Richard Merrick: Harmonic formation helps explain why phi pervades the solar system

Posted: April 4, 2014 by tallbloke in Astrophysics, Celestial Mechanics, Cycles, Energy, Fibonacci, Phi, solar system dynamics, waves

This is a repost of an article by Richard Merrick  published on the website. This is highly relevant to the research Stuart ‘Oldbrew’ and I have been doing to try to define the mechanism by which sufficient energy is being passed between planets and the Sun to account for the observations we have been making in our Why Phi? series:

Harmonic Formation
By Richard Merrick

How do harmonics form?

As waves reflect and resonate inside a container or cavity, they cross one another. As they cross, they exchange energy at specific locations called ‘damping wells.’ In quantum mechanics this is explained by Landau-Zener theory (1932).

Known as Landau damping, waves that pass through one another mostly transparently, avoiding a direct collision, are called ‘avoided crossings.’ In such cases, energy is exchanged in a ‘parameter zone’ where one wave pushes against another, creating a kind of spinning well or vortex action. Like a kind of switch, energy is passed ‘adiabatically’ (without heat loss) across the damping well in a kind of torque action.



We can think of the damping well as a kind of low-pressure zone much like those in our atmosphere that create storms, hurricanes and tornados. The surrounding pressure differential causes an implosion toward the center of the low-pressure zone, forming a vortex.

In the special case of a standing wave, damping wells form at constant locations at golden sections of the period of the prime resonant frequency. This is because the golden ratio constant, represented by the Greek letter _ (or Phi) and equal to the ratio of about 1 : 0.618033… or 1.618033, is non-reflective while having the unique ability to nest into itself infinitely. As a result, energy is exchanged between harmonics at Phi proportions while harmonic wave partials begin to ripple outward from the edge of the well.



Because of this, the damping well of a standing waves can be described as a Fibonacci spiral converging to Phi, the deadest location in a standing wave and thus the point of greatest torque and energy exchange between harmonics. This can be proven by using the Fibonacci series as a nominal solution for the second-order equation known as the ‘characteristic wave damping equation.’ In this proof, the golden ratio Phi becomes the ‘eigenvector’ and the Fibonacci series (1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …} become its ‘eigenvalues.’

In fact, we can understand harmonic formation as something called ‘phi-heterodyning’ where each harmonic emerges around the Phi eigenvector in nested golden sections like a fractal. A paper by Bovenkamp and Giandinoto entitled ‘Incorporation of the Golden Ratio Phi into the Schrödinger Wave Function using the Phi Recursive Heterodyning Set’ shows how this can occur, but there is an easier way to visualize it.



The Fibonacci damping well approximates the irrational Phi Recursive Heterodyning Set with rational harmonic waves. Above the proportion of 13:8, wave formation is damped and suppressed while below this in the range of {1 .. 12} whole number harmonics can form and resonate constructively with the prime resonant frequency. In general, waves that come too close to Phi proportions in the resonant frequency are killed while those furthest away resonate the most.



While this sounds like a new idea, it is actually part of the age-old practice of creating resonant chambers that suppress the formation of unwanted standing waves and reflection. Stradivarius was well-known for using golden sections in the design of his violins. The best rectangular concert halls use golden proportions in their dimensions, as do modern day speaker enclosures.



Standing wave reflection simply cannot be sustained when one of the three dimensions in a container is at or even near a golden ratio to another.

Further Reading

Interference – A Grand Scientific Musical Theory
, By Richard Merrick

Content courtesy of Richard Merrick
Copyright (c) 2011. All Rights Reserved.

  1. oldbrew says:

    More info on mean motion resonance in the solar system – with examples – here.

    Exoplanets also show signs of similar ‘behaviour’.

  2. Roger Andrews says:


    This is totally off-the-wall, but here goes anyway:

    According to the above results two superimposed waveforms generate a vortex that mimics a Fibonacci spiral with Phi = 1.618.

    Any numerical sequence constructed by summing the previous two numbers to get the next, including the Fibonacci sequence, gives Phi = 1.618.

    So a numerical sequence that sums the previous two numbers analogs two superimposed waveforms.

    But if we have three superimposed waveforms do we have to sum the previous three numbers, whereupon Phi becomes 1.839?

    And with four waveforms do we need to sum the previous four numbers, whereupon Phi becomes 1.928?

    And so on to an infinite number of superimposed waveforms, whereupon Phi converges on 2?

    How many superimposed “waveforms” are there in the solar system?

  3. tallbloke says:

    Roger A: Very good question. 2 is one of the constants in the linked paper, along with e and phi and pi

  4. oldbrew says:

    RA: re ‘How many superimposed “waveforms” are there in the solar system?’

    To answer a Q with a Q: how are we supposed to find out?

    My take on it was from the last line:
    ‘Standing wave reflection simply cannot be sustained when one of the three dimensions in a container is at or even near a golden ratio to another.’

    Maybe the apparent stability of the system means there are few ‘standing wave reflections’?

  5. tallbloke says:

    Maybe the entropic tendency means the system organises to minimise the amount of energy trapped?

  6. Roger Andrews says:

    OB: You ask, how are we supposed to find out how many superimposed “waveforms” there are in the solar system?

    Well, a simple way of looking at it is to assume that each planet in the solar system generates its own unique “waveform” as it orbits the sun, so if we ignore the asteroid belt and the planetary moons there are nine.

    TB: Maybe these nine waveforms interact to minimize trapped energy?

  7. oldbrew says:

    RA: planetary conjunctions i.e. 2 or more planets in line with the Sun, seem to be an additional factor?

  8. The planet pairs Jupiter /Saturn and Neptune/ Uranus
    resolve to 1 (fundamental ) resonance

    more importantly, the tip of this triangle apex also resolves to 1 ( fundamental)
    4.236: 1.618 = 2.618
    2.618: 2.618 = 1
    fundamental ( resonance)

    I am interested in the shape.. Resolving to an apex. of a triangle

    Merricks uses a rectangular box

    Asfar as l can remember .. standing waves must not exist because resonance would ‘break’ the structure’ Like the resonating bridges that break

    Which planetary pairs resonate and which are damping?

    144/89 is a common ratio found in your research. This is a damped ? ratio

  9. Roger Andrews says:


    I don’t know whether this is of any help, but it occurred to me that it would be interesting to try to quantify the relative strengths of planetary interactions rather than just comparing planetary orbital and rotational periods. So I did some simplistic calculations of how the gravitational attractions between the planets change as they go whizzing around the sun. I assumed circular and coplanar orbits and calculated gravitational attraction using the good old Newtonian m1*m2/R^2 equation.

    Here are two figures that use Earth as an example. The first shows the changes in gravitational attraction over time between Earth and individual planets with all of them lined up at day zero. Almost all of the “pull” is exerted by Jupiter and Venus, with Saturn and Mars providing very minor contributions and the other planets effectively none at all.

    The second sums the total gravitational attraction from all the planets. There’s a spike in maximum gravitational attraction once every ~5190 days, or ~14.2 years, basically when Jupiter and Venus realign. A 14.2 year period doesn’t ring a bell with me, however.

  10. 14.2 yrs for re-alignment? This article says
    Contrary to popular belief, conjunctions between Venus and Jupiter are not rare events; in fact they occur on average about every 13 months.
    Interestingly, the sidereal revolution periods (that is, the time it takes them to make one full orbit relative to the stars) of Venus, Earth and Jupiter are, respectively, 224.7008, 365.2564, and 4332.5894 days.

    So 39 such periods of Venus are virtually equal to 24 orbital periods of the Earth and 2 periods of Jupiter. For this reason, circumstances involving specific conjunctions between Venus and Jupiter repeat under almost identical conditions after a time span of just over 24 years.
    of interest here for me anyway is the ratio
    39 periods of Venus are ~= to 24 orbital periods of earth
    39:24 = 1.625

    Merricks identifies this ratio value of 1.625 as the borderline between damping and resonance
    39 (venus) : 24(earth) : 2(Jupiter)

    19 .5 : 12 : 1

    19.5 (venus) : 12 (earth) = 1.625

    1.625 : 1 = Merricks borderline between damping and resonance

    These 3 planets some sort of equilibrium? according to Merricks model?
    and further quote
    On March 5, 1988, Venus and Jupiter were in conjunction, with Venus passing 2.2 degrees north of Jupiter. A little more than ….24 years later…. brings us to next week. On March 13, 2012, Venus will pass 3 degrees north of Jupiter.

    Looking ahead, on March 22, 2036, these same two planets will come to conjunction once again, with Venus passing 4 degrees north of Jupiter.”
    end quote

    WC says..There appears to be a 24 yr cycle in the.. length of day data. with the maximum appearing when this phenomena occurs? and so 12 yr phases of peaks and troughs


    If you can link the planets to LOD frequency you can link to the 60 yr earth climate cycle

    So can this max gravity pull by Venus /Jupiter change the LOD of earth?

  11. oldbrew says:

    RA says: ‘There’s a spike in maximum gravitational attraction once every ~5190 days,’

    Almost 12 Chandler wobbles?

    WC says: ‘Contrary to popular belief, conjunctions between Venus and Jupiter are not rare events; in fact they occur on average about every 13 months.’

    Jupiter-Earth conjunctions occur every 13 months – J-V is near to 8 months. If you put ‘VEJ’ in the ‘search this site’ box there are some relevant posts by Ian Wilson.

  12. Paul Vaughan says:

    “[…] waves that come too close to Phi proportions in the resonant frequency are killed […] Stradivarius was well-known for using golden sections in the design of his violins. […] Standing wave reflection simply cannot be sustained when one of the three dimensions in a container is at or even near a golden ratio to another.”

    Ben Chao: “[…] baffling behavior of the Chandler wobble during ~1925 when it reached a near-zero amplitude and underwent a concurrent large phase jump […] The seemingly peculiar event was simply fortuitous by chance.”

    “simply fortuitous by chance.”

    For naive aggregations, surface temperature reversed correlation with solar activity at the 2 times indicated.

    SCL = solar cycle length
    JEV-J = JEV long cycle ~= 166 years
    6.4 years = polar motion envelope period

  13. NikFromNYC says:

    “Triggered by a moment of insight thirty years earlier, this theory explains how harmonics combine to form coherent geometrical patterns that our auditory system recognizes as simple shapes.”

    This brings to mind one of the most astonishing discoveries of modern times and one of the most neglected mostly due to Drug War red tape: most anybody who smokes the drug DMT (dimethyltryptamine) experiences a 3D “computer graphics”-like animated world of geometric shapes that dynamically and playfully transform along with thought itself, having been described as being linguistic objects by the late popular guru Terrence McKenna ( ). This strongly suggests that thought and consciousness has a geometric order to it at some level, and that these sculptural forms are in fact the brain in the process of thinking but the drug molecule as a probe has connected this inner working to the visual and kinesthetic/tactile system, or merely has removed the normal filtering process that blocks it from perception. Related to this, very strongly too, is the mostly unexplored nature of the geometric forms that mathematical savants all describe as being how they internally multiply and divide incredibly long integers, namely that two forms that represent each number are somehow combined and the resulting shape then represents the answer. Well, what *are* those forms? And where are the R&D guys hiring skilled artisans to sketch out descriptions of them? Where are the brain scanner guys, looking into this great new frontier of consciousness? Nowhere to be found! Instead they hand wave this stuff away as being retinal patterns. But a full room full of playful alien objects, dancing around, connected to the words you are thinking, well, that’s not just buzzy retinal patterns, is it?

  14. oldbrew says:

    @ NikFromNYC

    The IPCC crew must have been smoking the wrong stuff 😉

  15. Roger Andrews says:

    Upthread I posted two graphs showing the gravitational attraction between planets. The first was okay, the second wasn’t (it ignored vector components). So here are two replacement graphs that plot the axial vector components (i.e. those pulling Earth towards or away from the sun) of the gravitational impacts of Venus and Jupiter on Earth.

    The first graph shows the individual JE and VE components. (Note that the planets are lined up on day zero so that Venus will be pulling Earth in one direction and Jupiter in the other.) The JE component is effectively a sine wave with a period of 399 days, equal to the JE synodic period. The VE component shows a series of abrupt troughs, coinciding with the periods of closest approach, with a period of 741 days, equal to the VE synodic period.

    The second graph combines JE and VE. JE generates the regular fluctuations and VE generates the spikes at the bottom and a couple of dips at the top. The sequence of spikes repeats itself once every 5,190 days, or once every 7 VE synodic periods and 13 JE synodic periods. Sorry I can’t make it 8 and 13. 😉

  16. tallbloke says:

    Roger A: Maybe it’ll turn out nearer 9:13 if you use the correct VE synodic period of 583.92 days? 😉

    224.701 * 365.256 / (365.256 – 224.701) = 583.923

    399 * 13 / 583.923 = 8.883

  17. Roger Andrews says:


    Punched in 244.701 instead of 224.701 days for Venus rotational period 😦 Thanks for catching it.

    Anyway, third time lucky:

    There’s the 584 day VE synodic period. Now the second graph shows spikes every 1170 days (2 VE SPs) with a rightward shift of 568 days (close to 1 VE SP) just after 4000 days. There’s no obvious relationship with the JE SP except for a possible spike pattern repetition every 7,590 days (19 JE SPs).

  18. tallbloke says:

    Roger A: Easily done. I’m trying to understand your plots. I can’t see why the VE blue line in the first plot would ever cross the zero line. How would Venus ever be in a position to pull the Earth away from the Sun? I can see that Venus’ pull on the Earth is going to be very small when Venus is on the far side of the Sun from Earth, but it would always be pulling Earth towards the Sun, never away from it.

    It might help if I knew what your Y axis units were.

  19. Roger Andrews says:

    Problem with the sign of the cosine for VE. Fixed:

    Units are m1*m2/r^2 with mass normalized to mass earth = 1 and distance normalized to orbital distance of earth from sun = 1 (angstrom unit).

  20. tallbloke says:

    Angstrom unit? That’s Astronomical Unit. 🙂

  21. tallbloke says:

    Interesting plots. Try doing Venus as the perturbed body. I think you’ll find E&J have similar power pulling V outwards.

  22. Roger Andrews says:

    Working on it. Stand by.

  23. Roger Andrews says:


    Here are the plots with Venus rather than Earth as the “perturbed body”.

    The third plot includes the gravitational effects on Venus of all the planets in the solar system (except Pluto). Adding the other planets doesn’t make much difference.

  24. tallbloke says:

    Roger A: Thanks, very interesting. The third plot clearly shows the effect of the Jupiter-Saturn synodic the low amplitude wave. The Jupiter-Earth spikes approximately 1100 days apart appear to be somewhat synchronised with that J-S synodic cycle too. I wonder how tightly the phases are bound or whether they beat over a longer cycle.

    I’ll run some calcs. Maybe you could run the plot over a longer timespan too? Say 60 yrs.

  25. Roger Andrews says:

    TB: Plots extended out to 30K days (82 years):

    The low amplitude wave you see in the second plot is indeed the ~7250-day JS synodic cycle, but I can’t measure an accurate period from it. The spikes have a period of 1176 days, or 2.013 times the 584-day VE synodic period. (Why 2.013 and not 2.000? I’m pretty sure it’s because of aliasing. You may also notice that the spikes occur in superimposed sequences displaced by about 500 days from each other. I think this is an aliasing problem too. I wrote something about this in a comment on an earlier thread but don’t remember which one.)

    All the other planetary synodic periods, or multiples thereof, are also present in the second graph but they’re too small to see.

    Incidentally, according to my calculations the gravitational attraction between Jupiter and Saturn accounts for over 90% of the total gravitational attraction between all the planets in the Solar System, so if you’re looking for a stand-alone planetary pair JS is it.

  26. tallbloke says:

    Roger A: Many thanks for running the longer series. The average of the length from peak to peak of the triple conjunction spike cycles is 22.03yrs. Very close to the Hale cycle length. But they vary from 7300 to 8800 days, which is around the same degree of variation we see in pairs of solar cycle lengths. I think this is why Ching Cheh Hung’s JEV planetary index stays in phase with the solar cycle reasonably well. Timo Niroma found the colar cycle length clustered around 10.4 and 12 years. The JEV cycle basic period is 10.38yrs, the J-S synodic 19.86yrs and the Jupiter orbital period is 11.86yrs.

    7300days = 19.98yrs = ~Jupiter-Saturn synodic cycle
    8100days (average) = 22.03yrs =~ Hale cycle
    8800days = 2*12.05yrs = ~2 Jupiter orbits

    Not bad from just eyeballing the periods over the limited timeframe on your plot.

    When I took Roy Martin’s model replication of Hung’s index and tweaked it with Svalgaard’s solar windspeed reconstruction, it fitted the phase of the solar cycles very well:

    We would see more or less the same thing with Earth as the perturbed body in your plots. Jupiter and Earth are the planets with strong magnetospheres so I’m going to stick my neck out and say it’s predominantly the J-E-V interaction with the Sun which is modulating solar activity at the Hale solar-magnetic cycle timescale, modulated by The interaction of the gas giant pairs on the Jose cycle timescale as discussed in the McCracken et al thread.

    So we likely have several mechanisms in play here. The JEV system which is modulating solar activity through interaction with the interplanetary magnetic field plus Ian Wilson’s tidal-torquing effect, and the ‘sloshing’ mechanism brought about by the angular momentum modulation effect of the outer planets.

  27. Run them ( graphs)longer than 60 yrs cos’ want to fit in the 60 yr earth climate cycle
    What about longer than 200 yrs to fit in the de Vries as well.
    lets not miss ANYTHING

  28. great stuff Roger.

    profound statement
    “Incidentally, according to my calculations the gravitational attraction between Jupiter and Saturn accounts for over 90% of the total gravitational attraction between all the planets in the Solar System, so if you’re looking for a stand-alone planetary pair JS is it.

    I love the way al these independent investigation brings about similar conclusions regarding the relative importance of certain planets like J, S , V, E

    what about the sun’s role in gravitational attraction?

  29. tallbloke says:

    The Sun’s role is pretty constant, and central. 😉

  30. oldbrew says:

    TB says: ‘The Jupiter-Earth spikes approximately 1100 days apart appear to be somewhat synchronised with that J-S synodic cycle too. I wonder how tightly the phases are bound or whether they beat over a longer cycle.’

    It’s 11 J-S : 200 J-E or 55:1000 as a close approximation.
    There’s one extra J-E every 55 x 7 J-S (i.e. total J-E = 7001).

    TB says: ‘The JEV cycle basic period is 10.38yrs, the J-S synodic 19.86yrs and the Jupiter orbital period is 11.86yrs.’

    7 Jupiter orbits = 8 x 10.379 years (83.033y)
    13 E-V synods = 2 x 10.39 years (20.783y)
    16 J-V synods = 10.382 years
    19 J-E synods = 2 x 10.375 years (20.75y)

  31. Roger Andrews says:


    How much gravitational interaction is there between the planets and the sun? To find out I lined all the planets up with the sun (the zero line), pressed the start button and set them orbiting. Then I calculated the components of gravitational attraction between the planets and the sun along the zero line, and here’s what I got:

    The first graph shows a dominant Jupiter with weak contributions from Venus, Earth and Saturn and close to nothing from the other planets. Summing the components (second graph) gives a waveform that’s dominated by Jupiter and which has a period effectively equal to the period of Jupiter, i.e. 11.86 years.

    Here’s a table of the average strength of the gravitational attractions between the sun and the individual planets, expressed as a percentage relative to the sum of all sun-planetary attractions. Three-quarters of the total is contributed by Jupiter:

    Jupiter: 74.3
    Venus: 9.8
    Saturn: 6.6
    Earth: 6.3
    Mercury: 2.3
    Mars: 0.3
    Uranus: 0.2
    Neptune: 0.1

    Note also that the gravitational attractions between the planets and the sun are something like 25,000 times stronger than the gravitational attractions between the planets.

    Here’s a plot carried out to 1,000 years. There are no obvious longer-period cycles:

    What next?

  32. tallbloke says:

    Roger A: the thing about the Sun is that it’s gravitational force on the planets is more constant than the forces between the planets are. So although it dominates in terms of strength, it doesn’t cause the perturbations the other planets do to each other.

  33. Roger Andrews says:


    Assuming circular and coplanar orbits, which I did for simplicity, the sum of planetary gravitational impacts on the sun is indeed constant.

    However, the planets will exert a tidal influence of the surface of the sun depending on where they are relative to the sun, with the point on the sun’s surface closest to the planet being pulled outwards and the point farthest away being pulled inwards.

    Which means we have to consider the rotational period of the sun (24.47 days at the Equator).

    I start with all the planets lined up (my zero line) and assume that the point on the surface of the sun that this line intersects is the “zero point”. The first graph shows how the gravitational attraction between the sun and Jupiter changes at the zero point as Jupiter orbits the sun and the zero point moves around the surface of the sun. All you see is the rotational period of the sun. You can’t pick out Jupiter’s orbital period.

    But if you assume that the zero point stays in the same place you get the second graph:

    Not sure what to make of this.

  34. Roger Andrews says:


    Re your 5.20 pm comment about the relative gravitational contributions of the sun and the planets, it turns out that the 0.0489 eccentricity in Jupiter’s orbit generates a change in gravitational attraction between Jupiter and the sun that’s over 300 times larger than the changes in gravitational attraction between Jupiter and Saturn during an orbital cycle.

    Will look into this as time permits.

  35. tallbloke says:

    Roger A: Yes, I realised later that the eccentricity of Jupiter would cause a large magnitude change in gravitation over the orbital period (and solar cycle). I also said earlier that in its eccentricity cycle of ~67kyrs it was in a 3:2 with Earth’s eccentricity cycle (and glacial/interglacial periods). It’s also in a 2:3 with Earth’ ~41kyr Obliquity cycle, which was around the periodicity of glacial/interglacial cycles prior to the shift to 100kyr a couple of Myrs ago.


  36. Roger Andrews says:

    Jupiter has an appreciable “tidal” impact on the sun when I allow for orbital eccentricity.

  37. tallbloke says:

    Roger A: Don’t forget tidal forces drop off on an inverse cube law rather than inverse square. Also don’t forget that the force with with Jupiter ‘pulls’ the Sun towards itself is a lot less than the force with which the Sun ‘pulls’ Jupiter. Each body has its own gravitational force proportional to its mass. This is considering matter in the Newtonian framework, no-one knows how gravity actually works.

  38. Roger Andrews says:

    Yes, it’s difficult to model the effects of gravity when you don’t know exactly how gravity works 🙂

    But the “tidal force” is still there with an inverse cube relationship, just at lower amplitude.

    I’m sure this has been discussed before, but the 11.86 year Jupiter “tidal cycle” isn’t that far off the 11-year Schwabe cycle.

    Off traveling for a while. In touch periodically.

  39. tallbloke says:

    Roger A: Yes, and the schwabe cycle is more often around 10.4 or 11.9 yrs than it is around 11.05yrs. Enjoy the traveling.

  40. Brian H says:

    Repeated impulses cause convergence. Somewhere.