Category: Themes

guided tours behind the posts

Modularity of Seven

Sacred Geometry is metrical, it is based upon the interactive properties of “natural” (that is, whole) numbers and cosmic constants.

We live in a civilization where everything is thought to be functionally due to forces and laws, these all calculated using numbers and algebra. For this reason, it is hard to see the influence of numbers acting directly in situations to reveal that, geometrical forms are only possible due to numbers. One such form is the equal perimeter circle and square: this figuring heavily in my later books, as an ancient model, and in postings on this website (opens in new tab).

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The Cellular World of Twelve

The foot has twelve inches just as the British shilling had twelve pence. A good case can be made for twelve as a base like 10, since there are 12 months within the year and many ancient monuments can be seen to have employed duodecimal alongside decimal number, to good effect.

Until 1971 the currency in the United Kingdom of Britain was duodecimal, called pounds, shillings and pence.

This old system of currency, known as pounds, shillings and pence or lsd, dated back to Roman times when a pound of silver was divided into 240 pence, or denarius, which is where the ‘d’ in ‘lsd’ comes from. (lsd: librum, solidus, denarius). see historic-uk.com

There were 12 pence in a shilling and 20 shillings in a pound, that is 240 pennies. The change to a decimal (100) pence in a pound caused a lot of inflation during the changeover due to price opportunism, then part lasting recession. In British heads the skill of giving and taking change in a duodecimal arithmetic was soon lost. In the late 70’s my mother, when visiting the US, was amusingly referencing “old money”, alongside the exchange rate between a decimal pound and the decimal dollar, just as Greeks had problems with the Euro.

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The Stonehenge Crop Circle of 2002

One sees most clearly how a single concrete measure such as 58 feet can take the meaning of the design into the numbers required to create it. However, metrology of feet and types of feet can hide the elegance of a design.

photo by Steve Alexander of TemporaryTemples.co.uk

I received Michael Glickman’s Crop Circles: The Bones of God at the weekend and each chapter is a nicely written and paced introduction to a given years worth of crop circles generally in the noughties. The above is the second in proximity to Stonehenge reminding keen croppers of an earlier one. This cicle preceeded the late-season (August) circle at Crooked Soley that I have an analysis of soon to be posted, drawing on Allan Brown’s small book on it.

Glickman’s chapter 10 : Stonehenge Ribbons and Crooked Soley provided a tentative analysis of the Ribbons as having the ends of the ribbons measuring 58 feet. The design was observed as making use of a single half circle building block for most of the emergent six arms emerging from the center. Michael suggested that there were 13 equal units of 58 feet across the structure.

Figure 10.4 Showing thirteen divisions of one of the three diameters of ribbons. photo: Steve Alexander.

From this I was able to observe that clearly the divisions were not equal in size and the white ones were clearly smaller as was the central circle’s diameter. Scanning the picture and placing it in my Visio program, so that a rectangle of 58mm was equal to the diameter of the right hand ribbon end, it was possible to determine that the ratio between these lengths was 5 to 4, or 5/4, from which the shorter white length must be 46.4 feet and that the diameter can be seen as 9 units across, that is 104.4 feet. The unit is 104.4 feet divided by 9 which equals 11.6 feet, which is 10 feet of 1.16 feet, the root reciprocal of the Russian foot of 7/6 feet, that is 7/6 feet divided by 175/176 (= 1.16). Going down the “Russian” root led to the diagram below.

My analysis of Michael Glickman’s figure reveals a span of 580 Russian Feet.

There are parallax errors so I have had to show the ideal designed shortened across the left-hand of the design, but the design has many numerical aspects where each arm is 27 units so that two arms are 54 which, plus the center, gives 58 times 10 equaling 580 Russian feet. But then I noted that 58 feet, divided by 5, gave the unit as 11.6 English feet while 58 feet divides into the 58 unit diameter across the crop circle.

Now we see a set of multiples of 29 are there as numbers {29 58 87 116 145 174 203 232 261 … }. The reciprocal Russian at 1.16 feet and the unit of 11.6 feet are decimal echoes of the number 29. The formula of the Proto Megalithic yard is 87/32 feet and 261/8 inches.

To be continued

One sees most clearly how a single concrete measure such as 58 feet can take the meaning of the design into the numbers required to create it.

Music, part 1: Ancient and Modern

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

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

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

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

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

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

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

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

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

To be continued.

St Peter’s Basilica: A Golden Rectangle Extension to a Square

HAPPY NEW YEAR

above: The Basilica plan at some stage gained a front extension using a golden rectangle. below: Later Plan for St. Peter’s 16th–17th century. Anonymous. Metropolitan Museum.

The question is whether the extension from a square was related the previous square design. The original square seems quite reworked but similar still to the original square. The four gates were transformed into three ambulatories defining four circles left, above, right and centre, see below.

Equal Perimeter models at the center of St Peter’s Basilica

Equal Perimeter Models

The central circle can be considered as 11 units in diameter so that its out-square is then 44 units. The circle of equal perimeter to the square will then be 14 units in diameter and the difference of 3 defines a circle diameter 3 units. The 11-circle represents the Earth while the 3-circle represents the Moon, to very high precision – hence making this model a representative of the Mysteries inherited from deep antiquity; at least the megalithic age and/or early dynastic Egypt, when the earth’s size can be seen in Stonehenge and Great Pyramid. This inner EP model, is diagonal so that the pillars represent four moons.

An outer Equal Perimeter model is in the cardinal directions (this alternation also found in the Cosmati pavement at Westminster Abbey, and inner models are related to the microcosm of the human being relative to the slightly larger model of Moons). The two sizes of Moon define the circles at the center, around St Peter’s monument. The mandala-like character of the Equal Perimeter model give here the impressions of a flower’s petals and leaves.

Golden Rectangles

You may remember a recent post about double squares and golden rectangles, where a half-circle that fits a Square has root 5 diagonal radius which, arced down, generates a golden triangle. It is therefore possible to fit the square part of the original design and draw the circle that fits the half-diagonal of the square as shown below.

The golden extension of the Basilica’s Square Plan

By eye, the square’s side is one {1} and the new side length below is 1/φ and the two together are 1 + 1/φ = φ (D’B’ below) which is the magic of the Golden Mean. This insight can be quantified to grasp this design as a useful generality:

Quantifying how the golden mean rectangles are generating phi (φ)

Establishing the lengths from the unit square and point O, the center of the right hand side. OA’ is then √5/2. When this is arced, the square is placed inside a half circle A’C, BC is √5/2 + 1/2 = 1/φ.

The rectangle sides ACD’B’ are the golden mean relative to the width A’B = 1, the unit square’s side, but that unit side length A’B is the golden mean relative to the side of the golden rectangle BC. In addition the length B’D’ is the golden mean squared relative to BC, the side of the golden rectangle.

Commentary

It seems that the equal perimeter models within the square design of Bramante were adjusted. The golden mean was used to extend the Basilica (originally an Orthodox square building named after St Basil) into a golden rectangle. This could be done by adding the equivalent lesser golden rectangle, relative to the unit square through the properties of the out half-circle from O.

The series of golden rectangles can travel out in four directions, each coming naturally from a single unitary square. The likely threefold symbolic message, added by the extension seems to be the primacy of the unitary square, of St Peter (on whom the Church was to be founded) and of the Pope (as a living symbol of St Peter).

Double Square and the Golden Rectangle

above: Dan Palmateer wrote of this, “it just hit me that the conjunction of the circle to the golden rectangle existed.”

Here we will continue in the mode of a lesson in Geometry where what is grasped intuitively has to have reason for it to be true. It occurred to me that the square in the top hemisphere is the twin of a square in the lower hemisphere, hence this has a relationship to the double square rectangle. So one can (1) Make a Double Square and then (2) Find the center and (3) a radius can then draw the out-circle of a double square (see diagram below).

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