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| Patterns in the Harmonics |
Foreach harmonic H, the number of ways that it can be formed is called C(H). WhenC(H) is calculated out for higher numbers some very clear patterns emerge. Forthe patterns to emerge it is first necessary to see that there are two effectsgoing on that cause things to race off in opposite directions. By balancingthese two things we can see the beautiful music of the universe.
Thefirst effect has been mentioned for non-linear systems, and that is that energyis lost to harmonics, mainly the low order harmonics. Such a pattern is shownin the figure below. In the theory we assume only that the power in eachharmonic drops off as some inverse power of the harmonic number. It might be1/H or 1/H^2 or 1/H^3 or even a non-integer number. In every case the result ofgoing to higher order harmonics is a decrease in relative energy. But thisrecognises only one step in the process of making harmonics.
Thesecond effect is that numbers that are able to be factorised in a largernumbers of ways tend to be higher numbers rather than lower ones. This effectis in the opposite direction to the other effect but does not exactly cancel itout. Not all larger numbers have many ways of being factorised, but the largerthe number the greater the possibility as shown in the figure below.
Whatwe are really interested is, in any range what are the strongest harmonicswhich means the ones that can be factorised in the greatest number of ways forthe size of the numbers. These will be the waves with the most energy for theirsize and which will presumably show up to our senses or on scientificinstruments in preference to their weaker neighbors. The above two effects arecombined below for the first 100 harmonics by showing C(H)/H as a function ofH.
Although1/H works well in this range for detrending the rapidly rising C(H), it doesnot work so well for much larger H and C(H). This issue will be returned tolater. For now, let us observe some of the patterns that are present in theharmonics from 1 to 100.
Firstlywe see that multiples of 12 make stronger harmonics than other numbers and thatvery strong harmonics always have other strong harmonics at ratios of 2 aboveand below. By this is meant that if H is very strong then H/2 and H*2 will alsobe strong.
Nextwe may notice that there is a very strong musical pattern present. Inparticular, looking at the interval from H=48 to H=96 the very strongestharmonics are 48-60-72-96 which is a major chord in music (usually known as theratios 4:5:6:8 resulting when cancelled by factors of 12). In addition thenotes of the just intonation music scale are exactly represented in theharmonics 48-54-60-64-72-80-90-96. Additionally two extra notes labelled hereas Mi-flat and Ti-flat are quite strong. These are needed to make dominantseventh chords.
Thereason that music and the strong harmonics are so closely related is that bothare representations of frequencies which have as many small number ratiospresent as possible. As we look more deeply into these relationships we willsee why past great scientists have referred to the music of the spheres. Theunderlying energetic frequencies of the universe have relationships that areeverywhere musical, although as we go to higher numbers various different typesof music are found and some of it would be rather foreign to our ears.
| Outstanding Harmonics |
Tofurther delve into the pattern of the harmonics it is necessary to solve theproblem of detrending the upward sweep of C(H) which grows ever faster withincreasing H. For example, the harmonic 34560 has no less than 622592 ways ofbeing factorised, so if it were placed on the graph of C(H)/H above it would beat 18.01... which is way out the top and past your ceiling. And further upthere are numbers with 36 digits that have a number of ways of being factorisedthat has 57 digits so it would be more than out of the graph, it would be outpast Sirius!
Imight add as an aside that it takes a bit of computer time to calculate theseharmonics. When I was doing this in the early days of the Harmonics theory Ihad this new fast computer with an 80286 chip in it that couldcalculate harmonics up to 15 digits or so if left alone for a few days. Withthe advance in computer speed and a few programming and mathematical tricks itis now possible to calculate beyond 50 digit harmonics. It doesn't do allof them, only the strong ones and the other ones necessary to calculate thestrong ones.
It hastaken some considerable effort to find a formula that detrends this sweepinggrowth. After using a variety of techniques to separate the different parts ofwhat is going on, in March 2004 a formula that fitted the trend in thestrongest harmonics was found. It works reasonably well up to fifty digitharmonics but might require a little tinkering beyond there. The trend in C(H)for the outstanding harmonics is estimated as Cest(H) =(H^1.7323)/(10.18^sqrt(log10(H)). The following graphs have been detrended bydividing C(H) by this Cest(H). This leaves two other parts of the variations inC(H) untouched. They are the variations in C over several octaves of H, and thedetailed patterns within the octaves.
Theresult of this detrending is that we are free to look for the effects that arevisible to us in the energy patterns of the universe because we will becomparing the energy at similar scales and not trying to compare galaxies toatoms which is beyond the present scope.
Oneother thing will be done to allow the musical pattern to be more easily seen,and that is to use a log scale for the harmonic number H, meaning that eachdoubling of H will take the same horizontal space. This will also avoid thegraph going past Sirius in the horizontal direction. A doubling of H is what iscalled an octave in music, or going up from one note such as C to the next C.Now let's look at our harmonics in terms of whether they are outstandingharmonics in their range for harmonics from one to a million on a log scale.
It isclear that as the harmonic number gets higher the complexity increases. Thenumber of weak harmonics grows very rapidly and although the strong harmonicsstand out very much from these weak ones they are much further apart. There isless complexity among the strongest harmonics, with ratios of 2, 3, 4, 6, 12and 3/2, 4/3, 5/4, 9/8 and others being frequently present between strongharmonics. These are the musical ratios. This diagram is nothing more or lessthan an enormous chord spanning some 20 octaves and including within it all thecommonly known musical chords and scales. This aspect will be examined indetail later.
| Main Line Harmonics |
Theterm "main line harmonics" is used for the harmonics shown in largerfont in the above diagram. They are the series of harmonics starting from 1that are always related to each other by a prime number ratio and which are thevery strongest harmonics. There are some marginal cases to this definition aswe might have chosen either 5760 or 8640 to lie between 2880 and 17280. Eitherway the ratios would be the primes 2, 3 or 3, 2 but 8640 was chosen as being alittle stronger than 5760. Such a decision might be influenced by the formulafor detrending the harmonics.
Thepurpose in highlighting these main line harmonics and the prime ratios betweenthem is that the prime ratios contain a pattern which is almost regular. Theearly main line harmonics are 1, 2, 4, 12, 24, 48, 144, 288, 1440, 2880, 8640,17280, 34560, 69120, 207360, 414720, ... and these have ratios between them of2, 2, 3, 2, 2, 3, 2, 5, 2, 3, 2, 2, 2, 3, 2, ... and so on. It can be seen thatthere is one ratio of 3 for every two or three ratios of 2 and that the otherprimes occur much less often.
At onetime I conjectured that the primes occurred with inverse frequency p*log(p) andbecause it is difficult to calculate very high harmonics this conjecture wasused as a substitute for the actual calculations beyond the known limits.However with increasing computer speed and a few mathematical tricks it hasbeen possible to calculate the harmonics out beyond 10^50 now and to find thatthe conjecture is not correct and gets gradually less accurate at higher orderas the larger primes get further apart than expected. If this conjecture werecorrect, then a three would occur for every 2.38 twos, a five would occur forevery 5.8 twos and a seven for every 9.8 twos. On that basis a 7 should be dueto happen next and it does.
Theseproportions of 2s, 3s, 5s and 7s are quite consistent with musical practice.Pythagoras only used 2s and 3s in his musical scale, however Galilei, thefather of the famous Galileo, showed that a major third is really 5/4 and not81/64 as Pythagoras had assumed. Actually both of these ratios do occur but itis most often Galilei's ratio that happens in music. It has been expressed, forexample by the mathematician Euler, that the ratio 7 is a bit too harsh to beused in musical ratios. However it is used in Indian music and in Blues musicand should definitely be considered as a valid musical ratio, Part of thereason that it is not used more is not that it is naturally harsh but that itdoes not fit in with the present music notation which really is biased to theequitempered scale which only deals comfortably with ratios involving 2, 3 and5. The proportional frequency of occurrence of 2, 3, 5 and 7 according to theharmonics theory gives a meaningful answer to the question that Euler askedconcerning the concordance of sounds, a subject which has been studied by musictheoreticians since Euler's time.
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