Month: September 2022

On the Harmonic Origins of the World

Does the solar system function as a musical instrument giving rise to intelligent life, civilization and culture on our planet? This 2018 article in New Dawn introduced readers to the lost science of the megalithic – how our ancestors discovered the special ratios and musical harmony in the sky which gave birth to religion and cosmology. The musical harmonies were the subject of my book released that year, called The Harmonic Origins of the World.

After the ice receded, late Stone Age people developed the farming crucial to the development of cities in the Ancient Near East (ANE). On the Atlantic coast of Europe, they also developed a now-unfamiliar science involving horizon astronomy. Megalithic monuments were the tools they used for this, some still seen in the coastal regions of present day Spain, France, Britain and Ireland. Megalithic astronomy was an exact science and this conflicts with our main myth about our science: that ours is the only true science, founded through many historical prerequisites such as arithmetic and writing in the ancient near east (3000- 1200 BC) and theory-based reasoning in Classical Greece (circa 400-250 BC), to name but two. Unbeknownst to us, the first “historical period” in the near east was seeded by the exact sciences of the megalithic, such as the accurate measurement of counted lengths of time, developed by the prehistoric astronomers. With the megalithic methods came knowledge and discoveries, and one discovery was of the harmonic ratios between the planets and the Moon.

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Parthenon as a New Model of the Meridian

This was published as The Geodetic And Musicological Significance Of The Shorter Side Length Of The Parthenon As Hekatompedon Or ‘Hundred-Footer’ in Music and Deep Memory: Speculations in ancient mathematics, tuning, and tradition, in memoriam Ernest G. McClain. Edited by Bryan Carr and Richard Dumbrill. pub: Lulu. photo: Steve Swayne  for Wikipedia on Parthenon.

This note responds to Kapraff and McClain’s preceding paper, in which they discover a many-faceted musical symbolism in the Parthenon. Specifically,  Ernst  Berger’s  new measurements include the shorter side of the triple pedestal of the monument as an accurate length to represent one second of the double meridian of the earth. By applying a knowledge of ancient metrology, Anne Bulckens’ doctoral derivations of a root foot can resolve to a pygme of 9/8 feet, of which one second of latitude would contain 90 such feet. However, as a ‘hundred footer’, the foot  length  should  then be 81/80 (1.0125) feet, the ratio  of  the syntonic comma. This would indicate a replacement, by Classical times, of the geographical constant of 1.01376 feet  within the model of the earth since the original model, by the late megalithic, assumed that the meridian was exactly half of the mean circumference of the earth. These alternative geographical constants co-incidentally represent the ubiquitous theme in ancient musicology of the transition between Pythagorean and  Just tunings and their respective commas of Pythagorean 1.01364 … (in metrology 1.01376) and syntonic 81/80 (1.0125).

By Classical times the term hekatompedon or ‘hundred-footer’ had evolved, to describe the ideal dimensionality of Greek peristyle temples. One of the earliest, the Heraion of Samos, came to be 100 feet long by the end of the 8th century[1], in contrast to the surface width of the Parthenon’s stylobyte which had been established as in the range 101.141 (Stuart, c.1750) to 101.341 (Penrose in 1888) feet[2].

Recent measurements in 1982 by Ernst Berger[3] found that the top surface of the stylobyte was just over 101.25 feet wide4 and that the most frequently occurring length was 857.6 mm. Anne Bulckens’[5] corresponding foot measure for this would be a step of 2.5 feet, each of 9/8 (1.125) feet, to within
one part in 2500; a foot length called a pygme within historical metrology, after the size of small men first mentioned when Herakles was travelling back from India6. The shorter ends of the Parthenon’s stylobyte would then be 90 such feet across.

However, should the two ends be divided by 100, the required foot length of 101.25 feet becomes a microvariation of the English foot, namely 81/80 (1.0125) feet, a ratio identical with the syntonic comma. This is another ratio crucial to the history of ancient tuning theory; being found between pure Pythagorean tones (9/8) and their counterparts within just tuning (10/9); when string lengths are given specific whole number lengths to specify their pitches intellectually.

1. Hurwit, Jeffrey M., (1987), The Art and Culture of Early Greece, 1100-480 B.C., Cornell: Ithaca, 74-77
2. Berriman, A.E., (1953) Historical Metrology, London:
Dent. IX, 116-120.
3. Berger, E., ed. (1986) Parthenon-Kongress Basel, 2 Vols, Mainz: Philipp von Zabern.
4. an average noted by Berriman, 119.
5. Bulckens, A.M. (1999) The Parthenon’s Main Design Proportion and its Meaning, [Ph.D. Dissertation], Geelong: Deakin University, 269 pp. ; (2001) The Parthenon’s Symmetry in Symmetry: Art and Science (Fifth Interdisciplinary Symmetry Congress and Exhibition of the ISIS-Symmetry), (Sydney, 2001), no. 1-2, pp. 38-41.
6. Philostrates of Lemnos (c. 190 – c. 230 AD) Imagines Heracles among the Pygmies, see Loeb Classical Library

A recent article by Jay Kapraff and Ernest McClain[7] observes that the width of the Parthenon symbolically defined one second of latitude (taking surface lengths as linear fractions of latitude). This implies the double meridian length was known within 0.003% of its modern estimation.

A geodetic symbolism was apparently given to shorter side length of the Parthenon, making it smaller than it would have been if modelled on the circumference of the earth as one 3,600th of one 360th part of the mean earth. If so, this geodetic meaning of the Parthenon can be compared with monuments built two thousand years earlier, such as Stonehenge and the Great Pyramid of Giza, within which the relationship of the mean earth was specified, relative to the polar radius, using the same metrological system.

The ancient model of the earth, recovered[8] by John Neal[9] and John Michell[10], used three different approximations of π to model the distortion of
the rotating planet relative to its mean, or perfectly spherical, size. In that model, the Meridian was assumed to be half the circumference of the mean earth of 44 times 126 (131,383.296) feet or 24,883.2 miles. Had the Parthenon’s builders used this model then its ends would be 101.376 feet in width and one hundredth of this would be a foot of 1.01376 feet, the foot known as the ‘Standard Geographical’ Greek foot[11].

The mean circumference of the earth (24,883.2 miles) and the actual double meridian length (24,859.868 miles) are in the same ratio as the geographical foot of 1.01376 (3168/3125) and 1.0125 feet: the 81/80 foot measure that makes the Parthenon’s 101.25 feet a ‘hundred footer’. It is therefore reasonable to assume that, between the building of Stonehenge and Great Pyramid (by 2,500 B.C.) and the building of the Parthenon (designed by 447 B.C.), a more accurate
measurement of the Meridian had superseded the previous assumption, within the old model, that the Meridian was half the length of the mean earth circumference.

7. The Proportional System of the Parthenon, in preparation for the In Memoriam volume for Ernest McClain (1918-2014)
8. Michell by 1980 and Neal, fully formed, by 2000.
9. Neal, John (2000) All Done With Mirrors, Secret Academy, London.
10. Michell, John (1982) Ancient Metrology, Pentacle Books, Bristol, 1982; (2008 new ed.) Dimensions of Paradise, Inner Traditions: Rochester.

Further to this, one can see how the transition from Pythagorean to just tuning systems[12] is strangely present in the relationship between the mean earth circumference and the actual meridian length, since the geographical constant of 1.01376 is near identical to the Pythagorean comma of 1.0136433 while the (chosen) ratio of 1.0125 is the syntonic comma and this, times 100, is near identical to the actual length of one second of latitude which would be 100 times 1.0128 feet[13], just one third of an inch different from a more
modern result.

The Parthenon ‘Hundred footer’ was able to dimensionally reference one second of the Meridian by having its shorter sides one hundred feet of 1.0125 feet long. Aligned to north, this presented accurate Classical knowledge of the
Meridian’s length. The monument expresses other musicological features via its metrology: the 81/80 foot unit is 125/128 of the Athenian foot of 1.0368 feet, a musical interval called the minor diesis, also found within just intonation and equaling the deficiency of three major thirds to the octave

12 The latter prevalent in other aspects of the monument, see Kappraff, J. and McClain, E.G. (2005: Spring–Fall) The Proportions of the Parthenon: A work of musically inspired architecture, Music in Art: International Journal for Music Iconography, Vol. 30/1–2.
13 A non-harmonic 79/78 feet.

The Approximation of π on Earth

π is a transcendental ratio existing between a diameter/ radius and circumference of a circle. A circle is an expression of eternity in that the circumference, if travelled upon, repeats eternally. The earths shape would be circular if the planet did not spin. Only the equator is now circular and enlarged, whilst the north and south poles have a shrunken radius and, in pre-history, the shape of the earth’s Meridian between the poles was quantified using approximations of π as was seen in the post before last. In some respects, the Earth is a designed type of planet which has to have a large moon, 3/11 of the earth’s size and a Meridian of such a size that the diverse biosphere can be created within the goldilocks region of the Sun’s radiance.

It would be impossible to quantify the earth as a physical object without the use of approximations to π, a technique seen as emerging in Crucuno between its dolmen and famous {3 4 5} Rectangle where the 32 lunar months in 945 days was used, through manipulation of proximate numbers to rationalize the lunar month to 27 feet (10 Drusian steps) within which days could be counted using one Iberian foot (of 32/35 feet) as described here and in my Sacred Geometry book.

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π and the Megalithic Yard

The surveyor of megalithic monuments in Britain, Alexander Thom (1894 – 1985), thought the builders had a single measure called the Megalithic Yard which he found in the geometry of the monuments when these were based upon whole number geometries such as Pythagorean triangles. His first estimate was around 2.72 feet and his second and final was around 2.722 feet. I have found the two megalithic yards were sometimes 2.72 feet because the formula for 272/100 = 2.72 involved the prime number 17 as 8 x 17/ 100, and this enabled the lunar nodal period of 6800 days to be modelled and and the 33 year “solar hero” periods to be modelled, since these periods both involve the prime number 17 as a factor. In contrast, Thom’s final megalithic yard almost certainly conformed to ancient metrology within the Drusian module in which 2.7 feet times 126/125 equals 2.7216 feet, this within Thom’s error bars for his 2.722 feet as larger than 2.72 feet.

This suggests Thom was sampling more than one megalithic yard in different regions or employed for different uses. Neal [2000] found for Tom’s statistical data set having peaks corresponding to the steps of different modules and variations in ancient metrology, such as the Iberian with root 32/35 feet and the Sumerian with root 12/11 feet. It is only when you countenance the presence of prime numbers within metrological units that one breaks free of the inevitably weak state of proof as to what ancient units of measure actually were and, more importantly, why they were the exact values they were and further, how they came to be varied within their modules. However, the megalithic yard of 2.72 appears to outside the system in embodying the prime number 17 for the specific purpose of counting longer term periods which themselves embody that prime number.

The discipline of using only the first five primes {2 3 5 7 11} must have been accompanied by the perception that only if primes were dealt with could certain ends be served. This is crystal clear when we come to musical ratios in which the harmonic primes alone are used of {2 3 5} with an occasional “passenger” of the prime {7} as in 5040 which is 7 x 720, the harmonic constant.

Using 2.72 feet to count the Nodal Period

The first remarkable characteristic of 2.72 feet is that 8 x 17 in the numerator means that the approximation to π of 25/8 = 3.125 can, in (conceptually) multiplying a diameter, provide an image of 25 units on the circumference of a stone circle. For example a diameter of 2 MY would suggest 17 MY on the circumference, which is quite remarkable. Further to this, we know that the 6800 days of nodal cycle is factored as 17 x 400 and that the MY was shown (acceptably) to have been made up of 40 digits (in conformance to the general tradition within metrology that there are 16 digits per foot and 40 for a step of 2.5 feet, which a MY traditionally is). The circumference of 17 MY is then 17 x 40 digits which means that a diameter of 20 MY would give a circumference of 17 x 400 digits equalling 6800 digits as a countable circumference in digits per day.

This highlights how prime number factors played a role, in the absence of arithmetical methods, in solving the astronomical problems faced by the late stone age when counting time periods in days.

The Broch that Modelled the Earth

Summary

In the picture above [1] the inner profile of the thick-walled Iron-Age broch of Dun Torceill is the only elliptical example, almost every other broch having a circular inner court. Torceill’s essential data was reported by Euan MacKie in 1977 [2]: The inner chamber of the broch is an ellipse with axes nearly 23:25 (and not 14:15). The actual ratio directly generates a metrological difference, between the major and minor axis lengths, of 63/20 feet. When multiplied by the broch’s 40-foot major axis, this π-like yard creates a length of 126 feet which, multiplied again by π as 22/7, generates 396 feet. If each of these feet represented ten miles, this number is an accurate approximation to the mean radius of the Earth, were it a sphere.

The two ratios involved, 22/7 and 63/20, each an approximation to π, become 9.9 (99/100) when they are multiplied together, as an approximation to π squared.  Figure 1 shows that these two ratios, if 22/7 differently used as its reciprocal 7/22, also generates the ratio between the mean and polar radii of the Earth, since 63/20 x 7/22 = 441/440. The ancient Meridian length could be calculated from 396 when multiplied by using the most accurate rational π noted by Fibonacci as 864/275. The 396 units, of 10 miles per foot, was a practical distance to have realized in the megalithic without arithmetic, to store the 3960 mile mean radius of the earth, since the mile of 5280 feet is 4/3 of 3960; that is, 396 x 4/3 equals 528, implying that this model was conceived of within a decimal framework but without the base-10 positional notation of arithmetic. We show that the methods of calculation used can only have seen numbers-as-lengths as being composed of factors of just the first five prime numbers {2 3 5 7 11} and that this limitation upon numbers created a metrology in which fractional units of measure could manipulate lengths to multiply and divide them through addition and subtraction of the powers of these primes.

Marc Calhoun’s picture from the Island (picture from his blog)

Contents

  1. Summary
  2. Introduction.
  3. Main Thesis.
  4. Pre-arithmetic Calculation using Powers of {2 3 5 7 11}
  5. Combining Prime Number Composites.
  6. Appendix 1 Extract from Science and Society in Prehistoric Britain.
  7. Appendix 2: Preface: The Metrology of the Brochs.
  8. Metrological Bibliography.
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Mangroves and the Moon’s Maximum

photo: Ariefrahman for Wikipedia /_

Tides on earth are due to the sun and moon. During the year, the Sun reaches extreme solstice points and during the lunar month, the phases indicate where the sun is relative to the sun: their configuration relative to one another, leading to stronger or weaker tides.

The tides therefore vary but when the lunar orbit is in phase with the solar ecliptic path, the moon rises above and below that path and the moon becomes more extremely north and south than the solstice sun ever can be. Within a single year, the sun is at winter solstice in midwinter, and summer solstice in midsummer. But the moon takes 18.618 years to reach its maximum standstill, further south and north than the solstice sun.

Ancient cultures were aware of this cycle and sometimes thought to place monuments or burial places on an alignment with maximum moonrise or moonset, occurring north and south of east on the eastern or western horizon.

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