The Harmony of the Spheres

by Thomas Váczy Hightower.

The concept of harmony

It is general knowledge that the notion of harmony comes from the ancient Greek word, harmonia. It was a key word of the Pythagoreanism and meant primarily the joining or fitting of things together. Originally it was connected to the concept of cosmos. Its musical meaning was established by the early fifth Century B.C. according to professor Guthrie ("A History of Greek Philosophy", Vol. I,) first seen from Pindar. 

The numerical explanation of the universe was a generalization from the discovery made by Pythagoras himself and revealed the numerical ratios which determine the concordant intervals of the scale. We find it clearly in Aristotle's explanation of Pythagoras harmony of the spheres and also in Plato's statements. 

However, as Burnet notes, there was no such things as harmony in our sense. Harmonia meant tuning, scale or octave. Classic Greek music was melodic, modal, without use of harmonious chords as we are used to.
Plutarch (44-120 B.C.) stats that for Pythagoras and his disciples, the word harmonia meant "octave" in the sense of an attunement, which manifests within its limits both the proper fitting together of the concordant intervals, fourth and fifth, and the difference between them, the whole tone. 
Moreover, Pythagoras proved that whatever can be said of one octave can be said of all octaves. 

The essential point was that the three intervals of octave, fourth and fifth were regarded as primary and fixed, as the element out of which any musical scale or composition was build. The other notes was not fixed and could move accordingly to the different modes.  

Order and beauty - the concept of cosmos - was imposed by the three fixed notes with the ratio of 1:2, 2:3 and 3:4 on the chaotic range of sounds from the other untied notes. 
This is for me a graphic display of Harmonia; an integration of order and chaos. Harmony does not in my interpretation means pure harmonious tones, but permitting dissonance between the column of consonance creating harmony. 
The ancient Greeks had a profound understanding of balance and proper proportions. It was not accidental that they chose the formulation on the temple of Apollo as "Nothing too much" or "Observe limit". 

Chaos meant for the Greeks unlimited – evil.
Limit stood for order, moderation - beauty. 
The Greek genius in thought and art, represented the triumph of ratio, meaning on one hand the intelligible, determinate, measurable, as opposed to the fantastic, vague and shapeless. There had to be a proportion of things both in themselves and as related to the whole.

The integers 1,2,3 and 4 add up to 10, which was considered perfect and contained in itself the whole nature of numbers. This number was graphically represented by the figure known as the Tetratys, which became a sacred symbol for the Pythagorean's.
Numbers was responsible for "harmony", the divine principle that governed the structure of the whole world.

For the Pythagorean's the numbers had and retained a mystical significance, an independent reality. Phenomena were secondary for the only significant thing about phenomena was the way in which they reflected numbers. We have here an attitude that is utterly different from that of a mathematician of today. Mathematics had for them and also later for Plato a metaphysical as well as a purely mathematical significance.

The Harmony of the Spheres

The most remarkable feature of the Pythagorean cosmology recorded by Aristotle is that it displaced the Earth from the center of the universe and made it into a planet revolving in an orbit like the other planets. This idea was unparalleled in per-Platonic thought. Even Plato could not embrace this notion, which first later became accepted in the heliocentric theory.
The Pythagorean believed that the center of the whole system was occupied by the central "fire", which could not be seen. The relation to the Sun as the heat- and light giving body to the central fire was not explained by Aristotle, but later sources states it as a case of reflection like that of the moon's light from the sun.

A part of the cosmology of the Pythagorean is their extraordinary theory of the "harmony of the spheres", which caught later generations in the ancient world and the Renaissance. 
It is generally accepted by scholars, that Pythagoras himself was the first to formulate that concept, which reflects the whole cosmic plan and showed the intimate connxtion between the laws of mathematics and of music. 
Aristotle characterizes the Pythagorean as having reduced all things to numbers or elements of numbers, and described the whole universe as "a Harmonia and a number". 

Aristotle continued: "They said too that the whole universe is constructed according to a musical scale. This is what he means to indicate by the words "and that the whole universe is a number", because it is both composed of numbers and organized numerically and musically. For the distances between the bodies revolving round the center are mathematically proportionate; some moves faster and some more slowly; the sound made by the slower bodies in their movement is lower in pitch, and that of the faster is higher; hence these separate notes, corresponding to the ratios of the distances, make the resultant sound concordant. 
Now number, they said, is the source of this harmony, and so they naturally posited number as the principle on which the heaven and the whole universe depended."

Many scholars has pointed out: how could the Pythagorean s have supposed that all eight notes of an octave sounding simultaneously would produce a concordant and pleasing effect?
Others have answered: there is no question of harmony in modern sense, but only attunement to a perfect scale. When Greeks called certain intervals concordant, they were thinking primarily of notes sounded in succession. The word "harmony" means, in the Greek language, first "tuning" and then "scale".

Classical Greek music was melodic, not harmonic in modern sense. Further, the seven-stringed lyre had only three strings in a fixed tuning, the octave, the fifth and the fourth, which together made use of the first four integers forming the sacred tetractys. The other strings were adjustable to the type of scale required and therefore were called movable. It is therefore likely to assume that Pythagoras himself only had these concordant intervals in mind, which sounds very pleasant simultaneously, instead of the seven or eight strings.

Compared with the scattered and few information from the Pythagorean's themselves about the harmony of the spheres, it is staggering to see what an impact that notion has caused in later generations. 

Johannes Kepler

The most passionate lifelong study of the Music of the Spheres was done by Johannes Kepler (1571 - 1630). By the end of his career he had solved both the problem of what law governs the planetary distances and what law governs their velocities.
His solution of the first was geometrical, that of the second musical, and his Third Law of Planetary Motion (that the squares of the periodic times are to each other as the cubes of the mean distances from the Sun) provided the link between them.

He also changed the focal point from the Earth to the Sun by converting to the Copernican system. pondering the question of why there should be only six planets (instead of the canonical seven), and what governed their distances from the Sun.

←The solar system in a double logarithmic plot representing orbital speed and distance from the sun. 

Then he was visited by a mystical intuition that the secret of the solar system lies in the five regular or Platonic solids. In between the six orbits of Mercury, Venus, Earth, Mars. Jupiter, and Saturn, he then inscribed the five solids in this order: octahedron, icosahedron, dodecahedron, tetrahedron, and cube; and behold, not only was the number of six planets accounted for, but their proportional distances were closely in accord with Copernicus's figures. 
Yet the figures from Tycho Brahe's observations, as Kepler worked on them year after year, stubbornly refused to fit the regular solids.

Some other factor must be at work. It was the difficulties of calculating the orbit of Mars that led to the answer and to the overwhelming realization that the orbits of the planets are not perfect circles, nor even the elaborate combinations of pseudo-circles which astronomers from Ptolemy to Copernicus had used to keep the circularity of the heavens inviolate, but ellipses with the Sun at one of their foci.

From the point of view of time, these explain the variable speeds of the planets, which move faster as they approach the Sun. 
It remained to find a rationale for the ellipses being just so, and it was to musical harmony that Kepler turned to.
He compared each planet's angular velocities at perihelion (nearest to the Sun) and aphelion (furthest from the Sun), and expressed this ratio as a musical interval.

 This was not all. Kepler had broken new ground in listening to the Music of the Spheres not from the point of view of Earth, like all his predecessors, but from the Sun. Henceforth it is no longer a harmony made for the benefit of our own planet, but the song which the cosmos sings to its lord and center, the Solar Logos.
Like Eriugena's approaches to the harmony of the spheres with widely varied planets in speed and distance, (Eriugena was an Irish monk from the 9. Century) each of Kepler's planets has a variety of notes: there is no question here of a simple planet-tone scale of any type. Unlike Eriugena, however, Kepler lived in an age when musical polyphony was the norm. The planetary music must therefore be polyphonic, too.
As Eriugena had already pointed out, the musical possibilities of these sliding scales are almost inexhaustible. Most of the chords they make together are dissonant to one degree or another, but once in many years a five or six-part consonance will occur. The time-scale exceeds human imagination, even human life: a single journey of Saturn up and down a major third (5:6 is the ratio between its maximum and minimum) takes 30 years!

The harmonics of planets angular velocities, or the ratios – interval -  seen from the Sun are as valid today than they were in his time. In addition, the outer planet discovered after Kepler’s time, Uranus, Neptune and Pluto also fit into Keple's system.

His method was briefly as follows: The ratios compare the velocities of the planets at their fastest and lowest speed by calculating how far they go in 24 hours, (it can take many years before the planet in question have reach these positions) measured in minutes and seconds of arc as viewed from the Sun. These ratios are then simplified by octave reduction to give an interval between C and C’

An example of octave reduction: the ratio between Jupiter’s maximum and Mars minimum speed is as 5:24. That is equivalent to the interval of two octaves plus a minor third. The two octaves are eliminated by dividing 24 with 4, which gives the ratio of 5:6, a minor third.  
By placing the calculated ratios from the many combinations after their harmonic numbers of a fundamental, let us call it C, a display of a harmonically scale is produced. A major part of the planets tones belongs to the major triad CEG.

In ”Harmonia Mundi” Kepler present a very consistent theory of the music of the spheres. What was more important was his method. He used exact observations and formulated the data in a mathematical form, that could depict nature lawfulness. In this way he became the first modern scientist.

Frequencies of orbiting planets

With reference to the hierarchies of motion, which was explained in the previous part of The Sound of Silence, part I, we have oscillating bodies on any level of sizes. On the quantum level we have the double motion of spin and orbiting. On the level of molecules and cells only one motion - the pendulum to and fro motion exist.
When we reach into the size of the celestial bodies, the movements of orbiting and spin reappear. Our planet system behaves in that double motion, rotating about their axes while in the same time orbiting around our sun.

There is a basic acoustic law to calculate oscillation of a body into frequency. It is the relationship between the period of oscillation (the time it takes to complete the period or move back and forth) and its frequency. It is an inverse proportion: the period = 1/ frequency; (or frequency = 1/ period.) It means that the reciprocal value of a period of time represents its frequency (in seconds).  

Frequency range can be measured in a basic unit called octaves. In practice one can form an octave by multiply or divide a frequency with 2. The law of octaves raise or lower the fundamental frequency by factor 2 to any preferred level, e.g. to the audible range. It is still the same fundamental tone but in another octave.

The frequencies of our planets can be calculated with this formula, where the multiplication by 2³⁶ brings the fundamental to an audible range:  1 / period in seconds (1 day= 84400 sec.) * 2³⁶ (numbers of octaves) = frequency in Hz.

The approximate chromatic tones are picked from the below frequencies table for the equal tempered chromatic scale with the concert pitch of middle A =  440 Hz. The above display of tones does not constitute a musical scale as we can perceive. Other type of scales do not change much.

Note	0	1	2	3	middle C	5	6	7	8	9
C	16.35	32.70	65.41	130.81	261.63	523.25	1046.50	2093.00	4186.01	8372.02
C#	17.32	34.65	69.30	138.59	277.18	554.37	1108.73	2217.46	4434.92	8869.84
D	18.35	36.71	73.42	146.83	293.66	587.33	1174.66	2349.32	4698.64	9397.27
D#	19.45	38.89	77.78	155.56	311.13	622.25	1244.51	2489.02	4978.03	9956.06
E	20.60	41.20	82.41	164.81	329.63	659.26	1318.51	2637.02	5274.04	10548.08
F	21.83	43.65	87.31	174.61	349.23	698.46	1396.91	2793.83	5587.65	11175.30
F#	23.12	46.25	92.50	185.00	369.99	739.99	1479.98	2959.96	5919.91	11839.82
G	24.50	49.00	98.00	196.00	392.00	783.99	1567.98	3135.96	6271.93	12543.85
G#	25.96	51.91	103.83	207.65	415.30	830.61	1661.22	3322.44	6644.88	13289.75
A	27.50	55.00	110.00	220.00	440.00	880.00	1760.00	3520.00	7040.00	14080.00
A#	29.14	58.27	116.54	233.08	466.16	932.33	1864.66	3729.31	7458.62	14917.24
B	30.87	61.74	123.47	246.94	493.88	987.77	1975.53	3951.07	7902.13	15804.27

 

     It is apparently too demanding for us humans to expect, that our orbiting plants will present a neat musical scale just from their orbiting periods! It is more likely that the planetary orchestra is playing for the Sun. The frequency of the sun.

We have to remember, that time is different from celestial bodies that for humans. We perceive sound in seconds. For planets the sound unit is more in the category of years. For the Sun the concept of time is even more "inhuman". A lifetime of a human is just a "spark" for the Sun!

The time factor is the key to understand the notion of the music of the spheres. As demonstrated above the sound on the planetary scale has to be raised about 36 octaves to be perceived by humans.

Planetary conjunctions as "Chords"

There is other parameter than periods of orbiting planets to employ in the search for the music of the spheres. The planetary conjunctions is considered to be a powerful alignment of celestial forces. Using conjunctions of planets on a larger scale of time ex. 48 years, clues of a diatonic scale emerge.
Rodney Collin tried this in his book, "The Theory of Celestial Influence", chapter VI, the Harmony of the Planets. One of his examples displays one of the major cycle, where Earth, planets and sun are in exactly same position to the Milky Way or Zodiac.

Planetary conjunctions (Earth, Planet, Sun, Zodiac) as "Chords"

The time line (24 to 48 years) could be considered as a "string" on a musical instrument. Planetary conjunctions take place over time according to these same ratios, that divide up the length of the string in a Just Intonation temperament. 
Jupiter, e.g. has 3 conjunctions over the time span of 24 - 48 years and strike the notes do, sol and do₁.
There are minor cycles for just Earth, planets and sun producing a similar way other conjunctions as "chords".

Planets as tones in the diatonic scale

Several have through time made calculations, which place the distance of the planets from the Sun into a diatonic scale. That might be possible if one step down in octaves in regard to the outer planets, which are Jupiter, Saturn, Uranus etc. 
Joscelyn Godwin has in his book "Harmonies of Heaven and Earth" in the chapter The Music of the Spheres a fine historic account for different planetary scales.

The most known examples of planetary "musical scale" are the mentioned Pythagoras' and Kepler's. In addition, there is a number of other planetary scales, such as Thomas Michael Schmidt, who derives significant harmonies from the time-periods of the planets rotations. W. Kaiser composing a scale based on the mean distances from the Sun.  Emile Chizat, who makes the Titius-Bode numbers yield a perfect concord. Or Alexandre Dénéréaz working from the Golden Section of planetary distances.

The principles of order in the solar system

In the 18th century Titius discovered, that all the planets known to astronomers in his time possessed a special order in the mean orbital distances from the Sun. The astronomer Bode formulated that discovery few years later into a law, which, when the outer planets were later discovered, fit the whole solar system quite accurately. The aberration was not more than few present, except Mars with 6 %.

These orbits becoming progressively greater by an average factor of 1.68... Such progression with nearly the same factor could indicates a special relationship. Astronomers have throughout centuries contemplated over the planetary movement and intuitive felt a harmonious relationship, the Music of the Spheres or a hidden lawfulness.

The Golden Section

The beauty of this relationship became first clear, when John Pritchard in a mail told me about the findings of the Russian astronomer Butusov, who in 1978 found the answer. Butusov established, that the ratios of the adjacent planets cycle times around of the Sun are equal or the golden proportion 1,618, or its square 2,618.

The ratio between the planets is simply the PHI ratio, 1.618...or The Golden Section. The Solar system is a manifestation of the spiral of the golden section, which is so prominent in Nature and art.
 http://solargeometry.com/DistCalc.htm ;
The chart below use Mercury mean distance from the Sun as the base line, which fit best to proportion of the golden section.

There are other ways to calculate the relationship of planets. Below we can use the common way of calculation in AU with Earth = 1, by dividing each planets mean distance from the sun with the previous planets solar mean distance.

Similar order appears for the moons in our solar system, especially for the nearly circular orbits. The numbers are taken from a Danish astronomical book from 1979.

There are other principles of order in the solar system, such as the planets are orbiting close to the same plane and the elliptic orbit do not deviate much from each another.

NASA has released records of the "music" of our planets. From the Voyager program, e.g. the magnetosphere of Saturn emits waves which, transposed into audible sound by speeding up and played through a music synthesizer sounds as a slow, dreamy melody!
Those who have seen the movie, 2001, by Stanley Kubrick, will remember the music of Richard Strauss under laying the beautiful space pictures. That deep vibration sound in the "Also sprach Zarathustra" intuitively connect the idea of the vibrating universe.

Man as a microcosm

An explanation for the strong attraction to the concept of the harmony of the spheres may have something to do with the way man functions and regulates his body and mind. He is a microcosm where the endocrine glands serve as vital regulators and transformer of the life energy. The endocrine glands are: Thymus, Pancreas, Thyroid, Para-thyroids, Adrenals, Posterior Pituitary, Anterior Pituitary, Gonads and Pineal (function ?).
The interacting nervous controls are three fold: the cerebral-spinal, which serves conscious control; the sympathetic, which stimulates unconscious or instinctive functions; and the parasympathetic and vagus, which are explained as slowing down these instinctive functions.

Looking at endocrine glands placement in the body we can lay out a schematic diagram, where the heart is the center and the glands place themselves as an expanding spiral similar to that which is found to represent the line of forces or growth in many natural phenomena. 
Just as galaxy appeared to be an expanding spiral of suns, and the solar system an expanding spiral of planets, so could the spiral of glands in the human body give an impression of an expanding spiral of functions.

In the solar system it is evident that the sun is the source of life and energy. The Earth and its bio sphere can be viewed as a stock of passive compounds on which the planets serve as a forming, organizing and functional force producing the whole world of Nature. 
This hypothesis of the planets as the creators of form and function on Earth has no substantiate arguments in science, either has the harmony of the spheres. 

This is an attempt to give a hint to why the concept of the harmony of the spheres have tantalized the imagination of man for so many centuries and why there have been such a persistent belief in a link between planetary influence and human behavior and personality.
Astrology has unfortunately degenerated from its original objective study of correspondences and tendencies towards prognostication and fortune-telling.

The Theory of Celestial Influence by Rodney Collin presented for me the most coherent theory, where the endocrine glands possess an affinity for their respective planets due to the magnetism of the particular planet. Each planet produce by its spin different magnetic fields and the whole solar system can be consider as a great transformer of cosmic influence.

Collin made a detailed account on these matter, but here I will just conclude, that the endocrine glands are functioning as a receiving sets for planetary influences. The influence varying due to distance and angle of the Earth
Moreover, he did presented a very interesting theory of "planetary types", which has enriched my understanding of human psychology.

The bottom line is, as Collin states: "that the glands in the order of their distance from the heart, obey the same laws as the planets in the order of their distance from the sun. Created from the same design, the one responds to the other. Each gland is revealed as a sensitive instruments, which not only transforms human energy to the tension required for its function, but is tuned to a similar instrument on a cosmic scale and obeys its guidance."

What I want to point out is that basically each living being on every scale are oscillating. E.g. celestial bodies, nature and its numerous entities, humans, molecules and atoms - probably also the nucleus. See Dimension and Scales.

If one can imagine that notion, then each atom has its signature "sound" according to its number of electrons. The atoms "sing", as other bodies do on another and bigger scale.

Together all sing their different songs in union - the Uni-verse.

Thomas Hightower, 2003-5.

Main page The Sound of Silence part II.

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