The following is text has been produced from conversations that took
place in the Alexandria City in August 1996. To participate in the city
of Alexandria just send an email to "majordomo@world.std.com" with a body
of "subscribe alexandria" and be prepared to receive lots of email. The
discussion group is based on the city of Alexandria and so is a centre
for anything relating to philosophy, history, music, mathematics or cosmology
with particular reference to Greek knowledge. The main participants in
this particular discussion were the listmistress of Alexandria, Cynndara
Morgan This first page contains my presentation and the following 3 pages have
the discussion.
Ray Tomes wrote:
The octave was found to be a 1:2 ratio and what we today call a fifth
to be a 2:3 ratio. Pythagoras concluded that all the notes could be produced
by these two ratios as (3/2)*(3/2)*(1/2) gave 9/8 which is a second and
so on.
The problem was that after applying these ratios repeatedly he was able
to move through the whole scale and end up back where he started... except
that it missed by a bit, called the Pythagorean comma. After twelve movements
by a fifth (and adjusting down an octave as required) he got back to the
same note but it had a frequency of 3^12 / 2^19 [Note ^ means to the power
of] which is 1.36% higher in frequency than it should be.
Although Pythagoras did a wonderful job he did get it slightly wrong.
The correct solution was worked out by Galilei (the father of the famous
Galileo Galilei) who concluded that the best frequencies were in the proportions
It is interesting to look at the ratios between the notes. do-mi-so
are 24-30-36 which can cancel to 4:5:6. This same proportion links the
notes fa-la-do which are 32-40-48 cancelling to 4:5:6. Again, so-ti-re
(re from the next octave) gives 36-45-54 which cancels to 4:5:6 again.
So every note is linked to "do" by three major chords which have ratios
of 4:5:6.
However when music contains modulations, that is, changes of key, then
some of the notes need to change frequency. As many instruments cannot
do this it was necessary to make a compromise. Many systems were developed
for this compromise and it is called temperament.
Instruments such as pianos, guitars and trumpets have fixed frequencies
while violins and the human voice can vary to any note required.
An example of a chord which requires a change is re-la which have 27-40
above. This needs to change to the ratio 2:3 so either the 27 must become
26+(2/3) or the 40 must become 40+(1/2). Human voices and string quartets
do this adjustment automatically because they listen for the harmony. Guitars
and pianos just cannot do it hence the compromise.
Bach popularised a system called "equitempered" which is used almost
exclusively today. It is a compromise between all keys and uses a common
ratio between every semitone of 2^(1/12). This gives frequencies of:
Pythagoras and his followers and later Kepler were to consider that
these musical relations or harmonies had wider application in the universe.
This idea was almost forgotten or dismissed for many centuries. However
I will hope to show you that there is much evidence that the universe is
completely organised on a system of mathematical harmony and that it shows
up in every branch of scientific study.
After a while I noticed that the periods that I was using were all very
near exact fractions of 35.6 years. Also, other cycles existed at other
fractions of this period such as ~12 years and a fraction under 4 years.
The literature showed that there were other shorter cycles known as well
as longer ones. I acquired some weekly data to look for shorter cycles
and found that there were similar patterns at shorter periods and that
often they had proportions of 2 and 3 in them.
Then it struck me. These fractions of 35.6 years were in fact frequencies
of 4:5:6:8 which is exactly a major chord in music. Also, the shorter cycles
turned out to be exactly in the proportions of the just intonation musical
scale plus a couple of back notes (E flat and B flat if we are in the key
of C).
35.6/8=4.45 35.6/6=5.93 35.6/5=7.12 35.6/8=8.9 years
I realised that the Kondratieff cycle of about 54 years also fitted
in that 2*54 is very near to 3*35.6.
There was of course the question "Why 35.6 years?" and the answer almost
surely had something to do with causes from beyond the earth. For Jupiter's
orbital period is 11.86 years which is very close to 35.6/3 and the node
of the moons orbit takes 8.85 years to travel once around the earth. There
are other astronomical periods which fit also.
This was very weird and for some time I didn't tell anyone because I
was sure they would think I was weird. However, I heard about a place called
the Foundation for the Study of Cycles in the late 1980s and visited there
in 1989.
Edward Dewey had formed the Foundation in about 1940 and had unfortunately
died before I got there. He had left behind an enormous legacy of research
into cycles. In one of his articles I was to find the following diagram.
Dewey 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:-
Interestingly Dewey, using data from different countries, different
time periods and different fields of study had arrived at a table which
included a very good match to my figures. There was 35.5 years looking
at me along with 4.44, 5.92 and 8.88 years. Although this table didn't
show 7.12 years, his catalogue of reported cycles showed a clear concentration
of reports at this figure.
The above table shows several of the periods, such as 142, 53.3 and
17.75, 5.93 years, similar to those found by Chizhevski in the cycles of
war, namely 143, 53, 17.7, 6.0 years. However it doesn't show the 11 and
22 year cycles and some others. To find these it is necessary to introduce
a ratio of 5 just as was done by Galilei to Pythagoras' music scale. For
22.2 years is 5 times 4.44 and 11.1 is 5 times 2.22 years. When the above
periods are multiplied by 5 they also produce many other commonly reported
cycles such as 178 years which is found in the alignment of the outer planets,
in solar activity and in climatic variations.
It is worth mentioning that these cycles have been found in every aspect
affecting life on earth. Wars, economic fluctuations, births and deaths,
climate, geophysics, animal populations, social variables, stock and commodity
prices. We literally live inside a giant musical instrument which is playing
notes, chords and scales in such slow motion that only the Gods could hear
it.
Dewey wrote a very touching piece late in his life where he likened
himself to Tycho Brahe who gathered and catalogued the information about
the planetary motions. He said that he had so wanted to solve the riddle
but was then very old and knew that he was leaving it for some later Kepler
to explain.
In my next post I will stake my claim to being the Kepler or the Newton
of Cycles and you can be the judge.
The musical relationships are characterised by the quality that there
are as many small number ratios between the frequencies of notes as possible.
This indicated that the cause of it all was related to the formation of
harmonics. The word "harmonics" has a slightly narrower meaning in physics
than in music and means "frequencies which are a multiple of some fundamental
frequency".
It is well known in mathematics/physics that a non-linear system will
develop harmonics. Non-linear simply means "not exactly proportional".
For example gravity is non-linear because it is not proportional to distance.
In the real world almost everything is non-linear.
To begin with I assumed that some long period cycle existed in something
and then looked at what would happen as that something affected other things.
The universe is full of ways for things to affect each other and I was
not concerned with the details, just the broad idea. I quickly proved that
an initial long cycle could only ever produce other cycles which were harmonics,
that is, had multiples of the original frequency or fractions of its period.
That was fine, but it could produce any harmonic, not just the observed
favoured ones of 2, 3, 4, 6, 8, 12 etc.
Perhaps I should say something about the use of the word "frequency"
for cycles. Usually we use "period" for long cycles (as in period of 5
years) and frequency for short ones (as in frequency 440 cycles per second)
but it is equally valid to say a frequency of 0.2 cycles/year or a period
of (1/440) second. Frequency is used here because it makes the maths much
simpler.
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:
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.
What shows up is that the strongest expected harmonics in the range
48 to 96 are 48:60:72:96 which is our old friend the major chord 4:5:6:8.
Also the other strong harmonics match the other notes of the just intonation
scale. I have labelled the white notes and the two black notes which are
the same ones found in my cycles research.
There are some less important in between harmonics and these turn out
to be in just the places where there have been disputes (Pythagoras' 81/64
vs Galileo's 5/4) and where the extra notes in Indian music are.
So let me stress what this proves. The pattern of cycles found in every
field of study on earth, in astronomy and also in music are all explained
by a simple rule that says that a single initial frequency will generate
harmonics AND EACH OF THESE WILL DO THE SAME. Please excuse the caps, but
that is the important bit.
What then is the longest cycle? I already knew that there were some
very long cycles like 2300 and 4600 years in both climate and astronomy,
but also the Milankovitch cycles of 100,000, 40,000 and 25,000 years which
relate to the earth's orbit and axis and also determine ice ages. But not
so long ago someone reported a 27,000,000 year cycle in the extinction
of species and geologists find even longer cycles.
This all seemed to be leading towards a conclusion which I initially
joked about and then finally embraced; the fundamental cycle was the cycle
of the universe!
I had a false start in trying to calculate the very large harmonics
and at one time had to go back 2 years in my research and do it again.
However what came out of that is the realisation that there is an especially
important harmonic which is 34,560. This number 34,560 is 2*2*2*2*2*2*2*2*3*3*3*5
and you can see why Pythagoras and Dewey found lots of 2s and 3s but only
Galilei found the 5.
The harmonics theory predicts that, compared to the entire observable
universe taken as the fundamental oscillation, the 34560th harmonic will
be an especially important one. It also predicts that at further ratios
of 34560 in size there will be important oscillations and sizes.
To understand how harmonics divide space as well as time, consider a
stringed instrument. It can oscillate at a fundamental frequency which
has just one wave in the string. It can also oscillate at the 2nd harmonic.
In that case both the length of the string and the time of the oscillation
are divided by 2. Likewise, if we could get the 34560 harmonic going in
the string it would divide both the length and oscillation period by 34560.
Why do galaxies and stars form at these places? An analogous situation
is to toss a handful of sand on a drum and then beat it (not at the centre)
and you will find that the sand moves to certain places. These are the
nodes of the standing waves in the drum. My picture of the universe is
very similar. The standing waves are electromagnetic waves (which means
radio waves, light and x-rays etc).
Things are a bit oversimplified above. In fact there are many other
moderately strong waves predicted but the above ones are the super strong
ones. The other waves turn out to explain galaxy clusters and other things.
In each scale there are multiple strong waves and for the distances between
the stars for example they are 4.45, 5.93, 8.9 and 11.86 light years. These
are the same periods that were found by Dewey and I in cycles on earth.
They are indeed "influenced by the stars" but not in the way that astrologers
normally mean.
I reached this stage in about 1993. Since then I have found that the
detailed predictions of the Harmonics theory are confirmed by observations
in cosmology, geology, atomic physics, economics, climate, biology and
human affairs.
The universe is a musical instrument and everything in it is vibrating
in tune with the larger things that contain it. I believe that there are
no other laws in the universe than this. All the other laws of physics
appear to be the result of the wave structure that leads to the Harmonic
law.
Previously I mentioned that the harmonics calculated in the 48 to 96
range exactly fitted the just intonation musical scale and that the strongest
of these; 48, 60, 72, 96; are a major chord (ratios 4:5:6:8). Further examples
of major chords happen at other places in the harmonic structure.
There are also minor chords found. These happen in the transition zones
between the places where the major chords are found. It is this transition
which I believe gives the minor its quality. The graphic: "http://www.vive.com/connect/universe/ha-idji.gif"
shows the harmonics from 20 to 360 and shows some of the strong harmonics
240, 288, 360, 480 which makes a minor chord (ratios 10:12:15:20).
If the electromagnetic zone around the earth vibrates, it does so with
a frequency of 7.5 Hz because the speed of light is 300,000 km/s and the
circumference of the earth is 40,000 km. Therefore the predicted strong
harmonics of this vibration should have frequencies of 7.5 Hz times the
various harmonics numbers. Interestingly, the frequencies resulting exactly
match those used in Indian music.
Redshifts are what astronomers use to tell how far away galaxies are
and are believed to be based on the velocities of galaxies relative to
us and caused by the big bang. I don't believe in the big bang or that
redshifts are due to velocity. The harmonics theory predicts that the redshifts
of the galaxies should favour the following values which are in km/s:
Two years ago I put a message in the sci.astro usenet group which predicted
that galaxies should come at these favoured redshifts. I knew that the
72 km/s value had been observed but not any of the others. Those observations
were not taken seriously by most astronomers because they could not reconcile
them with their beliefs in the big bang theory. As a result of that post,
someone directed me to the work of W G Tifft who had observed the following
redshift quanta (or tendencies for redshifts to come in multiples):
144, 72, 36, 24, 18, 16, 9.0, 8.0, 6.0, 3.0, 2.67 km/s
You can see that they match almost perfectly. I hadn't included the
16 and 2.67 km/s values in my original list because they were slightly
weaker values but Tifft had found them anyway.
If Tifft's observations were really the results of "noise" in the data
as most astronomers believe, or if my theory was not correct about the
universe, then the chance of such a good match between the numbers would
be 1 in about 1,000,000,000,000,000,000. In other words, most astronomers
believe in something that is incredibly unlikely.
Similar calculations show that the stars should favour certain distances.
The following histogram is based on the distances between all pairs among
the nearby stars. Each "*" is one star distance.
Likewise the distances of the planets favour multiples of two distances,
near 10 and 0.35 astronomical units (1 a.u. = earth-sun distance). The
accuracy of this agreement is shown in a graphic.
These two distances correspond to waves that oscillate in 160 minutes and
5.8 minutes. The sun has a strong oscillation at 160 minutes and a set
around 5 to 6 minutes. This shows that such waves exist in the solar system
even though we cannot directly detect them.
The Harmonics theory also works at the atomic and sub-atomic scales.
In 1994 at a lecture at Princeton I predicted that there should be a particle
with a mass 68 times an electron or 1/27 of a proton. In 1995 just such
a particle was discovered and it was unexpected and unpredicted by any
other theory.
Last year while travelling by plane I noticed some very regular cloud
formations, like ploughed fields. As near as I can estimate the distances
between the rows were 1/34560 of the earth's circumference, or about 1.16
km. More research is needed into regular structures on the surface of the
earth and other bodies.
It seems appropriate to close on a note related to the ancient Greeks.
As mentioned in a previous post in reply to Mary Lynn Richardson, it seems
that neolithic people and ancient greeks used a system of measurements
which had many ratios of 2 and 3 and also of 12. Their units include feet,
yards, chains and such and the entire pattern of these is extremely similar
to the pattern of wave sizes predicted by the harmonics theory. After finding
some rocks near my home that have exactly these dimensions (yards, cubits,
spans, feet etc) I am now convinced that these ancient units were based
on naturally occurring dimensions which reflect the electromagnetic wave
sizes in the universe.
Harmonics, Pythagoras, Music and the Universe
Part 1: Musical Background
After researching what notes sounded pleasant together Pythagoras worked
out the frequency ratios (or string length ratios with equal tension) and
found that they had a particular mathematical relationship.
do re mi fa so la ti do
1 9/8 5/4 4/3 3/2 5/3 15/8 2
Which may be represented as whole number proportions as
24 27 30 32 36 40 45 48
These proportions are called the Just Intonation music scale and are the
most pleasing proportions for note frequencies for any one key. The differences
from Pythagoras are small, so that mi is 5/4 (=1.250) rather than 81/64
(=1.266).
equitempered 1.000 1.122 1.260 1.335 1.498 1.682 1.888 2.000
just int. 1.000 1.125 1.250 1.333 1.500 1.667 1.875 2.000
which are nearly right as you can see. Bach popularised this tuning by
some very clever pieces such as the well-tempered clavier and so on. As
pointed out to me by a friend, this piece is full of musical puns. In fact
many times the puns have three possible meanings. My friend was reduced
to rolling about the floor laughing when he attempted to play guitar chords
along with a piano playing this piece.
Part 2: Cycles Background
Back in 1977 I was using computers to try and predict various economic
variables for corporations in New Zealand. In the course of doing this
I found that many aspects of the economy showed quite clear cycles. After
designing a method to search out the most consistent cycles they turned
out to be ones with periods of 4.45, 5.9, 7.15 and ~9 years. These worked
well for making forecasts.
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.
TURN! TURN! TURN!
Words: Book of Ecclesiastes
Adaptation and Music: Peter Seeger
To everything (Turn, Turn, Turn)
There is a season (Turn, Turn, Turn)
And a time for every purpose under heaven.
A time to be born, a time to die;
A time to plant, a time to reap;
A time to kill, a time to heal;
A time to laugh, a time to weep.
A time to build up, a time to break down;
A time to dance, a time to mourn;
A time to cast away stones,
A time to gather stones together.
A time of love, a time of hate;
A time of war, a time of peace;
A time you may embrace,
A time to refrain from embracing.
A time to gain, a time to lose;
A time to rend, a time to sew;
A time to love, a time to hate;
A time for peace, I swear it's not too late.
Part 3: The Harmonics Theory
After finding that Dewey had observed very similar cycle periods and also
the same musical/harmonic relationships between the cycle periods to those
that I had found I knew that it was real effect and not some delusion on
my part. I found several other sources for similar observations and in
all cases they fitted the same periods. The question to be answered was
why?
1 --> 1 2 3 4 5 6 7 8 9 10 11 12 13 ...
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 --> 9
10 --> 10
11 --> 11
12 --> 12
13 --> 13
etc
Now what is immediately obvious here is that some frequencies are produced
in many more ways than others; 4, 6, 8, and especially 12 are produced
often while 11 and 13 aren't.
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
There are some much nicer versions of this graph one showing the result
of calculations for harmonics up to See 1,000,000
and one showing the detail of harmonics 20 to 320.
ratios
V V V V V V V V V V <--of 34560
A A A A A . . A A A <--things
Univ. Stars Moons Cell Baryon observed
Galaxies Planets Atom
When we do the calculations from the size of the observable universe we
find that the 34560 harmonic predicts the correct typical distance between
galaxies. When we divide this by a further 34560 we get the typical distance
between stars, then next time we get the distance between planets and so
on. Eventually we get the typical distance between cells, atoms and nucleons
(protons and neutrons). So the entire structure of the universe is predicted
from this one simple principle. A table shows
the values predicted by a repeated ratio of 34560.
Part 4: Predictions and Verifications of the Harmonics Theory
This is the last article in this series except to answer any questions
arising. Anyone who finds it interesting can find a lot more material under
the Harmonics theory. This part will just briefly describe some of the
detailed findings and give references for more.
Harmonic h 48 54 60 64 72 80 90 96
Frequency h*7.5 Hz 360 405 450 480 540 600 675 720
Indian note pa dha ni sa ri ma ga pa
Western scale F G A Bb C D E F
The modern standardised scale has A=440 Hz and the others adjusted according
to the equitempered scale which does not quite fit this table. However
the trend has been for A to increase with time and it had got to 450 Hz
before the standard was set. Based on Indian music, the earth's natural
resonance, a study of the rhythm speed for great composers and on other
evidence, I believe that 450 Hz is the true and correct A. It is in harmony
with the earth. For indian scales relationships see graphic.
144 72 36 18 9
48 24 12 6.0 3.0
(16) 8.0 4.0 2.0
(2.67)
The prediction of galaxy redshift distributions is shown in a graphic.
Number Of Star Pairs At Distance *
*
*
* *
** * *
** * ****
* * * ** ** * * ****
* * * * * * *** **** * *** ******
* * * **** * ***** **** ************** ***** ******
0 1 2 3 4 5 6 7 8 9 10 11 12
A A A A A A
4.45 5.93 7.12 8.9 9.6 11.86
Distance between star pairs in light years --->
Also shown (by A's) the expected universal harmonics, and the
common cycles periods from Dewey's catalogue.
It is quite clear that stars do favour the distances predicted by the harmonics
theory and that these distances in light years exactly match the period
of common cycles on earth as reported by Dewey.
