Harmony of the Spheres | Dance of the Planets

James Ferguson’s (1710-1776) representation of the apparent motion of the Sun, Mercury, and Venus from the Earth, based on similar diagrams by Giovanni Cassini (1625-1712) and  Roger Long (1680-1770). Taken from the "Astronomy" article in the first edition of the Encyclopædia Britannica (1771; Volume 1, Fig. 2 of Plate XL facing page 449). This geocentric diagram shows, from the location of the Earth, the Sun's apparent annual orbit, the orbit of Mercury for 7 years, and the orbit of Venus for 8 years, after which Venus returns to almost the same apparent position in relation to the Earth and Sun. In Arabic, Venus is called “El Zahra” - the flower. See HERE + HERE + HERE + HERE

Earth - Mercury Cycle.
This and all following graphics by John Martineau.

Earth - Venus Cycle:
Earth = 8 years x 365.256 days/year = 2,922.05 days
Venus = 13 years x 224.701 days/year = 2,921.11 days (ie. 99.9%)

Earth - Mars Cycle.

Earth - Jupiter Cycle.

Saturn - Uranus Cycle.

Jupiter - Saturn Cycle.

Venus - Mars Cycle.

The radius of the Moon compared to the Earth's is 3:11

Radius of Moon = 1,080 miles = 3 x 360
Radius of Earth = 3,960 miles = 11 x 360 = 33 x 1 x 2 x 3 x 4 x 5
Radius of Earth plus Radius of Moon = 5,040 miles = 1 x 2 x 3 x 4 x 5 x 6 x 7 = 7 x 8 x 9 x 10
The ratio 3:11 is 27.3%, and the orbit of the Moon takes 27.3 days, which is also the average rotation period of a sunspot. The closest to farthest distance ratio that Venus and Mars each experiences in the Mars-Venus dance is also 3:11. The Earth orbits between them. The sizes of the Moon and the Earth is drawn to scale in the last illustration above, where the perimeters of the dotted square and the dotted circle are of the same length: The perimeter of the dotted red square is 4 x Earth’s diameter = 4 x 7,920 miles = 31,680 miles. The circumference of the dotted blue circle is 2 pi x radius = 2 x 3.142 x 5040 miles = 31,667 miles (ie. 99.9%).

The Measure of the Circle | Math for Mystics # 2

Pi (π) is the ratio of the circumference of a circle to its diameter. 
As a fraction, its closest approximations are 22/7, 333/106 and 355/113.

Projection on the plane of the ecliptic of the parabolic
orbits of 72 comets, 1802. Engraving by Wilson Lowry after
Johann Elert Bode.

Circle of Fifths, and relationship of relative
minor keys to major key signatures.

 

"The circle is one of the noblest representations of Deity, in his noble works of human nature. It bounds, determines, governs, and dictates space, bounds latitude and longitude, refers to the Sun, Moon, and all the planets, in direction, brings to the mind thoughts of eternity, and concentrates the mind to imagine for itself the distance and space it comprehends. It rectifies all boundaries; it is the key to information of the knowledge of God; it points to each and every part of God's noble work."

John Davis (1845): The Measure of the Circle [p. 12].

The World Is Sound | Joachim-Ernst Berendt

Joachim-Ernst Berendt (1987) - According to the Law of the Octave the duration of a planet's rotation, that is, the time a celestial body takes to revolve around its own axis and/or the time it needs for one orbit around the Sun, can be transposed into tones and colors. The tones and colors are analogous to rotation and revolution. In order to arrive at the frequency in Hertz (vibrations per second) from an astronomic period, the reciprocal value has to be formed of the duration (expressed in seconds) [...] The Earth, for instance, has a rotation period of 24 hr, or to be more precise, of 23 hr. 56 min, and 4s, totaling 86,164s. If one takes the reciprocal value, that is, divides 1 by this number, a frequency of 0.00001160577 (an inaudible G) is obtained. Though this G is below the hearing range (which starts at about 16 Hz). transposing it by 24 octaves will create an audible G. 

[...] Tones exist, whether we hear them or not. Any music lover knows that a melody can resound within even when it is not being played. A composer hears the music within while notating it and before any sound has been made. For this reason, transposing by octaves is a legitimate process. Even scientists are using it (for instance, to transpose sound of deep sea fish and bats from the ultrasonic range into human audibility or to better understand signals of pulsars and other stars). The octave (1:2) is the most frequent relationship in the universe - not only in music, but anywhere in nature, from the micro- to the macro-cosmos. We use the same names for tones that are octaves apart [...] When a cell divides in mitosis, it chooses the "position" of the octave. The result is the "same cell" again. An octave may vibrate at twice or half the rate (or in powers of two or one-half) but it still is the same tone. It may split the one in two parts or double it, and the result is the same again. Its frequency may be completely different from the basic tone, many Hertz above or below it, but the result is still the same tone again. The octave is the most convincing symbol of unity that we can find in nature. And in nature, it is omnipresent.

[...] Because the Law of the Octave is universal, one can continue transposing by octaves to reach the electromagnetic vibrations of colors. From the tone of the Earth (194.71 Hz) another 36 octaves are required to reach 700.16 Nm (Nanometer), which is analogous to the color of orange-red (also analogous to the tone G and to the rotation of the Earth around the Sun). However, the range of human vision is limited to only one octave compared with the ten octaves of the hearing range [...] The tone of the Earth is the most important tone for all living beings on this planet, whether we leave it inaudible or make it audible by transposing it into higher octaves. It is with this tone that we rise in the morning and go to bed at night; to this tone we do our work, we get hungry, and we love. But other planetary vibrations and tones, especially those of the Sun, the Moon, Venus. Mars, and Jupiter, also vibrate directly into our earthly existence. This is why I call them primordial tones [...] For millions of years, longer and more steadily than any other comparable vibration, the Earth. Sun, Moon, and the planets have been vibrating in cosmic space. Our genes and those of all living beings have experienced these vibrations so often that the processes and mechanisms of genetic programming must have stored them long ago.

[...] The period from Full Moon to Full Moon (the "synodical month") lasts 29 days, 12 hr, 44 min and 2.8s; a total of 2,551,442.8s. In order to transpose the corresponding frequency into the average range of human hearing, we have to transpose it by 30 octaves. The result is a tone of 420.837 Hz (G sharp), a tone of no great importance to our Western music today, but during the Baroque and early Classical periods, it was of major importance. Mozart's tuning fork, for example, had 421.6 Hz. At its pinnacle, Western music was directly connected with the tone of the Moon. Concert pitch started to rise in the middle of the 19th century, striving for the superficial effect of making the music sound brighter. Thus Western music started to turn away from the moon's field of resonance, but the Moon, in all traditions, is responsible for the arts and the artists, being the planet of sensitivity and creativity. In the 20th century, major American symphony orchestras kept raising the concert pitch tone more and more. In doing this, they have banished Western music from its cosmic relationship to the celestial body of the arts and the artists.

[...] The tone of the Sun results from the tropical year lasting 365.242 days or 31,556,926s, and it is C sharp. We can hear it at 136.10 Hz. In Indian classical music, this C sharp is still the fundamental tone. It is called sa or sadja, the "Father of Tones." Bells (e.g., temple bells and gongs) are often tuned to this tone, not only in India but also in Tibet, Japan, and on Bali. The prime word OM, the holiest of mantras, has been chanted to the sa more often than to any other tone. Today classical Indian music remains in a relationship to the Sun, as Western music of the Baroque, Classical and Early Romantic periods was formerly in relationship to the Moon.

 

Quoted from:
Joachim-Ernst Berendt (1987) - Primordial Tones: Meditation on the Archetypal Energies of Celestial Bodies.

See also:
Joachim-Ernst Berendt (1991) - The World Is Sound: Nada Brahma.

 

Sound, Light, Color, Heat = Different Manifestations of Energy.

The Same, The Other, And The Essence │ Theology of Arithmetic

“One, two, three [...] Let me tell you then why the creator made this world of generation. He was good [...] He took the three elements of the same, the other, and the essence, and mingled them into one form, compressing by force the reluctant and unsociable nature of the other into the same. When he had mingled them with the essence and out of three made one, he again divided this whole into as many portions as was fitting, each portion being a compound of the same, the other, and the essence. And he proceeded to divide after this manner: 

First of all, he took away one part of the whole [1], and then he separated a second part which was double the first [2], and then he took away a third part which was half as much again as the second and three times as much as the first [3], and then he took a fourth part which was twice as much as the second [4], and a fifth part which was three times the third [9], and a sixth part which was eight times the first [8], and a seventh part which was twenty-seven times the first [27]. After this he filled up the double intervals [1, 2, 4, 8] and the triple [1, 3, 9, 27] cutting off yet other portions from the mixture and placing them in the intervals, so that in each interval there were two kinds of means, the one exceeding and exceeded by equal parts of its extremes [1, 4/3, 2, in which the mean 4/3 is one-third of 1 more than 1, and 1/3 of 2 less than 2], the other being that kind of mean which exceeds and is exceeded by an equal number. Where there were intervals of 3/2 and of 4/3 and of 9/8, made by the connecting terms in the former intervals, he filled up all the intervals of 4/3 with the interval of 9/8, leaving a fraction over; and the interval which this fraction expressed was in the ratio of 256 to 243. And thus the whole mixture out of which he cut these portions was all exhausted by him.

This entire compound he divided lengthways into two parts, which he joined to one another at the center like the letter X, and bent them into a circular form, connecting them with themselves and each other at the point opposite to their original meeting-point; and, comprehending them in a uniform revolution upon the same axis, he made the one the outer and the other the inner circle. Now the motion of the outer circle he called the motion of the same, and the motion of the inner circle the motion of the other or diverse. The motion of the same he carried round by the side to the right, and the motion of the diverse diagonally to the left. And he gave dominion to the motion of the same and like, for that he left single and undivided; but the inner motion he divided in six places and made seven unequal circles having their intervals in ratios of two-and three, three of each, and bade the orbits proceed in a direction opposite to one another; and three [Sun, Mercury, Venus] he made to move with equal swiftness, and the remaining four [Moon, Saturn, Mars, Jupiter] to move with unequal swiftness to the three and to one another, but in due proportion.” Timaeus - Plato (360 BCE)

Johannes Kepler knew that "ubi materia, ibi geometria" (where there is Matter, there is Geometry), and "that the geometrical things have
provided the Creator with the model for decorating the whole world". In Harmonices Mundi (The Harmony of the World, 1619) he related musical
consonance and the angular velocities of the planets, for example, 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. From his studies of planetary harmonics Kepler also arrived at the bold conclusion that between Jupiter and Mars must
exist an unknown planet: "Intra Jovem et Martem posui planetum." (Between Jupiter and Mars I put a planet.") Some 170 years later the so-called
asteroid belt was found in the corresponding place. 

On Harmony And Beauty

The Pythagoreans averred that mathematics demonstrated the exact method by which the good established and maintained its universe. Number
therefore preceded harmony, since it was the immutable law that governs all harmonic proportions. Summarizing the relationship between the
human body and the theory of architecture, Marcus Vitruvius Pollio (80–15 BC) wrote in his De Architectura: "Since nature has designed the
human body so that its members are duly proportioned to the frame as a whole, it appears that the ancients had good reason for their rule,
that in perfect building the different members must be in exact symmetrical relations to the whole general scheme. Hence, while transmitting
to us the proper arrangements for buildings of all kinds, they were particularly careful to do so in the case of temples of the gods, buildings
in which merits and faults usually last forever. Therefore, if it is agreed that number was found out from the human fingers, and that there is
a symmetrical correspondent between the members separately and the entire form of the body, in accordance with a certain part selected as
standard, we can have nothing but respect for those who, in constructing temples of the immortal gods, have so arranged the members of the
works that both the separate parts and the whole design may harmonize in their proportions and symmetry." 

"Harmony is a state recognized by great philosophers as the immediate prerequisite of beauty. A compound is termed beautiful only when its parts are in harmonious combination. The world is called beautiful and its Creator is designated the Good because good perforce must act in conformity with its own nature; and good acting according to its own nature is harmony, because the good which it accomplishes is harmonious with the good which it is. Beauty, therefore, is harmony manifesting its own intrinsic nature in the world of form. The universe is made up of successive gradations of good, these gradations ascending from matter (which is the least degree of good) to spirit (which is the greatest degree of good). In man, his superior nature is the summum bonum. It therefore follows that his highest nature most readily cognizes good because the good external to him in the world is in harmonic ratio with the good present in his soul. What man terms evil is therefore, in common with matter, merely the least degree of its own opposite. The least degree of good presupposes likewise the least degree of harmony and beauty. Thus deformity (evil) is really the least harmonious combination of elements naturally harmonic as individual units. Deformity is unnatural, for, the sum of all things being the Good, it is natural that all things should partake of the Good and be arranged in combinations that are harmonious. Harmony is the manifesting expression of the Will of the eternal Good." Secret Teachings of All Ages - Manly P. Hall (1928)

Credits: Samuel Colman (1912): Nature's Harmonic Unity - A Treatise on Its Relation to Proportional Form

The Pythagorean Definition of God

St. Bernard of Clairvaux: "God is breadth and length, height and depth."

By That Pure, Holy, Four Lettered Name On High | Pythagoras

The Pythagorean Oath mentions the Tetractys:
"By that pure, holy, four lettered name on high,
nature's eternal fountain and supply,
the parent of all souls that living be,
by him, with faith find oath, I swear to thee."
Credits: Michæl Paukner

Albert Mackey (1873) - "The Greek word 'Tetractys' signifies, literally, the number four, and is synonymous with the quaternion; but it has been peculiarly applied to a symbol of the Pythagoreans, which is composed of ten dots arranged in a triangular form of four rows."   

The first four numbers symbolize the harmony of the spheres and the Cosmos as: (1) Unity (Monad); (2) Dyad - Power - Limit/Unlimited; (3) Harmony (Triad); (4) Kosmos (Tetrad) - The four rows add up to ten, which was unity of a higher order (The Dekad).  The Tetractys symbolizes the four elements — fire, air, water, and earth. The Tetractys represented the organization of space: the first row represented zero dimensions (a point). The second row represented one dimension (a line of two points). The third row represented two dimensions (a plane defined by a triangle of three points). The fourth row represented three dimensions (a tetrahedron defined by four points). 

The Pythagorean musical system is based on the Tetractys as the rows can be read as the ratios of 4:3 (perfect fourth), 3:2 (perfect fifth), 2:1 (octave), forming the basic intervals of the Pythagorean scales. That is, Pythagorean scales are generated from combining pure fourths (in a 4:3 relation), pure fifths (in a 3:2 relation), and the simple ratios of the unison 1:1 and the octave 2:1. The diapason, 2:1 (octave), and the diapason plus diapente, 3:1 (compound fifth or perfect twelfth), are consonant intervals according to the tetractys of the decad. The diapason plus diatessaron, 8:3 (compound fourth or perfect eleventh), is not.

Planetary Hours

One fundamental principle of cognition is the scission of the Monad into two parts of symbolic opposites: hot and cold, light and dark, hard and soft, raw and cooked, good and evil. The division of the day into planetary hours is based on this way of thinking: the cycle of the day is split into a dark and a light part. The light part is defined by the length of time between sunrise and sunset and the dark part comprises the hours between sunset and sunrise. And since the length of day and night is only equal at the equinoxes, whereby its opposite, the longest day and shortest night or vice versa occurs at the solstices, William Lilly wrote in his Christian Astrology:

1 Planetary Hour = (Sunset - Sunrise) / 12

“It is very true, some of the Ancients have Winter and Summer, made the day and night to consist of equal hours. I mean every hour to consist of sixty minutes, equally; but Astrologists do not so, but follow this method, viz. according to the motion of the Sun both  Summer and Winter, so do they vary their hours in length or shortness.” One measures the time between sunrise and sunset and divides it into 12 equal parts. These are the planetary day hours. The same may be done with the night hours, measured from sunset to next day’s sunrise to find out the length of each of the planetary night hours.

Watch rulers of days, hours and signs, especially beginnings of Sun
and Moon hours as well as rise, culmination and set of planets.

“You must understand that as there are seven days of the week [...] there are seven Planets [...] We appropriate to each day of the week a several Planet; as to Sunday the Sun, to Monday the Moon, to Tuesday Mars, to Wednesday Mercury, to Thursday Jupiter, to Friday Venus, to Saturday Saturn.” 

This order is known as the “Chaldean Order”, derived from the planets' relative mean speeds which are important in horary astrology (HERE).

Calculation of the planetary hours played a certain role in Renaissance astrology and magic. Astronomical tables published in the late 15th or during the 16th century often included a table of planetary hours with their significations.

Eternal Recurrence of the Same | Friedrich Nietzsche

» I am all the names in history. «

» What, if some day or night, a demon were to steal after you into your loneliest loneliness and say to you: ‘This life, as you now live it and have lived it, you will have to live once more and innumerable times more; and there will be nothing new in it, but every pain and every joy and every thought and sigh … must return to you — all in the same succession and sequence — even this spider and this moonlight between the trees and even this moment and I myself. 

The eternal hourglass of existence is turned over again and again — and you with it, speck of dust!’ Would you not throw yourself down and gnash your teeth and curse the demon who spoke thus? Or have you once experienced a tremendous moment when you would have answered him: ‘You are a god, and never have I heard anything more divine!’ If this thought were to gain possession of you, it would change you as you are, or perhaps crush you. The question in each and every thing, “do you want this once more and innumerable times more?” would lie upon your actions as the greatest weight. Or how well disposed would you have to become to yourself and to life to crave nothing more fervently than this ultimate eternal confirmation and seal? «

Aphorism § 341 - Die fröhliche Wissenschaft

Friedrich W. Nietzsche

1882