Fibonacci Analysis
Page 13
Frequency is represented with the letter f, so f equals 261 Hz in my example. Music is a logical matrix-based notation. If f = 261 Hz, then the harmonic series created from this fundamental frequency of 261 Hz is f 2 = 522, f 3 = 783, and so on to f n. A harmonic interval is the ratio of the frequencies of two tones. For example:f3/f2 = 783 Hz/522 Hz.
The higher of the two values is placed over the lower because we are considering an upward harmonic interval. To complete this example,f3/f 2 = 783 Hz/522 Hz,
= (3 × 261 Hz)/(2 × 261 Hz),
= 3/2 or, written as the ratio, 3 : 2.
We have already visually experienced a perfect fifth, or the ratio 3 : 2, in the geometry of the nautilus shell. We have also experienced a perfect fourth, which has the exact ratio of 4 : 3. But now that you have an appreciation of the significance of these ratios, they are best illustrated in a more practical way as they occur within a monthly chart for the S&P 500 in Figure 8.4.
The S&P 500 chart in Figure 8.4 shows that the nearby highs at the end of 2007 are pressing up against more than just a zone of resistance. The three most critical pivots precisely connected by harmonic proportion in this market are the 1987 low, the 2002 low, and the nearby highs. The current market high is an exact perfect fifth harmonic interval relative to the 2002 price low. The ratio of the 1987 S&P 500 price low and the 2002 price low is an exact perfect fourth harmonic interval. When you want to know the mathematical start of a market move, use the harmonic proportion as the relevant milestone. Therefore, we should note that the S&P 500 market has placed great importance on the 1987 price low and the current market highs, suggesting this is a high-risk pivot zone for a market correction. This harmonic geometry added to our technical indicators becomes a powerful tool for us.
FIGURE 8.4 S&P 500 Futures Respecting a Harmonic Perfect Fourth and Perfect Fifth Interval
Connie Brown, www.aeroinvest.com. Source: Copyright © 2007 Market Analyst Software
While Figure 8.4 shows the S&P 500 futures respecting a harmonic perfect fourth and fifth interval, the chart is showing much more. The juxtaposition of harmonic intervals throughout the chart as marked on the left side of the chart is a specific harmonic series known as the Pythagorean diatonic scale.
Why do we technical analysts and traders have any interest in the Pythagorean diatonic scale? Not only do we find the perfect fourth and fifth intervals within the monthly S&P 500 market, as shown in Figure 8.4, we see that this market respects all of the harmonic intervals that define the Pythagorean harmonic series. This harmonic series, then, shows us which Fibonacci confluence zones have the greatest weighting. While not every market will build its harmonic grid according to the Pythagorean scale, this scale is the most important one you will need as you begin your study of market harmonics within North American equity charts.
While most recognizable harmonic music scales are created from a base prime number raised to a series of exponential whole numbers, Euclid created a harmonic scale of fourteen proportions that only needed Philolaus’s five different length ratios of 4/1, 2/1, 3/2, 4/3, and 9/8 to create the complete harmonic scale. Plato mentions Philolaus on three occasions in a dialog entitled Phaedo,8 but does not credit him in Timaeus when he reveals the diatonic tetrachords.
To suggest we can apply harmonic intervals to market analysis charting and ignore the conceptual “weight ratios” and resulting “vibration ratios” is not unique. Creative theorist-musicians like Euclid, Ssu-ma Ch’ien9 (also known as Sima Qian) (163-85 B.C.), Ptolemy (c. 100-165 A.D.), Al-Kindi10,11 (d. c. 874), and Al-Farabi12 (d. c. 950), simply ignored weight ratios and vibration ratios in their geometric work. All of these writers based their work on the premise that frequency ratios are a function of length ratios. Therefore, it would be correct to apply their conceptual work to price swing and market ratio analysis.
Harmonics show which Fibonacci confluence zones should be given greater weight or importance. Harmonics also help us understand how confluence zones are related, or not related, to one another within our charts. We can examine the S&P 500 monthly chart in Figure 8.5 to see how matrix math can further enable us to combine harmonic ratios. We can apply the harmonic intervals work of Philolaus in our chart analysis. As an example, the perfect fourth and perfect fifth combine to make one full octave (3 : 2 × 4 : 3 = 2 : 1). This is of tremendous interest to us in current times. The S&P 500 market position at the end of 2007 is at a very critical and significant harmonic inflection point because it is at the full octave interval from the 1987 lows. Only time will tell if the market will respect this zone of resistance in the S&P 500. Oscillators confirm the market is overbought, but oscillators alone cannot tell us the magnitude of the reaction that may follow.
FIGURE 8.5 S&P 500 Futures—Combining Harmonic Ratios
Connie Brown, www.aeroinvest.com. Source: Copyright © 2007 Market Analyst Software
While the Fibonacci confluence zones are not displayed in Figures 8.4 or 8.5, to keep the harmonic scale uncluttered and easy to examine, clearly not all harmonic ratios within a musical scale will mirror the Fibonacci confluence zones we create. We are looking for an overlap between the harmonic series and the Fibonacci confluence zones. In other words, when a harmonic ratio lands within a Fibonacci confluence zone, it is a target area of tremendous interest.
There is more about the Pythagorean diatonic harmonic series to discuss. Notice how, in Figures 8.4 and 8.5, intervals are marked on the far left. The difference between a perfect fifth and perfect fourth (3 : 2 - 4 : 3) is the interval the ancient Greeks called the whole tone, which is the harmonic ratio of 9 : 8. The Pythagorean scale displayed in Figures 8.4 and 8.5 has five whole-tone intervals and two semi-tone or halftone intervals. Very quickly a pattern emerges that is best represented with a rhythmic wave diagram. Our earlier discussions about the logarithmic proportions within a frog and about the Roman Colosseum have prepared us to use a more advanced rhythmic wave diagram to evaluate the market.
Figure 8.6 is a complete rhythmic wave diagram, which has become a tool to map all the harmonic intervals within the S&P 500. What you need to conclude from this diagram is that harmonic intervals do not create a linear series. For example, follow the arcs, or oscillations, defining the perfect fifth intervals (3 : 2) within the series. Observe how they must skip over some nodes. The nodes are the arc crossover points through the midline. Notice how a simple rhythmic wave diagram was added to Figure 8.5 to help you see how to read the harmonic intervals in a simple application. Now notice how a few harmonic intervals can only form when the harmonic series begins to repeat beyond the first octave range. The concepts of harmonic analysis apply to both the price and time axes. The idea of fixed-period time cycles is shattered by these concepts. Imagine: A cycle low defined as a major price low may not have a comparable cycle low until x harmonic units away, which warns you to skip intervals immediately ahead as irrelevant or noise.
FIGURE 8.6 Rhythmic Wave Diagram
Source: Connie Brown
The Pythagorean scale and others are created from a prime number in the base that you will find in more detail in a video I donated to the Market Technicians Association.13 Therefore, further examination of prime numbers is relevant within the context of harmonic analysis.
During a scientific meeting in 1963, the mathematician Stanislaw Ulam began a doodle with similarities to the Square of Nine configuration of a Gann Wheel. The only difference between Ulam’s original spiral and a Gann Wheel is Ulam’s placement of the number 2 to the right of the first number, as shown in Figure 8.7. A Gann Wheel Square of Nine places the number 2 to the left of the first number. As the spiral is created, the number 10 must move one place unit outside the first square, and then the spiral pattern can resume until the number 25, at which point another jump must occur to start the next outer box.
FIGURE 8.7 Ulam Spiral
The whole number spiral in Figure 8.7, which mathematicians know as the Ulam spiral, was not original, but what Ulam did next was. He began to
circle just the prime numbers. On the upper portion of the figure is the number sequence forming the spiral. On the lower portion of the figure, all numbers have been removed except the prime numbers.
When we have a computer create this grid from the numbers 1 to 40,000 and then keep only the prime numbers, we begin to see diagonal lines appear in the results, as shown in Figure 8.8. This is the hint of another pattern taking form that can only be fully revealed when the computer is then asked to remove all primes, except those that are directly touching two other primes. The remaining primes can touch one another on either side or touch another directly above or below the column or row forming a triangle relationship.
FIGURE 8.8 Ulam Spiral Pattern—Prime Number Pattern
FIGURE 8.9 Ulam Spiral Pattern—Subset of Primes
What remains from the original prime number series is the pattern demonstrated in Figure 8.9. As we now know, musical scales have a prime base number raised to an exponential whole number, and we can conclude that Fibonacci numbers are part of a larger mathematical set involving harmonics.
Harmonic intervals form between Fibonacci confluence zones, but not all Fibonacci confluence zones are connected by harmonic intervals or related to one another sequentially. We have only touched upon the study of harmonics within our markets and conceptually opened the door to a fascinating field. The work I am doing will continue, however, and with your help we will push our industry further into testing and developing a deeper understanding of how harmonics flow within markets. Our industry is never stagnant. To do well, we always have to push ourselves as the markets demand we do. It is my hope these methods help you come closer to your goals. But I also hope some of the concepts in these last two chapters begin to drive you a little crazy so that they may ignite a fire of curiosity and motivate you to join the research. After all, it is only through a study of harmonics that we will finally be able to put to rest the burning question, “Why did that market stop exactly on a Fibonacci confluence zone again?!”
FIGURE 8.10 Sunflower Head
Source: Connie Brown
My best wishes to you.
Connie Brown
December 31, 2007
CHAPTER NOTES
1 Barker, 46-52.
2 Barker, 245-269.
3 Nicomachus, The Manual of Harmonics (translated by Levin); Madden.
4 Chalmers.
5 Barker, 303-304.
6 Aristoxenus, The Harmonics (translated by Macran), 198.
7 Burkert, 394. “These considerations, along with the archaic terminology, allow us to regard Philolaus’s Fragment 6 as one of the oldest pieces of evidence for Greek music.”
8 Plato, Complete Works (edited by John Cooper), 60d, 61d, and 61e.
9 The oldest source that gives detailed calculations for this twelve-tone scale is a book by Ssu-ma Ch’ien (163-85 BC) entitled Shih Chi (Records of the Historian), written c. 90 BC.
10 Lewis, 225. For a history of Arabian music, read “The Dimension of Sound,” by A. Shiloah.
11 Lachmann. Ishaq Yaqub ibn Ishaq al-Kindi (c. 801-873 CE) was the first and last example of an Aristotelian student in the Middle East. He viewed the neo-Pythagorean mathematics as the basis of all science. The oldest source on Arabian music is his work entitled Risala fi hubr ta’lif al-alhan (On the Composition of Melodies).• Al-Kindi’s twelve-tone scale is the first tuning that uses identical note names to identify the tones of the lower and upper “octave.” In his text, al-Kindi specifically states that the musical “qualities” of tones separated by an “octave” are identical.
• This is the first mathematically verifiable scale that accounts for the Pythagorean comma. In his ud tuning, al-Kindi distinguishes between the apotome [C sharp], ratio 2187/2048, and the limma [D flat], ratio 256/243.
• This is the first mathematically verifiable example of a Greek tetrachord on an actual musical instrument.
12 The most famous and complex Arabian treatise on music is a work entitled Kitab al-musiqi al-kabir (Great Book of Music) by Al-Farabi (d. c. 950). See also Farmer (page 28), whose work consists of an annotated bibliography of 353 Arabian texts on music from the eighth through the seventeenth century.
13 Presentation by Connie Brown, Great Market Technicians of the 21st Century: Galileo, Beethoven, Fibonacci. Please contact the MTA at www.mta.org. All proceeds go to the MTA to help rebuild the library.
APPENDIX A
FIGURE A.1
(Left) Alvin 7½ in. Lightweight Duraluminum Proportional Divider, Part Number: ALV-450;
(Right) Alvin 10 in. Proportional Divider, Part Number ALV-458 Manufacturer: Alvin and Company
APPENDIX B
Common Errors
WHILE EVERYONE SEES charts in their own way, there are a few common errors that nearly everyone makes when they attend one of my seminars. This appendix will try to make you sensitive to a few of these common errors so you do not make the same mistakes.
The first example is the chart in Figure B.1. This is a 3-day chart for a bank stock on the Australian exchange. The first error is for a trader to start the range to create the Fibonacci retracements in the wrong direction. Start from a high and work down when you want to find support. Work from a low and move up, to determine resistance. In this chart we want to begin to find support, so the correct start is from near the high. Traders new to this method will also run into trouble when they start at the top and then end the range at the capitulation spike low at point 1. If you subdivide the range from the high to point 1, all the Fibonacci subdivisions for 68.1 percent, 50 percent, and 38.2 percent will fall above the price low at point 2. The error traders make is forgetting why they are making the calculation in the first place. Support levels must be below the current closing price. If the price range selected causes the Fibonacci subdivisions to fall above the current close, support has not been identified.
Point 3 is extremely common and a deadly error. Pay close attention to this one. Traders start the Fibonacci range at the high and then drag their mouse down to consider subsequent price lows. That is correct. However, do not use any low that has been retraced by a more recent price swing. If it has been retraced, as we see in the decline from the high to point 1, all the support levels behind and above it (those enclosed in the black box) are violated. So why use or consider them? Instead, scan down to point 1 and then immediately drag the cursor over to point 3, and continue to a lower price low to end the range that will be subdivided. Keep in mind, confluence must occur when different Fibonacci ratios overlap, so you would continue to increase the size of the range subdivided by moving down to the next pivot, gap, or strong thrust price bar.
FIGURE B.1 Australia and New Zealand Banking Group Ltd. 3-Day Chart
Connie Brown, www.aeroinvest.com. Source: Copyright © 2008 Analyst Software
In Figure B.2 we are looking at a 2-week chart for the Hong Kong WEB. The chart includes three subdivided ranges. A common error for a novice is to forget that multiple ranges will always start from the same price level. The top of the chart was truncated, and ranges that end at price lows 1, 2, and 3 were drawn. The confluence zone near 20.00 was respected by the market. This means the zones that form under this area are correct. When I truncate the start, often the lows that form spikes are not truncated. The market internals will tell you which is correct. However, do not pick ranges because of the internal alignments. The Nikkei 225 Stock Average will help you see why in the next example.
The chart in Figure B.3 is a 3-day chart of the Nikkei. I want to find resistance in this example. My start for defining resistance must be a price low, and then I consider multiple price highs. In this chart there are two subdivided ranges. The first range ends at level 1 and the second at level 2.
Notice the ranges and subdivisions do not extend to the left. I have not selected these ranges based on any data behind the current market swing down. Once people start to catch on to the concepts of confluence zones, they begin to pick ranges because they can make the marke
t fit. Do not do this—you will fall into a huge trap. Here is why. The first confluence zone defining major resistance is at horizontal level 3 and price 14,269. I have extended the confluence zone slightly to the left in Figure B.3. If you looked at price lows marked a and b, you might think you made a mistake and then try to adjust the ranges. The next chart shows you why this would be a serious error to change the ranges to create what may appear to be a better confluence zone.
Figure B.4 shows the Nikkei 225 Index in a 3-week chart. You would normally never see the data extend back to 1992 in a single 3-day chart. But for this chart the x-axis was compressed to extend the historical view and the horizontal line created in the prior discussion was extended to the far left. It maps the exact lows at point 3 and 2 as support. Point 1 is the start of a major leg down failing at the same area as resistance. Keep in mind the horizontal line was the first confluence zone identified from short horizon data. Do not try to impose your will on the Fibonacci grids as they fall into place because you feel you have a better idea.
FIGURE B.2 Hong Kong Index 2-Week Chart
Connie Brown, www.aeroinvest.com. Source: Copyright © 2008 Market Analyst Software
FIGURE B.3 Nikkei 225 Stock Average 3-Day Chart
Connie Brown, www.aeroinvest.com. Source: Copyright © 2008 Market Analyst Software
FIGURE B.4 Nikkei 225 Stock Average 3-Week Chart
Connie Brown, www.aeroinvest.com. Source: Copyright © 2008 Market Analyst Software