Ancient musical knowledge came to Just tuning long before Greek music, in Babylonia. It now seems likely that two sources of musical information, were involved in an early tradition of musical tuning by number: firstly, the early number field is the original template upon which musical harmony is based; and secondly, the prehistoric geocentric astronomy which preceded the ancient world had been comparing counted astronomical time-periods, and had discovered the rational tone and semitone intervals between the lunar year, Jupiter and Saturn . Ernest G. McClain identified a harmonic parallelism within ancient texts in which the anomalous numbers found within mythic narratives inferred a unique array (a matrix) of whole numbers, shaped like a mountain, which could explain plot elements, events and characters of the narrative, as intended parallels to such harmonic mountains. McClain’s matrices allowed the author to locate the harmonic intervals found between planetary synods as a reason why religious texts should have employed harmonic numbers, these relating to planetary time as gods alongside ancient systems of tuning. Based on a talk delivered at ICONEA2013, 5th December at Senate House, University College London.
Month: September 2020
Geometry 4: Right Triangles within Circles
This series is about how the megalithic, which had no written numbers or arithmetic, could process numbers, counted as “lengths of days”, using geometries and factorization.
My thanks to Dan Palmateer of Nova Scotia
for his graphics and dialogue for this series.
This lesson is a necessary prequel to the next lesson.
It is an initially strange fact that all the possible right triangles will fit within a half circle when the hypotenuse equals the half-circles diameter. The right angle will then exactly touch the circumference. From this we can see visually that the trigonometrical relationships, normally defined relative to the ratios of a right triangle’s sides, conform to the properties of a circle.
Geometry 3: Making a circle from a counted length
The number of days in four years is a whole number of 1461 days if one approximates the solar year to 365ΒΌ days. This number is found across the Le Manio Quadrilateral (point N to J) using a small counting unit, the “day-inch”, exactly the same length as the present day inch. It is an important reuse of a four-year count to be able to draw a circle of 1461 days so that this period of four years can become a ouroboros snake that eats its own tale because then, counting can be continuous beyond 1461 days. This number also permits the solar year to be counted in quarter days; modelling the sun’s motion within the Zodiac by shifting a sun marker four inches every day.
Our goal then is to draw a circle that is 1461 day-inches in perimeter. From Diagram 1 we know that a rope of 1461 inches could be divided into 4 equal parts to form a square and from that, an in-circle to that square has a diameter equal to a solar year of 365ΒΌ days. Also, with reference to Figure 1, we know that the out-circle will have a diameter of 14 units long relative to the in-circle diameter being 11 units long, and this out-circle will have the perimeter of 1461 inches that we seek.
For this, the solar year rope (the in-circle diameter) needs to be divided into 11 parts. Start by choosing a number that, when multiplied by 11, is less that 365 (and a 1/4). For instance, 33. A new rope will be formed, 11 x 33 = 363 inches, marked every 33 inches to provide 11 divisions. Through experience, we discover we need 2 identical ropes so as to make practical use of the properties of symmetry through attaching ropes to both ends of the solar diameter rope.
Place one rope at the West side of the in-circle diameter and swing it up until it touches the in-circle. Place the other rope at the East side of the in-circle diameter and swing it down until it touches the edge of the in-circle. Now connect the 33 inch marks between the 2 ropes. This will divide the 365 1/4 diameter into 11 segments.
Seven of those segments are the new radius to create the 1461 inch outer-circle.
This novel application of the equal perimeters model, rescued from Victorian textbooks by John Michell and applied by him most memorably perhaps to Stonehenge and the Great Pyramid (in Dimensions of Paradise) is a general method for taking a counted length and reliably forming a radius rope able to transform that counted length into a circle of the same perimeter as the square, easily formed by four sides ΒΌ of the desired length.
The site survey at the start, drawn by Robin Heath, appeared in our survey of Le Manio.
Preface: The Metrology of the Brochs
feature picture: Broch of Mousa. The broch on the island of Mousa is the best-preserved of the many brochs in northern Scotland. It is thought to be some 2000 years old
credit: Anne Burgess / Broch of Mousa / CC BY-SA 2.0
I wrote this preface for Euan MacKie who had resurrected his work on measures found within the brochs of Scotland. Euan was almost a lone voice in support of Alexander Thom’s work on metrology in the megalithic, and also the long distance alignments in the Western Isles of Scotland. When he met John Neal at the latter lecture in Glasgow, at which I was present, they appear to have entered into a review of the data and John Neal came back with an interesting theory which would make a full range of historic measures to have been employed in one area of northern Scotand, in the Iron Age. I sent Euan a summary of what ancient metrology appeared to be as a system of ratios and why Neal’s finding within MacKie’s data would be important. It became the preface for the article called The Roundhouses, Brochs and Wheelhouses of Atlantic Scotland c.700 BC-AD 500: Orkney and Shetland Isles Pt. 1: Architecture and Material Culture (British Archaeological Reports British Series) which I have recovered from a partial proof copy.
Preface
by Richard Heath
John Neal has demonstrated elsewhere [All Done With Mirrors, John Neal, 2000] that ancient metrology was based upon a “backbone” of just a few modules that each related as simple rational fractions to the “English” Foot. Thus a Persian foot was, at its root value, 21/20 English feet, the Royal foot 8/7 such feet, the Roman, 24/25 feet and so on. By this means, one foot allows the others to be generated from it.
These modules each had a set of identical variations within, based on one or more applications of just two fractions, Ratio A = 176/175 and Ratio B = 441/440. By this means ail the known historical variations of a given type of foot can be accounted for, in a table of lengths with ratio A acting horizontally and ratio B vertically, between adjacent measures.
In the context of what follows, this means that each of the differently-sized brochs analysed by Neal appear to have used a foot from one or other of these ancient modules, in one of its known variations. That is, the broch builders seem to have chosen a different unit of measure rather than a diffΓ©rent measurement, as we would today, when building a differently sized building. Furthermore, these brochs appear to have been based upon the prototypical yet accurate approximation to pi of 22/7, so that – providing the broch diameter would divide by seven using the chosen module – then the perimeter would automatically divide into 22 whole parts.
Thus, John Neal’s discovery that broch diameters divide by seven using a wide range of ancient measures implies that the broch builders had – (a) inherited the original system of ancient measures with its rational interrelations between modules and variations within these, from which they could choose, to suit a required overall size of circular building, often the foundations available: (b) were practicing a design concept found in the construction of stone circles during the Neolithic period.
These measures, used in the brochs, are not often found elsewhere in Britain, but are historically associated with locations hundreds if not thousands of miles distant. This suggests that the historical identification of such measures is only a record of the late use of certain modules in different regions, after the system as a whole had finally been forgotten, sometime after the brochs were constructed.
Such conclusions, if correct, are of such a fundamental character that they present a compelling case for ancient metrology and its forensic power within the archaeology of ancient building techniques.
—x—
Throughout Scotland and the Scottish islands there are in excess of 200 major broch sites. The following analysis is taken from, what I believe to be, the accurately measured inner diameters of 49 of them as supplied by Professor Euan MacKie. The modules are expressed in English feet although the original measurements were taken in metres and converted to feet at the rate of 3.2808427 feet to the metre. The range of diameters extends from the smallest, at Mousa, 18.897654ft, to the greatest at Oxtrow at 44.816311ft. John Neal’s original work on this can be found in this article, from this website’s earlier incarnation which also included a version of Appendix 2 of Sacred Number and the Origins of Civilization – soon also to be added, for reference.
Read 145 times by September 2012 10:28
Sacred Number and the Lords of Time
Back Cover
ANCIENT MYSTERIES
βHeath has done a superb job of collating his own work on the subject of megaliths with the objective views of many other researchers in the field. I therefore do not merely recommend reading this book but can state unequivocally it is a must read.β
–John Neal, British metrologist and researcher and author of Measuring the Megaliths and The Structure of Metrology
βIn Sacred Number and the Lords of Time we have an important explanation of how megalithic science was developed. This book is a long-overdue wakeup call to a modern culture that has abandoned this fully developed and astonishingly rich prehistoric model of the physical world. The truth is now out.β
–Robin Heath, coauthor of The Lost Science of Measuring the Earth and author of Sun, Moon and Earth
Harmonic Explorer app
The work of the late Ernest McClain [1918-2014] was based upon a technique in which ever higher numbers appear to have been studied in antiquity as to the harmonic field of intervals
McClain’s work was very briefly introduced in Precessional Time and the Evolution of Consciousness (2011) in its Chapter 3 on the Age of Aries, when such calculational musicology/theology seem to have replaced the megalithic activities in Age of Taurus. After then collaborating with Ernest for a few years, after his death I attempted to make an accessible book on Ernest’ work in my 2018 book Harmonic Origins of the World.
Ernest had a fine sensibility for the Ancient Near East (or A.N.E.), its number science and symbolisms. Through many email exchanges I wanted to be able to generate his tonal diagrams for myself; to both learn more and answer specific questions. I was able to make a web page so that people like myself could more easily enter his world of tonal yantras (that look like mountains) and their corresponding tonal mandalas (the octave circle); without having to learn the method of “limiting the products of powers of three and five and then doubling all numbers to the limiting number”. This Harmonic Explorer app enables ridiculous freedom over the previously required manual calculations, including walking through limiting number space with prime factor buttons, multiplying/dividing by 2, 3, 5, 10 or 12. It can also be very useful for screen grabbing when discussing a given tuning system as in:
TRY IT NOW HERE: Harmonic Explorer
Ernest’s website has study pdf’s of his books and papers plus other resources.