The trouble you are pointing out suggests the theory of planetary aspects such as trine and square is without basis. Of course this is part of the notorious scandal of astrology, that it has built a large industry on foundations which to date lack any systemic evidentiary proof, and which are often in flat contradiction to each other and to existing scientific knowledge. Science concludes from this scandal that astrology is obsolete. Phil Plait and all who oppose bad astronomy are quite right to poke fun at this industry, and even express alarm that it diverts attention from reality, while it fails to reconcile its claims with physics. I am suggesting however, that these effects, if they exist at all, such as the four monthly rhythm you question, are so weak that previous methods used to detect them have been inadequate. I can see why superficially you would view any effort to apply mathematics to this problem as tendentious numerology, but I argue there is room to apply scientific method, both in the mathematics of the harmonic theory I have presented here, and in empirical analysis of data.

New statistical approaches are needed to assess these claims, including ways to present data to look for periodic cycles. At http://www.bautforum.com/attachments...ndar-dates.gif I show the monthly cycles for 2007 in an empirical ephemeral format, with the sun passing through one row per month and all planets shown with actual perceived path over the course of the year. Any four-monthly rhythm would produce a ‘topology’ of this empirical diagram. Using any large data set of natural events, such as rain, plant growth or hospital records, a test for a four monthly cycle would require aggregation of data by element. By this I mean that over a long period of time the total data for events occurring during each of the fire, earth, air and water elemental group periods (if I may retain these objectionable labels for convenience) would be tested for statistical significance. I do not know if this has ever been done, but the small study I presented at Planets and Rain discusses suitable methods.

You are right to note that my recent post did not make clear the reason to apply the quarter and third division to the different fractional sine waves. I have claimed in this thread that the sign cusps are the trines of equinoxes and solstices. Therefore the twelve-fold cycle of the signs is derived separately from these sine waves as presented. My purpose in presenting them was to look at what is happening to these fractional waves against the sign cycle. The answer is that for the ¼ sine wave there is a regular 1/3 pattern corresponding to the qualities, while for the 1/3 sine wave there is a regular pattern corresponding to the elements. This is physics, not mysticism.