Hornblower, you have been far too hasty here. My second sentence said "Cardinal, fixed and mutable are the periods shown by dividing the ¼ year sine wave in three." Contrary to your reading, this is entirely compatible with the following statement, "Cardinal is where this sine wave moves from inflection past the positive turning point, fixed where it moves across the downward inflection point, and mutable its rise back to the upward inflection point."

The quarter year sine wave has frequency of three months. Therefore, dividing this wave in three will produce three monthly segments in a recurring pattern exactly as I described it. Looking again at the sine wave overlay, the pink wave has period six months and the blue wave has period four months. I used the six month wave instead of the three month wave in this diagram, but the three month wave has exactly double the frequency of the six month wave and is directly analogous to it. In the six month wave, the points marking entry into cardinal, fixed and mutable periods are the points at which the four month wave has stationary and inflection points. Hence in the six month wave, cardinal is when the wave is moving away from inflection, fixed where it turns, and mutable where it moves towards inflection.

This material is more than a mathematical curiosity because it provides the astrophysics of the astrological signs. Mainstream astronomy assumes that no such astrophysics exists, but my explanation here refutes that assumption.

Your question responded to my statement “the cyclic meaning is real and corresponds to the four quarters of a sine wave function”. Your question of the effect of the elements is a request to prove the existence of the signs of tropical astrology. At #87 I said “The goal is to show that the twelve signs of the zodiac have an empirical effect, ie that there are discernable average differences and similarities between them. Population level epidemiological study of births, deaths and marriages could yet demonstrate the existence of the annual sign cycle. Such findings have so far eluded research, showing only that the signs are too weak, if they exist at all, to be proven. http://www.rudolfhsmit.nl/s-crit2.htm is a good paper from a site critical of astrology that shows how a massive effort to find sign data failed to convince critical statisticians. ... Michel Gauqelin demonstrated the existence of planetary effects but found no evidence for the signs.”

My use of the four elements as names for these periods assumes that the meaning ascribed to these wave functions in tropical astrology has an empirical basis, ie that a 1/3 temporal harmonic of the annual sine wave with frequency four months demarcates periods with elemental difference. However, statistical effort has not yet demonstrated such sign effects. My use of this material here is intended to show that the structure of the Great Year matches this annual cycle. The lack of evidence shows that the claim is empirically weak against current knowledge, but does not falsify the claim. More sophisticated analysis could yet prove that the mathematical analogy between temporal and musical cycles has a physical effect. My efforts here are directed towards putting such analysis on a scientific basis.

Yes, it was intended to be, as it was in response to someone with no physics education telling an ex-professional physicist that he does not understand harmonics.

Do you see what I did there, I mocked your patronising tone with one of my own? It seemed pretty appropriate.

It was not even as though I'd written anything that even implied that I was a bit rusty on the subject.

To get to the point: the tropical year is an obvious cycle with obvious consequences such as the seasons. As with all sine waves it has four obvious "landmarks": the maximum, the minimum, and the two points with value zero.

What is there with a period of four, eight, or sixteen months that would permit a partitioning into one, two or four months from which we could get the cardinalities from? Please, no pretty pictures, just identify some real events that occur with at least one of the indicated periodicities.

And, why subdivide by three? Why not two or four or six, or even five or seven? (Chinese astrology divides by seven to obtain 28 divisions. Can you guess why? Hint: they call these subdivisions "lunar mansions".)

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Sorry for offending you, my comment about understanding harmonics was intended with specific reference to my claim that planetary cycles have complex resonant patterns. It was in response to your comment which seemed to indicate lack of effort to understand the issues at hand. You seem to have engaged with this thread from an arrogant perspective that assumes my claims are rubbish. I am used to that, but on the substance of the argument, I have explained in simple terms how a sine wave resonates with its integer harmonic multiples, and you have asserted that a non-integer division is somehow relevant. To repeat, you may be discussing a completely different field of harmonic theory with your suggestion that a sine wave can be broken into 6.3 natural pieces, but this 6.3 division seems to me to be irrelevant to the question of whether harmonic resonance can be postulated in temporal cycles.

It was still wrong, I understand harmonics very well.

And if you perceive arrogance in people's replies, it is likely because of such foolishness as this;

If you can't even be bothered to understand the differency between frequency and period right, how can you expect to be taken seriously?

I recommend that you spend some time to actually study some physics. It is a fascinating subject, and one that you might enjoy learning about.

This thread is about waves with extremely slow periods/frequencies, eg one cycle per 25765 years. Saying the 1/4 year sine wave function has frequency of three months is a legitimate description of its period, when discussing a wave function with no commonly used denominator, such as the year or the day. I could have said it has frequency of one cycle per three months, but I thought the meaning was clear enough. Thank you for pointing out that period is the correct term.

You are welcome.

When people come up with somewhat "off the wall" ideas, it is these little errors that give the rest of us a clue about how we should pitch our responses. When someone posts a word salad (and I assume that you understand what this phrase means), it helps us to understand if it is because the ideas themselves are muddled, or just because a poster has not communicated the idea well.

With your posts, the fact that you seem to know virtually nothing about actual standard mainstream physics, I am afraid, suggests the former. You seem not to understand people's responses which point out to you that your responses assume your conclusions. You have picked what is an arbitrary division of your arbitrarily defined sine wave, to match the number of cycles that you want for your theory, and have justified it with reference to that very theory.

To put this plainly again, your division of the sine wave into X pieces has so far not been justified by you. Can you please explain why you believe that this split is valid, without assuming the correctness of the theory that you seem to be cooking up?

Robert, a lot of what you say doesn't mean what you really mean, which I think is a major issue in your threads. This isn't the only term that you've misused. Bad terminology makes it hard for us to understand what you're really saying and it seems to indicate that you don't understand what you're saying. Phrases like "the ecliptic a sinusoidal function of [the celestial equator]" are nonsense. The meaning of "ecliptic," "function," and "equator" do not fit that context at all. You should really learn the proper meaning of the terminology before you try to apply the terminology and the concepts the refer to.

You are far too harsh here. The problem is that I am discussing issues which have not been addressed much before. With a wavelength of three months, there is no conceptual problem in treating the period as the frequency, given that this area of study lacks any commonly agreed denominator such as second or minute. The ecliptic is in fact a sinusoidal function of the celestial equator. A map of the sky with the equator as a straight line shows the ecliptic as a sine wave with zero points/equinoxes at the equator.

I disagree with your analysis here. Small departures from convention are rather used as an excuse to mount a fallacious argument to the effect 'here is a questionable use of terminology, therefore the entire theory can be ignored.' Yes, I do have enough knowledge of physics to understand the responses here, but the problem is that this discussion of harmonics emerged from questions about the nature of the tropical signs. If we assume as a hypothesis that the signs exist, then this harmonic theory explains what they are. I have acknowledged that there is no accepted scientific evidence that the signs have any reality, so in that sense this material could be considered an exercise in abstract pure mathematics, built upon the empirical data of the earth's temporal structure. I am however confident that further research will provide empirical correlations. I will respond to your last question together with the previous question from Celestial Mechanic.

Your explanation can be depicted at this diagram of the zodiac sine wave showing the “landmarks” at the equinoxes (the zero points at the celestial equator) and the solstices (maxima and minima). Apologies for including the diagram linked above, but it helps to reference the discussion. The periods of four, eight and sixteen months obviously do not fall on the ‘landmarks’, which are a duple division of the twelve month natural cycle. My claim is that in addition to this duple division, which produces the cardinal points of the solstices and equinoxes, there is also, at least mathematically, a triple division of the annual cycle. The question of the real effects of this postulated triple division can, I argue, be approached through harmonic theory.

Following Kepler, I am postulating that the solar system exhibits a harmony of the spheres. Looking for such harmony of the spheres in the sine wave function of terrestrial time, it seems entirely reasonable to me to explore the analogy of musical harmony, to see if findings in harmonic resonance between musical notes can be informative for the study of time.

In music, the strongest harmonic is duple, producing the octave. The next strongest harmonic is the triple, producing the perfect fifth. Hence the analogy with the tropical year is that the sine waves based on points at three, six and nine months stand in octave relation to the start and end, while the periods at four, eight and sixteen months stand in perfect fifth relation to the annual wave function. By this I mean that imaginary sine waves with 4, 8 and 16 month periods will have frequency in perfect fifth relation to the frequency of the year.

The duple division produces the four cardinal points of the compass, and the other qualities of fixed and mutable. The triple division is responsible for the four elements, which match the four parts of its sine wave. Hence in answer to your question, the four and eight month waves, involving a triple division, have nothing to do with cardinality.

I’m tempted to say cardinality is a function of duplicity. While perhaps technically correct, this terminology may be a less than clear way of putting it.
My proposed sand and water mandala experiment is an easy way to see how this triple wave function, operating on a circle with four cardinal points, produces twelve ‘nodes’ at the sign cusps. The question here is the relative harmonic power of the integer subdivisions of a wave. The division in two has the strongest resonance with the original wave, illustrated by the close harmony between a note and its octave. The division in three is next strongest, illustrated by the perfect fifth, and so on, with each successive integer division, 4, 5, 6, 7, …n, having weaker harmonic resonance.

Now, when all these n hypothetical sine waves are compared, the strongest high n waves are those with the most factors. For example, the sine wave with frequency 12 Hz resonates with waves of frequency 1, 2, 3, 4, and 6, while the waves of 11 or 13 Hz do not resonate strongly with other whole number frequencies. Where the 1 Hz wave is considered as analogous to the year, the 12 Hz wave is analogous to the tropical month, and the 3 Hz wave to the four seasons. In seeking natural temporal divisions of the year, the twelve tropical signs emerge as strong harmonic factors of the annual cycle.This mention of the moon points to a further reason why the one month period is of particular interest, compared to say the 6, 4, 3, or 2 month periods or those of 1/5 or 1/7 of a year. The lunar orbit (29.53 days) is close to 1/12 of a year (30.44 days). There may be entrainment operating between the lunar cycle and the tropical signs, with the moon’s orbit reinforcing the power of the month as a determinant temporal unit for the annual cycle of the earth.

To conclude, I will summarise some main points in this thread.

The stars Canopus and Vega provide useful markers for the precession of the equinox, sitting respectively opposite the current positions of the South Celestial Pole and the North Celestial Pole on the circular paths traced by the Poles as a result of precession. This motif of the polar path matches the interpretation of the Vedic Yuga as a 24,000 year (actually 25765 year) cycle of terrestrial time, indicating that ancient Indian astronomers almost certainly had a roughly accurate model of the apparent movement of the stars due to precession. Of course they lacked a dynamic model, hence my surmise that they erred in seeing these polar markers as a ‘binary’ cause for precession.

This Vedic stellar model of time also appears in Western tradition, especially with the depiction of the holy city in Revelation 21 as having a wall 12000 stadia across, with, by old orthodox story, its twelve foundations seen as the twelve zodiac signs in reverse starting with Pisces. The width of the holy city matches the Vedic theory of the 12000 years from the high point of the Yuga cycle (when Canopus and Vega are near the Poles in about 13000 CE) to its low point around 500 CE when these stars are furthest from the pole. The hopeful theme in this cosmology is that the earth passed the cyclic low point more than a millennium ago and is improving, if slowly.

The model of the earth I presented shows the empirical structure of precession, with the addition of interpretive themes derived from astrology. I have sought to justify this interpretation as empirically possible, suggesting a complex emergent resonance between periodic cycles, beginning with the daily and annual rhythm of the earth and extending to harmonic cycles of the Great Year. Respondents have focussed on the physics of this wave theory, arguing that periods of time do not exhibit resonant interaction analogous to musical harmonics in the way described.

I acknowledge there is little empirical evidence for the claim that the earth follows such complex resonant temporal cycles, recognising that such evidence would constitute partial proof for astrology. Without such evidence it will be hard to make this analysis persuasive, even though it is mathematically and physically possible. However, I have been able to provide mathematical explanation of the nature of the signs in terms of the elements and qualities of traditional tropical astrology, showing that these old ideas have a coherent empirical derivation. Discussion of the four traditional elements is constrained by the assumption in modern science that these ideas sought the same status as atomic elements, even though these old elements are primarily cyclic in nature, and can indeed be understood as the four parts of a sine wave. I have argued this cyclic pulse can be seen as like a second temporal dimension alongside duration, providing the basis for what Kepler understood as the music of the spheres.

A key idea for this overall claim is that the solar system is a unit with its rhythms modulated by the planet Neptune. This hypothesis emerges from the observation of the alignment between Neptune, Saturn and Jupiter in their 179 year conjunction cycle, which is 1/144th of the Great Year, and which also creates the main permanent wave structure of the solar system centre of mass, a wave structure exhibited in micron tides on earth. These three gas giants meet in overlapping conjunction families, with each family having twelve occurrences in a Zodiacal Age, each occurrence separated by 30° of arc, one in each of the twelve signs of the zodiac. In the biblical motif of the holy city, this JSN cycle matches the unit of the cubit.

A wealth of ancient astronomical teaching is encoded in mythology. This model of time provides a prism to interpret texts from a wide range of sources, from Jason and the Argonauts to the Epic of Gilgamesh, the Finnish Kalevala, the Vedas, the Bible, and Egyptian and Norse mythology. For example, some astronomers are familiar with the reductio ad absurdum joke that the universe is turtles all the way down. I have discovered by making this model that the Large Magellanic Cloud, located at the centre of earth’s southern spin cone, is a very good candidate for the myth of the turtle at the bottom of the universe. Similarly, as explained in my related thread on Argo, the star Canopus has an intriguing place in mythology, usurped as the star of Osiris by Orion in European interpretation of Egyptian myth. The invisibility of Canopus from Europe can be traced in mythology, with Argo losing its prow ornament (ie Canopus) on passage through the Bosphorus in the myth of the Golden Fleece.

I apologise that I have not found time to answer all questions within the thirty day limit, and also that my expression is often too condensed. The discussion of harmonic theory was a legitimate scientific point of engagement, and may prove a useful theme for further future discussion.

Bold mine.

The biggest problem with your idea is that it only applies to Earth. If this were truly connected to a universal truth of reality, it would have to apply everywhere in the universe.

The reason that I bolded the "wavelength of three months" part is to ask, "Who's three months are we talking about?" Keeping it just in our solar system, we have nine to choose from. This is not an unreasonable question, as the zodiac is established on the celestial sphere and thus, if we were to accept your ideas, applicable to all bodies within that sphere. By claiming that only the three month period of Earth is valid puts Earth in a unique place in the universe, which current mainstream physics rejects. Next, consider Mercury. Since we define a year as one revolution around our sun and a day as one rotation about an axis of spin, then Mercury has a day that is longer than it's year. What does that do to your three month cycle?

Since the zodiac is established with the Earth as it's center, then how does your idea transfer to, say, the Andromeda Galaxy? Or the Large or Small Magellanic Clouds for that matter?

No, for your idea to be truly a part of how reality works then it must be applicable everywhere and it cannot single out one pale blue dot in the whole of the universe.

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Thank you AstroRockHunter. The validity of the three month period only applies for earth. The three month period (eg from equinox to solstice) is a specific function of our planetary rhythm, part of the structure of terrestrial temporality in its long term stable geocentric cycles. All parts of the universe have complex pulse structures, but each place is unique in the wave functions of its temporality.