Harmonics, Pythagoras, Music and the Universe
(condensed from the theories of Pythagoras and modern neo – Pythagorean writings in books and on the net)

1) Musical Background
Pythagoras worked out the frequency ratios (or string length ratios with equal tension) and found they had particular mathematical relationships. 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 notes could be produced by these two ratios as (3/2)x(3/2)x(1/2) = 9/8 (which is a second) and so on.

When 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 almost got back to the same note, but it was 1.36% higher in frequency than it should have been.

A better solution was achieved by Galilei (the father of Galileo) who modified the major scale as follows:
 do------re------mi-------fa-----so------la-------ti----do
1/1----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 the Just Intonation scale and represent 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). 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. 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 slightly change pitch (frequency). Instruments such as pianos, guitars and trumpets have fixed frequencies while violins and the human voice can vary to any note required automatically because they listen for the harmony; hence the compromise.

The Bach system of equal temperament (used almost exclusively today) is a compromise between all keys and uses a common ratio between every semitone of 2(1/12). This gives frequencies of:
Equal temp: 1.000 - 1.122 - 1.260 - 1.335 - 1.498 - 1.682 - 1.888 - 2.000
Just inton: 1.000 - 1.125 - 1.250 - 1.333 - 1.500 - 1.667 - 1.875 - 2.000

Pythagoras, and later Kepler, considered that these musical harmonies had wider application in the universe. This idea was almost forgotten or dismissed for many centuries.
2) Cycles (and the work of Ray Tomes)
Ray Tomes predicted various economic variables and found that many aspects of the economy showed quite clear cycles. The most consistent cycles turned out to be ones with periods of 4.45, 5.9, 7.15 and ~9 years. He noted that these periods were all very nearly exact fractions of 35.6 years. Also, that other cycles existed at other fractions of this period such as ~12 years and a fraction under 4 years.

His theory is that 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 were exactly in the proportions of the just intonation musical scale plus a minor third and a minor seventh (interestingly the two main dissonant notes in a ‘blues’ scale).

 (35.6/8=4.45 35.6/6=5.93 35.6/5=7.12 35.6/4=8.9 years)

He also believes that the Kondratieff cycle of about 54 years also fitted in that 2*54
is very near to 3*35.6. Jupiter's orbital period is 11.86 years, which is also very close to 35.6/3. The node of the moons orbit takes 8.85 years to travel once around the earth. There are other astronomical periods that also seemed to fit.

Edward Dewey founded the Foundation for the Study of Cycles in about 1940. He has left behind an enormous legacy of research into cycles. 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:

142.0---213.9---319.5---479.3
 71.0---106.5---159.8
(x2) 35.5---53.3 (x3)

 17.75

(/3) 5.92---8.88 (/2)
 1.97---2.96---4.44
0.66---0.99---1.48---2.22
0.22---0.33---0.49---0.74---1.11


The underlined figures are commonly occurring cycles. Several of the periods, such as 142, 53.3 17.75 and 5.93 years, are similar to those found by Chizhevski in the cycles of war, namely 143, 53, 17.7, and 6.0 years. To find others it is necessary to introduce a ratio of 5 (just as was done by Galilei to Pythagoras' music scale) which produces other common cycles such as 178 years which is found in the alignment of the outer planets, in solar activity and in climatic variations.

These cycles have been argued to affect every aspect of life on earth: wars, economic fluctuations, births and deaths, climate, geophysics, animal populations, social variables, stock and commodity prices.

 3) Harmonics Theory
Musical relationships are characterised by small number ratios between the frequencies of given notes, known as harmonics. The word harmonics has a slightly narrower meaning in physics, being 'frequencies which are a multiple of some fundamental frequency'. It is accepted in mathematics/physics that any non-linear system will develop harmonics. In the real world almost everything is non-linear.

The universe is full of ways for things to affect each other. An initial long cycle can only ever produce other cycles which have multiples or fractions of that cycle. The term 'period', ‘cycle’ and ‘frequency’ are synonymous, meaning events per period of time.

Consider an initial frequency 1 in such a system. One then generates 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 that will be frequencies of:


01 -->--1----2----3----4----5----6----7----8----9----10----11----12----13 (etc.)
02 -->--------2----------4----------6---------8----------10------------12--------------14- - -
03 -->--------------3----------------6----------------9-------------------12- - -
04 -->--------------------4---------------------8--------------------------12- - -
05 -->--------------------------5---------------------------10--------------------------------------15- - -
06 -->--------------------------------6-------------------------------------12- - -
07 -->--------------------------------------7----------------------------------------------14- - -
08 -->-------------------------------------------8- - - - - - - - - - - - - - - -
09 -->-------------------------------------------------------10- - - - - - - - - -
11 -->----------------------------------------------------------------11- - - - - - -
12 -->-----------------------------------------------------------------------12- - -
13 -->---------------------------------------------------------------------------------13- - -(etc.)

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. 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). When the spectrum of this function is examined it produces strong frequencies that have relationships exactly in the proportions of major chords in music, and moderately strong frequencies in exactly the proportion of the just intonation musical 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 <--'white' notes
---------56-----------------------------84 <--'black' notes
C-----D-Eb-E--F-------G-------A-Bb-B--C <--scale of C/Cminor

The strongest expected harmonics in the range 48 to 96 are 48:60:72:96 which is the major chord 4:5:6:8. Also the other strong harmonics match the other notes of the just intonation scale.

There are some less important ‘in between’ harmonics which have been disputed (Pythagoras' 81/64 vs Galileo's 5/4) and where the ‘extra notes’ in Indian music are.

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.

There are some very long cycles like 2300 and 4600 years in both climate and astronomy. The Milankovitch cycles of 100,000, 40,000 and 25,000 years relate to the earth's orbit and axis and also determine ice ages. 27,000,000 years appears to govern the extinction of species.

 The fundamental for all these cycles, Pythagoreans argue, is the cycle of the universe.

There is one especially important harmonic, which is 34,560 (2x2x2x2x2x2x2x2x3x3x3x5). Pythagoras and Dewey found lots of 2s and 3s but only Galilei found the 5. The harmonics theory predicts that, if the observable universe is taken as the fundamental oscillation, the 34560th harmonic will be an especially important.

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.

 V-----V------V-----V-----V-----V-----V-----V-------V-----V <--of 34560

A----- A-----A -----A-----A- - - - - - - - - - A-----A------A <--things observed
Univ---------Stars--------Moons- - - - -Cell- - - - - Baryon
-------Galaxies-------Planets- - - - - - - - - - -Atom

When we do the calculations we find that the 34560th harmonic predicts the 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, Pythagoreans suppose, the entire structure of the universe is predicted from this one simple principle.

An analogous situation is to toss a handful of sand on a drum, beat it and watch the sand move. These are the nodes of the standing waves in the drum. This picture of the universe is very similar. The standing waves are electromagnetic waves (which means radio waves, light and x-rays etc). 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 Tomes in cycles on earth.

Since 1993 Tomes found that the detailed predictions of the Harmonics theory are confirmed by observations in cosmology, geology, atomic physics, economics, climate, biology and human affairs. He believes, as did Pythagoras, that the universe is a musical instrument and everything in it is vibrating in tune with the larger things that contain it; and that there are no other laws in the universe than this.

And that all the other laws of physics appear to be the result of the wave structure that leads to the Harmonic law.

The thinking of the Neo Pythagoreans is poised between mathematical inspiration and not a little wishful thinking. Nevertheless, their predictions and verifications of the Harmonics Theory should be of interest to anyone seriously looking at the implications of Sound.
Certainly the derivation of harmonics numbers that exactly match the frequencies used in Indian music is quite compelling.

 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
Diatonic Separation....(.......t.........t...........s.........t............t.........t.........s..)
(tone..semitone)

Similarly that a modern standardised scale has A=440 Hz which has reversed a trend for A to increase with time (it had crept up 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, that 450 Hz is supposed by some to be the true and correct A because it is in harmony with the earth.

Whether quackery or not the Harmonics theory also works at the atomic and sub-atomic scales. In 1994 at a lecture at Princeton, Tomes 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, unexpected and unpredicted by any other theory.

 To conclude. Neolithic people and ancient Greeks used a system of measurements that had many ratios of 2 and 3 and also of 12. Their units were similar to feet, yards, chains and other imperial measures. The entire pattern of these is extremely similar to the pattern of wave sizes predicted by the harmonics theory. Tomes contends that these dimensions (yards, cubits, spans, feet etc) were based on naturally occurring dimensions which reflect the electromagnetic wave sizes in the universe.

 Which is maybe what I think my Grandma was trying to say when they brought in decimal currency and metric measures, in so many words..