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| Families of Harmonics |
When several main line harmonics in a row are at ratios of 2 relative to each other,those families of harmonics do not stop just because they are not in the mainline. For example, looking at 144 and 288 in the main line, we can trace asmooth curve back through 72, 36, 18 and 9 and forwards through 576, 1152, 2304and further on into the weaker harmonics jungle. These harmonics that are linked by ratios of two are called families of harmonics and they can help usto see the fascinating structure present. In music all of the notes linked byratios of two are given the same name such as C, although they may be calledC4, C5 and C6 so that we know which C is meant. The families are just like themusical notes, and adjacent families, which are different by only a ratio of 3,are like the most common musical modulations of adding or taking away a sharpor flat in music.
The first family goes 1, 2, 4, 8, 16, ... or just the powers of 2. Initially theseare the strongest harmonics but at 8 the harmonics dip below the next family,their cousins, 3, 6, 12, 24, 48, 96, 192, ... which are all powers of two times3. Again these are the strongest harmonics from 12 to 48 but then 96 fallsbelow the line of the next family. This process keeps repeating with families that are powers of two times 3, 3^2, 3^3, 3^4 and so on. It can be seen that thepeaks of these coloured parabolas occur between 12 and 24 then 144 and 288 andso on. This supports the view that a ratio of 3 happens for every 2.38 ratiosof 2 because that also would fall between those same numbers.
However something else happens to our coloured lines as we move from left to right, thepeaks of these parabolas are themselves making a parabola and as a result theystop being the highest harmonics of all. A new unmarked family of 720, 1440,2880, 5760, ... takes over the lead. This family is of the form 2^n*3^2*5 andsecond cousin 5 has joined the fray.
The curved lines are getting a bit more crowded now, so the new families, which allhave a factor of 5, have been added in darker colours. These new families againeach make parabolas and the cousin families again extend this with a parabolaof parabolas. At the right edge of the graph some of the strong harmonics arenot members of any of the coloured families, but belong to a new cousinsinvolving the ratio 7.
It is possible to make parabolas through harmonics that are linked by ratios of 3rather than 2 also. An example would be 16, 48, 144, 432, ... and many others.Such groups fall away a little faster in each direction than the families basedon ratios of 2. If we do the same thing with ratios of 5 or 7 then they fallaway faster still. That is why repeated ratios of 2 and 3 happen in music but not with the higher primes.
| Flow of Energy |
These patterns help to see what is going on as regards how strong or energetic eachharmonic is and its relation to others, but they do not make clear why ithappens. For an understanding of why this happens it is helpful to look at theflow of energy between harmonics as described in the harmonics axiom. Eachharmonic loses energy to exact multiples of that harmonic.
The following diagram does not attempt to be an exact representation or to showevery harmonic and every flow of energy, just the flows to multiples, 2, 3, 4,and 6. Remember that more energy flows to the lower harmonics, especially 2times. The thickness of the blue lines is representative of the amount ofenergy flowing. Those harmonics with little input obviously must have corresponding output.
The diagram does show that even though energy is mostly lost from harmonic 1 alongthe first line and only a small proportion goes downwards, eventually thatproportion accumulates because it never comes back upwards. So by the time weget to 8 the next line at 24 is already stronger. Further along the leakage hasmade the third line predominate and so on. The green circles show the strongestharmonics which are called the main line.
Remember that some energy is also going to ratios other than 2, 3, 4 and 6 as shownhere. Some of these ratios are within the diagram such as 8, 12, 16, 24, 9 ...but the direction of travel of energy is always from left to right and top tobottom. The extra flows not shown include 5, 10, 15 and others with the ratio 5.To include these in the diagram we would need to include a third dimension,putting another layer below the first with harmonics 5 times higher and anotherwith 25 times higher. Such a diagram would then show why eventually thefamilies related by ratios of 3 also fall away, because the family with oneratio of 5 then takes over in the main line.
Between these different ways of looking at things it should now be clear that althoughthe initial assumptions of the harmonics theory are extremely simple, the patternof energy made by harmonic number, or frequency, is quite elaborate. Thequestion to be ultimately asked is whether that elaborate pattern bears anyrelationship to the real world around us. If it doesn't then this exercise would still be interesting in terms of mathematics and music but would not haveserved its purpose which is to explain the structural scales at which theuniverse manifests energy.
Before looking at these relationships to the universe it is necessary to look atextremely large harmonics, going not just up to a million but through more than170 octaves to numbers with over fifty digits in them. This is required becausethe scales of the universe that have been observed range from tens of billionslight years down to around 10^-17 meters which is below the scale of naturally(at least in our little corner of the universe) occurring sub-atomic particles.The ratio between these extremes is about 10^43 and we cannot just assume thatthe largest scale so far observed is the size of the universe, because thedistance that we have observed to keeps on going up over time.
| Really Big Numbers |
During March 2004 I decided to extend the harmonics calculations which I had doneyears before. Computers had got much faster since then and I put a littleeffort into speeding up the calculations by taking some additional mathematicalshortcuts. 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. WhatI did was recognizing 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 ofpowers of these numbers. The first speed up in calculating the number of ways each number can be factorized 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 thesmaller harmonics. We start with H(1)=1 not because 1 has any factors otherthan itself, but because if we don't nothing else works. Where harmonic onecomes 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 itis made from H(1) and H(2). H(6) is 3 because it gets a share of H(1), H(2) andH(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 theeight primes 2, 3, 5, 7, 11, 13, 17 and 19 just to visit each number once, andanother 8 nested repeats within those to add up all the harmonics that are thenumbers factors. That adds up to a really big number.
In the latest runs I reduced the calculating time by incorporating two time savingfactors. The first is to recognize that 2^8*3^3*5^1 can be factorized in thesame 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 ithas done before in a different order then it takes a wee peak at the result andpretends it didn't copy off someone else. This doubled the speed ofcalculation.
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) isalready 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 happensincredibly often in the calculations, it is extremely difficult to generalizehow to do it. However I recognized one very common situation which is whengoing up by a factor of 2 and just did that one. It improved the speed by afactor of ten and enabled me to calculate to beyond 10^50 which was a desired result.
Here is a graphic showing all the stronger harmonics up to around 10^54. Notice that at the topend the pattern goes a little funny, which is due to incompleteness of mycalculations in that vicinity. This graphic and the file are reasonablycomplete to around 10^51 for even moderately strong harmonics and complete forthe strongest ones to 10^53.
The parabolas showing the families of harmonics with ratios 2 are visiblethroughout 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 generalwaviness in the main line harmonics. Given that a detrending formula has beenused it is reasonable to ask to what extent this waviness is real and to whatextent it is a product of the analysis. The answer is that the broad sweep ofthe graphs, which includes a slight upturn at the very high end, may be aresult of the analysis. However the waviness on the scale of several powers often is most definitely a real thing although its exact measurement may besubject 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 curvesin. Secondly, the very high peaks are always associated with places where theone family stands out strongly from any other families which means that themaximum energy flow is in that particular family at the peak. Thirdly, alternativedetrending methods that take a less global perspective still show these peaks.Fourthly, and perhaps most importantly, the major peaks occur at the placeswith 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 mainline sequence, then the energy is split equally between two different familiesof harmonics and so none can stand out from the others. This may be understoodin the "Flow of Energy" diagram where for harmonics around thevicinity of 6 and 8 the energy is taking different paths and they come together again at 24 making a big peak.
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