing note's gravity into the equation, we would necessarily have to factor in ALL tones of a given tonal order - and all TO's would result in SEVERAL consecutive tones having the tonical weight of 21.

Notes that will not be used in a given tonal order need not be factored in. I will post my specific observations in regards to this, but first I'll give you opportunity to see what results you get by REMOVING the non-scale tone from your calculations.



Ben:

LC Order of TG .... Its initial origin derived from how each of the progressively more outgoing tones of this order contributed its unique color to (and is featured by) the 7 vertical chord producing (parenting) Principle scales.

Have I been mistaken all this time thinking that GR claimed that each of the "out" notes was actually more distant in fifths? Does "progressively more outgoing" mean distant in fifths, or distant in dissonance?

LC Order of TG is not a horizontal matter

(setting LCS argumentation aside for a moment) I see HTG and VTG, not as two separate forces, but as two different manifestations of one force - tonal gravity. Maybe this is wrong, but I can't help but feel this is valid, since, as GR taught, "the vertical aspect creates and understands the horizontal aspect in all and in everything".

The strongest, most "ingoing" horizontal movement that can take place is from a Lydian Scale to another Lydian Scale one fifth in the sharp direction. That knowledge is encapsulated in the Lydian Scale itself, since PMG1 (and in fact all PMG's) have their strongest resolving tendency to CMG's Vh and IIIh, representing I and VI of the next sharp-lying LS.

Maybe I'm wrong, but since both the 11TO and 12TO, theoretically present us with that "8-tone ladder" (12TO if rooted on lowest note, 11TO if rooted on second lowest note), it seems like that arrangement, strongly resisting simultaneous vertical existence, forces us to choose one LS or the other, and then remove the offending note. Since the "resistance" to that vertical state is greater when the lowest tone is LT than when the second lowest is LT, the "pressure" is pointing in the sharp direction.

Of course, no one ever uses these tonal orders in this explicit form, but my feeling is that they are the underlying cause of the horizontal force - much like the way any physical system kind of "desires" to return to a state of lower energy, and in the case of physical gravity, that can be accomplished by "moving" (falling) the distance between one elevation, and a lower one.

Take, for example the classic V7-I progression, II-Vh in LCC terms. If the actual chords were G7-C, they would represent F MG II resolving to that scale's CMG Vh. V is a major triad, primarily parented by C Lydian, in which it is MG I. Continuing to use F along with the C major triad is outgoing (11TO), and sounding C Lydian's F# in context of F Lydian is even more outgoing (12TO). The POTENTIAL outgoingness in either situation COULD be lived with, if you like cacophony, but is far more often yielded to, by choosing one LS or the other, and dropping the offending tone to that LS.

So, while having two different BEHAVIORS, I tend to think of VTG and HTG as just opposite manifestations of one and the same FORCE of gravity. When you think about it, the strongest vertical interval is the P5, and the interval characterizing the "horizontal scales" is the P4 - the same interval, but not rooted on it's tonic. That's why I see the FLOW of VTG as going counterclockwise in the circle of fifths, and the FLOW of HTG as going clockwise in the same circle. The outgoing vertical stack of seven fifths being most effectively discharged by a "movement" by one fifth to a sharp-lying LS.

Possible? Plausible? Correct? Incorrect? Thoughts?

[strachs]

Chespernevins:


To continue with our comparison of approaches: If we remove the tone of a tonal order from our reckoning that does not actually occur in a typical scale of that order, we get the following weights (comparing LA and Lb7)

C Lyd + Ab (G# - creates LA)

b6....Ab.....C4 3 D6 1 = 4
b2
+4....F#.....Ab2 5 C6 1 = 6
7.....B.....F#1 6 Ab3 4 = 10
3.....E.....B1 6 F#2 5 Ab4 3 = 14
6.....A.....E1 6 B2 5 F#3 4 Ab5 2 = 17
2.....D.....A1 6 E2 5 B3 4 F#4 3 Ab6 1 = 19
5
1.....C.....D2 5 A3 4 E4 3 B5 2 F#6 1 = 15

C Lyd + Bb (creates Lb7)

+4.....F#.....Bb4 3 C6 1 = 4
7
3.....E.....F#2 5 Bb6 1 = 6
6.....A.....E1 6 F#3 4 = 10
2.....D.....A1 6 E2 5 F#4 3 = 14
5.....G.....D1 6 A2 5 E3 4 F#5 2 = 17
1.....C.....G1 6 D2 5 A3 4 E4 3 F#6 1 = 19
4
b7.....Bb.....C2 5 G3 4 D4 3 A5 2 E6 1 = 15

However, in so doing, the LA is revealed to have more weight on it's II MT than on it's LT. Doesn't that indicate that Lb7 is more ingoing than LA? Kind of/kind of not.

This is certainly true for PMG I. But the other PMG's all have more weight in LA.

I has greater weight in Lb7 than LA.
VI has greater weight in LA than Lb7
II has greater weight in LA than Lb7
+IV has greater weight in LA than Lb7

What this demonstrates to me, is that figuring out the tonical weight of a SCALE is not necessarily the same as the tonical weight for all of it's MG's. The scale's outgoingness is NOT EVENLY DISTRIBUTED among it's MG's, as you would think.

In LD, two of the PMG's (VI and +IV) have a greater weight than they did in Lydian, and two have less (I and II), but have varying results relative to Lb7 and LA:

C LD scale

b3....Eb....C3 4 G4 3 D5 2 A6 1 = 10
b6
b2
+4....F#.....Eb3 4 C6 1 = 5
7.....B.....F#1 6 Eb4 3 = 9
3
6.....A.....B2 5 F#3 4 Ab5 2 = 17
2.....D.....A1 6 B3 4 F#4 3 = 13
5.....G.....D1 6 A2 5 B4 3 F#5 2 = 16
1.....C.....G1 6 D2 5 A3 4 B5 2 F#6 1 = 18

I has greater weight than than LA but less than Lb7.
VI has same weight as LA but more than Lb7.
II has less weight than either LA or Lb7.
+IV has greater weight than Lb7, less than LA.

It seems there is no numerical way to unequivocally support one view or the other. One thing seems certain, though: Tonal Order does not uniformly affect all of the scale's MG's.

My conclusion on the matter is rather inconclusive.

The LSC does, in many respects, represent an in-to-out order as it affects the chordmodes of the representative parent scales. Not perfectly, but in a general way, yes. It does not adhere strictly to an in-to-out MEASUREMENT in fifths, but perhaps that ideal just cannot be realized in a way that perfectly complements vertical expansion of chordmodes. Maybe that's not really what GR was claiming.

The polymodally-influenced system of in-to-out measurement that I have proposed does adhere striclty to an in-to-out measurement in fifths, but again, falls short in describing these tonal order's effect upon MG uniformly, other than to say that each MG gains a tone that normally represents X number of modes over.

My position at this moment is to appreciate and respect the benefits and limitations of each method. I still favor my own system, of course, being that it corresponds with like measurements between Lydian Scales and serves to measure the levels of Polymodality, as well. However, I agree with bobappleton that "Down with the LCS" is not warranted.

I hope this discussion has been at least insightful on some levels, and continue to welcome input from as many as possible.

[chespernevins]

Strachs,

Well, I just don't know.

In my original post I used "tonical weights" and in my "simplified" post I used a simple number of fifths, with no weights/distance factored in. I think they both were sufficient to illustrate my point, but neither may be a more universal truth.

These examples came to me in a flash - as a way to illustrate my thinking, but I did not spend time testing the ramifications or other proofs of "tonical weights". I think it's a concept that makes sense at a certain level, but don't know how universally it predicts certain characteristics. I tried both the "tonical weights" and the "simple number of fifths owned" with 3 note combinations and they did not always predict the tonic that I thought was common sense.

I started with the way you are doing it now, but it seemed to me to make more sense to treat it as a question of tonal order rather than scales. I wanted to gauge the influence a single interval (with the LT) outside the ladder of fifths (or tonal gravity field) would have on the ladder. And to test one non-lydian tone at a time seemed like the most consistent way to judge anything about those tones. The scales method (modifying lydian notes) seemed to make sense up to b7, but not as much on 4 or b2. After all, the scales are Aux Dim and Aux Dim Blues at that point. It's not strict note substitution. I liked something ML said in his deleted post about the symmetrical scales at the higher orders. They seem to dodge more identifiable tonalities.

Perhaps my overall premise is close but I have not found the right reasoning. Maybe not having the exact reasoning invalidates the idea to a certain exent. Or perhaps my overall premise is way off.

I think we are hearing the message from Sandy and Ben about George's approach to the Tonal Orders. This is a good reminder - even though it still seems to make sense to want to reconcile the Ladder of Fifths, Interval Tonics and the Western Order.

It's amazing what we learn just grappling with this stuff though.

[chespernevins]

You're correct in talking about how tonal centers relate to each other in terms of fifths - this is what the Circle of Close to Distant Relationships and the Law of Resolving Tendencies is based on. However, the LC Order of TG is not a horizontal matter - rather it represents a contained Vertical organization which places (grades) the 12 chromatic tones in their vertical ingoing to outgoing order relative to one key center, to one Lyd Tonic. Its initial origin derived from how each of the progressively more outgoing tones of this order contributed its unique color to (and is featured by) the 7 vertical chord producing (parenting) Principle scales. Anytime we talk about a relationship between keys, we're addressing the level of HTG. And you are correct in stating that movement in a sharp direction cancels the previous Lydian Tonic. Conversely, movement in a flat direction sounds suspended.

I'm hearing you Ben, on all points.

[sandywilliams]

I can remember GR at the piano, playing the Principal Chordmodes of the eight PMGs. To be truthful, it reminded me so much of the harmony and voicings on Kind Of Blue. This made me do some thinking about the listening and thought process that may have gone through his head when putting together the LCC. The WOTG is really about ‘passive unity’ with vertical structures, the ‘basic chord catagories’ and PMGs. You can experience this yourself by playing the examples starting on page 23. “The skipping of the interval of a fifth between the seventh and eighth tones of the Lydian Chromatic Scale allows the five basic chord categories of Western harmony to be assimilated by its Nine-Tone Order, Semi-Ingoing Level, in the logical order of their development in Western harmony and the Lydian Chromatic Scale.â€

[sandywilliams]

How 'bout this:
http://forums.allaboutjazz.com/showpost ... stcount=99

[strachs]

sandywilliams wrote:

In VTG the main criteria has to do with notes that create or add to various modal genres, and are in unity with them.

and then wrote:

If you play an A minor chord what note would most compromise the tonal integrity of the VI MG?

Ben Schwendener wrote:

Its initial origin derived from how each of the progressively more outgoing tones of this order contributed its unique color to (and is featured by) the 7 vertical chord producing (parenting) Principle scales.

chespernevins wrote:

it still seems to make sense to want to reconcile the Ladder of Fifths, Interval Tonics and the Western Order.

George Russell wrote (and was quoted by sandywilliams):

“The skipping of the interval of a fifth between the seventh and eighth tones of the Lydian Chromatic Scale allows the five basic chord categories of Western harmony to be assimilated by its Nine-Tone Order, Semi-Ingoing Level, in the logical order of their development in Western harmony and the Lydian Chromatic Scale.â€

[Ben Schwendener]

"One thing I would still ask for input on is: do you have any theories about how exactly "the vertical aspect creates and understands the horizontal aspect in all and in everything"? "

Sure. The reason the major scale resolves to its tonic is because there are actually 2 vertical entities in its structure. In the case of the C major scale, you have a flat lying F major chord (whose primary vertical Lydian Tonic is F), placed in the context of the sharp lying C Maj tonic station chord - which is the horizontal center of Tonal Gravity the scale is based on. But why is it 'horizontal' ; why does the whole scale resolve to the Ionian 'Do'? Answer - because the C major chord IS C lydian, which is sharp lying to the actual Vertical center of tonal gravity of the whole scale (F). In other words, we hear C maj as a final sounding Tonic Station only because it is actually occurring within and horizontally influenced by the 'Vertical' presence of F Lydian. Conceptually, this is noted in the LCC as Vh.

In short, the complete spectrum of resolving possibilities between LC Scales is only fully realized by first understanding and identifying what primary vertical modalities are, and then reconciling these in terms of the Law of Resolving Tendencies (and of course keeping Harmonic Rhythm of the phrase in mind).


Hope that helps

[strachs]

Thank you, Ben.

I am 100% comfortable with the fact that the vertical Lydian scale also contains the materials for horizontal resolution by means of it's CMG's, representing sharp-lying Lydian Scales.

My question stems from the fact that the Major Scale is also viewed as an eleven-tone-order scale.

Some questions arise:

1. Does the 11TO always represent two vertical entities, or just when in the form of an explicit major scale?

2. If it does represent two vertical entities, isn't that second entity a flat-lying entity (rather than a second-to-last, 'semi-outgoing' order of ONE vertical entity)?

3. Is it the AVOIDANCE of vertical outgoingness that suggests or provokes/encourages/motivates the horizontal movement to a neighbouring lydian scale?


In a nutshell, my questions have to do with reconciling the equal validity of the major scale as a "duality" (manifestation of two LS's) and as a singular outgoing verticality.

[strachs]

My mind goes crazy with questions!

Here's another:

Could you say that the GCE for VTG is the PMG's of a scale, and the GCE for HTG is the CMG's of a scale?

[Ben Schwendener]

its both. relative to the lydian tonic as a horizontal tonic station, any presence of the 4th creates the duality necc for resolution. that said, the actual major chord the lydian tonic creates and represents has the 4th degree as a semi-outgoing (11to) vertical element. VTG and HTG are two separate levels, as I mentioned earlier - they are understood only by what you're centered on, a chord or a tonic station. the 4th creates the resolving quality of the major scale octave, while existing as a pretty outgoing tone, vertiacally speaking, to the Lyd Tonic I Major chord.

[Ben Schwendener]

no. the GCE of VTG is the chord of the moment. The GCE for HTG is a prevailing tonic station.