CHAPTER 7
Rhythmic Wave Diagrams
I INTRODUCED RHYTHMIC WAVE DIAGRAMS in Chapter 4, in the discussion of the Roman Colosseum and how Fibonacci ratios form in the horizontal, vertical, and diagonal axes. The flowing arcs visually connect points that have proportional relationships. But a man-made structure is only a copy of the geometry found in nature. Now we are going to see how biology, physics, and proportional geometry come together in ways you might not have contemplated before. This discussion assumes you have made the connection that mathematical ratios and the resulting proportions found in nature are mirrored and replicated within financial market data.
In prior chapters, we discussed how to reveal the confluence zones that a market uses to build future price swings. The confluence zones formed from the cluster of ratios developed from subdivisions of various price ranges. It was important to use certain internal milestones that consistently created confluence zones, such as gaps and the start of strong third waves. The result was a mathematical grid that overlay the entire data set that could be used in various ways to forecast future measured moves. Now, we are about to find that nature creates a grid of confluence zones from internal structures as well. The conceptual conclusion you need to derive from this chapter so you can grasp the final discussion is this: The points where multiple ratios intersect are not related to one another in a simple linear series. Because markets mirror the laws of nature, you can deduce that the relationships between Fibonacci confluence zones within market price data do not create linear series either.
FIGURE 7.1 Nautilus Shell Geometry
Source: Connie Brown, © 2005 Aerodynamic Investments Inc., www.aeroinvest.com
The last diagram in Chapter 4 revealed the geometric figures that evolve from within a nautilus shell. Figure 7.1 demonstrates the golden spiral form of the nautilus shell as well. While the geometry was revealed, the question of how to map a spiral and the numerous geometric figures onto a two-dimensional chart was left unanswered. Now is the time to address this question.
Traders familiar with Fibonacci ratios within nature are well aware that plants, fish, and butterflies all exhibit 161.8 percent and 61.8 percent ratios within their structures. It is hard to write a book about Fibonacci ratios without some sort of bug or plant as a point of discussion. But this time our goal is a deeper discussion, rather than mere proof these ratios exist in nature. Now is the time to consider how the points that form the extreme and mean proportion relate to one another.
Notice that I have mapped the skeleton of a frog using a rhythmic wave diagram in Figure 7.2. The last time I looked at a frog this closely was in high school biology, and I hate to admit how long ago that was. But this rhythmic wave diagram will help you see how examining the frog’s proportional sections will further our discussion about the relationships between confluence price zones.
The rhythmic wave diagram on the right side of an actual frog’s skeletal form details the proportional connections along line sm. View these connection points as confluence zones. Line sm is the maximum stretch between the longest finger above the forearm and the longest digit of the foot. The rhythmic waves or confluence zones connect the primary joints and body extremities. As we study the internal string of subdivisions along the line sm, it becomes clear a logarithmic proportional relationship is present that connects the smaller parts within the whole. The analysis of greatest interest to us is how the Fibonacci ratio 0.618 appears within this proportional diagram.
Observe the wave proportion between confluence points s, n, and m. You will find a sine wave that begins at point s and swings out to the left. It then cuts back and swings through line sm at point n, which aligns with the frog’s knee. The proportional wave continues and swings out to the right and reconnects at point m. For each proportional wave there is a mathematical relationship between the connected ranges, but not all segments exhibit Fibonacci ratios. This last point is extremely important. It can be suggested that Fibonacci ratios are only part of a larger subset.
FIGURE 7.2 Frog Skeleton with Rhythmic Wave Analysis
Source: Connie Brown, © 2005 Aerodynamic Investments Inc., www.aeroinvest.com
If we want to find the anatomical joints or extremities that form a Fibonacci ratio, we have to break away from measuring the whole range sm. For example, the confluence zone at point p aligns with the frog’s abdominal joints. If we measure the distance from p to m with a proportional ratio compass, we will find that the Fibonacci ratio 0.618 falls right at the knee, at point n. Proportional line A, on the far right of the diagram, will help you visualize this extreme and mean proportion. Study the joints at p and n very closely, as if you were an engineer evaluating thrust points of stress. The point that would bear the greatest force as a frog leaped forward would be the joint at the knee, which would create the force that is jammed into the abdominal joint at p. The joint at p creates an angle at which the bones must transfer the thrust to the upper portion of the backbone and upper body frame.
Had we measured a range from the head to the extreme at m, the proportional Fibonacci ratio would fall at point o, which aligns with the frog’s tail. Anatomical physics makes more sense to connect the Fibonacci ratio with the head and body extremes than it would to connect the head with the abdominal joints. The more you study these ranges and consider mechanical physics in light of the anatomy and its function, the more astounding the diagram becomes.
If you measured the range sm as 1.618, you would find that no confluence zone or joint falls on the 38.2 percent or 61.8 percent ratio. Only when you measure a range that makes sense in terms of anatomical mechanics or body proportion do you find the intersecting body joints with Fibonacci ratios. Fascinating. Proportional lines A, B, and C on the far right of the diagram map only three extreme and mean proportions. There are several others. For example, use a proportional divider to measure the length of the body from the top of the head to the tail. Then find the 0.618 relationships within the body.
Consider the proportional math found in nature further. Point o, which defines the tail end of the frog’s body, has no proportional connection to points p or n. However, point o is a link between these two anatomical points that logarithmically connects the whole.
Measure from the ankle and point r to s. The only joint creating the extreme and mean proportion resides at point o. Now consider the distance of line ro and line np and that mechanical thrust is dissipated between each set of points. While my knowledge of body mechanics is only related to my experiences as an international swimmer years ago when computer analysis was becoming common, I understand enough to recognize that proportional math shows that each logarithmic section is important to the whole, but some proportions have greater weighting due to specific mechanical functions. The key point is that mathematical relationships within the skeleton of the frog are not simply linear relationships; nature is not linear. The ratios of greatest interest do not fall one after another in a linear sequence. As markets follow laws found in nature, the relationships between the confluence zones are not linear either.
What relationship, if any, exists between the confluence zones that do not create extreme and mean proportion in a logarithmic series? It is important to realize proportion is used to solve problems involving ratios. We must push ourselves beyond the mere identification of confluence zones, and venture into a new field of proportional analysis that shows that harmonic intervals develop within the grid of confluence zones. We will look at this new field of study for chart analysis in the final chapter.
CHAPTER 8
Harmonic Unity Within Market Price and Time
THROUGHOUT THIS BOOK we have subdivided price ranges and discussed how markets respect the zones in which these Fibonacci ratios cluster together. In this chapter you will study how these confluence zones relate to one another. To do this, you need to have a deeper understanding of how proportion is evaluated so you can solve problems involving ratios.
You will recall from an ear
lier chapter that proportion is a repeating ratio that typically involves four terms, 4 : 8 :: 5 : 10, stated as “4 is to 8 as 5 is to 10.” The Pythagoreans called this a four-termed discontinuous proportion. The invariant ratio here is 1 : 2, repeated in both 4 : 8 and 5 : 10. An inverted ratio will reverse the terms, so it can be said that 8 : 4 is the inverse of 4 : 8, and the invariant ratio is 2 : 1.
The Pythagoreans knew of three different ways to identify proportional relationships between elements. These remain the three most important ways for us to study proportion: arithmetic, geometric, and harmonic.
Arithmetic Proportion
A 2-week chart of Centex is displayed in Figure 8.1, as it is easier to visualize data ranges than it is to study algebraic equations. Notice that the ranges drawn in this chart begin and end with values, as we have discussed, that create confluence zones. Do not try to read a market projection method into this discussion.
FIGURE 8.1 Centex Corp. (CTX) 2-Week Chart—Arithmetic Proportion
Connie Brown, www.aeroinvest.com. Source: Copyright © 2008 Market Analyst Software
You will see that range a is subdivided into 8 equal units. The length of 1 unit within range a allows you to see that price range b is 6 units in length, range c is 12 units in length, and d is 10 units in length. The algebraic expression to define arithmetic proportion is a - b = c - d. The difference between the first two elements must be equal to the difference of the second pair of elements. The ratios are not the same; that is to say, a/b does not equal c/d. But the arithmetic differences are both 2 units as illustrated with black boxes on the right-hand side of Figure 8.1. An arithmetic proportion would not have been demonstrated if range c had been measured from the price pivot low behind it. Similarly, you would not have an arithmetic proportion if range b had ended at the price high. Using internal structures, as we have done throughout this book, reveals an arithmetic proportion that would have been disguised with a surplus or shortage if we had assumed the price highs and lows were always the most significant pivots to use. If you think about it, a simple series of whole numbers, such as 1, 2, 3, 4, 5, is an example of a continuous arithmetic proportion because the difference between each whole number is always 1.
Geometric Proportion
While the arithmetic proportion only has equality in the differential comparison of the elements, a geometric proportion must have a proportional ratio in the range segments and create a ratio between the differences. The algebraic equation for geometric proportion is a : b = c : d. This is the same as using a fraction that is written as a/b = c/d. The nominators and denominators can be divided or multiplied to easily find geometric proportion. For example, you can create the ratios of 4 : 8 and 2 : 4 from Figure 8.2, viewing the left-hand side of the diagram. The ratios of 4 : 8 and 2 : 4 are divisible, and they can be written 4 is to 8 as 2 is to 4 (4 : 8 :: 2 : 4). Geometric proportion allows the elements to be checked by multiplying the inner elements and the outer elements for equality comparison. For example, 8 × 2 = 4 × 4.
You may recall from the first chapter that the geometric mean between two numbers is equal to the square root of their product. Therefore, it can be said that the geometric mean of 1 and 9 is √(1 × 9), which equals 3. This geometric mean relationship is written as 1 : 3 : 9, or inverted, 9 : 3 : 1. The 3 is the geometric mean held in common by both ratios. The Pythagoreans called this a three-termed continuous geometric proportion. The proportion of the Golden Ratio is a three-termed continuous geometric proportion.
Harmonic Proportion
Arithmetic proportion is about the differences between elements. Harmonic proportion is about the ratios of the differences. In Figure 8.2 the harmonic proportion is illustrated on the right-hand side of the illustration, which shows the resulting harmonic ratio of 1 : 2. If four elements had been shown, the middle elements would always be the same number. Harmonic proportion is the mathematical ratio of musical intervals. Developing a better understanding of harmonic proportion and the mean relationships are most pertinent to building upon our prior discussions in this book, because harmonic relationships connect the major confluence zones in our market data.
FIGURE 8.2 Geometric and Harmonic Proportions
Source: Connie Brown
Harmonic Intervals
The mathematical comparison of market price swing lengths is conceptually no different from the work of ancient Greek and Arabian geometricians who studied string length ratios to define musical intervals. The Greeks primarily used verbal terms to describe the mathematical ratios of musical intervals. Furthermore, the Greeks never employed fractions such as 1/2, 2/3, 3/4, and 8/9, to identify musical intervals. The ratios of Greek music always express quotients that are greater than one. For example, in Figure 8.3 the ratio of the octave (or diapason) is diplasios. Since the prefix di means “two,” and plasma means “something formed,” diplasios means “twofold,” or “double.” Therefore, the octave is a musical interval produced by a string that is twice as long as a similar string; hence, the resulting length ratio is 2/1 or 2 : 1. Similarly, the term for the ratio of the fifth (or diapente) is hemiolios. Since the prefix hemi means “half,” and olos means “whole,” hemiolios means “half and whole”: 1/2 + 1 = 1/2 + 2/2 = 3/2. The Greek term for the ratio of a fourth is epitritos. Since, in a mathematical context, the prefix epi describes the operation of addition, and tritos means “third,” the term epitritos denotes “one third in addition,” and connotes “one and one third”: 1 + 1/3 = 3/3 + 1/3 = 4/3. Therefore, the fourth is a musical interval produced by a string that is by a third longer than another string. Similarly, the term for the ratio of the whole tone is epogdoos (or epiogdoos), which connotes “one and one eighth”: 1 + 1/8 = 8/8 + 1/8 = 9/8. Therefore, the whole tone is a musical interval produced by a string that is by an eighth longer than another string.1 It was Nicomachus2 who abandons the concept of length ratios to study harmonic intervals in Chapter 6 of his Manual of Harmonics.3 He tells the story of Pythagoras comparing various weights suspended from the ends of strings and discovers that the harmonic intervals for the octave, fifth, and fourth vary directly with the tension of the string. The octave was produced from the sound of two similar strings of equal length when one was pulled by a 12-pound weight and one by a 6-pound weight. The tension on the strings created the harmonic interval. The ratio is 2 : 1. The fifth was duplicated by weights of 12 and 8 pounds, or 3 : 2. The fourth was created from 12- and 9-pound weights, or 4 : 3. You will recall it was Pythagoras’s study of hammer weights striking an anvil that led to the discovery of the ratio 1.618. As market technicians very familiar with the ratios phi and Phi, we are not venturing that far from the first apple tree where these concepts originated.
FIGURE 8.3 Musical Intervals Expressed as String Length Ratios
Source: Connie Brown
Greek Harmonic Intervals, Tetrachords, and Scales
Musicians who played Greek lyres tuned the four strings to tetrachords,4 or to simple scales emanating from the interval 3 : 2, or the fifth. Ptolemy (c. 100-165 AD), and Archytas ( fl. c. 400-350 BC) identified three different kinds of tetrachords: the diatonic genus, the chromatic genus, and the enharmonic genus.5 These tetrachords can be found in a treatise entitled The Elements of Harmony by Aristoxenus6 (fl. fourth century BC).
The intervals of greatest value to the ancient Greeks, such as Philolaus, are the intervals of value to us today in analyzing market swings and Fibonacci clusters. Philolaus defines7 three intervals:Syllaba = 4/3, Di’oxeian = 3/2, Dia Pason = 2/1
He went on to describe the following mathematical harmonic intervals or relationships that help us see matrix math being applied to further the capability of working with harmonic intervals:Harmonia = 4/3 × 3/2 = 2/1
Epogdoic ratio = 3/2 divided by 4/3 = 9/8
Two whole tones = 9/8 × 9/8 = 81/64
Philolaus called the smallest interval of the diatonic genus, ratio 256/243, a diesis. With the exception of Philolaus, all ancient Greek and modern theorists refer to the rati
o 256/243 as a limma, or the “remainder.” This is the interval that remains after one subtracts two whole tones from a fourth.
You may have realized that the harmonic intervals between two notes (or confluence zones) form ratios that utilize the first seven numbers of the Fibonacci number series (0, 1, 1, 2, 3, 5, 8). Examples of the musical notes with fixed harmonic relationships to one another are listed in Table 8.1.
TABLE 8.1 Harmonic Intervals from Fibonacci Numbers
Harmonic Series
We are interested in more than the interval between two notes. That would be like having one major price swing in place off a bottom and the first pivot, then not knowing what will happen next. But market swings form a series of harmonic intervals. This is how a harmonic series is developed: A fundamental frequency is a selected vibration, such as a tuning fork that vibrates or oscillates at precisely 261.626 Hz per second. This frequency in Western music is known as the note middle C on a piano, or C4 in scientific pitch notation. The piano has a range of seven octaves plus a few extra notes, and the human ear can hear nearly eleven octaves. The frequency of a vibrating string at 523.251 Hz is one full octave above middle C (C5), and it is vibrating at twice the rate of the lower C (C4), so the harmonic interval of an octave is 2 : 1. Picture a fundamental frequency in your mind as a price milestone like the start of a strong move, a gap, or a significant market pivot. Conceptually, you might also consider 261 Hz to be the price low of the 1987 crash in the S&P 500. Now we want to know what Fibonacci confluence zones have a harmonic proportional relationship to this market pivot so we can identify mathematically where the next major market reversal may occur if the market is building upon the same price grid.
Fibonacci Analysis Page 12