This graphic demonstrates how the inner geometry within numbers can point to significant aspects of Celestial Time or here Space regarding the relative sizes of the Earth and the Moon, namely 11 to 3 according to pi as 22/7.

In some ways one cannot understand numbers without giving them some kind of concrete form as with seeing them as a number of identical units. Sixteen units can make a square of side 4 since the square root of 16 is 4 and 6 is factorial 3 (3! = 1 x 2 x 3 and 1 + 2 + 3) which is triangular, so together they make 22, and if the triangle to placed on top of the square, like a house and its roof, then the house is 7 tall. If you want an accurate approximation to pi of 3.14159 … (pi is transcendental), the 22/7 is good and the house defines it.

This adds another mystery to this form of pi often used in the ancient world where numbers were best handled as whole numbers and ratios of these. This pi allows a circle of diameter 11 to be set within a square of side 11, whose perimeter is then 44. This can be seen in the diagram as made up of 16 yellow squares and 6 blue ones, centered on the circle and making 22 squares in all.

If one looks to the end of the 7th square, as a radius, then 22/7 will deliver the dashed circle (red) of circumference 22 and hence equal to the house number (16 +6 = 22) just 1.5 units beyond the first circle (green). This is called the equal perimeter geometry and a small circle radius 1.5, diameter 3, will “orbit” the inner green circle and the ratio between the circles is obviously 11 to 3, and this is exactly the ratio between the mean Earth and the Moon.

It was thought, by John Michell, that the model was well known in the megalithic since simple experiments in geometry, as above, delivers the relationship between a circle’s diameter and its circumference with very small whole numbers. My own work finds it is indeed prevalent within the design of later buildings, for example in domes, circular windows, and sacred pavements. If so, such buildings became sacred spaces as models of the Earth and moon. Many examples are explored and interpreted in my Sacred Geometry: Language of the Angels.

The oral world of early numeracy was rather like number theory, where numbers can be observed as being related to the geometries of square, triangle and hexagon. The Islamic world of the Sufis appears to have continued this form of numeracy.

A recent book about possible Platonic numeracy in the Quran, Plato and the Quran, suggests the numbers 3 to 9 were stated as a puzzle inviting both the addition and multiplication for seven consecutive numbers, to generate two significant numbers, 33 and 20160, where 33 reminds us of the solar hero period of 33 years and 20160 is twice 10080, the diameter of the equal perimeter model of the Earth and the Moon.

Many centuries later, an early poem of Sufi master Ahmad Yasavi, in present day Khazakhstan, expressed a similar additive formula; that one should add the numbers 4 to 8 together and, when done, this generates the number 22. Twenty two was important in the ancient world and was seen to form the geometry of the equal perimeter square side 11 and circle diameter 14, which, can represent the relative sizes of the Earth and Moon. The geometry is a manifestation of a useful approximation to pi, as 22/7 = 3 + 1/7 or 3.142857, instead of the transcendent number 3.14159 … .

If one looks at the sequence, there are four numbers starting with four and so part of 22 is here 4 x 4 = 16, a square number. In addition there are the added ones of enumeration.: 4 + 1 = 5 + 1 = 6 + 1 = 7. These add up to 1 + 2 +3 = 6, a triangular number which one famously sees in the Tetractys of 1 + 2 + 3 + 4, then usually expanding downwards from 1, and this then adding to 6 + 4 = 10.

The lesser triangle of 6 can sit on top of the square of 16 to equal 22 while looking like a house roof for the square. The whole structure is seven units tall and I am looking at calling this a house number, but perhaps it is known somewhere in the literature – please let me know.

The first house number must be 5, a single 1 above 4 = 2 x 2.

The second house must be 12, a triangle of 3 above 9 = 3 x 3.

The fourth is 22.

The fifth is 35, a triangle of 10 above the square of 25 = 5 x 5.

The sixth is 51 , a triangle of 15 above the square of 36 = 6 x 6.

In each case, the triangle’s bottom row can be seen to share the top row of the house’s square and the triangular roof is most simply equilateral.

I wish happy celebration to my worldwide visitors, between the solstice and new calendar year; inviting you to see this “house number” as a “room” with a celestial “roof”.

There are two things we can count in this world, one is the number of objects on the Earth and the other is the number of time periods between events in the Sky.

photo: The Moon, with Jupiter and Mars, on 11th January 2018. (see end for interpretation)

Objects are counted in an extensive way, from one to an almost infinite number, the count extending with each addition (or multiplication) of a population.

Time periods appear similar but in fact they emanate from measurable recurrences, such as phases of the moon, and these derive from the behaviour of celestial objects as they divide into each other.

For instance, the unit called the day is created by the rotation of the earth relative to the Sun and the lunar month by its orbit around the Earth relative to the Sun, and so on.

Thus, time originally came from the sky. Furthermore, it largely came from the zodiacal band of stars surrounding the Earth within which the planets, Sun and Moon progress eastwards. The Earth’s own orbital motion is superimposed upon those of the other planets and the inner planets (Mercury and Venus) also appear to orbit a Sun that appears to orbit the Earth once a year.

The zodiacal band is naturally divided up into a number of constellations or stars and about three thousand years ago it became popular to follow the Sun throughout the year into 12 constellations whilst the Moon tends to create 27 or 28 stars (nakshatras) where the Moon might sit on a given evening. When the moon is illuminated by the sun, the primordial month has 29 1/2 days and twelve such in less than a year hence perhaps first defining the 12-ness of our months within the year.

All celestial cycles recur and this has formed our notion of eternity, that the sky world is made up of cyclic time rather than extensive time – every year being the same cycle seen again but then numbered so that they can be referred to as to when something happened in the past. The intensive reality above our heads is the polar opposite of extensive counting of time we see in History where numbered years and days within named months provide an unbroken continuum of time and famous people are said to have made history through their actions at a given date.

Whilst on Earth we might measure feet or meters between objects, above we effectively measure angles and angular rates to arrive at a synthesis between intensive and extensive time we call a calendar, an inevitable necessity for an organised civilization. And the moon and then the sun gave rise to the early calendars that naturally led to the arising of history as a human phenomenon. The oldest myths were connected to the sky, and were less than historical because the language of the sky had not been formalized in a way we would recognize.

Myths speak of eternal patterns that repeat rather than of existential events, on earth. The sun, moon and planets were seen as gods whose generative functions were hailed as emerging from their interactions with each other.

It has been widely assumed that “primitive” thought was premature, fantasizing planetary gods out of thin air with an as yet unripened grasp on logic and reason. But a simpler explanation, for the equation of planets with super beings, was their finding of special numbers linking the planetary cycles when these were counted and compared. This quantification of celestial time evolved from knowing the days in a year and a month, into a running calendar – of various sorts. The Maya Long Count is an example where numbers could interact through week lengths of 13 and 20 days to give a sacred calendar of 260 days whilst in historical times the 7 day week emerged, tied to Saturnian time. In this way, a calendar could add weeks adapted to societal events such as having a market every Tuesday.

This is a big subject where we have all the sky data but do not spend time understanding it. In the past, the sky was our constant companion between few man-made spaces. The sky sits within the horizon and so was like a primordial cave for humans and, the sky became an early teacher through its phenomena.

Jupiter and the Lunar Year

The lunar month is like the common denominator of what happens inside time. The sun illuminates the phases of the moon during its month so that, the month combines the movements of the moon and the sun to form a synthetic (combined) period of 29 1/2 days and twelve of these months fit inside the solar year as the lunar year of 12 1/3rd months (354.367 days). Jupiter has its own relationship to the sun in that, when the sun is opposite the moon, Jupiter describes a loop amongst the stars, and strangely there are 13 1/2 lunar months between loops (Jupiter’s synodic period of 398.88 days). 13 1/2 months divided by 12 months is the ratio 9/8, a musical whole tone.

But in the image above, of Jupiter and the Moon, the moon would be full if Jupiter was going to loop (as earth “overtakes” Jupiter on the “inside lane” – the planets inspiring ancient racetracks). Mars is another “outer planet” which loops in the same way and so Mars is also not looping.

But without understanding these matters, the picture cannot be understood. The phase of moon shows where the sun is. The planets have been in conjunction. If Venus had been present, then it has a 4/3 ratio to Mars but has to remain close to the sun to appear first as an evening star, then a morning star, in a cycle 8/5 years (584 days) long compared to Mars synod (between loops) of 780 days. Less accurate than Jupiter to the Lunar year, by a day. This is what I mean by being inside time, where all the celestial bodies have relationships to one another, when these are seen by us from earth.

This is how I started, with my first book Matrix of Creation. The musical ratios and their entrance into ancient stories was explored in Harmonic Origins of the Earth. How ancient humans counted time was discussed in Lords of Time and a unified treatment made in Language of the Angels. Used alongside archaeology, more can be understood about the prehistoric and early civilizations since astronomy was the first real subject for the human race.

In March next year my new book will be released (see publisher website). The book has been typeset and is out for printing, having been favorably reviewed by a peer group. (The original date was February, but printing schedules have had to be adjusted.)

figure: the punctuation of towers and western outlook. Possibly a funerial building for the king, it could be used as a living observatory and complex counting platform for studying the time periods of the sun, the moon, and even the planetary synods.

Some new material was added during production, including chapters on the design of Angkor Wat (chapter 9) and St Peter’s basilica in Rome (chapter 10), and some early articles on these can be accessed on this site, most easily through the search function.

As you can see, my books partly emerge through work presented on this website. This has been an important way of working. And whilst I am providing some ways of working that could be duplicated by others, at its heart, my purpose is to show that the celestial environment of our living planet appears to have been perfectly organized according to a numerical scheme.

My results do not rely on modern techniques yet I have had to avail myself of modern techniques and gadgets to work out what the ancient techniques arrived at over hundreds if not thousands of years.

My basic proposal is that ancient astronomers learned of the pattern of time in the sky by counting days and months between events on the horizon or amongst the fixed stars. Triangles enabled the planetary motions to be compared as ratios between synodic periods.

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 12^{th} 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

12^{th} Century Musical Theory

In the 10^{th} 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 12^{th} 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, 12^{th} 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.