Harmonics Theory, Part 12: Really Big Numbers

by Ray Tomes

Harmonics Theory: Really Big Numbers

This blog series from FSC Science Director Ray Tomes shares the fundamentals of physics in layman's terms, showing how present theory must inevitably lead to all waves losing energy and forming harmonically related waves. The end result is a very specific detailed structure that matches the observed universe and explains many previously mysterious observations. This series was previously published.

Really Big Numbers

During March 2004 I decided to extend the harmonics calculations that I had done years before. Computers had got much faster since then, and I put a little effort into speeding up the calculations by taking some additional mathematical shortcuts. The speed of calculation improvement can be broken into three steps, one of which was undertaken years ago in order to even do 15-digit numbers. If I had told my computer to lay out a table like the ones in the chapter "Many Harmonics" twelve years ago, then it still wouldn't have completed the calculations that I did then. So what I did was recognising that numbers with lots of powers of 2 and less each of 3, 5 and 7 were present, I made a different shaped table with just products of powers of these numbers. The first speed up in calculating the number of ways each number can be factorised is to discover this formula:

H(C) = sum H(I) over all I that are factors of H, except H itself

An example is always worth a thousand words, so for example H(12) = H(1) + H(2) + H(3) + H(4) + H(6), which doesn't help us very much until we have calculated the smaller harmonics. We start with H(1)=1, not because 1 has any factors other than itself, but because if we don't, nothing else works. Where harmonic one comes from is not part of the definition, it is an assumption that will no doubt lead to speculations about God and many other things.

We know that H(2) and H(3) are 1 because 2 and 3 are primes. H(4) is 2 because it is made from H(1) and H(2). H(6) is 3 because it gets a share of H(1), H(2) and H(3). So now we can add them up and get the answer H(12)=8.

When this process is done for number to 10^50 it requires 8 nested repeats for the eight primes 2, 3, 5, 7, 11, 13, 17 and 19 just to visit each number once, and another 8 nested repeats within those to add up all the harmonics that are the numbers factors. That adds up to a really big number.

In the latest runs I reduced the calculating time by incorporating two time saving factors. The first is to recognise that 2^8*3^3*5^1 can be factorised in the same number of ways as 2^3*3^8*5^1 and 2^8*3^1*5^3 and so on. So rather than calculate all of these, if the program finds a pattern of prime indices that it has done before in a different order, then it takes a wee peak at the result and pretends it didn't copy off someone else. This doubled the speed of calculation.

The last improvement to speed was to see that when doing C(24) we get H(1) + H(2) + H(3) + H(4) + H(6) + H(8) + H(12) and that H(1) + H(2) + H(3) + H(4) + H(6) is already known and is the same as H(12) because that is how we calculated H(12). So H(24) is just 2*H(12) plus H(8). Now although this type of thing happens incredibly often in the calculations, it is extremely difficult to generalise how to do it. However, I recognised one very common situation, which is when going up by a factor of 2 and just did that one. It improved the speed by a factor of ten and enabled me to calculate to beyond 10^50 which was a desired result.

The output of my 25th March calculation was a 463 MB tabulation. This is too big to share and not a lot of use on its own. So I make available on my website several files in CSV (Comma Separated Values) format which is plain text numbers separated by commas. It can be used as input to spreadsheets by anyone who wants to play around with these numbers and calculations. One file is the mainline harmonics which is quite a small file and another is harmonics selected for being strong in their range which is rather large because otherwise it is not a lot of use.

Here is a graphic showing all the stronger harmonics up to around 10^54. Notice that at the top end the pattern goes a little funny, which is due to incompleteness of my calculations in that vicinity. This graphic and the file are reasonably complete to around 10^51 for even moderately strong harmonics and complete for the strongest ones to 10^53.

The parabolas showing the families of harmonics with ratios 2 are visible throughout the range 1 to 10^53 as are the other common musical relationships. However there are some other things going on too, most noticeably some general wave-ness in the main line harmonics. Given that a detrending formula has been used it is reasonable to ask to what extent this waviness is real and to what extent it is a product of the analysis. The answer is that the broad sweep of the graphs, which includes a slight upturn at the very high end, may be a result of the analysis. However the waviness on the scale of several powers of ten is most definitely a real thing although its exact measurement may be subject to some refinement yet. This is attested to by four different things.

Firstly, the detrending has only two terms which does not allow it to put so many curves in. Secondly, the very high peaks are always associated with places where the one family stands out strongly from any other families which means that the maximum energy flow is in that particular family at the peak. Thirdly, alternative detrending methods that take a less global perspective still show these peaks. Fourthly, and perhaps most importantly, the major peaks occur at the places with are furtherest away from the larger primes in the main line sequence.

This last reason makes sense, because if we are right on a large prime in the main line sequence, then the energy is split equally between two different families of harmonics and so none can stand out from the others. This may be understood in the "Flow of Energy" diagram where for harmonics around the vicinity of 6 and 8 the energy is taking different paths and they come together again at 24 making a big peak.