Build the Solar System with the golden Section and music intervals

I can hardly believe it! but having spent over 20 years playing around with these numbers I have not only got Phi values but also Pi! Now I wasn`t looking for Pi but surely this is too good not to be true?

Panning for gold or a process of distillation?

H.H. Voigt in his book breakdown of astronomy states that Copernicus showed that the earth and the planets circle the Sun, Kepler showed how the planets circle the Sun and Newton showed why the planets circle the Sun. I am just a wannabe Kepler in search of Keplers 4th law of planetary motion.As to why my equations should mean anything I have`nt a clue! I will at some point need to find  a far smarter poor mans Newton to work out why! Untill then I am quite content to play around with my pocket calculater searching for equations that I, at least find beutifull.                                                                                                To begin with it was like panning for gold, just trying to find rough equations that I would soon abandon in favour of better ones applying the simple logic of accuracy on the one hand against elegance on the other. In constructing a model B, it seems this  is more a process of distillation, such that I won`t claim that my answers are right, more that I have come up with some sort of raw material that needs further destillation to find any kind of truth.

N.A.S.A. has calculated planetary distances for two different time periods.I will be using the 3000 BC to 3000 AD values unless otherwise stated.

Here below are the Mars and Venus values that I will be using in my model B.

The picture below is a simple illustration of one astronomical unit, from which all the N.A.S.A. numbers come from and the pentagram, the most obvious  depiction of Phi, the golden section.

It had always annoyed me that my Mars and Venus Phi relationship was based on the music interval 1.875 and not 1.5 and 1.25, because it is these two intervals along with 2 the octave that all other music intervals derive.Splitting my equation into three levels though, clearly shows that I am indeed using 1.5 and 1.25.The badly drawn picture below illustates what I mean and I have squared both factors to show there are 3 levels.

 

Although all of the above equations could be seen as pure coincidence, I find it remarkable that in correcting the music interval power equation to equal Phi squarerd then the resulting changes improve the 2nd equation!

So without any changes the discrepancy was 1.000088909% and after the changes 1.00001185% this is amost a decimal place better.It might be fifty fifty as to it being higher or lower but being more accurate, on a discrepancy so low is something else. The 76th root of the discrepancy would correct the equation.

Below is a picture showing a connection between Phi and musiic intervals, 3.2 devided by 2 is 1.6

Here is another coincidence, which I can't find a use for but it is worth mentioning

Well 1.2 the minor 3rd, is by no means an obscure interval and I could use these values in model B or another model. The problem here is that I would have to use the whole thing as a power which would be even more obscure and complicated

The following equations show how I have defined the value for Jupiter

Here is perhaps a better example of the harmonius numbers in this relationship, this is compatable with model B and it is clear to see how easy it is to factor down.

Here is an equation with a natural number sequence for the powers.

I have decided to ajust my model B  values to include this equation

Here is the equation that defines Saturn.

Model B                                                                         

So here are the values for my model B whereby I have gone for accuracy over elegance. I have alterd the values of Saturn, Uranus and Neptune sinnce I found better equations.

Mercury

This definition of Mercury may prove better than I first thought.

I am unsure as to how important these natural number sequences are but here is one with the first four planets (The earth is one!) and it is compatable with model B. I will in future refer to these four equations as my Rocky Planet equations.

So what am I to understand with these chance number sequences? in some ways this is the best pattern I ever found! Could one say that by using the the two inner planets  I have discoverd the first of the outer planets? and perhaps using  Mercury and Venus I could find the next planet Jupiter in a similar manner? Or should I understand that I have found Mercury with the use of Mars and Venus the two planets between which Kepler placed the icosahedron and the dodacahedron? Well this is what I am working on just now!                               

It seems almost unfair that in my model B it is the female planet Venus that carries most of the discrepancy but even if I start with the real value of Venus, then Mars comes out at 1.523728738 or 1.000010703% away from the Nasa value and Mercury with .387097415  which is 1.000002622%. So even choosing the least favourable position to start with, Venus allows Mars a discrepancy of a 93000th part and Mercury with a 381000th part error.

As an after thought, there is a special quality to a Phi sequence such as those above with Mercury but not in the world of devision and multiplication but in the exotic world of addition and subtration. Phi power 1 + Phi power 2 equals Phi power 3 and Phi power 2 plus Phi power 3 equals Phi power 4  this is of course true for all Phi powers. Here is a picture to illustrate what I mean.                                                                 

The law of unintended consequences

It is of course inevitable that in constructing  equations that I find elegant, I will also end up with equations that I do not find elegant and it is only fair to point out some of them! 

Another question is, of course, why am I playing around with music intervals when I have found a Phi value without them? The music intervals are only there to spot patterns that otherwise could or would remain undiscoverd. One may also ask why am I looking for Phi powers? Because of their natural beauty! Keper described the golden section as a precious jewel and I wonder if he meant a diamond, which of course can be used as a cutting tool. It could be the case that, in my speculations, Phi is not so much the answer but more the cutting tool to find the answer! My model B may be perhaps a house and the Phi powers more the scaffolding. Or perhaps another  analagy may be the search for a geometric shape that is only possible with the structure of graph paper. Here is the relationship between Jupiter Mars and Mercury expressed in three equations.   

All three equations are compatable with Model B.The first with music intervals and Phi powers, the second with Phi powers but without music intervals and the third without either. The problem is where is the elegance? 

The above equation is the best Jupiter Mercury relationship I have found up to now, compatible with Model B, maybe there is something more elegant! 

The Reunion of Broken Parts

The Arabic word Algebra, when first translated into Latin, was defind as the reunion of broken parts and I find it nice to think of my Mars and Venus 1.875 equation of two broken parts of Phi squared! The question remains however as to could my Mars and Venus values on their own, each have a Phi value as in the first part of my blog whereby I found the 52nd root of Phi power 35 as an approximation the Venus value? Well to my suprise it turns out that my Venus value in Model B is the 1649th of Phi power 1110.This is by no means elegant but I am glad that it breaks down to two rational numbers! Mars it turns out is 5 times 1649 and thus the 8245th root of Phi power 7216.

A photo of myself at age about 30 from about 1984