Harmonics Theory Overview |
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Overview
Key Words: Harmonics, Non-Linear systems, Chaos, Theory of Everything
Preconditions: (1) In any system which has non-linear relationships, harmonics will develop, and for many systems there will be power in all or most harmonics, usually with some law of diminishing power with higher order of harmonics (eg y=exp(x) y=ln(x) y=1/(k+x^2)).
(2) If the system also has some form of resonance, then individual harmonics may be selected by parts of the system, and in turn generate further harmonics, and so on.
Once the above two conditions are satisfied, it is not very important what the nature of the system or the functions are, because the result is almost totally independent of them.
Consider an initial frequency 1 in such a system. It will generate harmonics of frequencies 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, etc.
Now consider each of these frequencies in turn. They will each create harmonics of themselves which will be frequencies of ...
1 --> 1 2 3 4 5 6 7 8 9 10 11 12 ... 2 --> 2 4 6 8 10 12 ... 3 --> 3 6 9 12 ... 4 --> 4 8 12 ... 5 --> 5 10 ... 6 --> 6 12 ... 7 --> 7 ... 8 --> 8 ... 9,10,11,12 ---> 9 10 11 12 ... etc
Now notice that some numbers occur much more often than others:-
Number 1 2 3 4 5 6 7 8 9 10 11 12 ... Occurs 1 2 2 3 2 4 2 4 3 4 2 6 ...
This process can be carried on for after 3 steps and 4 steps and so on. Eventually we find that the total ways of producing each harmonic is given by the number of ways each number can be factorised (+1).
The result looks like this:- Number 1 2 3 4 5 6 7 8 9 10 11 12 ... Factors 1 1 1 2 1 3 1 4 2 3 1 8 ...
The number of ways each number can be factorised is a measure of how much power we can expect to find in that harmonic (after allowing for the general drop-off in power for higher level harmonics). It turns out that when the spectrum of this function is examined (AT ALL SCALES) it produces strong frequencies which have relationships exactly in the proportions of major chords in music, and moderately strong frequencies in exactly the proportion of the musical scale (the old just intonation scale, not the modern equitempered scale). An example is the range of harmonics from 48 to 96 shown below with relative power after allowing for the drop off with higher harmonic number.
I I I
I I I I relative
I I I I I I I I power
I I I I I I I I I I I I I I
I I I I I I II I I I I II I II I I I I I
IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII
48 60 72 96 <--MAJOR CHORD
48 54 60 64 72 80 90 96 <--SCALE white
56 84 black
C D Eb E F G A Bb B C <--scale of C
Exactly these type of relationships of frequency have been found by
myself and others in economic and other time series. Specifically,
the most powerful cycles found have frequency relationships in the
proportion 4:5:6:8 = Major Chord, and less powerful ones with
frequencies matching the white notes plus Bb. Dewey[1] found many
relationships with proportions 2 and 3 in cycle periods starting
from a period of 17.75 years, in an enormous variety of different
time series. His table of periods in years is:-
142.0 213.9 319.5 479.3
-----
71.0 106.5 159.8
-----
35.5 53.3 x2 x3
---- ---- \ /
17.75 \ /
-----
5.92 8.88
---- ---- / \
1.97 2.96 4.44 / \
---- ---- ---- /2 /3
0.66 0.99 1.48 2.22
---- ---- ----
0.22 0.33 0.49 0.74 1.11
---- ---- ---- ----
Underlined figures are commonly occurring cycles. Notice that the
upper right and lower left are not underlined. This is expected from
my theory.
It is possible to calculate out the power in individual harmonics to very high orders using a computer. The reason for doing this is that the observed pattern of harmonics in nature can be traced from many thousands of years down to days and less. It was speculated that the whole pattern might result from just one fundamental frequency, which would be the cycle of the universe. This will be demonstrated to be so, and along the way a whole host of previously mysterious things will be clearly explained.
When the high order harmonics power is calculated in the same method as above, after allowing for the general drop off in power with higher harmonic number, the pattern that is found is crudely represented by the following diagram. The vertical scale is again power and the horizontal still harmonic number, but both now are log scales.
I I I I I
I I I I I I I I I I I
I I I I I I I I I I I I I I I I I I I
I I I I I I I II I I I I I I I I I I I I I
I..I.II.II.I..I.II..II.I..I..II.II..I.II.II.I..I..I.I..I..I.
1 2 34 68 1 2 34 79 1 2 57 11 2 45 81 1 3 6 1 2 4
2 4 68 26 4 8 72 14 8 37 61 7 4 9 0 0 1
4 8 60 54 8 26 45 2 5 1 3 7 4
20 0 00 02 8 6 2 6 3 7
0 0 0 0 8 6 2
0 0 0
The harmonic numbers are written vertically downward so that they can
be fitted in. Note the parabolic shapes of each series 1,2,4,8 then
3,6,12,24,48,96 then 36,72,144,288,576,1152 and so on. This diagram
shows only the very strongest harmonics in any range. An especially
powerful harmonic occurs at 34560 and its relatives. Very powerful
harmonics also occur at 2, 24 and 2880.
I have defined the "Mainline of Harmonics" as those most powerful ones each of which is the previous one times a prime. This gives the series
1 2 4 12 24 48 144 288 1440 2880 5760 17280 34560 69120 .. x2 x2 x3 x2 x2 x3 x2 x5 x2 x2 x3 x2 x2 ..Notice that in the line showing the prime ratios, that 2 is the most common and then 3 with 5 making just one appearance. This is the explanation for the corners in Dewey's diagram being missing. The ratio 2 should occur more often than 3.
I have computed harmonics to over 10^20 and studied the relationship of the power to the prime factorisation. I have determined an empirical formula, but it needs a mathematician to prove it. The formula says that each prime occurs with a relative frequency of 1/(p log(p)) where p is the prime. This gives relative frequencies and spacings (1/freq) for the primes as follows (using 2 as a base).
2 3 5 7 11 13 17 19 23 1.0000 0.4206 0.1723 0.1018 0.0526 0.0416 0.0288 0.0248 0.0192 1.0000 2.3774 5.8048 9.8257 19.0269 24.0529 34.7434 40.3553 52.0210Using this empirical formula it is possible to calculate the harmonics out to as high an order as desired. When this is done, it turns out that there are some additional interesting features to the power vs frequency diagram. We already saw the little parabolas which showed a peak for each power of 3 (when 2 had just the right power to go with this, the harmonic is more powerful), and how an especially powerful harmonic occurred after the first power of 5. It turns out that after every power of 5 there is such a strong harmonic, and they occur at roughly powers of 34560. The first is 34560, the second 34560^2x7/2 and so on. Every so often the other primes must be incorporated.
Now here is the juicy bit. If we plot the observed scales in the universe of the spacings of all the most noticeable types of things guess what happens? Log scale again of course...
Another thing that happens with the primes is that they conspire to all want to come together and make the most powerful harmonics of all in the vicinity of 10^40. Actually I should say all but 13 (and 23). We have by then 19^1 17^1 13^1 11^2 7^4 5^9 3^~21 2^~52 The funny pattern of the primes with 17 and 19 near each other with a big gap below and above (with only 23 between 19 and 29) is the cause of this. For those not aware of it, the number 10^40 is the large number in the large number theories which notes the coincidences:-
It was clear that something was going on here. The answer lies in the harmonics of the universe.
Many tests are possible of this hypothesis, and some of these have been made with generally very good results. I have ignored one or two problem areas, and even more good results in the above. In following papers I will deal with all the different areas of application.
Quite a few existing mysteries will be cleared up, so stay tuned for the next exciting instalment!
[1] Dewey, E R 1967 The Case for Cycles. Cycles 18:186
[2] Castle, D S 1956 Stock Market Cycles Fit the Musical Scale.
Cycles 7:336-337.
[3] Tomes, R J 1990 Towards a Unified Theory of Cycles.
1990 Cycles Conference Proceedings.
(available in proceedings and videotape)
[4] Tomes, R J 1994 The Ultimate Cause of All Cycles.
1994 Cycles Conference.
(available as diagrams and audio tape only)
Above all obtainable from:
Foundation for the Study of Cycles, Inc.
900 West Valley Road, Suite 502
Wayne, Pennsylvania 19087-1821
Tel (610) 995 2120 Fax (610) 995 2130