The picture below is a composite of three things

- The Chartres eleven level labyrinth discussed in chapter seven.
- The iconography of Thoth as Pi within the circle (from
*Temple of Man*). - The hexagonal number 19 as circles

The 22 units of the 21 unit sector of Thoth’s fathom correspond to 19 “cogs” of the circumference of the Chartre labyrinth.

It is fabulous that the cogs are used to define, by their centres, the perimeter as the unit called *ped manualis *by the builders*, *according to John James (the foremost investigator of that cathedral’s construction order – see his website).

Whilst the *ped manualis* is a Royal Foot, 8/7, in Neal’s *Standard Geographical *variation (times 126/125 and times 176/175), it is also close to 22/19 feet (different by 8 thousandths of an inch and 99.94% accurate). Thus whilst 19 cogs equal 22 units, the cogs are 22/19 which times 19 is 22 feet – plus 19 is a hexagonal number and there is the motif of six petals in the centre.

The entire circumference is, like the iconography of Thoth, 6 x 22 = 132 feet long. Using Pi at 22/7, the 22s cancel and the result is a diameter of 6 x 7 or 42 English feet. Like the Scottish brochs, the units directly interpret the ideal value of Pi itself as 22/7, employing as it does the prime numbers 11/7 that also define "Ancient Model of the World".

]]>*Published in Nexus Magazine in 2004*

When understanding the origins of human knowledge, we tend not to look into the everyday aspects of life such as the calendar, our numbering systems and how these could have developed. However, these components of everyday life hold surprising clues to the past.

An example is the seven day week which we all slavishly follow today. It has been said that seven makes a good number of days for a week and this convenience argument often given for the existence of weeks.

Having a week allows one to know what *day of the week* it is for the purposes of markets and religious observances. It is an informal method of counting based on names rather than numbers. Beyond this however, a useful week length should fit well with the organisation of the year (i.e. the Sun), or the month (i.e. the Moon) or other significant celestial or seasonal cycle. But the seven day week *does not fit in* with the Sun and the Moon.

Whilst some historical cultures ran a 360 day year, within which a 10 day, 6 day, 12 day or even 8 day week would fit, seven does not divide into 360. Neither does it divide into 365, the number of whole days in a solar year.

Seven would divide into a year of 364 days as the familiar 52 weeks in a year, only exactly instead of one day out. This is why our own 365 day year leads to days of the week moving forward one day every most clear when birthdays and Christmas are on different weekdays. The seven day week’s “fit” to a 364 day year leads to some familiar numerical logic, for there can then be 13 months of 28 days, each month then having four seven day weeks. In such a calendar, the days of the week within the year are kept synchronised by having a special extra day. More important, with a 364 day year there is then some justification for having a seven day week.

We know that this calendar of 364 days must have been practiced within living memory for the expression “King for a year and a day” hails from the time when society was centred around women rather than men. It is quite clear that matriarchy and not patriarchy once ruled domestic and tribal politics. This natural fact of life, emerging out of the stone age, ran into the Neolithic: As humankind developed a more settled agrarian economy the “gatherers”, within the hunter gatherer partnership, were the home builders and the creators of new humans.

There is a connection between the seven day week and this age of different sexual politics shown by the archaic use of a “Saturnian” calendar in Crete.

In the modern age there are always attempts to say that four weeks of seven days is a lunar month, but the month is twenty nine and a half days long according to the Moon’s phases and not twenty eight. The lunar orbit of the Earth (a hidden aspect) is twenty seven and one third of a day long and it is unlikely the ancients “rounded up” that invisible time period. In other words, there is *no fit between the lunar month and the week*,and yet this wrong idea is quite widespread. The origins of the seven day week are not with the Moon’s periodicity.

Another accepted premise for the week is that the Babylonians and possibly the Sumerians before them used it. These cultures of the fertile crescent hosted one of the earliest city state cultures and they were keen astronomers, but surely that just means that they would likely have an astronomical reason for having a seven day week. In those days, astrology was indistinguishable from astronomy and the five visible planets were added to the Sun and Moon to obtain seven, leading to the “astrological” week with planetary day names.

Two accepted historical channels for receiving a seven day week from the Babylonians are:

**The Greeks brought back the seven day week from the conquests of Alexander the Great and gave it to the Romans.**The Romans moved from a ten day to a seven day week with their assimilation of Christianity, which itself was partly a Greek system of thought.**The Jews adopted the week from the Babylonians after their captivity.**Theirs was a different version however since planetary deities could not represent the days (Saturday = Saturn’s day) as with most of the other cultures that have this week. The number seven was especially sacred in the Jewish tradition. For instance the Babylonian epic of a seven day creation starts the Bible as one of the earliest stories of the original Pentateuch and the sacred measures (in cubits, etc) often expressed the number seven. Seven in the Jewish week was sacred but not planetary like that of the Greeks.

In the undocumented times we call prehistory, traditions like the week could have been the “diffusion” of something innovated in a single place like Sumaria. On the other hand such ideas can also come from a common experience such as the astronomical observation of time periods. When the latter is the case, arguments for diffusion only look good until we find the same tradition, but out of the required timing for diffusion to have taken place. It looks as though the seven day week did not need to have come from the East; it was already in the Mediterranean in bronze age Crete.

To find an astronomical cause appears initially difficult because, as stated above, the periods of Sun and the Moon, the year and the month, do not divide by seven days and neither does Venus, the most visible planet.

And why should any astronomical length of time, such as the day on Earth, fit any other astronomical periods anyway? The whole premise of modern science is that the planetary system merely “settled down” into a set of planets and that, within certain limits, there can be no detailed order relating the rotation and orbit of the Earth with the periodicity, seen from Earth, of another planet. However this is exactly what is found, and there are such exact relationships between celestial periods seen from Earth: This is the subject of my book *Matrix of Creation, Sacred Geometry in the Realm of the Planets* (Inner Traditions, Vermont, 2003). What follows is new and complementary material to *Matrix of Creation *{only partially found in my second book on *Sacred Number and the Origins of Civilization* in 2007}.

Traditionally it is the planet Saturn that is associated with the number seven, and of course the Jewish Sabbath is Saturn’s day or Saturday. Also interesting is the fact that the Jewish calendar is lunar throughout, as is the Islamic, and so the arrangements of seven days into a four week month of twenty eight days seems perfect, so much so that the Nazarenes are reputed to have had four such weeks, ending in a sabbath at the end of each major phase of the Moon: New, Half waxing, Full and Half waning, so as to make seven work with a month longer than twenty eight.

In *Matrix of Creation *I point out that in 29 Practical Years of 365 days there are 28 synodic periods of Saturn,where a synodic period is the time taken for Saturn to again be opposed the Sun seen from Earth, like a full moon but seen as a loop of Saturn every year in the sky. So 28 is found within the behaviour of Saturn but not yet in a way that yields a 7 day week. However, larger numerical coincidences are, as we shall see, based upon the numerical interrelationships found between smaller periods of time.

The evolution of sky observation into a calendar and our week is perhaps as simple as counting itself.

Alexander Marshak in *The Roots of Civilisation *illustrates many examples of stone age markings, often on bones, that appear to be keeping a tally of the Moon’s phases. The counts typically run over two lunar months, probably because the month is itself 29 and one half days long: a double count gives a whole number of 59 days and is quite accurate. Since the processes of the sky are essentially circular, returning to the same condition to rejoin the beginning of the cycle then the natural tendency, when the medium will allow it, is to draw the cycle as a circle of marks.

Such counting is a *measurement* from which a number emerges associated naturally with the celestial cycle in question. It marks the achievement of knowledge but not necessarily the ability to *use it*. To synchronize life to a celestial cycle, beyond observation in the sky, requires that the new knowledge be translated into a *model *on Earth.

Now we know that from Megalithic times and into the bronze age, many large models of calendric knowledge were being built throughout Europe. Many of these were directly observational, such as stone circles and their alignments with solsticial sunrise, sunset, and lunar maximum. These seem to form a continuity with the bone count measurements, yet they also form an *operational* calendar. Something new then became possible, a calendrically based building that could contain observances connecting to the gods of celestial time phenomena.

The creation of numerical rings would then allow a further possibility: that a numerical ring could simulate celestial phenomena and, to a degree, become detached from the sky as an abstract system more akin to a clock. If and when a series of celestial cycles were found to be interrelated, these would have allowed the creation of an orrery or planetarium which outputs the condition of a number of different celestial phenomena through the relatively simple *activity* of counting smaller time periods.

Our clocks today have evolved from such roots by employing gear wheels as numerical rings and the activity of counting is automated in the form of a spring-driven escapement, that produces a regular advancement of the gears according to the numerosity of these cogs, rotating in circles. Thus, a clock or an orrery is based upon the relative counting of cogs cut into wheels but is essentially no different to what can be achieved by the manual movement of markers in rings of holes moved in time with a regular celestial cycle, with the day being the simplest choice.

If the ancients wanted to create an orrery, their best option was to use a ring of holes and indeed this has been suggested as one of the uses for the circles of post holes, most notably the Aubrey Circle of 56-holes around Stonehenge. Fred Hoyle showed how the Aubrey Circle could be driven as an orrery to accurately track the Sun, Moon and eclipse nodes around the ecliptic stars, using a procedure further refined by Robin Heath.

In fact a bronze device using gears has been found from no later than 80 BCE, near Crete. It is a planetarium designed to simulate multiple celestial periods, called the“Antikythera Clock”, and it used advanced mathematical knowledge of “continuing fractions” (reported in June 1959 Scientific American, p.60-7, see first page below.) It remains an anomaly that undermines the consensus view that clock mechanisms were a product of our industrial revolution.

The main point here is that the use of numbers to model the movements of celestial objects, relative to each other,should not be seen as unlikely back in the bronze age or even within the later Neolithic period, that encompasses the Megalithic. There are objects displaying the capacity and desire of those peoples to do just this. A counting device only requires the identification of a time period that divides well into one or more larger time periods, as a whole number of counts. It will be shown that the day and week are just perfect for this purpose – a fact since forgotten inits applications but retained as our seven day week associated with Saturn, The King of Time.

When visiting Crete, the southernmost Island of Greece and home to the Minoan Culture between 2500 and 1450 BC, it is obligatory to visit the Heraklion Museum filled with Minoan artefacts. In the final room relating to Knossos, the famous Minoan complex, one comes across item 2646, a “perforated utensil” of a sort that might be interpreted as an incense burner/diffuser from the period.

As seen below, it is made of spun pottery, painted with a seven-fold wave pattern and covered in circles of holes. When counted, the number of holes count *1:15:22:38:62, *and the holes seems to have been punched into the clay in a slightly ragged way. I will spare the reader a longer discussion of this disk’s construction, available elsewhere here, and get to the point that concerns us here, the origin of the week of seven days.

The central motif is “seven-rayed” which seems to point towards Saturn. Looking first at the 15 hole ring, I discovered that the ratio of the lunar year to the Saturn synod is 15:16. This is surprising new information since Jupiter has a ratio of 8:9 relative to the lunar year and, most significantly, these two ratios are both musical and correspond to a pure major halftone and tone respectively. The unit of time involved in the 15:16 ratio is *exactly *4/5^{th} of a lunar month, as can be seen by dividing 378 days by 29.53 to get 12.8 or 12 and 4/5^{th }lunar months.

The 22 hole ring seems obscure but the 38 ring hole, at 2 times 19, is reminiscent of the Saros cycle of 19 eclipse years between the recurrence of lunar and solar eclipses, lasting 223 lunar months or just over 18 years. In fact the decoration around these 38 holes is appropriate to indicating eclipses. The reason for 38 and not 19 holes was that the eclipse year is made up of two eclipse seasons. An eclipse season is the time between the crossings of the Moon’s orbit, by the Sun, the only time at which eclipses can occur during “seasons” lasting 34 days. [whilst solar eclipses are rare, there is usually one lunar eclipse per year]

If we divide the eclipse season period of 173.31 days by 22 we get a period of time 63/7 days or 7 days plus 7/8^{th }day. This period is exactly one third of 4/5^{th} lunar month, and so there is the implication that the 22 count contributed to tracking the eclipse seasons and that the 15 count showed in some way a relationship between Saturn and the Moon. (The 7 7/8^{th }day period is in a whole tone relation of 8:9 to the seven day week.) But if so, how was the counting done and what unit was being counted?

The remaining 62 ring seems very close to the number 63 found in 63/7. Indeed, if the 378 days of Saturn’s synod is divided by six, the result is 63 days. It seemed therefore that the unit of time, 63/7 days might be related to the outer ring of 62 if the central hole is included in the count to make 63 holes.

*The outer track of holes can simply count days.*

If the period sought had been 8 days rather than 7 7/8^{th} days, then 64 holes or days would give 8 periods, but the sought after period is 1/8^{th} of a day less than eight full days. By including the inner hole, the outer ring can count 63 whole days which, after one round would contain eight periods of the required length. Such a use of the central hole symbolises the end of one-sixth of the Saturn synod, a day that may have been special in this Saturn calendar – hence the central hole.

After eight days, i.e. 8 holes, the count should have lost 1/8^{th} of a day and by counting in an anticlockwise fashion, the time of day of this “falling behind” can be tracked exactly as if reading a 24 hour clock face. Figure 8 shows how after starting the count, the period of 7 7/8^{th} days looked for will end very close to where the marker now stands on the “24 hour” clock face. The marker will deviate from perfection in this regard but will generally always be showing in which three hour period the end of the 7 7/8^{th} day period occurs. This means there would be no need to show this explicitly in the decoration since it could be read accurately in this direct way. It also means that, theoretically, this ring can, with the central hole, measure the required periodicity *within three hours *without any further techniques or technology. Incidentally,this is an ingenious version of the Vernier technique, invented in 16^{th }century by Pierre Vernier, in which two slightly different scales interact to yield a more accurate reading.

This disk thus leads directly to a very complete and accurate calendar (see Figure 10) that can track the motion of Saturn, the Moon and the eclipse seasons. The calendar employs the simplest unit of measure available on Earth, the Day. It also employs units based upon the week, because Saturn’s period divides perfectly by seven days, that is

Our present seven day week would be the natural choice for people operating the Saturnian calendar.

This explains the seven day week’s adoption, both practically, as a calendric device, and evidentially, as an historical reality predating classical Greece. We know this calendar would have been contemporaneous with Egypt and other parts of the Minoan sea-trading network. It also connects to biblical history since Moses and Aaron could have encountered it in Egypt, and this could have lead to its adoption by the Jews alongside other knowledge including Jewish sacred measures and building techniques.

Other cosmic relationships appear within this new calendar (see Figure 10), relationships quite surprising in that they indicate that time on Earth is simpler that it should be. The main rival of Chronos, Zeus, the planet Jupiter, is shown to have a new relationship since their 378 day and 399 day periods have the ratio of 18 to 19 units of 21 days. This unit is exactly three weeks of seven days. (see Figure 9)

The chance of the only giant planets visible to naked eye observers both having periods that divide by seven seems unlikely and hence this would have seemed a significant fact to ancient peoples. The adoption of a seven day week would naturally be confirmed as a logical part of a sacred calendar.

In the mythology of Zeus, Chronos is accused of swallowing his own children and perhaps we can see in this a reference to *a system of time* that, if followed, effectively denied (swallowed) all the other celestial cycles/ planets. In fact the ancients could have “got hung up” on such a simple system of time. Zeus, a child of Chronos, is saved from such a fate by his mother and is brought up in a Cretan cave hidden from his father who might hear his cries. This implies that the drama is one being played out in Crete, with Chronos just down the road rather than in some heaven and routinely omniscient.

Most significantly, Zeus grows up to depose his father and become the god of the classical world from which western culture has largely evolved.

This calendar, implicit in the Disk of Chronos, evidently fell out of use and was replaced, probably with those for which there are historical records. It therefore seems likely that the overthrow of Chronos by Zeus was related to these calendrical practices and that Chronos was related to some fixed religious regime associated with the older Saturnian calendar. Since the calendar is simpler than it “should be”, that is, because time periods should not match so simply (in days) the periods of Saturn, the Moon and eclipses, then no further development of time was likely when living under such a calendar. The God of Time would have dominated thought and the religious precincts of the bronze age, until deposed and replaced.

All of this confirms the basic tenets, expressed in *Matrix of Creation*, that there was an ancient science that employed numerical arts to discover order in the world and also build monuments relating to their discoveries and that science. Lying behind it all was a world view that numbers actually defined how the world was built.

Such a belief in numerical creation could have been “seeded” in the fabric of the solar system as the numeric relations that are to be found, seen from Earth. In other words, our ancient awakening to the numerical relationships in the world could have been a cause for the evolution of human understanding itself. In the presence of an apparently designed world, a religious sentiment would have been a natural one. The original meaning of the word “religion” is, after all, not based upon *beliefs*but on *reconnection*, presumably to the truths of the cosmos.

It is obvious that there must be many artefacts and monuments with further facts to transmit to us from the past, facts that would restore our connection to the whole as Cosmos, and cause little understood traditions such as the week to become re-rooted in their original context.

For seven day week, try www.webexhibits.org/calendars/week.html

For a great book on calendars, try

*Mapping Time: The Calendar and its History*by E.G. Richards, Oxford, 1998

For more on the significance of numerical astronomy, read my book

*Matrix of Creation: Sacred Geometry in the Realm of the Planets*Inner Traditions Press, Vermont 2004

A Synod is always a repeat cycle time relative to the Sun, see from Earth. The Saturn synod is 378 (378.09) days whilst that of Jupiter is 399 (398.88) days.

- If the basic time period of 63/7 is being counted, then the Saturn Synod of 378.09 days is achieved as 6 times 8 times 63 divided by 8 as 378 days yielding an accuracy to the Saturn synod of 99.976%
- The Lunar Year of 12 lunations is then tracked as 15 times 3 times 73 divided by 8 equalling 354.375 days versus 354.367 days giving an accuracy of 99.998%
- The period between Eclipse Seasons is tracked as 22 times 63 divided by 8 or 173.25 days versus an actual period of 173.31 days giving an accuracy of 99.965%
- The eclipse season is plus or minus 17 days and 17 days corresponds to two holes of the 22 holes used for counting its periodicity. This means that the eclipse season can be expected in the area of fulfilment of the counting in the 22 hole ring, plus or minus two holes of the end of the count. If a lunar eclipse should occur with the marker outside this range, then the 22 hole marker could simply be moved to the nearest in-range hole, making the system self-correcting on the basis of simple observation.
- As stated in the text, the 63 hole count automatically gives an implicit reading of how many 1/8
^{th}days should be removed from the whole day, advancing the time of day at which the count is actually progressing. This does not effect the actual movement of the day marker, but indicates to high precision of a few hours, the exact moment referred to by all the markers in their different track rings. - Three periods of 7 7/8
^{th}days equal 4/5^{th}of a lunar month to the very high accuracy of 99.998%, repeating the result in 2 above.

We have to ask “Would this type of accuracy be useful?” and the answer appears to be that it would have been. The simplicity of it means that, once established,the level of skill required to track time, focused on the chosen time periods, would have been on a par with skills compatible with our knowledge of other bronze age activities.

]]>Using the lowest limit of 18 lunar months, the commensurability of the lunar year (12) with Saturn (12.8) and Jupiter (13.5) was “cleared” using tenths of a month, revealing Plato’s World Soul register of 6:8::9:12 but shifted just a fifth to 9:12::13.5:18, perhaps revealing why the Olmec and later Maya employed an 18 month “supplementary” calendar after some of their long counts.

By doubling the limit from 18 to three lunar years (36) the 13.5 is cleared to the 27 lunar months of two Jupiter synods, the lunar year must be doubled (24) and the 32 lunar month period is naturally within the register of figure 1 whilst 5/2 Saturn synods (2.5) must also complete in that period of 32 lunar months.

One can also see Plato’s World Soul in the Pythagorean Tetraktys (see Keith Critchlow’s *Foreword *and Robin Waterfield’s *Introduction *to *The Theology of Arithmetic*, Phanes, 1988), as a development of the Lambda diagram as per figure 2.

Keith Crichlow demonstrates that the Lambda diagram is none other than the *Tetraktys*, the latter probably a schematic missing in Plato’s *Timaeus* 35b&c, cryptically referred to by Plato since the uninitiated lacked it. The Tetaktys gives a beautiful organisation of the Pythagorean numbers involving only 2 and 3 as factors perfectly represented by an equilateral triangle growing organically out of ONE.

The Tetraktys of figure 2 differs from the mountain of figure 1 in that

- the vertical sense is not due to increasing multiplication by five, but rather decreasing multiplication by 2 or 3, and
- the elements were not normalised to a single octave.

The role of 36 within the Tetraktys of figure 2 corresponds to the number of lunar months chosen for the limit in figure 1, and so one can choose to normalise it, through doubling numbers where required, into the octave range 18:36.

On the bottom register, 27 already fits the octave whilst 12 and 18 must be doubled to 24 and 36, and 8 quadrupled to 32. At that point all that is above shares the numerocity of what lies in the bottom register, being dependent only on their powers of three.

The numbers 32:24:36:27 are identical to the white register of figure 1 and this may well indicate that the secret knowledge of the Pythagoreans, said to be summarised in the Tetraktys in particular, could well have been the astronomical harmonic relations to be found between the double and triple lunar year periods, the double synod of Jupiter and the 32 lunar month period which is also both 945 solar days *and* 2.5 Saturn synods long. This would have been a very simple means for the Demiurge to have employed, close to the harmonic origin story from one, two and then three.

If the right hand three elements are 12:18:27, and this is reminicent of the numbers generally used to designate three of the four Elements (water, air and fire) by the medieval period, and the four Elements were often shown then(after Plato), as is the case here, as being separated one from the other by fifths – exactly as Plato describes it in Timaeus (ref). However, most schemata differ from the allocation found in the decorated crypt of the Pope’s summer palace, the Anagni Cathedral (figure 4, right).

In the planetary matrix of synodic periods, Saturn is distinguished as the cornerstone having no other factors than the powers of two with respect to the lunar year. Hence it is harmonically commensurate (5/2) with the 32 lunar month period which is 8 merely doubled twice to fit the limit of 36. So it would seem that Saturn represents the Earth element. It is quite obvious that the lunar year of 12 lunar months is Water and that half the triple lunar year, of 18 months, is the element Air. The element of Fire is then the double synod of Jupiter (27).

**8:** Since 32 lunar months is 945 days,
then the earth which continually rotates towards the east is the cycle of
barrenness, according to Plato, is like the number 2 of octave doubling which
provides a container for intervals as a womb enables children to arise. Saturn
is also considered “plumbous” or “weighty” and so is naturally represents the
solid state of objects and, traditionally, the giver of boundaries being the
outermost visible planet.

**12:** The Moon’s links to water come
directly through the tides as first of many traditions such as that it fills
with water until “full” and thereafter starts to “empty”.

**18:** The lunar year also collaborates
with the near anniversary of three solar years equalling 37.1 lunar months.
This periodicity was studied by the megalithic astronomers of Carnac as a
right-angled triangle whose counted base was 36 lunar months symbolised by 36
stones in the kerb marking the base ay Le Manio’s Quadrilateral.

**27:** After two loops of Jupiter, the middle
of the retrograde loop will be punctuated by a full moon (at maximum retrograde)
since the Sun is then in line with the Earth and Jupiter. This fact enables the
loops of outer planets to be counted in lunar months, as required. The
association of Jupiter with fire seems natural since his primary weapon is
lightning, the cause of natural fires.

- The numerical Tetraktys,
- the account in Timaeus of a harmonic Creation and,
- the harmonic realities of Saturn and Jupiter relative to the lunar year,

are so closely related as to suggest the first two were a plan and description of the latter. The numbers referred to are the natural ones, in lunar months and this affects the history of ideas in sourcing Plato’s cosmology (influential in the Arabic Golden Age (800-1100) and subsequent western Quadrivium of numerical arts) on concrete astronomical facts ascertainable from simple observation and counting of time periods, inherited by the Pythagoreans.

The Pythagoreans considered themselves privy to a secret doctrine, and central to this was the school of “theoretical” harmonists in Greco-Roman times, recognizable today under the rubric “Harmony of the Spheres”. Whilst many cunning versions of such a thing have been proposed, the core of the matter rests in the simplicity of the numbers in figures 1 and 2, of the synodic periods of the outer planets in months relative to the lunar year. To think otherwise requires an explanation as to why this is not obviously a simpler and more probable fact. It is easiest to extrapolate megalithic astronomy as having observed and counted the synodic periods of the outer planets and, on comparing these with the lunar year length, discovering the planetary harmonic ratios, hence leaving a legacy of harmonic numbers referring to the heavenly world ancient literature and crafts.

The above is a further development of the theme of my recent book *The Harmonic Origins of the World*.

Jupiter reaches its maximum retrograde motion half way in the loop, after 60 days from its standstill in the sky. If, at that point, there is a conjunction of the moon and Jupiter then the moon must be full since the sun will be opposite both the moon and Jupiter. This means that, when a full moon is conjunct Jupiter at mid-retrograde, it will set to the west and one can start counting lunar months until the same phenomenon occurs.

We know a single synod of
Jupiter is 398.88 days and so there are *exactly*
13.5 lunar months in each synod since 398.88/ 29.53059 = 13.5073, just over 5
hours longer than 13.5 months. It is therefore true that, if one counts between
full moons occurring 60 days into successive retrograde loops of Jupiter, it
will be two whole synods before a full moon occurs in the same visual offset to
Jupiter at maximum retrograde in the sky. The lunar counting in between will be
27 whole months and at that point, the synod is known to be 13.5 months long
and 9/8 times longer than the lunar year.

In this way, not only could the Jupiter synod be found easily, using megalithic horizon astronomy, but knowing its length relative to the lunar year would reveal the remarkable harmonic ratio of the Pythagorean whole tone between Jupiter synod (9) and the lunar year (8 units of 1.5 months). This would have introduced the megalithic to that uniquely simple category of ratios responsible for the highly-ordered world of musical harmony.

The same procedure applied to the Saturn synod would require five Saturn synods (378 days) to complete since only then do 12.8 lunar months per synod yield an integer number of 64 lunar months. The loop of Saturn is smaller than that of Jupiter, 6.5 degrees compared to Jupiter’s 10 degrees. The reverse is true though, of the days spent by each planet in its loop, Saturn taking 140 days rather than 120. And the ratio of the Saturn synod to the lunar month is another crucial musical interval, the semitone of 16/15.

Tones and semitone are crucial to the formation of musical scales and so my proposal is that megalithic astronomy, once aware of these intervals, would have started to investigate practical music and instruments that make musical intervals. The best possible are string instruments where, if one uses a single string, one can measure the lengths of strings just as one can measure the lengths of synodic periods and find these ratios with which to form musical scales and a deeper musical tradition, inherited by the ancient Near East and other civilizations of the third millennium BC.

]]>How can an immortal god die? Especially Zeus who was not just a god but head of the Olympians, a new breed of gods that had replaced the Titans and their “despotic” ruler, Chronos. A Rome holding to Zeus/Jupiter perhaps rejected the Cretan tradition of the god’s death with the well-broadcast adage “All Cretans are liars”.

But we all *should
*know that mythology uses contradictory, or at least inconsistent, versions
of the same story, to express alternative perspectives and to transmit more
knowledge in the process, rather than “a lie”.

The importance of the death of Zeus is that the story emerges exactly from that point in time and cultural transformation in which Zeus is also born and at that time it was familiar for a vegetative god, representing nature blooming in spring and dying in autumn, to die and be re-born within the immortality of the eternal round of the year or yearly daemon.

There were other norms too, including the birth of men and their world of form, out of the Earth and from within The Cave, as a natural sacred space created by the Mother or earth goddess. Directly symbolic of her womb, form emerges as shapes in formation like dreams, travelling towards the definite order found on the surface.

It was a Cretan cave where Rhea, the sister and wife of Chronos, secretly gave birth to Zeus with the help of the earth goddess. This cave can be found on the plain of Lasithi, beneath the mass of mount Dikti (figure 1). Used since paleolithic times as a place of offerings, it is filled with impressive stalagtites and stalagmites that appear like emergent statues and friezes, exactly as if the earth were creating prototypes and narratives, a world coming into existence.

Another format of ancient observance was the very tops of mountains, elevated santuaries close to the workings of the celestial world and looking out over creation. It should not be too surprising then that through the south gate of the Minoan palace of Knossos, the mountain where Zeus “dies” can be seen, mount Yiouchtas(pron. “Yuktas”), upon which the Minoans had a small santuary (figure 2). From the west, this mountain is traditionally viewed as the reclining form of the dead god (figure 3) and this long, ridged mountain dominates the whole landscape south of the present city of Heraklion, attractively rich in olive groves and vineyards.

At the northern descent ofthe Yiouchtas ridge a very unusual piece of archaeology has also beenachieved: A small Minoan temple in which human sacrifice was interrupted by anearthquake that killed the priest and two attendants with falling masonrythough the child was already and just dead. Was this a Minoan norm or was thisan extreme measure designed to forstall a series of such disasters destined todestroy the grip of this great maritime empire of the Minoans on theMediterranean? Whatever the motives were, the tradition of child sacrifice isnow firmly associated with Zeus’ death within this landscape by this discoveryat the head of the mountain, below its peak sanctuary.

In the semitic version, Abraham’s near-sacrifice of Isaac is a story that tells us what made the evolving godhead of Jehova different from his Middle Eastern neighbours. Christianity has that god sacrifice his own son “for our sins” and Rome equates Jesus with God, but a death place for God himself is inconceivable.

It seems that there is a direct confrontation here between the recurrence of Eternity, born out of observation of nature, and Historic narrative, contradictory to the essence of myth. This is seen when the classical commentators call the Cretan story of Zeus’ death “a myth”, which it is, when they mean by that “a lie”. The core meaning of myth is “Traditional narrative usually involving supernatural or imaginary persons and often embodying popular ideas or natural and social phenomena” [Oxford] whilst the usage “A fictitious person, thing or idea” has in practice come to mean “a lie”.

Cultural imperialism
characterises all mythic narratives outside of its own as being a lie and its
own as being a fact, and this has become a dominant phenomenon since the birth
of Zeus, acting naturally against any past and present alternative myths. Thus
the definition of myth above, as “embodying popular ideas or natural and social
phenomena” has been inverted where
mythical stories are used to *define* what ideas are to be popular and
which natural phenomena are to be studied and why. Therefore they act to *control*
social phenomena.

No sooner does Zeus gain ascendancy over Chronos, his father, but the god gains ascendancy over his own death, that is, his own story, and becomes a system of belief not based on nature but rather upon the existence of the god within the minds of the people. Except, that is, in Crete, where he dies and the Cretans all become liars.

It is as if the role of myth has been subtly altered from being a mechanism of receptivity towards nature, which then becomes “superstitious fantasy”, into a mechanism of social action and belief, whose imperative is the mythos that must be believed as fact and the technos of manipulating nature and dominating others.

Therefore this new active mode of social action should be called Technos, whilst the old, receptive mode can be summed up as Cosmos. Within Technos, the powers of the male in particular are in the ascendant and the manifestation within the world at this time, the Bronze Age.

It is therefore not only the godhead who is transformed in Crete since the bronze age itself was born here and can be seen as the transformation of the ancient smiths or astronomers called the Cyclopes, who built only monuments to the Cosmos. It is they who were always being “locked up” in Tartarus, the celestial depths, by the preceeding godheads Uranos and Chronos and who wore the concentric rings of the Sun and inner planets or similar, upon their brow as a caste mark, hence their “one eyed” characterisation.

Zeus releases the Cyclopes to depose Chronos whereupon they become Hephaistos, the Olympian smith god, equivalent also to Prometheus, the Titan who stole the fire from the gods that would “smelt the bronze” of human invention. Other characters such as Daedelus and Talus invent all the innovations of their age in myth, and strange new symbols such as the labyrinth, within which a half-Taurus, half-man is imprisoned. Hephaistos may be bottom of the heap in Mt Olympus but he is also head boffin in the research and development of the bronze age.

Perhaps it is the mythos of an interpenetrating Cosmos that died in the birth of Zeus but became represented as his death, so near his birth place, in Crete. It is the old ways that were dead.

According to Plato, the world was created using the rules of musical harmony in a scheme involving perfect fifths of 3/2, fourths of 4/3 and tones of 9/8, leaving “leftover” semitones of 256/243: a rudimentary musical scale. This only used prime numbers 2 and 3 and multiplications with themselves and each other; a system called Pythagorean tuning. The Kaaba incorporates another prime number 5, called the human number, this enabling two more large intervals called thirds, the major third of 5/4 and minor third of 6/5. Using 5 enables more and better scales to be formed and fills in the gap between 4 and 6 to show all the large intervals in the first six numbers, 1:2:3:4:5:6. This fuller tuning system has been found in the ancient Near East as long ago as the Sumerians, in their tuning texts on cuneiform (c. 3000 BC onwards).

Ernest G McClain wrote on the proportions of the Kaaba[2]:

“What follows is an adventure in imagination which aims at grounding the Kaaba’s proportions in the sacred sciences of earlier civilizations. Since Islam already claims a very great antiquity for the Kaaba, any explanation which appears plausible will merely have the effect of supporting the Islamic claim. My argument is based on the very interesting coincidence between the Kaaba’s proportions—not its absolute measures—and those of the Temple of Poseidon in Plato’s myth of Atlantis, a myth which has proved capable of very detailed explanation by the methods and metaphors of the Pythagoreans. The temple of Poseidon is described as involving the ratios “6:3 plethra (full)” meaning 6:5:4:3. The Kaaba’s proportions are precisely these:

Meditations through the Quran. Ernest G McClain. Maine: Nicolas-Hays 1981. 78-79.

Width | Length | Height | |

Measures (in meters) | 10 | 12 | 16 |

Proportions (in smallest integers) | 5 : | 6 | |

3 : | 4 |

We can say that the Kaaba” embodies” the ratios 3:4:5:6, the numbers 3 and 6 being an “octave identity” in Pythagorean harmonic theory. We are investigating notthe visible geometry of the Kaaba but the proportions which have their meaningin the invisible sonar implications of a number theory which the whole ancientworld shared in common for several millennia before Muhammad.”

My own measures for the monument (derived from
the previously mentioned plan) *seem* different
in giving a width of 11 yards rather than 10 metres but 11 yards are 10.0584
metres and McClain’s figures (after Titus Burckhardt) were only approximate. My
length is 39 feet which is 11.89 metres, that is, nearly 12. My geometrical proposal
is illustrated below in figure 1.

The walls of the Kaaba can be considered as blocks
one-yard square. The inside floor area would then be 9 x 11 = 99 square yards, the
number symbolic of the ninety-nine names of Allah. When seen as a set of nested
rectangles, the sequence 13:11:9:7:5:3:1 arises, in which 7 is the middle term
and the sum is 49 or 7 squared, seven being the number that terminates the *senarius*. The walls of the Kaaba also
seem to have a meaning in inches since the outer perimeter is 12^{3} = 1728
inches, symbolic in numbers given for near-eastern arks, whilst the inner
perimeter is 1440 inches, symbolic of Adam when his letter numbers of 1.4.40
are placed in decimal position notation [McClain, *The Myth of Invariance*, p126].

Figure 2 Inner and outer perimeter lengths in inches

Sacred buildings are obviously related to arks and salvation whilst the Biblical figure of Adam was further developed in the Prophet’s new narrative; that the Kaaba was a building created by God when Adam was made, whilst Abraham and Ishmael, his first son, built the earliest version made by human hands. Mohammad himself participated in a major rebuilding of the Kaaba after flood damage, which could have been designed to the present Kaaba’s proportionality and later works all based on that footprint.

It is the perimeter lengths of the Kaaba’s
nested rectangles (figure 1) which tell the story of the *senarius*.

half perimeter | perimeter |
senarius
| interval |

3 + 1 = 4 | 4 × 2 = 8 | 8/8 = 1 | Unison (1:1) |

5 + 3 = 8 | 8 × 2 = 16 | 16/8 = 2 | Octave (1:2) |

7 + 5 = 12 | 12 × 2 = 24 | 24/8 = 3 | Fifth (2:3) |

9 + 7 = 16 | 16 × 2 = 32 | 32/8 = 4 | Fourth (3:4) |

11 + 9 = 20 | 20 × 2 = 40 | 40/8 = 5 | Major Third (4:5) |

13 + 11 = 24 | 24 × 2 = 48 | 48/8 = 6 | Minor Third (5:6) |

Table 1 Outer perimeters of nested rectangles indicating the Senarius

McClain’s intuition has continued to be valid,
that the Kaaba enshrined the *senarius*
or “creation out of six” of musical harmony. It seems likely that the building
employed ordinary feet and yards to embody successive perimeters, with inches
providing additional meanings congruent with McClain’s published work on
ancient near-eastern harmonic code numbers and the Bible. This innovative and previously
unknown way of using nested rectangles and adjacent odd number-pairs, to symbolise
the *senarius*, should give weight to
the idea that this focal monument for Islam symbolised a harmonic basis for the
creation of the world on which it sits.

[1]
https://en.wikipedia.org/wiki/Kaaba#/media/File:Kaaba-plan.svg

[2] A uniform spelling of the Kaaba has been imposed. Continuing, he wrote: *Muhammad, as an ” unlearned” man, has no share in this ancient science; he is merely the agent—as he conceived himself to be—by which ancient knowledge came to new life in Arabia. Before embarking on our new study of the Kaaba, let us first notice the role which number plays in the Quran. My whole study would be invalidated— at least for a Muslim—if the Quran itself did not virtually ignore geometry and laud, instead, number and proportion.*

The diatonic scale is … an abstractum; for all we have is five tones and two semitones a fifth apart [until] we fix the place of the semitones within the scale, thereby determining a definite succession …, [and] we create a mode. [Levarie. 213].

Musical Morphology,.Sigmund Levarie and Ernst Levy. Ohio:Kent State 1983. 213.

One can see that the tones are split by the major diatonic into one group of two (T-T) and one group of three (T-T-T), so the semitones are opposed (B-F) towards the tonic C as in figure 1.

Letters such as C are called note classes so as to label the tones of a diatonic scale which, shown on the tone circle, can be rotated into any key signature of twelve keys including flattened or sharpened notes, shown in black in figure 1. We will first show how these black notes came about naturally, due to two aspects of common usage.

The note classes arose from the need of choral music to notate music so that it could be stored and distributed. When we “read music” today, the tablature consists of notes placed within a set of five lines with four gaps, and two extendable areas above and below in which only seven note classes can be placed, seven being the number of note classes in the modal diatonic and the number of white keys on the keyboard, which is the other aspect of usage.

One can see that the note classes (A-B-C-D-E-F-G) do not correspond with the white keys of the keyboard correspond to C-Major (C-D-E-F-G-A-B) and this can only be because the first keyboards, used to control groups of bells, or organ pipes, and then clavichords and harpsichords, were naturally playing the major diatonic, which today starts with our first note C. The notational system of classes was developed to help choirs learn plainsong as do-re-me-fa-sol-la after Guido of Arezzo, who lived from 992 until after 1033[1]), regarded as the father of modern musical notation but not of the note names. Guido’s system forms the basis of the solfeggio system, our “do-re-me-fa-sol-la-si-do”, still widely used in non-English speaking countries. The note Do is invariably the start and end of our C major scale and hence, solfeggio was remarkable compatible with the later notion of key signatures in which each key is rooted in the major scale.

When the church wanted to play different modal scales in keyboard instruments, black notes were added between the white notes of existing keyboards[2]. These black notes arrived in a distinctive order as one tries to play Greek modal scales from a common starting point such as C. They came to be designated sharp or flat, which implies a process by which they came into existence. In the case of the Dorian, the note B-flat will replace B and so a black key was added between A and B. Whilst the black notes appeared naturally within keyboards, when church modes needed to be performed, modern music has abandoned those scales exactly because the keyboard and musical notation treated the major scale as the primary scale. The eponymous system of *key* signatures, starting from any key in the major scale including the black notes, does not find note C as first key of the keyboard’s notes. Keyboards evidently started with C, having only the white notes we call C major. So why should A be our first note class rather than C?

The idea of notes as letters was first proposed
by Boethius (c. 480-524) who started with the A of the alphabet but having sixteen
note letters – that is, not constraining himself to a single octave scale. “A”
was nothing more than a label for the lowest note but Notker Babulus (d. 912) first
used A to refer to what we now call C, the home key of the white note major
scale. But Boethius was eventually followed since, in his system of note
letters, the initial tetrachord after A was tone-semitone-tone (T-S-T), which
is the tetrachord of the symmetrical Dorian scale T-S-T-T-T-S-T. The Dorian was
probably chosen by Boethius because this is the scale of the Pythagorean
heptachord, resulting from tuning by pure fifths and fourths within the octave.
A is the first note below C that can play that tetrachord on the white notes of
the keyboard. Choosing A locates D as the white note centre of symmetry amongst
the population of black notes on the keyboard and D is the natural Dorian key
having T-S-T *either side of it*, with A
below and G above.

This gives us a reasonable historical likelihood, with three important initial conditions,

- Boethius invented note letters starting with the symmetrical tetrachord of Dorian, whilst
- it was the major scale that ruled early western music, upon which the solfeggio registered its notes as do-re-me-fa-sol-la-si-do, and
- early keyboards had only the major scale as white notes but developed black notes to allow modal scales to be played.

Having created chromatic keyboards, there was a natural pattern of white and black notes which did not follow the idea of the first note do being C. The Pythagorean heptachord, created using fifths and fourths when tuning lyres, harps, clavichords and harpsichords, evidently started Boethius’ note letters, and this meant C must move from being the first white note of the keyboard to being the third (after the semitone of a tetrachord T-S-T of the notes A-B-C-D). Because Dorian has both an initial and final tetrachord of T-S-T, this created the present keyboard which is symmetrical about D.

The key signature system was then conceived as
being from white key C, since the note classes of C-Major key are all unmodified
letters, with no sharps of flats. However, whilst the major diatonic can be
played starting from different keys this can be hard to learn since more or
less black notes replace the white notes of the major diatonic in different
keys. For example, if one is to play the major diatonic sequence of
T-T-S-T-T-T-S in the Dorian-in-C tone circle of figure 2 (left), then one would
start with black note B*b* and hence the
key Bb major. Similarly, if you wanted to play the Dorian on the right-hand
tone circle of figure 2, that is in C-major, you would start with D and could
still play all white notes.

This gives us a clue as to how, whilst key
signatures filled the key-board with major scales, the modal scales were still
available within each key once black notes were added to the keyboard. The
primacy of the major scale probably drove the adoption of key signatures and
the evolution of modern practices of modification within key signatures called minor,
seventh, augmented and so forth. These modifications have the advantage of
being similarly available in each key signature, modifying the major diatonic in
a lesser fashion than playing other modal scales draws on many of the different
tonal features found in the antique world of scales. Once the fingering for a
given key was mastered, similar melodic and chordal adjustments could be
similarly found in every key, allowing *transpositions*
of musical ideas between the key signatures.

Another feature of key signatures is made clear in tablature where key signatures are associated with a number of sharpened or flattened note classes that then run on along the stave’s (a.k.a. the staff’s) lines and gaps. This is supplemented by accidentals before notes, to achieve note classes not within the major key. Perhaps more importantly, as with the Dorian in C of figure 2, the key signatures starting from C develop increments of one sharp or one flat to form a natural sequence as in figure 3.

To travel from C major to the key with just one flat, the key must start a descending fifth from C, that is in F-major. Similarly, the key with a single sharp is G-major, an ascending fifth from C. This mechanism of travel, around the circle of note-key classes continues algorithmically when adding two flats with Bb-major being a descending fifth from F; and adding two sharps in D-major, moving the tonic through an ascending fifth from G. This continues until the twelve key-signatures are exhausted[3]. This process of generation, moving the tonic through fifths requiring new flats or sharps, impacts how the black note classes are called since every player relates to the black notes primarily through the key that requires them as chromatic[4] flats or sharps.

This naming of chromatic notes is particularly clear when non-musicians try to name them “wrongly”, that is not according to their emergence due to the key signature system. The system of keys is an algorithm for ordering the keys by their complexity of sharps and flats, and the consequent separation of the keys so arranged (by descending and ascending fifths) is fundamental to the process whereby the old Greek-style scales have been side-lined by modern music through the convenience of playing in the major scale starting with each possible key of the keyboard.

Of course, the evolution of such a system would never have been possible in practice without the attempt to temper the tones and semitones away from being pure numerical ratios so that each key should have uniform tones and semitones of one defined size. This search for equality of tones at the expense of exactitude to the ratios of ancient tuning systems was crucial to being able to write most of the recent Classical music within the key signature system. Our key signature system today works perfectly, based upon equal temperament in which each twelve chromatic semitones are equal in ratio being to the twelfth root of the 2 of octave doubling.

One is led to infer four influences upon the eventual system of key signatures:

- Firstly, key signatures were a potential held between the twelve chromatic notes, arising naturally in modal scales but then having a natural process of development through sharps and flats spaced a fifth from each other.
- Secondly,the note letters innovated by Boethius and alternate solfeggio notation of Guido, directed musical practices dominated by the major scale.
- Thirdly, the early keyboard instruments, with white keys but then black chromatic keys placed A before C in Boethius’ final tetrachord for D, the natural axis of symmetry for the keyboard, corresponding with the Heptatonic.
- Finally, the tempering of strings using beats established a non-theoretical basis for equal temperament.

[1] He published Micrologus, 1026. The original letters used by Guido were the “ut–re–mi-fa-so-la” syllables are taken from the initial syllables of each of the first six half-lines of the first stanza of the hymn Ut Queant Laxis in which the first syllable of each line was sung in the order of the major diatonic. Ut was changed to Do by Giovanni Battista Doni: He convinced his contemporaries to make the change by arguing that “Do” was easier to pronounce and was an abbreviation for “Dominus,” the Latin word for The Lord, who is the tonic and root of the world.

[2] On early instruments now only existing in art. Church influences led to keyboards being played within art by angels, choristers or saints.

[3] apart from special keys that are different ways of looking at one of these twelve keys.

[4] The chromatic notes were initially the five notes not required by the major scale, but chromatic music more loosely means music using twelve notes to the octave rather than just seven.

]]>The ancient notion of tuning matrices, intuited by Ernest G. McClain in the 1970s, was based on the cross-multiples of the powers of prime numbers three and five, placed in an table where the two primes define two *dimensions*, where the powers are ordinal (0,1,2,3,4, etc…) and the dimension for prime number 5, an upward diagonal over a horizontal extent of the powers of prime number 3. Whilst harmonic numbers have been found in the ancient world as cuneiform lists (e.g. the Nippur List circa 2,200 BCE), these “regular” numbers would have been known to only have factors of the first three prime numbers 2, 3 and 5 (amenable to their base-60 arithmetic). Furthermore, the prime number two would have been seen as not instrumental in *placing* *where*, on such harmonic matrices, each harmonic number can be seen on a harmonic matrix (in religious terms perhaps a holy mountain), as

- “right” according to its powers of 3.
- “above” according to its powers of 5.

An inherent duality of perspective was established, between seeing each regular number as a whole integer number and seeing it as made up of powers of the two odd two prime numbers, their harmonic composition of the powers of 3 and 5 (see figure 1). It was obvious then as now that regular numbers were the product of three different prime numbers, each raised to different powers of itself, and that the primes 3 and 5 had the special power of both (a) creating musical intervals within octaves between numerical tones and (b) uniquely locating each numerical tone upon a mountain of numerical powers of 3 and 5.

The familiar form of an octave is created by prime-2 doubling of the tone used as an octave’s tonic. Inside an octave, intervals provide pathways between the notes, these usually seen as modal scales, created by the primes of 3 and 5 because *only a larger prime number*, employed as a denominator, can break open the octave interval range equal to 2; into three parts or five parts, or multiples thereof such as 15 parts, etc. Since each brick in the right hand mountain of figure 1 is a unique composition of powers of 3 and 5, then all the possible tone numbers, translated onto a tone circle, will carry with them a unique composition of those primes. It is interesting here to follow visually how this works within a Tone Circle.

If we locate the tonic “do” as the modern note class (or letter) D: the symmetrical disposition of white and black keys around it on the keyboard reveal the symmetrical scale formed as the modern Dorian scale. The black chromatic keys indicate where the primes 3 and 5 are transported to *within* *key signatures* *and scales other than C-major and Dorian*.

The lowest regular number capable of forming five of the Greek scales (as well as some chromatic tones) is D equal to a limiting number of 720. The powers of 3 and 5 are unaffected by raising the harmonic root of 45 by 2^4 (=16) and the populated octave will then form around 45 as the darker bricks, on the first three rows in figure 1. These rows are shown in figure 2 and all the three rows in figure 1 have merely been brought into the range 360:720 as integer numbers, using as many powers of 2 as it takes, but their locations, on this “hill of primes” (figure 2), of each tone number, *are fixed by the powers of 3 and 5 they embody*.

All the tones in figure 2 can be transposed, as factors of 3 and 5, to the tone circle for limit 720 as follows,

Between the primes are the component intervals (figure 4), and these are entirely due to the inevitable exchanges, in powers of three and five, between the tonal numbers adjacent to each other on the tone circle.

The tone circle for a limit such as 720 (figure 4) produces little of much direct use to modal music and hence, in the past, this tone circle appeared to be unrelated to practical music. But if one attends to the prime number transfers between adjacent tones one can, by re-hydrating the resulting ratios using prime number 2, figure out the intervals. The interval low-D to e-flat loses a three and a five giving 1/15 which, times 2^4 (=16), makes the interval 16/15, the just semitone. The next interval (e- flat to e) loses one three and gains two fives giving 25/3, which divided by 2^3 (=8), makes the interval 25/24, the chromatic semitone.

In figure 5 one sees that these two intervals combine, as 16/15 times 25/24 equals 10/9, the just tone – then an interval used in modal scales and music-making. The interval between e and E in figure 4 gains four threes whilst losing one five, an interval of 81/5 which becomes 81/80, the syntonic comma. The syntonic comma links Pythagorean and Just tones, and adding it to the just tone of 10/9 leaves 9/8 (as two threes one five and a two cancel out), the Pythagorean tone (figure 5).

Curt Sachs’ notion of the Indian *srutis* was used in my book *HarmonicOrigins of the World* (2018), using a similar approach: finding 22 *srutis *could arrive at all of our modal intervals that are made up of just three types of interval worth 1, 2 and 3 *srutis of different size*, forming the more familiar tones and semitones of our modal scales. The results of figure 6 exactly correspond with the work of K.B.Deval which must be where Sachs gained his data (see end note**).

Based upon the prime number composition of tone numbers, a change of tonic will involve the movement of the location of D upon the mountain so that, A in the above limit of 720 will install D at the current location of A. Doing that places the tonic (4320 in figure 6) amongst the harmonic numbers associated with ancient Indian cosmology, such as those beginning with a “head number” 432, multiplied by prime 2 such as 864 and 1728 or by prime 5, then raising its location to, for example,the number of the flood heros: **8,64**0,000,000. In this way, as Ernest McClain proposed in *The Myth of Invariance (1976) *and other writings, an ancient music theory was integrated with ancient cosmological ideas, found in texts as references to “harmonic numbers” containing primes 2, 3 and 5.

**Curt Sachs, *The Rise of Music in the Ancient World*, New York: Norton 1943. *165*. The standard work on this is Mark Levy’s *Intonation in North Indian Music*. Chapter 3 starts with K.B. Deval’s work (1910, 1921) in which the Sa, Ri, Ga, Ma, Pa, Dha, Ni, Sa scale is given the ascending intervals in cents of 182, 112, 204, 204, 182, 112, 204, where 112 is the semitone of 16/15, 182 is the just tone of 10/9, 204 is the Pythagorean tone of 9/8, and the scale is modern Dorian.

Interpreting *Lochmariaquer *in 2012, an early discovery was of a near-Pythagorean triangle with sides 18, 19 and 6. This year I found that triangle as between the start of the Erdevan Alignments near Carnac. But how did this work on cosmic N:N+1 triangles get started?

Robin Heath’s earliest work, *A Key to Stonehenge* (1993) placed his **Lunation Triangle** within a sequence of three right-angled triangles which could easily be constructed using one megalithic yard per lunar month. These would then have been useful in generating some key lengths proportional to the lunar year:

**the number of lunar months in the solar year,****the number of lunar orbits in the solar year**and**the length of the eclipse year in 30-day months.**

all in lunar months. These triangles are to be constructed using the number series 11, 12, 13, 14 so as to form N:N+1 triangles (see figure 1).

n.b. In the 1990s the primary geometry used to explore megalithic astronomy was N:N+1 triangles, where N could be non-integer, since the lunation triangle was just such whilst easily set out using the 12:13:5 Pythagorean triangle and forming the intermediate hypotenuse to the 3 point of the 5 side. In the 11:12 and 13:14 triangles, the short side is not equal to 5.

Each triangle could then have an intermediate hypotenuse set at the 3:2 point of the shortest side, so as to form the eclipse year (11.37 mean solar months) and solar year (12.368 in lunar months), plus the lunar orbits in a solar year (13.368 orbits). The 12 length is **the lunar months in a lunar year** but also **the mean solar months in a solar year** and the length 13 is **the length of another type of lunar year** (in lunar months) and **the number of orbits in a 12 month lunar year**.

Robin’s triplet of triangles was a fantastic early result in geometry that really did not prove to be useful or likely. There is no need to have linked triangles and the megalithic appears to have found other ways to geometrically link and compare different counted lengths. It is interesting also that day-inch counting was not seen as the initial technique leading to such geometries in the megalithic, until around 2008. Whilst counted lengths were frequently interpreted within monuments, the idea that they were relics of actual time counts using constant lengths was overlooked and very hard for me or Robin to see, the Le Manio Quadrilateral proving a decisive confirmation.

It came to me that the **18-19-6 tripl**e was interesting in that the amount by which it deviated from the Pythagorean rule (whole numbers for all three sides) causes **the 18 side to be 18.027 long**, if the triangle is to be right angled. This number (18.027) was familiar to me as being **the length in solar years of the Saros eclipse period**, of 18.030 solar years. This seemed wonderful, **as if nature was somehow shaping reality to suit a near-Pythagorean triangle**. It was also interesting because **the Saros is 19 eclipse years**, by definition, so the base of the triangle can be metrological shifted to units in **the ratio solar year to eclipse year**, so that both the longer sides are then, each 19 (different) units long, eclipse years on the base and solar years on the hypotenuse. This is to be found in the angle of the Erdevan Alignments near Carnac see "Erdeven Alignment’s counting of Metonic and Saros Periods").

This triangle itself became eclipsed by new progress. One Christmas (1993 or 1994), Robin and I found the single unit that divides into both the eclipse and solar years, later revealing that the moon’s nodal period, **the solar year and the eclipse year are normalised to the tropical day**, through the rate by which the eclipse nodes move slowly retrograde. (the nodes are where the orbit of the moon crosses the sun’s path in the year, due to which eclipses can only occur when the sun is on one of these two nodes.)

The number of days it takes for the lunar nodes to move one DAY, in angle, (the angle the sun moves in a single solar day) is 18.618 days. This makes the eclipse year equal 18.618 x 18.618 days, the solar year 19.618 x 18.618 days, *exactly because* the difference between these two types of year is then being 18.618 days and the nodal period being 18.618 solar years and 19.618 eclipse years long. (This was further explored in Robin Heath’s Wooden Book called *Sun, Moon and Earth,* or see Figure 7 for an early summary) Because of this the eclipse year is the square of 18.618 or more properly, the square root of the eclipse year is 18.6177.

Out of this relationship comes the VERY important triangle found at Le Menec (the angle of its Alignments) and Mane Lud/Locmariaquer, with longer sides 18.618 and 19.618 and a third side 6.184.

Obviously, this is just larger than the 18-19-6 triangle mentioned above and its sharp angle is less, 18.36 instead of 18.40 degrees.

The third similar triangle in our title is that produced by **the diagonal of a triple square** and this similarity was understood by Robin through his work on **Alexander Thom** geometry for a **Type B flattened circle** – a circle whose perimeter has been reduced in length through using arcs of variable radius.

Robin noted that, implicit within the Type B design, an 18.618:19.618 triangle can be inferred, though it is actually the angle of the diagonal of a triple square, which is 18.43 degrees and only 1 1/2 minutes of a degree different to the angle of an 18.027:19:6 triangle.

Within the last decade, **Howard Crowhurst** discovered a whole system of megalithic alignments, using multiple squares, between sites near Carnac (see his *Megalithes* book, 2007). I could see this probably also applied to time measures, as lengths within megalithic constructions directly using triple squares (figure 5), most notably at Locmariaquer (figure 6).

The above three triangles of figures 2, 3, and 5, are

similar and near congruentand yet their smallest angles are slightly different and this may be measurable to say which geometry was used.

If the* Tumulus d’Er Grah* ended after a **Saros period of day-inch counting from Er Grah** this would mean that a point **19 years of day-inch counting lay directly north of Er Grah** and it would be would be 6 years of day-inch counting away from the (original but now sadly altered) northern tip of the *Tumulus d’Er Grah*. The significance of this is brought out in the dolmen of *Mane Lud* where, observing the possible parallelism noted in my *Sacred Number and the Lords of Time (pages 56-63)*, the north end of the Tumulus d’Er Grah (the broken grand menhir) corresponds with the end of the west passageway.

There are obviously further questions about the whole matter that await some practical measurements. However, by exploring what the astronomical and geometrical facts are and by accepting day-inch counting as the first feasible means for the monuments around Locmariaquer to have been built, we become capable of glimpsing, for the first time, the earliest use of geometry as a tool.

]]>The word Alignment is used in France to describe its stone rows. Their interpretation has been various, from being an army turned to stone (a local myth) to their use, like graph paper, for extrapolation of values (Thom). That stone rows were alignments to horizon events gives a partial but useful explanation, since menhirs (or standing stones) do form a web of horizon alignments to solstice sun and to the moon’s extreme rising and setting event, at maximum and minimum standstill. At Carnac the solstice sun was aligned to the diagonal of the 4 by 3 rectangle and maximum and minimum standstill moon aligned to the diagonal of a single or double square, respectively.

It seems quite clear today that stone rows at least represented the counting of important astronomical time periods. We have seen at Crocuno that eclipse periods, exceeding the solar year, are accompanied by some rectalinear structures (Le Manio, Crucuno, Kerzerho) which embody counting in miniature, as if to record it, and it has been observed that cromlechs (or large stone kerb monuments) were built at the ends of the long stone rows of Carnac and Erdeven. Sometimes, a cromlech initiated a longer count,with or without stone rows, that ended with a rectangle (Crucuno). The focus on counting time naturally reveals a vernacular quite unique to this region and epoch. We have seen that the Kerzerho alignments were at least a 4 by 3 rectangle which recorded the 235 lunar months in feet along its diagonal to midsummer solstice sunset. After that rectangle there follows a massive Alignment of stone rows to the east,ending after 2.3 km having gradually changed their bearing to 15 degrees south of east. Just above the alignments lies a hillock with multiple dolmens and a north-south stone row (Mané Braz) whilst below its eastern extremity lies the tumulus and dolmen,”T-shaped passage-grave” (Burl. Megalithic Brittany. 196) called Mané Groh.

If we take the hint that the Kerzerho rectangle ("Kerherzo Rectangle near Erdeven & Crucuno") at the western extremity of the Erdevan alignments has the length 235 feet along its diagonal, and remember that terminating monuments often form a micro-site, a clue to the whole provided by an oral culture, like a textbook,then we can expect the Metonic of nineteen years within which 235 lunar months complete, to be present in the larger monument. We also know that the 47 lunar month approximation/reminder for 4 eclipse years was also involved since 235 =5 x 47, and the Crucuno count to the south, between dolmen and rectangle, was 4 by 3 with units of 47 feet.

It is possible that the alignments (figure 1), which were once “probably complete” (Lukis, 1888) and gaps now widened by agricultural necessity, sought to avoid the small hill of dolmen called Mané Braz whilst providing the alignments as a counting corridor-like cursus. Below the eastern extremity of the alignments lies the tumulus and dolmen of Mané Groh and one can see its bearing,on Google Earth, as being 18.4 degrees south of east whilst its distance, from stone left of the entrance to Kerzerho, divides by 235 feet 32 times. Thus, if a unit of 32 feet is used, then the distance to Mané Groh is 32 times the distance across Kerzerho’s diagonal and it too marks the Metonic period.

The angle to Mané Groh, at 18.4 degrees relative to east, is that of an 18-19-6 near-Pythagorean triangle and, since east and west are so accurately maintained at Crucuno and Kerzerho rectangles, then the 18 side can be assumed, there is no monument (today) at the right angle but one can see in figure 3, the 18 side can be taken down to area where the alignments terminate on the east, in their attempt to avoid the hill of Mané Braz whilst probably providing a counting corridor for the Saros period of just over 18 years.

The Erdeven alignments were evidently a single monument including the dolmens necessary for marking out the triangle (Mané Groh), conducting the astronomy (Mané Braz) and the rectangles at the western extremity.

There was a full article relating to two series of three Pythagorean triangles ("Story of Three Similar Triangles") in which my discovery of a near-Pythagorean triangle with shortest side 6 units and longest side 19 units (the Metonic in years) leads to a base side of √(192 / 62) = 18.0278 units whilst the Saros period (of 19 eclipse years or 223 lunar months) equals 18.030 solar years, just 19 hours longer.

This means that the 18-19-6 triangle is a very
excellent model of the Saros to Metonic ratio, of 223 to 235 lunar months,
providing the 19 and 6 sides are integer: the 18 side will be longer by the required
amount. Having counted
223 units of 32
feet directly east, one would reach the right angle of the triangle as shown in figure 3. So can one find (without looking
too hard) the reference
length of 223 feet within the Kerzerho
rectangle? Figure 4 shows that we can and in this is both reminds me of the multiple
square geometries AAK found associated with monuments around Carnac and directly illustrates it in the symbolism of Kerzerho.

*Note the raised yellow
length of 223 feet, between
large stones A and B, crosses line of 235 feet, and it is the beginning of the 235 x 32 = 7520 foot alignment to Mané Groh*

Two large stones define a length of 223 feet along the alignment to Mane Groh, which became the 235 months of 32 foot months, equalling 7520 feet in all. The starting stone A then defines the west-east line of latitude along which the abstract base of the larger triangle 32 times larger than the upper triple square length in figure 4.

The 32 foot units of length probably reveal something about the metrology practiced by the astronomers. At Crucuno ("Lunar Counting from Crucuno Dolmen to its Rectangle") I found a similarly large aggregate of 27 feet which, divided by 32/35 becomes 29.53059 Iberian feet. This means that one can use aggregate units of 27 feet to count lunar months, all the while able to also count days as individual Iberian feet within such months. In this case, of 32 feet, that length equals 35 Iberian feet since the 32 cancels leaving 35. In fact, 32 feet also equals 28 Royal feet (relevant in this context,having identified the royal cubit of 12/7 engraved upon Gavrinis’ stone C3). We are faced with the probability that such equivalences, between whole numbers of commensurate measures, may have been how measures were seen rather than, as we do today, see feet other than the English as a ratio of it.

The larger the unit of measure used to make a large count of lunar months, when applied to counts which consist of an whole number of lunar months (as eclipse cycles do), reduces the task to counting full or half moon’s within a smaller scale counting process, then to move on eclipse markers along a Saros encoded alignment. It would soon be found, within the frame of the Metonic’s 20 eclipse years (of 5 x47 lunations) lay the Saros period of 19 eclipse years. Easily then recognised as a superior eclipse period, repeating near identical eclipses at full moon, a very accurate value for the eclipse year (of two semesters) would emerge – far beyond what might be expected possible in the Neolithic period. Hence the significance of the Erdeven alignments to the history of humanity and of science. It was possible for an exact astronomical science to evolve before the historical appearence of exact sciences in the Ancient Near East.

One would need an eclipse monument to process the pattern of eclipses over a long period so as to establish the consequences of the invisible invisible lunar nodes, where the moon’s orbit crosses the path of the sun during the year. The sun has to be at one of these nodal crossings for an eclipse to occur and, because the sun moves slower than the moon, the moon can often pass opposite the sun when it lies behind a lunar node. Having established there are 223 lunar months in a Saros and 235 in a Metonic, one can build the right angled triangle 235-223-(74.1) by recognising the Metonic as 19 solar years long and seeing the other sides as 18 years and 6 years, but also that 18 needs to a just that bit larger. The angle can be known as the diagonal of a three square rectangle and already, at Carnac, the single, double square and 4×3 rectangle were familiar.

To construct the needed larger eclipse
monument to refine the Saros and eclipse
year lengths, in days, the angle of a triple square was projected south of east
in a count of 235 lunar months each equal to 32 feet, at which point the
tumulus and dolmen of Mané Groh was built.

*Figure 5 Design of Mané Groh as “micro-site” containing useful
elements of the greater monument*

The length of the corridor appears to be 32 feet and the symmetrical pairs of chambers which form the transcepts form part of a nested 4 by 3 design whose diagonal points to north relative to an otherwise non-planetary alignment to the southeast. The transcept design strongly mirrors the horizontal shape of the existing western extremity of the Kerzerho monument beside the road. If intentional, this may point towards that monument not having been encroached so much in the intervening periods. Conforming to the greater pattern of the alignments, indicate the main stones are in their original positions.

Another strange feature of 32 feet is revealed in having 384 day-inches within it. The triple square geometry relates eclipse to solar years as well as solar years to the lunar year of 13 lunar months (the “embolismic” year required to calculate Easter after the Synod of Whitby).

It is easy, since the romanticism of the 19th century, to talk of the megaliths as “rude monuments”, associated with tribalism, “rituals”, “sacrifices”, and “healing stones”. This indicates that when something significant goes beyond the event horizon of living memory or current ways of doing things, they enter the world of human imagination that whilst entertaining is completely wrong-headed. These monuments were actually pre-religious and highly technical in approaching astronomy without modern or even ancient numeracy.

If you are conducting a long count in which multiple objects mark past eclipses, one needs a long but not necessarily straight pathway on which to conduct the counting. To keep accurate track of the passing of the year, months, lunar orbits one needs alignments to the horizon exactly as one finds expressed in Mané Bras where the surviving dolmen point in various way to the horizon.

If you will not take into account the counting of time between horizon events, the use of exact measures and geometrical methods then it is the void then created that , in order to have any explanation of monuments, we are left with a primitive race and their rude monuments. How much better to see disparate monuments coalesce into a common purpose as at Erdeven and Crucuno.

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