Cubes: The Ancient Division of the Whole

Volume as cubes reveal the wholeness of number as deriving from the unit cube as corner stone defining side length and “volume” of the whole.

The first cube (above left) is a single cube of side length one. One is its own cornerstone. The first cubic number is two to the power of three, with side length two and volume equal to eight cubes that define the unit corner stone.

In modern thinking, and functional arithmetic, volume increases with side length but the cube itself, as archetype of space, is merely divided by the side length of the unit cornerstone, which is 1/8th the volume and therefore reciprocal to the volume of 8 leaving the cube singular.

This may not seem important but, by dividing a whole cube, one is releasing more and more of the very real behavior that exists between numbers, within the cube. For example the number 8 gives relations between numbers 1 to 8, such as the powers of two {1,2,4,8} and the harmonic ratios {2/1, 3/2, 4/3,5/4,6/5}. These can give an important spine of {4/3, 5/4,6/5} which equals 6/3 = 2 of the yet to (numerically) be octave of eight note classes. Moving to side length 3, the cube of three is twenty seven (27), as seen in figure above, top right. To obtain it, the corner stone must be side length 1/3, and volume 1/27 so that, in these units, the volume of the cube is 27.

If one were to reciprocally double the 1/3 side length, each cornerstone unit would have 8 subunits, so that the volume of 27 would be times 8 which equals Plato’s number of 216. Another view is then that the cornerstone side length has divided the bottom right cube into six units which number, 6, cubed, is 216 a perfect number for Plato.

By accepting the cube of one as the whole, this form of thinking reciprocally divides that whole side length to generate an inner structure within the whole cube of one, equal to the denominator of the reciprocation. The role of the whole is then to be the arithmetic mean between a number and its reciprocal. This procedure maintains balance between what is smaller than the whole (the reciprocal) and what is larger than the whole (in this case the volume).

In ancient tuning theory this was expressed by the two hexchords descending and ascending from the tonic (we might call do), expressed by the two hands. The octave of eight and the cube are both wholes to be broken into by numbers greater than one by means of reciprocation.

Ernest G. McClain revealed the scale of such thinking was massive, whilst also but secretly reciprocal, so that a limiting number could express how different wholes will behave due to their inner diversity of numbers at work within them.

In this example, a musical code for planetary resonance is revealed within the metrology of the Parthenon (above), implied tone set (right) and octave mountain of numbers below 1440 (left bottom). In this case the number 24 has been multiplied by 60 to give a limiting number of 1440. The cornerstone in this case is bottom left of the mountain = 1024, a pure power of 210.

By simply quoting a limiting number, in passing, ancient texts could, in the hands of an initiate, create an enormous world of tonal and impied religious meaning – through a kind of harmonic allusion.

It is only by using the conceptual approach of the ancients, that their intellectual life can be recovered – just by adding the waters of number and some powers of imagination.

Harmony of the Biblical Patriarchs

This extract from The Harmonic Origins of the World (p58-62) shows how what are taken to be arbitrary numbers, in the narrative of the Patriarchs, expressed knowledge of planetary resonances.

above: Diagram of the Bible code of “holy mountains”: Products of the powers of five upwards and of the powers of three across, each mountain within a limiting number if note D is {60, 120, 45, 180, 720} .

Female archetype Eve, “mother of all living,” becomes Abram’s wife Sarai. Childless, Abram was encouraged (by Sarai) to have first son Ishmael by concubine Hagar, but the Lord God (the mountain god whose number sums to 345) renames Sarai as Sarah and Abram as Abraham. At a miraculous 90 years of age she gives birth to Isaac (“he laughs”), then reportedly dies at 180 years old. It is harmonically relevant that (a) the giving of heh = 5 to both Sarah and Abraham elevates them from their former selves onto the second row, “stepping up” like the god Ea in Sumeria, and that (b) Adam’s number 45 has now been doubled to 90 and can form an octaval womb in which Isaac can be born, his life to then end at the doubling of Sarah’s 90 years to 180 years.

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Medieval Solfeggio within the Heptagonal Church of Rieux Minervois

This paper responds to Reichart and Ramalingam’s study of three heptagonal churches[1], particularly the 12th century church at Rieux Minervois in the Languedoc region of France (figure 1a).

image: The Church in situ

Reichart and Ramalingam discuss the close medieval association of the prime number seven[2] with the Virgin Mary, to whom this church was dedicated. The outer wall of the original building still has fourteen vertical ribs on the inside, each marking vertices of a tetraheptagon, and an inner ring of three round and four vertex-like pillars (figure 1b) forming a heptagon that supports an internal domed ceiling within an outer heptagonal tower. The outer walls, dividable by seven, could have represented an octave and in the 12th century world of hexachordal solmization (ut-re-mi-fa-sol-la [sans si & do])[3]. The singing of plainchant in churches provided a melodic context undominated by but still tied to the octave’s note classes. Needing only do-re-mi-fa-sol-la, for the three hexachordal dos of G, C and F, the note letters of the octave were prefixed in the solmization to form unique mnemonic words such as “Elami”.It is therefore possible that a heptagonal church with vertices for the octave of note letters would have been of practical use to singers or their teachers.

The official plan of Rieux Minervois

12th Century Musical Theory

In the 10th Century, the Muslim Al-Kindi was first to add two tones to the Greek diatonic tetrachord of two tones and single semitone (T-T-S) and extend four notes to the six notes of our ascending major scale, to make TTSTT. This system appeared in the Christian world (c. 1033) in the work of Guido of Arezzo, a Benedictine monk who presumably had access to Arabic translations of al-Kindi and others [Farmer. 1930]. Guido’s aim was to make Christian plainsong learnable in a much shorter period, employing a dual note and solfege notation around seven overlapping hexachords called solmization. Plainsongs extending over one, two or even three different hexachords could then be notated.

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Astronomy 4: The Planetary Matrix

The re-discovery of the ancient planetary matrix, seen through three my three books: Matrix of Creation, Harmonic Origins of the World and Sacred Geometry: Language of the Angels.

Harmonic Origins of the World inserted the astronomical observations of my previous books into an ancient harmonic matrix, alluded to using the sacred numbers found in many religious stories and the works of Plato, who might have been the savior of what Pythagoras had garnered from ancient mystery centers circa. 600 BC. According to the late Ernest G. McClain*, Plato’s harmonic technology had been widely practiced in the Ancient Near East so that, to the initiated, the stories were technical whilst, to the general population, they were entertaining and uplifting stories, set within eternity. Ancient prose narratives and poetic allusions conserved the ancient knowledge. Before the invention of phonetic writing in Classical Greece, spoken (oral) stories were performed in public venues. Archaic stories such as those attributed to Homer and Hesiod, gave rise to the Greek theatres and stepped agoras of towns. Special people called rhapsodes animated epic stories of all sorts and some have survived through their being written down. At the same time, alongside this transition to genuine literacy, new types of sacred buildings and spaces emerged, these also carrying the sacred numbers and measures of the megalithic to Classical Greece, Rome, Byzantium and elsewhere, including India and China.

* American musicologist and writer, in the 1970s, of The Pythagorean Plato and The Myth of Invariance. website

Work towards a full harmonic matrix for the planets

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Primacy of low whole numbers

  1. Preface
  2. Primacy of low whole numbers
  3. Why numbers manifest living planets
  4. Numbers, Constants and Phenomenology
  5. Phenomenology as an Act of Will

Please enjoy the text below which is ©2023 Richard Heath: all rights reserved.

What we call numbers start from one, and from this beginning all that is to follow in larger numbers is prefigured in each larger number. And yet, this prefigurement, in the extensive sense {1 2 3 4 5 6 7 etc.}, is completely invisible to our customary modern usage for numbers, as functional representations of quantity. That is, as the numbers are created one after another, from one {1}, a qualitative side of number is revealed that is structural in the sense of how one, or any later number, can be divided by another number to form a ratio. The early Egyptian approach was to add a series of unitary ratios to make up a vulgar* but rational fraction. This was, for them, already a religious observance of all numbers emerging from unity {1}.  The number zero {0} in current use represents the absence of a number which is a circle boundary with nothing inside. The circle manifesting {2} from a center {1} becomes the many {3 4 5 6 7 …}.

The number one manifests geometrically as the point (Skt “bindu”) but in potential it is the cosmological centre of later geometries, the unit from which all is measured and, in particular, the circle at infinity.

Two: Potential spaces

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Angkor Wat: Observatory of the Moon and Sun

above: Front side of the main complex by Kheng Vungvuthy for Wikipedia

In her book on Angkor Wat, the Cambodian Hindu-style temple complex, Eleanor Mannikka found an architectural unit in use, of 10/7 feet, a cubit of 20/21 feet (itself an outlier of the Roman module of 24/25 feet, at 125/126 of the 0.96 root Roman foot).

She began to find counted lengths of this unit, as symbols of the astronomical periods (such as 27 29 33) and of the great Yuga time periods proposed within Vedic mythology. Hence Mannikka’s title of Angkor Wat: Time, Space, and Kingship (1996). Whilst the temple was built by the Khymer’s greatest king, their foundation myth indicates the kingly line was adopted by a matriarchal goddess tradition.

Numerically Symbolic Monuments

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