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.

Hexachords conceptually overlapped (figure 2, left); another starting when the previous hexachord reached fa, the fifth, or when the melody again reached a given hexachord’s do of G, C or F. The Solmizations, prefixed with their note letters using Boethius’ Gamut system (Starting with our G (for Gamma ut G to e”) then A, etc., as we do today). For example, e’ would be uniquely called “Elami” since it was the note E, and the solfege la, for hexachord of G, whilst also mi, in the hexachord of C.

In contrast, the modern solfége of key signatures, without note letters, refers us to the major diatonic scale when equal tempered keyboards enable modulation of key signatures. By retaining the note letters, the hexachordal world could still reference the octave as a locational framework whilst also loosening the grip of do as tonic, as with modulation. The white notes of our keyboard were the basis of solmization with one exception: the minor hexachord starting at F had to impose the major diatonic T-t-s upon the T-t-T sequence of the diatonic scale by the solitary chromatic Bb.

Figure 2 The relations between Hexachords and the Octave of note names.
[on left, Willi Apel, 1969]

The solmization code created a namespace of unique composite words[1]. By combining note letters and hexachord positions, notes became unique words like Elami. Each note became linked to Beothius’ Gamut from G to e”, the solomised names explicitly identifying their context in the octave as well as the hexachords they belonged to (Figure 3).

Figure 3 The Solmization namespace combining Boethius’ note letters and Guido’s Solfeggio [Willi Apel, 1969]

When melodies exceeded the hexameter within which they were currently set; “In order to accommodate melodic progressions exceeding the compass of one hexachord, two (or more) hexachords were interlocked by a process of transition, called mutation”, since “in medieval theory the compass of tones was obtained not by joined octaves but by overlapping hexachords” and “tones of higher or lower octaves were not considered ‘identical’ within a Boethian scale of G to e””. [Willi Apel. 1969]

The Church as Octave within Solmization

If do of the “natural” hexachord (C) is placed on the (exactly) northern outer vertex of the fourteen vertices, then the three round pillars land, using Just intonation, in the midst of the Pythagorean tones of the major diatonic whilst the four vertex-like pillars coincide with the uniquely Just tones and semitones[1] (figure 4). The southern door marks the tritone between fa (F the minor hexachord) and sol (G, the hard hexachord). The walls of the church could therefore have usefully symbolised the intervals and note classes of the major scale[2] during the perambulation of the hexachordal plainchant, verbalized using Solmization. That is, if the church symbolized the successive octaves of the tonal world notated using hexachords, the building might have been a regional school for training singers, outside the customary cathedral and monastic schools of the 12th century. Guido’s method (staff notation and solfeggio and solmization) rapidly became famous and was widely adopted throughout north Italy and elsewhere[3]. When built, 12th century Languedoc and northern Italy was strongly populated by Cathars, so triggering the crusade from Rome and hence the subsequent confiscation of the church from its feudal owner.


[1] The practical scale of the day would have been the major diatonic since its three major thirds (between do and mi, fa and la and between sol and si) are achieved using the fifths and fourths of Pythagorean tuning in combination with the major thirds. This automatically generates the different tones and semitone found in Just intonation: T = 9/8 and t =10/9 form, in combination, the major third of T × t =5/4, short of the perfect fourth by the new Just semitone, s = 16/15.

[2] the natural scale for Just intonation when tuned using fifths, fourths and major thirds

In numerical tuning theory, the Virgin Mother would be the perfect symbol for an heptagonal church since the world of music springs from an octaval womb (whose number symbol is 2); only the male numbers (3 and 5) can reach into and divide the octave to create octaves of Pythagorean and Just intervals, then symbolic of Christ’s birth. The seven intervals and the notes of the diatonic scale provide a framework within which the magic of hexachordal singing expresses melody with a suppressed Ego or tonic. Hexachordal music strays across many tonic contexts. Numerical harmonists may have viewed tonics as titular deities of the limiting numbers required to theoretically generate Just Intonation[1], like the demiurges creating worlds but becoming an enemy of melodic freedom within them, by seeking to reference everything to their tonic. Arguably the natural tension, between static tonics of the octave and developmental movements like those found in hexachordal music, manifested the Classical traditions of sonata, concerto and symphony.

Drawing the intervals within the Church

If the two types of tone are each given a span of two or three sides of the tetradecagon, and the semitone a span of only one side, the total would be 3 + 2 + 1 + 3 + 2 + 3 + 1 equalling 15 sides rather than 14. But if one respects the natural symmetry of the tone circle about Re, as the (modern) Dorian scale, then one can make the initial tone of 9/8 symmetrical with the following tone of 10/9. In practice, nothing is lost since the church is only loosely a tone circle, with no imposition of logarithms except for those native to the ear, that hears intervals of the same size as the same size irrespective of pitch. Modes other than major could then have similarly been expressed by choosing other starting notes and vertices explicitly given within the fixed solmization words as the note letter prefix[2].

Figure 3 The encoding of intervals within the church

In the arrangement proposed, the disposition of round pillars coincides with the disposition of Pythagorean tones (of 9/8) on the outer wall, whilst the vertex-like pillars face the Just tones of t = 10/9 and s = 16/15. Pythagoras saw these now-eponymous tones of 9/8 as divinely perfect and hence a circular form is appropriate: The pure tones 9/8 are born (in numerical tuning theory) only by the divine male prime number 3 and the female octaval number 2 seen in 9/8[1]. In addition, Just tone 10/9 and semitone 16/15 require the humanly-male prime number 5 to birth them within the womb of the octave’s tone circle. The northern round pillar would also identify the necessarily shortened whole tone as Pythagorean, despite its being shortened, thanks to the association of pillar shapes with either type of whole tone.

As stated above, one can imagine that in a church, designed to represent an octave in the round, one could conduct the choir in Solfeggio.

My book on the role of musical theory in terms of both the number involved and ancient cosmological thinking is called The Harmonic Origin of the World. It came about through a virtual apprenticeship with Ernest G. McClain whose books The Myth of Invariance and The Pythagorean Plato revolutionized the subject (both books can be read in pdf at his posthumous website.)

References

Apel, Willi. Harvard Dictionary of Music. 1969.

Farmer, Henry George. Historical facts for the Arabian Musical Influence. 1930.


[1] Three Heptagonal Sacred Spaces by Sarah Reinhart and Vivian Ramalingam, pages 33-50 in Music and Deep Memory: Speculations in ancient mathematics, tuning, and tradition. in Memoriam Ernest G. McClain. ICONEA Publications 2018.

[2] which cannot join with any other number below ten or even twelve.

[3] then known as Ut–re–mi-fa-so-la-Sa-Io after the mnemonic “Ut queant laxis, resonare fibris, Mira gestorum, famuli tuorum, Solve pollute, labii reatum, Sancte Iohannes”: So that your servants may, with loosened voices, resound the wonders of your deeds, clean the guilt from our stained lips, O Saint John.

[1] Ernest G. McClain, The Pythagorean Plato 1978.

[4] a namespace arises when each name is unique whilst shared elements common to the other words, such as note letters and the solfege within hexachords.


[2] This transpositional modality is reminiscent of our later key signatures to which solfeggio is now applied.

[3] https://en.wikipedia.org/wiki/Guido_of_Arezzo.

[1] Ernest G McClain The Myth of Invariance, 1976



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

In my first book (Matrix of Creation,) I had not yet assimilated McClain’s books, but had identified the musical intervals between the lunar year and the geocentric periodicities of the outer planets. To understand what was behind the multiple numerical relationships within the geocentric world, I started to draw out networks of those periods (matrix diagrams) looking at all the relationships (or interval ratios) between them. This revealed common denominators and multiples which linked the time periods through small whole numbers. For example, the 9/8 relationship of Jupiter’s synod to the lunar year could be more easily grasped in a diagram to reveal a structural picture, visualized as a “matrix diagram” (see figure 1).

Figure 1 Matrix Diagram of Jupiter and the Moon. figure 9.5 of Matrix of Creation, p117.

One can see the common unit of 1.5 lunar months, at the base of the diagram, and a symmetrical period at the apex lasting 108 lunar months (9 lunar years referencing the Maya supplemental glyphs). I re-discovered the Lambda diagram of Plato (figure 8.7), and even stumbled upon the higher register (figure 2) of the Mexican Quetzalcoatl (figure 8.1) made up of {Mercury, the eclipse year, Tzolkin, Mars, Venus}, Venus also being called the feathered serpent. These periodicities are of adjacent musical fifths (ratio 3/2), which would eventually be shown as connected to that of the outer planets, using McClain’s harmonic technology, in my 5th book, Harmonic Origins of the World (see figure 3).

Figure 2 Incomplete discovery of the Maya Quetzalcoatl, in fig. 8.1 of Matrix of Creation. I had not noticed that 390 days, times 2, is 780 which is the synod of Mars! This is in fact 41.8949 Node Days, which might be significant.

Also called the flying serpent in Pharaonic Egypt, this set of musical fifths, apparently undocumented in the near east, was part of the Mexican mysteries of the Olmec and Maya civilizations (1500 BC to 800 AD). The serpent is flying harmonically, 125/128 above the inner planets – for example, the eclipse season is 125/128 above the lunar year: 354.367 days × 125/128 = 346 days, requiring McClain’s harmonic matrices to integrate these two serpents, in Harmonic Origins of the World (figure 9.3).

Figure 3 The two harmonic serpents of “Heaven” and “Earth”

By my 6th book, Sacred Geometry: Language of the Angels, I had realized that the numerical design surrounding our “living planet” sits, is a secondary creation – created after the solar system, yet it was discovered first, before the heliocentric, exactly because the megalithic observed the planets from the Earth. I therefore propose an alternative timeline for the ancient mysteries. Instead of proposing a progenitor civilization such as Atlantis, as per Plato’s Timaeus: of an island destroyed by vulcanism. https://en.wikipedia.org/wiki/Timaeus_(dialogue)

My working hypothesis is that Atlantis and similar precursor solutions, simply “kick the can down the road” into an as-yet-poorly-charted prehistory for which there is no strong evidence. In contrast, the sky astronomy and earth measures found in use by the megalithic can only be the product of that singular megalithic culture. There is clear evidence of megalithic monuments recording an understanding of the cosmos then found in the ancient mysteries. Megalithic evidence can show the geocentric world view as being their achievement, based upon the numbers they found using geocentric observations, counting lengths of time, using horizon events and the mathematical properties of simple geometries.

Geocentrism was the current world view until it was superceded, by the Copernican heliocentric view. The new solar system, held soon found to be held together by naturals gravitational forces between the large masses, forces discovered by Isaac Newton. The subsequent primacy of heliocentrism, which started 500 years ago, caused humanity to lose contact with the geocentric model of the world, which had the planets in the same order relating the two serpents of outer and inner planets. All references to an older and original form of astronomy, based upon numerical time and forged by the megalithic, was dislocated and obscured by the heliocentric physical science and astronomy of the modern day – which still knows nothing of the geocentrically order that surrounds us.

Figure 4 The Geocentric Model by 1660

The geocentric model entered Greek astronomy and philosophy at an early point; it can be found in pre-Socratic philosophy … In the 4th century BC, two influential Greek philosophers, Plato and his student Aristotle, wrote works based on the geocentric model. According to Plato, the Earth was a sphere, stationary at the center of the universe.

Wikipedia: “Geocentric model” https://en.wikipedia.org/wiki/Geocentric_model

Primacy of low whole numbers

  1. Preface
  2. Primacy of low whole numbers
  3. Why numbers manifest living planets
  4. Phenomenology as a Native Skill

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

From one {1} springs two {2}, to which we owe all forms of doubling as reproduction without sexual division, called “barren” by Plato, yet giving the possibilities of complex worlds of different scale, in terms of their limiting number. This is the first true number of Creation which gives the quality of polarity between the two halves of (as yet) nothing, halves of a world that will create the beginnings of an everything. Super dense, as an initial Form of forms, all things will come to rotate around this Axis of axes*. (Axes, when pronounced with a long e, is the plural form of the word axis, meaning imaginary lines that run through the middles of things. The word axe is derived from the Old English word æces, the axe which divides into two. ) This is the birth of duality, as with the centre and circumference of a circle or positive and negative (opposite) charge, and the medium of the wave or vibration, which gave birth to dynamic systems, such as planetary rotation of an axis or an orbit.

When number was incarnated in our own planetary creation, it was Saturn who visibly delimited the outer limits of the visible planets. His name is close to Seth and Satan (as the necessary adversary of the heroic Horus) and he was seen as limiting unbounded growth within existence. Saturn expresses 5 synods of the planet Saturn in 64 {26} lunar months (but this is to jump in numbers, though not too far, to the planetary double octave {24 48 96} lunar months. Sixty-four governs the “eye of Horus”: a government deriving from the mythical conflict between the god Horus with his rival Set; in the context of two eyes, here the right eye was torn out but then restored, to then see the role of two within the creation, in the “afterlife”.

Figure 1 The Egyptian icon of the Eye of Horus as the components used to represent vulgar fraction as a series of powers of two.

https://en.wikipedia.org/wiki/Eye_of_Horus: It derives from the mythical conflict between the god Horus with his rival Set, in which Set tore out or destroyed one or both of Horus’s eyes and the eye was subsequently healed or returned to Horus with the assistance of another deity, such as Thoth.

The Moon finds an exact reunion with the earth day after exactly 945 days, which equal 32 lunar months of 945/32 (29. 53125) days, very close to the actual lunar month of 29.53059, being effectively exact as 57 seconds different.

While the number two can, in being divided, create new areas of interaction (including cosmic octaves), its mere extension {2 4 8 16 32 64 128 …) forms only a backbone of potentialities, these then (see later page) borrowed by higher limiting numbers such as 720, a number containing favourable numerical factors for the creation an ideal “family” of limits, metaphorically presented by Adam and the Patriarchs of the Bible.

Figure 2 The vision of the Godhead asleep on a pre-creational ocean (of his sweat) attended by the Goddess until he awakes for a new creation.

The number {2} as dyad manifests as the geometry of the line. The line {2} from a center {1} as a rotational vector becomes the geometry of the circle.

Three and four: Actualization

The actual creation has a different planetary symbol, the equilateral triangle with three sides, seen also as the capital Greek delta, whose value is 4 because the planetary system is an Activity involving forces. These numbers are reconciled as 3 x 4 which equals 12, the number of Autocracy * and balanced action. We are told in myth that Jupiter’s twelvefold nature was “the receiving of the measures” from Saturn, as 4. Jupiter is the planetary demiurge which Plato describes carving out the World Soul “octave” {6 8 9 12}, using ratios involving Three, the cubit and its reciprocal {3/2 4/3} * (Timaeus). Only when we reach the lunar octave {24 27 30 32 36 40 45 48} will the diatonic scale of eight notes emerge, the first and last being the same note, doubled.

The first true doubling {2 3 4}, between 2 and its square, holds the first type of penetration of the octave, by Three {3}. With three, the Demiurge forms his World Soul using intervals involving only Two and Three {3/2 4/3}, which can create the fifth and fourth notes (“dominant and subdominant”) of an octave.

The number {3} gives form to the first geometry of area, manifest in the triangle which, given a right angle, becomes trigonometrical, as the functional mediation between the line and the circle.

The number four connects the relatedness of the Triad (3) with the existentially actual to provide an engine in which Form can become Substance through an intermediate pair of terms that fulfill the gap {2} between form and the actual situation. One could say this is the first instance of filling the octave with tones {8}, intermediate between 2 and 8. Four is the first square number which in geometry is called square as an area equal to 4 has sides equal to 2

Five: Vitality and Life

Coming next, Five {5} will also be able to divide the coming “octave” {3 4 5 6} in a superior way than three and two can alone, by redefining a new tone {10/9} for Just intonation and a corresponding semitone {16/15}, these cancelling the excessive powers of three produced by tuning only with three {3}, called the cycle of fifths, which used successive fifths and its inverse{3/2 4/3} because the ear can best define the larger musical intervals. The octave between three and six defines the framework of Just intonation where three intervals span the octave {4/3 5/4 6/5}, summing to doubling {2}.*

*This was probably referred to as “the three strides of Vishnu”, Trivikrama (‘having three steps’) being one of his 1,000 names.

The planet Venus brings a new type of harmony, which is also the sixth note {8/5 (= 1.6)} of a diatonic octave (see this page) since her synod of 584 days is 8/5 of the practical year of 365 days. The Fibonacci series allows whole number approximation to the Golden Mean {φ} between adjacent members obtained as being the sum of the two preceding numbers {0 1 1 2 3 5 8 13 21 34 55 …} unlike the ordinal set {1 2 3 4 5 6 7 8 9 …}, the latter instead obtained by more simply adding one more unit {1} to the preceding number. Unlike the musical tone and semitone of Jupiter and Saturn relative to the lunar year, Venus is resonant with the Earth’s orbital period of 365 whole days, and this type of orbital resonance, with each other, is mutually attractive, providing the lowest and most stable energy between the two planets. The inner orbital diameter (semi major axis) divided by the difference in orbital diameters, equals 2.618, or phi squared {φ2}*. (See later page for more on their orbits) Structures of growth, based upon Fibonacci ratios, are commonly found within living bodies, which must achieve this algorithm in which their present size added to their digestion of previously eaten food results in the sum of the two.

Figure The Fibonacci series in two dimensions are common forms of living growth.

The Venus synod will be seen to fit inside the octaves of the Moon because 20 lunar months is 590.6 days which is less than the synodic comma {81/80} of her 583.92 day synod*.

*The synodic comma is the exact ratio connecting Pythagorean and Just versions of the same note. One of the Indian temple designs is a nine-by-nine square grid which makes the number of equal-sized sub-squares {81) divided by the count where the central square is not counted gives the ratio of the synodic comma {81/80}.

Music: Child of the First Six Numbers

The larger intervals of numerically larger octave doublings are in this way foreordained in the first six numbers {1 2 3 4 5 6} and their relative size to each other, are five musical intervals {2/1 3/2 4/3 5/4 6/5}.  Doubling has led to the pillars of Plato’s world soul {3/2 4/3} and three when doubled {3 4 5 6} has led to the three strides {4/3 5/4 6/5}, both sets summing to Two {2}.

The first six numbers, creating all the large tones of musical harmony, punctuated by Seven.

Between the five musical intervals, the tones and semitones of Just intonation are to be found {9/8 10/9 16/15} so that, as a tuning system, the Just system leads automatically to the tones and semitone of the seven modal scales, in both melodic and polyphonic harmony.

When the World Soul {6 8 9 12} is twice doubled {24 48} and doubled again {48 96}, the two octaves express the world numbers of Gurdjieff {24 48 96}, but now these numbers correspond to lunar months and, as with music when heard, all of the possible intervals are compresent in the instrument, the Moon illuminated by the Sun, since one can count from any lunar month, over any number of lunar months, to achieve any of the larger and smaller intervals between these octaves. And it is now true that the three principle planets of Jupiter, Saturn and Venus are present at the second, fourth and sixth notes, each of these relevant to Gurdjieff’s theory of octaves as stated by him in Russia, and his cosmic epic Beelzebub’s Tales. And J G Bennett continued to build on what Gurdjieff had expressed, without knowledge of the astronomical references, to populate his own Dramatic Universe, in 4 volumes and many compendia (see Bibliography). Of particular importance is how human beings figure within the cosmic vision, without which a planetary cosmos involving consciousness and creativity would be meaningless. If one resists the modern functional view of cosmogenesis: music, or other forms of harmony, can be seen to redeem the creation of a world like ours, through the short cuts numerical systems naturally provide for us, through a gravitational environment that can provide these.

Coming soon: Why numbers manifest living planets

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

Interpreting a monument using its metrology can be contentious. For example, in the megalithic period the established position has been that there was no metrological tradition and, to be found proposing one can cause your work to be ignored if not exiled from peer-reviewed journals, as was eventually the case with Prof. Alexander Thom.

At Teotihuacan, Japanese professor Saburo Sugiyama proposed an architectural unit of 83 centimeters was used, since the monumental complex would then clearly have numbers of these units corresponding to significant celestial periods, as if periods had been counted out within the City: the eclipse half year of 173 days at the Moon Pyramid, the Tzolkin of 260 days at the Sun Pyramid, and the Venus synod of 584 days at the Quetzalcoatl pyramid’s compound. More such day lengths and a well-known harmonic matrix were also seen in my Harmonic Origins of the World.

Astronomical counting within Teotihuacan (adapted from fig. 8.9)

Sugiyama did not reply to my message that his Teotihuacan Measuring Unit of 0.83 meters was the 2.72 foot length of Thom’s megalithic yard, implying some connection between Olmec/Maya Mexico and megalithic Europe. This was probably not welcome. Wikipedia’s editors of the “Megalithic Yard” page also objected to my mentioning this since it was I that had noticed this correspondence.

Over a 20 year period, Eleanor Mannikka found a numbers that were symbolic** or actual long counts of the solar and lunar years. In her thesis, these numbers were embodied as a ritual background for visiting pilgrims, whose steps corresponded to numbers – the megalithic yard being a metrological step of 2.5 feet. Her eventual counts emerged by a protocol that skipped thresholds, ran beyond, or started before a threshold, the counts were being human walkways but also excellent surfaces for doing accurate metrology.

**Her rule-based system that revealed numbers may well be a later function of the eventual monument, made to correspond with the numbers found in Hindu epic stories, since these are lavishly illustrated within extensive bas-reliefs, visible to pilgrims, depicting major Hindu myths. Statues of the gods punctuate the building’s many walkways to express the Indian practice of parikrama, of circumnavigating holy sites (such as around Mount Kailash or the great dome of Sanchi).

The Temple as AN Observatory

The symbolic use of numbers could only have become established through cosmic measurement in which astronomy (before our own) counted the actual numbers of days or months between repeating cycles of celestial alignment, and the differences and ratios between these. That is, ancient symbolic numbers originated in the Sky, where number-laden events measured in days or months generate whole numbers that were only then held to be sacred. One might think Angkor Wat too recent to have been constructed to suit this ancient sort of astronomical work. But the temple’s explicit orientation, to the west, was suited to just that. This made the temple perfect for observing and counting all sorts of time-counts, repeating measurements made millennia before using megalithic monuments.

That is, Angkor Wat is a current-era megalithic monument to the sky gods, these illustrated using the famous tableau of Vedic and later Indian myths.

The sun and moon set to the west**, each having a maximum range north or south of west. The sun at winter and summer solstice defines a fixed range within the solar year, depending on the latitude of a given site. In contrast, the Moon ranges over the horizon when setting over one orbital period of 27 1/3rd days. However, the moons orbit is skew to the sun’s path (ecliptic) so that the moon rises above and below, except at its nodes where eclipses can take place. These nodes move backwards so that the moon’s range on the horizon expands and contracts over 18.618 solar years.

**Looking west is very convenient since the sun or moon approach the horizon rather than suddenly appearing as they do in the east.

As a consequence, there are seven key points on the western horizon, the maximum standstill to north and south, the minimum standstill to north and south, the solstice extremes of the sun in summer (North) and winter (South), plus the equinox sunrise**. It is possible to calculate these alignments for the virtually flat terrain of Cambodia as in Figure 2.

**The Equinox sunset is a very exact point to measure since the sun appears to move rapidly on the horizon, between sunsets.

Figure 2 The alignments of Sun and Moon to the west (Left) around 1000 CE at the latitude of Angkor Wat using the Processing.org framework.

The notion of alignments seems to throw light upon the highly specific elements of Angkor Wat (see figure 3), if these alignments were viewed from the north eastern and south eastern corners of the raised temple enclosure.

Figure 3 Viewing the alignments of Sun and Moon, to the west (on Left), from the eastern corners.

There is a natural north-south symmetry, where the alignments to the solstice cross in the pream cruciform (see figure 4). The punctuation of the towers of the temple, seen from the eastern corners, would provide landmarks to calibrate the movement of (a) the sun in the year and (b) the moon within the lunar orbit, as the 18.6 year nodal movement expands and contracts the lunar range.

Figure 4 The Alignments seen within the plan of the temple complex.

The cruciform terrace outside the walls and nine fold cruciform within, could relate to the crossings of alignment and the periodicity of these cycles which would be countable in days using units of length.

The maximum moon alignments near 1000 BCE were 30 north and south or west, and one can plot those alignments over a flat Cambodia to the boundaries with Thailand which are, in contrast, significantly mountainous (see dark green areas at end of yellow alignments in Figure 5.

Figure 5 Google Earth view of the mountains at the end of both maximum moon alignments.

Parallels with the Megalithic near Carnac

The basic idea of such an observatory is a stone square instead of a stone circle. Alignments can be built-in, between back-sight observation points and fore-sight marker stones, marking the horizon location of an extreme event such as solstice. An observatory location can also look to an horizon event for which a distinct natural feature exists on the horizon, from that location. The stone perimeters of Carnac, called cromlechs, are various shapes but at Kerlescan, the cromlech is a rounded square, where the western perimeter is concave towards the east. That is, it faced rising events on the eastern horizon instead of setting events to the west.

Figure 6 Alexander Thom’s survey of the Kerlescan cromlech.

Otherwise, the “setup” is conducive to the observation of the sun and moon possible at Angkor Wat. Below I show how the observatory could work for the epoch 4000 BCE. The red lines are solar extremes and green lines are lunar maximum and minimum extremes. Equinoctial events at Spring and Autumn complete the inherently seven-fold nature of such phenomena.

Figure 7 Possible use of the Kerlescan cromlech, as an observatory facing east rather than west (at Angkor Wat).

Developmental Roots below 6

Square roots turn out to have a strange relationship to the fundaments of the world. The square root of 2, found as the diagonal of a unit square, and the square root of 3 of the diametric across a cube; these are the simplest expressions of two and three dimensions, in area and volume. This can be shown graphically as:

The first two roots “open up” the possibilities of
three-dimensional space.
Continue reading “Developmental Roots below 6”

Music, part 1: Ancient and Modern

We would know nothing of music were it not that somewhere, between the ear and our perceptions, what we actually hear (the differences between different frequencies of sound, that is, different tones) is heard as equivalent musical intervals (such as fifths, thirds, tones, semitones, etc), of the same size, even when the pitch range of the tones are different. This is not how musical strings work, where intervals of the same size get smaller as the pitch at which tones occur, grows larger. On the frets of a guitar for instance, if one plays the same intervals in a different key, the same musical structure, melodic and harmonic, is perfectly transposed, but the frets are spaced differently.

The key is that human hearing is logarithmic and is based upon the number two {2}, the “first” interval of all, of doubling. This can only mean that the whole of the possibilities for music are integral to human nature. But this miraculous gift of music, in our very being, is rarely seen to be that but, rather, because of the ubiquity of music, especially in the modern world, the perception of music is not appreciated as, effectively, a spiritual gift.

Music is often received as a product like cheese, in that it is to be eaten but, to see how this cheese is made from milk requires us to see, from its appearance as a phenomenon, what music perception is made up of . Where does music come from?

Normally a part of musicology, that subject is full of logical ambiguities, confusing terminology, unresolved opinions, and so on. Those who don’t fully understand the role of number in making music work, concentrate on musical structures without seeing that numbers must be the only origin of music.

The ancient explanation of music was that everything comes out of the number one {1}, so that octaves appear with the number two {2/1}, fifths from three {3/2}, fourths from four {4/3}, thirds from five {5/4} and minor thirds from six {6/5}. Note that, (a) the interval names refer to the order of resulting note within an octave, (b) that intervals are whole number ratios differing by one and that, (c) the musical phenomenon comes out of one {1}, and not out of zero {0}, which is a non-number invented for base ten arithmetic where ten {10} is one ten and no units.

Another miracle appears, in that the ordinal numbers {1 2 3 4 5 6 7 8 9 etc.} naturally create, through their successiveness, all the larger intervals before the seventh number {1 2 3 4 5 6 7} leaving the next three {8 9 10} to create two types of tone {9/8 10/9} and a semitone {16/15} thereafter {11 12 13 14 15 16}: by avoiding all those numbers whose factors are not the first three primes {2 3 5}. Almost the whole potential of western music is therefore built out of the smallest numbers!

This simplicity in numbers has now been obscured, though the structure of music remains in the Equal Temperament form of tuning evolved in the last millennium. By having twelve equal semitones that sum to the number two, we can now transpose melodies between keys (of the keyboard) but we have pretty much lost the idea of scales. Instead, each key is the major diatonic {T T S T T T S} (where T = tone and S = semitone intervals) starting from a different key. The fifth is called dominant and fourth subdominant and the black notes (someway fiendish to learn) required to achieve the major key in all keys but C which is all white keys.

The old church scales are achievable by over ruling the clef with accidental notes, and the reason for different keys sounding different is that they contain aspects of what were the scales. So a pop song, for example, is usually in a scale. “Bus Stop” by the Hollies was in the Locrian scale.

Equal Temperament enabled the Western tradition to create its Classical repertoire but it has made ancient musical theory very distant and has abandoned the exact ratios it used to use since every semitone is identical and irrational. Plato described this kind of solution as the best compromise, where every social class of musical numbers has sacrificed some thing of their former self in order to achieve the riches versatility bestows upon modern musical composition.

To be continued.