Interval Tonics - Their Effect Upon the LCS

[strachs]

I came into the LCC world looking for unity and consistency of understanding. However, there are two statements on page 3 of LCC that I have always seen as contradictory.
The first is: "An ascending order of six consecutive intervals of a fifth offers, more than any other order of intervals, the most scientifically sound basis upon which to form an objective theory of music."

I agree with this (despite the minute differences in thirds I have discussed in past threads).

The second is: "ALL tonal phenomena are graded on the basis of their close to distant relationship to" the Lydian Tonic.

If you have read anything I've ever posted in this forum, you'll know that I don't feel this ideal is realized in the Lydian Chromatic Scale, since ignoring what happens at the 8th tone in the ladder is not "scientifically sound", as the Concept strives to be.

I have struggled with this issue for many months and from many angles. Some recent posts by Motherlode and Chespernevins provided me with some crucial insights that helped me sort all this out. I have responded already to Motherlode's insights, but here is what I gained from Chespernevins' focus on interval tonics (see LCC pg 6):

The tonal orders of the LCS do not introduce any new intervals that were not already in the Lydian scale, they just put them in LOCATIONS that were not available in the Lydian Scale - non-native locations, you could say. Putting them in non-native locations increases the span of the SCALE beyond six fifths. However, the span of the INTERVAL itself remains constant.

It is universally agreed (and audibly obvious) that the greatest SPAN that can be achieved for a scale is one that involves the NEXT tone higher in such a ladder, the m2 interval. But to achieve the widest spanning scale, have we REALLY only lengthened the ladder by ONE fifth?

The concept of INTERVAL TONICS, which chespernevins recently drew attention to (in his excellent renditions of the TG chart), provides an essential clue. The tonic of the m2 interval (as shown on pg 6 of LCC), regardless of where it occurs, is it's upper tone. The span of that interval, regardless of where it occurs, is five fifths.

The complementary interval to the m2 is the M7. The two add up to an octave, but the M7 is also a span of five fifths. The only difference between these two intervals, is which one is considered the tonic.

You could say that all intervals have two defining characteristics: 1. a SPAN in fifths, and 2. an indigenous POLARITY, whether tonical, or modal. The five interval pairs conform to this grading as follows:

http://www.4shared.com/file/136580572/e ... arity.html

The interval identities are even more visually evident in the circle of fifths:

http://www.4shared.com/file/136580628/7 ... arity.html

The arrows indicate the direction of VTG, always pointing toward a given interval's tonic.

Notice that all of the tonical intervals are on the right side and all of the modal intervals are on the left. The corresponding interval pairs are each the same distance in fifths away from 1.

Now, here comes the part that affects MEASUREMENT of VERTICAL SPAN of a scale or chordmode. In Russell's LCS, the m2 interval is seen to arise as the NEXT HIGHER TONE in a ladder of fifths (all from the clockwise direction).

But is the span of a minor second seven fifths? Is the lower tone the interval tonic? No and No. Not in the Lydian scale, so not in any scale. The minor second spans five fifths, and it's upper tone is it's tonic (upper within octave, lower withing ladder of fifths) regardless of context.

Let's apply this to a scale and a tonal order we actually use much more frequently: the Lydian Diminished scale, representing the 9TO of the Lydian Chromatic Order, or LCS.

Russell's ordering considers this tonal order and representative scale to come about by extending the ladder up (clockwise in COF's). The interval superimposed on the Lydian Scale by the 9TO ad LD scale is the m3. Is the span of the minor third really eight (or nine) fifths? Is the lower tone the interval tonic? Again, no and no. The minor third spans three fifths, and it's upper tone is it's tonic, regardless of context.

So, my conclusion is, the concept of interval tonics makes it much more consistent and intuitive to consider larger-than-lydian scales/chordmodes/etc to gain their increased span from the FLAT direction. This model naturally places the m2 interval in the model's "12TO", without having to disobey the ladder of fifths. As noted in earlier posts, this also is consistent with measurement of polymodal pairing of two lydian scales, as well as measurement BETWEEN single lydian scales in HTG.

Of course, abandonment of the LCS and it's prime order requires a re-shuffling of the TG chart, which I have done (laid out in fifths, then with interval pairs together):

http://www.4shared.com/file/136575557/1 ... Again.html

I have always thought the ladder of fifths model was really useful, and yet could not really live with the contradiction to it that the LCS requires. I feel that this explanation is a satisfactory way to reconcile the truth of the ladder layout with the loud warning bells that go off whey you try to expand the lydian scale in the sharp direction.

Whether you consider larger-than-lydian structures to be a pairing of more than one Lydian scale, or of an expansion of the Lydian scale itself into higher orders, the identity of the intervals themselves must remain intact.


More to come, but thank you Motherlode (for opening my eyes to Polymodality) and Chespernevins (for opening my eyes to Interval Tonics). These two contributions have helped me to understand and articulate what has been bothering me from the beginning. Now I am comfortable with all aspects of the Concept as I see fit to use it, and am looking forward to exploiting this understanding fully in analysis, performance, and composition.

I would love to hear if this makes sense to any of you, and if it doesn't, PLEASE let me know why you feel the LCS and it's order is more objective and scientifically based (pg 3 LCC).

[sandywilliams]

In HTG there is no abandonment of the Circle of Fifths, which GR called the Circle of Close to Distant Relationships. In VTG the main criteria has to do with notes that create or add to various modal genres, and are in unity with them. If you arrange things FCGDAEBF#, what have you gained in terms of creating chords( with F as the LT)?

[chespernevins]

Strachs,

I'm not adding anything new here, but I like your presentation of the tonic intervals vs the modal intervals. It does seem to reinforce why there is a distinction between the 7 notes of the Lydian scale and the 5 non-lydian notes.

Looking at the ladder of fifths starting on C:

++4 = G (redundant)
+7 = C (redundant)
------
+3 E#, tonic = E# (or F)
+6 A#, tonic = A#
+2 D#, tonic = D#
+5 G#, tonic of C and G# = G#
+1 C#, tonic of C and C# = C#
------
+4 F#, tonic = C
7 B, tonic = C
3 E, tonic = C
6 A, tonic = C
2 D, tonic of C and D = C
5 G, tonic of C and G = C
1 C

[strachs]

Sandy:

With much respect, I think that maybe you misunderstand me (then again maybe I misunderstand GR). I am not saying that F# is more ingoing with F lydian than Ab or Eb. I am saying that the ladder DOES NOT EXTEND UPWARD beyond the 7th tone in the ladder, and so therefore F# (Gb) is FIVE steps beyond the bounds of the Lydian Scale - not five steps in the sharp direction, but five steps in the FLAT direction.

My underlying point with all of this, is that the scientific principle of tonal gravity as it applies to intervals in isolation does not then behave differently when applied to scales and chords.

For all intervals in the Lydian scale (P5, M2, M6, M3, M7 and TT), tonal gravity flows DOWN via fifths to the INTERVAL TONIC. All of these intervals have a span in fifths, and so have a "close to distant" measurement that can be applied to them, indicating how many 'steps' in fifths that tonal gravity must 'flow' down to rest on that tonic.

Tonal gravity cannot and does not apply differently to the modal intervals. They are the SAME INTERVALS. A m2 is a M7 (span of five fifths), just not rooted on it's interval tonic. A m7 is a M2 (span of two fifths), just not rooted on it's interval tonic. The tonal gravity still flows DOWN FIVE FIFTHS in the case of the m2, and DOWN TWO FIFTHS in the case of the m7.

This is why the left and right sides of my tonal gravity chart mirror each other - the left is almost redundant because they are the SAME INTERVALS as those in the right side, just rooted on the tone that is not the interval tonic. That's what makes them MODAL intervals - the very term implies rooting something on other than it's rightful root (basically like chord inversions).

This same principle applies when creating scales and chords (just collections of intervals). Lydian is the prototype for this. Look at ANYBODY'S tonal gravity chart.

Where are the M7's in lydian? The two M7's in F lydian are F-E and C-B. They are the two places where a span of five fifths can fit - on the LT and on the first modal tonic (V).
Where are the m2's in lydian? The two m2's in F lydian are E-F and B-C. They are the two places where a span of five fifths can fit, but this time the interval is rooted on it's upper tone. So the intervals are rooted instead on MT's VII and +IV.

Now let's look at a scale with a WIDER SPAN - Lydian Diminished. Look at ANYBODY's TG chart. Also look at the row of fifths below, and notice that F lydian is on the right, and the five remaining tones are on the left, all spaced by fifths.

Gb Db Ab Eb Bb F C G D A E B
(-II +V -III -VII IV I V II VI III VII +IV)


Now, what new locations are available in LD for the m2 and the M7? The m2 is available on II (G) and the M7 is available on -III (Ab).

As in the Lydian scale itself, the measurement for BOTH of these intervals is five fifths. In the LCS, how does that span in fifths manifest itself? Is it valid in Lydian, but invalid in the higher orders?

In my model - still a ladder of fifths - the Ab-G interval still measures five fifths, agreeing with the measurement in the lydian scale itself. The span of this scale is three fifths larger than the lydian scale - call it Q3, call it the 10TO, whatever. The important thing is that the intervals are measured with the same rod that the scale is measured with, and ALL instances of a given interval retain this measurement.

Chord/scale unity is not affected at all, we just arrive at a more objective system of MEASUREMENT for scales and chords - one that is consistent with the measurement of intervals themselves.

I think that the LCC makes a good start by founding itself on a scientific, objective foundation, but then yeilds to common practice, rather than continuing to base the ENTIRE SYSTEM on the interval of fifths.

On the one hand, Russell claims that the tonal orders are a measurement in fifths of close-to-distant relation. In practice, he only applies the ladder of fifths up to the 7th tone, then yeilds to the historical evolution of chordal expansion, while still claiming that this ordering rests on the tonal gravity, measure-by-fifths foundation, which it does not.

Chespernevins:

You've KIND OF got it - the modal intervals in your post (upper ones) show the correct interval tonic. However, it is more consistent with the interval's distance to the Lydian Tonic if those intervals progress DOWNWARD from C, like this:

+4 F#, tonic = C
7 B, tonic = C
3 E, tonic = C
6 A, tonic = C
2 D, tonic = C
5 G, tonic = C
1 C, tonic = C
------------- (notes above are Lydian Scale, notes below expand scale)
4 F, tonic = F
-7 Bb, tonic = Bb
-3 Eb, tonic = Eb
-6 Ab, tonic = Ab
-2 Db, tonic = Db

In all cases, the intervals are being formed with C, and the interval's lower tone (lower in the ladder, at least) is it's tonic. Tonal gravity ALWAYS flows in a downward direction (VTG anyway, HTG flows in the opposite direction - remember the river current?). Regardless of a SCALE's tonal order, or span in fifths, the INTERVALS always, always retain their measurement, and the largest span for a single interval is six fifths (the TT).

Does that make it clearer?

[bobappleton]

Hi Strachs:

"Down with the LCS" is naieve and inappropriate language on George Russell's website.

You're going more than a little over the top here. It happens on the Form. Don't feel bad. But please tone it down a little. And listen to the the great feedback you get.

Thanks,

Bob

[strachs]

Hi Bob,

I appreciate your point, and so renamed this thread. My enthusiasm regarding my own personal discoveries seems to have momentarily outweighed my respect for GR and those in the Forum. My apologies to any whom I may have offended.

That said, I would love to hear your feedback on the subject, if you have any thoughts. So far, no one has endeavored to convince me that the LCS fulfills the what is claimed on the last paragraph on page 3 - which encapsulates one of the most attractive and idealistic features of the Concept.

A question I would pose to those who feel I'm way in left field here: If the LCS is just a natural extension of the principles embodied by the Lydian Scale itself (7TO), how does the principle of interval tonics complement or fit in with the other four tonal orders?

[chespernevins]

If the LCS is just a natural extension of the principles embodied by the Lydian Scale itself (7TO), how does the principle of interval tonics complement or fit in with the other four tonal orders?

I'm writing up some thoughts on these questions - give me some time. But in short, I think we need to take the effect of the interval tonics (especially of the "non-lydian" tones) into account in relation to the "tonal gravity field" of the ladder of 6 5ths.

For example, the interval of C - Ab in C Lydian. We can't necessarily conclude that because Ab is the tonic of the C-Ab interval that Ab negates the "tonical" quality of C in relation to the ladder of fifths built on C.

The question is: HOW MUCH does the pull of the Ab (being the tonic of C) influence the tonal gravity field of the ladder of fifths built on C? HOW MUCH does the pull of the Ab challenge the tonical authority of C over the 8 tones of C Lyd + Ab? Is it enough to pull us out of the key of C? Or is the key of C still intact, with the Ab sounding some kind of "dissonance" or "turbulance" within the key of C?

[strachs]

Right. I'm not suggesting that Ab negates the tonical authority of C in a Lydian scale built upon it. Like you said, the C Lydian scale is added to, but not ultimately changed from C Lydian.

I do think, however, that interval tonics are a basic manifestation of VTG, and therefore all vertical structures obey the tonal gravity of their component intervals.

So, while a chord may be primarily parented by the C Lydian scale, the superimposition of a modal interval on that scale simply influences the flavor of the chords in a manner consistent with the mode represented by that interval. It doesn't CHANGE to another lydian scale.
I only disfavor the LCS because I see it as contradicting the stated ideals and claims made the LCC itself.

The objectivity of the Lydian scale is the first thing that drew most or all of us to the LCC. The idea that all of the 12-tone chromatic universe can likewise be measured, mapped , and used at will by simply expanding on that ladder, is the other main draw.

On page 11, GR claims that "Tonal Gravity is the central author, authority, governing force, and foundation of that tonal organization" (the LCS).

However, rather than actually basing the LCS's tonal organization PURELY on tonal gravity, as it exists in the Lydian scale, GR allows "augmented and diminished chord colors ... to dictate the position of the remaining five tones of the LCS".

On page 12, the criteria for member scales are given. These do not provide any basis, though, for GRADING such scales on an in-to-out continuum (other than their alignment with the already determined tonal orders of the LCS). Their order (page 13) simply reflects the sequence of their derivation from the tonal orders of the LCS.

So, there is no necessity for chord parenting ability to factor in the in-to-out order. Basing the order, even in part, on the chronological succession of chords used in Western Harmony is a departure from the objectivity that is claimed on pages 3 and 11.

The desire to attach a smaller number to tonal orders that produce fundamental chord types (page 16) is not objective. Anyway, the augmented chord type exists in more than one tonal order (both Lb7 and LD/#2), so it cannot be a determinant in how outgoing a scale is. Claiming that augmented and diminished BELONG to MGI is not objective.

Page 17: "this structure (based on western development) does not alter the essential qualities of a tonal gravity field created by the sequential series of fifths."

I disagree.

Page 53: "However, in order to accommodate the evolution of the five main Western chord types ... the LCS skips the seventh fifth".

The LCS forces us to stray from the stated ideals, and my ongoing contention has been that there has to be a way to make the whole system conform to the close-to-distant thing introduced by the Lydian Scale. A way that is as objective as the Lydian scale itself. I don't see the historical development of chords to be the right source for that objectivity. The expansion of MG's through the TO's need not fit neatly with the historical development of chordal expansion, but should conform to the underlying laws and principles that govern the Lydian Scale itself, and even intervals themselves.

The entire justification for the LCS seems to be based on the ability to house the Augmented chord colour within a lower-numbered Tonal Order. I don't consider this objective. If anyone can point out a better reason than that for a departure from the actual ladder, I would be more than open to considering it. But if not, and it all boils down to accommodating historical usage of chords, do YOU see that as reason enough to switch up the order and make life more complicated?

I look forward to hearing your conclusions, chesper and all.

[chespernevins]

Hi all,

This is my current way of thinking about "Tonal Gravity" - how it works and why the tones are found in the order they are.

To try to illustrate tonal gravity in a more tangible way, I made up a figure which I'll call "tonical weight". This is just a number that tries to quantify the "gravitational pull" of the tonic of a given interval.

If we have an interval of:

G
C

C is the tonic. C to G is only one fifth apart. Because C and G are so close on the ladder of fifths, I give C a large "tonical weight" of 6. Like the gravitational pull the Sun might have on Mercury...

If we have the interval of:

F#
C

C is again the tonic, but C - F# is 6 fifths apart on the cycle of 5ths, or the ladder of fifths, so C exerts a weaker "tonical pull" on F# than it did on G. Thus, in this situation, I give C a low "tonical weight" of 1. Like the gravitational pull the Sun might have on Pluto...

So the closer the interval on the ladder of fifths (for example, a Perfect 5th or Major 2nd), the more weight (or strength, or gravitational pull) the tonic of that interval has. The larger the interval span around the cycle of fifths (for example, a Major 7th or tritone), the less weight the tonic has.

Here's a chart of all the tonical weight values:

The tonic of the interval of 6 5ths (tritone) gets a tonical weight of 1
The tonic of the interval of 5 5ths (Major 7th) gets a tonical weight of 2
The tonic of the interval of 4 5ths (Major 3rd) gets a tonical weight of 3
The tonic of the interval of 3 5ths (Major 6) gets a tonical weight of 4
The tonic of the interval of 2 5ths (Major 2nd) gets a tonical weight of 5
The tonic of the interval of 1 5th (Perfect 5) gets a tonical weight of 6

Or simply:

6 5ths = Weight 1
5 5ths = Weight 2
4 5ths = Weight 3
3 5ths = Weight 4
2 5ths = Weight 5
1 5ths = Weight 6 <-- the tonical weight of "6" is an attempt to quantify the pull that the tonic of the interval has.

If I had more time, I'd try to explain this better, but try to follow me here. I want to now jump to analyzing the weight of a given tone within a field of many notes. To do this, I am looking at the intervals of all notes to each other, noticing which notes hold a tonical role within each of these intervals, and how strong a pull the tonic of that interval is exhibiting.

C is, of course, the tonic of all the intervals within the 6 note ladder of fifths that include C . For example, C - G, C - D, C - A, etc.

F#
B
E
A
D
G
C

In addition, G is the tonic of the intervals G - D, G - A, G - E, G - B, G -F#. And A is the tonic of intervals A - E, A - B and A - F#, and so on.

Here are the "tonical weights" as applied to all the tones in the 7 note ladder of fifths (the lydian scale).

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

+4...F#...C6 1 = 1
7.....B.....F#1 6 = 6
3.....E.....B1 6 F#2 5 = 11
6.....A.....E1 6 B2 5 F#3 4 = 15
2.....D.....A1 6 E2 5 B3 4 F#4 3 = 18
5.....G.....D1 6 A2 5 E3 4 B4 3 F#5 2 = 20
1.....C.....G1 6 D2 5 A3 4 E4 3 B5 2 F#6 1 = 21

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

Let me explain the bottom line of the chart above, as an example:

1.....C.....G1 6 D2 5 A3 4 E4 3 B5 2 F#6 1 = 21

This line shows the tones that "belong" to C, the tonic. "G1 6" shows that G is "owned by" C (meaning that G is the non-tonic tone in the interval of C-G) and is one fifth up from C, therefore giving C a "tonical weight" of 6. "D2 5" shows that D is "owned by" C and is two fifths up from C, therefore giving C a tonical weight of 5, etc.

The last number on the line, "21", shows a sum total of all the "tonical weight" figures, giving a sense of the weight that C holds in this collection of tones.

In the second line from the bottom (on the chart above), I do the same thing for the note G.

5.....G.....D1 6 A2 5 E3 4 B4 3 F#5 2 = 20

I list all the non-tonic notes of the intervals which have G as a tonic, and grade G according to the "tonical weight" that is given by each interval.

Moving on to the "higher order" tones:

When we get to the non-lydian tones (+5 b3 b7 4 b2), we notice that the Lydian Tonic is no longer the tonic of the stand-alone interval (of the LT and the tone). For example, if we have an interval of C-Ab in C Lyd, the Lydian Tonic C is not the tonic of the interval - Ab is. Does this mean that C is no longer the Lydian Tonic of the key we are exploring? Not necessarily. This is why we are calculating the "tonical weight" of each tone. We want to see what amount of pull adding a foreign note, like Ab, has on the key of C Lydian. Just because Ab is the tonic of the C - Ab interval, does not mean that the Ab is necessarily strong enough to unseat C as the primary tone in the "tonal gravity field" that is the 6 note ladder of fifths. Ab will certainly add some degree of "turbulence" to the placid structure of the Lydian scale, and hopefully the "tonical weight" number will help us quantify its influence in the tonal gravity field.

We will add each "non-lydian" tone (Ab Eb Bb F Db) to the Lydian scale (or tonal gravity field) and grade the tonical weight of the tones. If the tonical weight of this non-lydian tone is relatively low, then it will not challenge the authority of the Lydian Tonic C very much, and will be heard as a relatively "ingoing" note. If the non-lydian tone has a tonical weight close to or equal to the actual lydian tonic (C), then it challenges C's authority as the Lydian Tonic, and will therefore be heard as an "outgoing" note in the C LCS.

Note: I'm not, for the purposes of this post, placing any importance on whether the "non-lydian" tones are represented as being at the top of the ladder or at the bottom. Consider that the ladder just keeps repeating around the cycle of fifths.

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

C Lyd + Ab

b6....Ab.....C4 3 G5 2 D6 1 = 6 <-- 6 is a low weight compared to C's 21
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.....G.....D1 6 A2 5 E3 4 B4 3 F#5 2 = 20
1.....C.....G1 6 D2 5 A3 4 E4 3 B5 2 F#6 1 = 21

Note that this chart could have just as well been written as follows, with the exact same result:

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.....G.....D1 6 A2 5 E3 4 B4 3 F#5 2 = 20
1.....C.....G1 6 D2 5 A3 4 E4 3 B5 2 F#6 1 = 21
4
b7
b3
b6....Ab.....C4 3 G5 2 D6 1 = 6


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

C Lyd + Eb

+4.....F#.....Eb3 4 C6 1 = 5
7.....B.....F#1 6 Eb4 3 = 9
3.....E.....B1 6 F#2 5 Eb5 2 = 13
6.....A.....E1 6 B2 5 F#3 4 Eb6 1 = 16
2.....D.....A1 6 E2 5 B3 4 F#4 3 = 18
5.....G.....D1 6 A2 5 E3 4 B4 3 F#5 2 = 20
1.....C.....G1 6 D2 5 A3 4 E4 3 B5 2 F#6 1 = 21
4
b7
b3.....Eb.....C3 4 G4 3 D5 2 A6 1 = 10 <-- Eb has a somewhat greater tonical weight, challenging C a little more

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

C Lyd + Bb

+4.....F#.....Bb4 3 C6 1 = 4
7.....B.....F#1 6 Bb5 2 = 8
3.....E.....B1 6 F#2 5 Bb6 1 = 12
6.....A.....E1 6 B2 5 F#3 4 = 15
2.....D.....A1 6 E2 5 B3 4 F#4 3 = 18
5.....G.....D1 6 A2 5 E3 4 B4 3 F#5 2 = 20
1.....C.....G1 6 D2 5 A3 4 E4 3 B5 2 F#6 1 = 21
4
b7.....Bb.....C2 5 G3 4 D4 3 A5 2 E6 1 = 15 <-- Bb getting more challenging to C's authority

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

C Lyd + F

+4.....F#.....F5 5 C6 1 = 6
7.....B.....F#1 6 F6 1 = 7
3.....E.....B1 6 F#2 5 = 11
6.....A.....E1 6 B2 5 F#3 4 = 15
2.....D.....A1 6 E2 5 B3 4 F#4 3 = 18
5.....G.....D1 6 A2 5 E3 4 B4 3 F#5 2 = 20
1.....C.....G1 6 D2 5 A3 4 E4 3 B5 2 F#6 1 = 21
4.....F.....C1 6 G2 5 D3 4 A4 3 E5 2 B6 1 = 21 <-- F has equal tonical authority to Lydian Tonic C. F is heard as very dissonant in the key of C!

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

C Lyd + Db

b2.....C#.....C5 2 G6 1 = 3 <-- Db has very little tonical authority of its own, BUT...
+4.....F#.....C#1 6 C6 1 = 7
7.....B.....F#1 6 C#2 5 = 11
3.....E.....B1 6 F#2 5 C#3 4 = 15
6.....A.....E1 6 B2 5 F#3 4 C#4 3 = 18
2.....D.....A1 6 E2 5 B3 4 F#4 3 C#5 2 = 20
5.....G.....D1 6 A2 5 E3 4 B4 3 F#5 2 C#6 1 = 21 <-- Db's existence has created a ladder of fifths from Db down to G (or G up to Db), giving G equal authority to C!
1.....C.....G1 6 D2 5 A3 4 E4 3 B5 2 F#6 1 = 21

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


So, a summary:

C's tonical weight in C Lyd is 21.

Ab's tonical weight in C Lyd is 6.
Eb's tonical weight in C Lyd is 10.
Bb's tonical weight in C Lyd is 15.
F's tonical weight in C Lyd is 21, equal to C's
Db's tonical weight in C Lyd is 3, but bestows upon G a weight of 21, equal to C's.

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

I don't think it is coincidental that the two notes one step sharp and one step flat of the ladder are the most dissonant:

C# <-- creates a ladder of fifths built on G - very challenging to the key of C Lyd.
---
F#
B
E
A
D
G
C
---
F <-- creates a ladder of fifths built on F - very challenging to the key of C Lyd.

[chespernevins]

oh, ML, I dug your response.

[strachs]

motherlode:

It may have been a rant, but was very motherly and responsible of you to reinforce bob's reminder about respect for the man and where he was coming from with all of this. I do wish you had left your comments in.

chespernevins:

Wow. I'm speechless (if you can imagine). Your approach is so similar to mine, and yet you did an excellent job of articulating your viewpoint. Awesome indeed.

I loved how scientific, numerical, and empirical your approach is. The book is after all subtitled "the art and SCIENCE of tonal gravity". The art is number one, of course, but GR was totally into the science of it all as well, as are you and I and others, too.

Let me digest your analysis a bit more before I reply to it specifically, but I wanted to first congratulate you on exactly the kind of reasoned response I have been hoping for. Well done indeed.

[sandywilliams]

I’m prone to oversimplifying but I’d like offer this thought about the –2 being the caboose in the LCS: If you play an A minor chord what note would most compromise the tonal integrity of the VI MG?
I think the Concept might prove much more useful to you if looked at like it was a cookbook instead of physics book.

[chespernevins]

I’m prone to oversimplifying but I’d like offer this thought about the –2 being the caboose in the LCS: If you play an A minor chord what note would most compromise the tonal integrity of the VI MG?

I agree that looking at the minor is helpful. But things that seem obvious to a seasoned musician like yourself may not be obvious to everyone.

I think the Concept might prove much more useful to you if looked at like it was a cookbook instead of physics book.

Not sure if you're addressing me specifically here, but personally, I do some of both, and feel that they inform each other. I also have approaches to music that can't be quantified by music theory.

However, concerning my post above, I don't think that it's scientific, which implies impartial observation and measurement, hypothesis and reproducable proofs. All the numbers and lists are more an attempt to clearly communicate an inutitive idea that I already had about "tonal gravity".

Do you have a specific objection to exploring the idea of "tonal gravity" in this way?

After all, many people have made the point that it's confusing when George says that the LCC is scientific and based on the tonal gravity of the ladder of fifths and then skips a fifth. That's how it comes across in the book, anyway, and it puts off people who started exploring the LCC with an open mind. But I tend to think that George knew what he was talking about but just didn't explain it in enough detail for the rest of us to follow logically. To me, that seems to naturally lead to where we are now.

[Ben Schwendener]

Remember that equal temperament is a concession to the overtone series, that human beings are not machines and that art is about beauty, energy and communication. George addressed the evolution of western harmony in setting up the LC (Western) Order of TG and presented the most ingoing Prent Scales ( of origin and unity) for all the traditionally definable chords. Then he wrote some amazing music that is physical, emotional, and intellectual as well. The LCC is about creating new music, and is solidly based on the principle of Tonal Gravity - which is indisputable. So, lets make music....

[strachs]

Humans are not machines ... but that doesn't make the science that relates to music - and governs the materials we create with - irrelevant, or relegated to machines. If it did, then why does Tonal Gravity and the Lydian scale even matter? If the stated ideals in the LCC appeal to you (they sure do to me), why not explore their application down to the details?

I compare this tonal gravity based theoretical modeling to the science behind the periodic table of elements. Some creators of synthetic substances would say "who cares how the table comes together, just work off of it". Others are fascinated by the incredible atomic laws and order that govern the behavior of atoms and inevitably result in the table.

I feel the same way about the tonal orders, the tonal gravity chart, and so on. I could just take it, and use it, and get on with life. But my personality gets as much "high" from the fascinating underlying order as from the end results.

If it all bores you, just don't read these threads. If it bothers you ... oh well.

Whether such exploration results in legitimate questioning some of GR's conclusions, or ends up reinforcing their rightness - doesn't the Concept only stand to benefit? As chespernevins points out, perhaps this kind of discussion could remove what currently stand as barriers to wider acceptance and appreciation for the Concept. That's my hope, anyway.