Tag: Ernest McClain

American Musicologist who decoded the cryptic numerical ciphers in Plato’s Dialogues

Cubes: The Ancient Division of the Whole

Volume as cubes reveal the wholeness of number as deriving from the unit cube as corner stone defining side length and “volume” of the whole.

The first cube (above left) is a single cube of side length one. One is its own cornerstone. The first cubic number is two to the power of three, with side length two and volume equal to eight cubes that define the unit corner stone.

In modern thinking, and functional arithmetic, volume increases with side length but the cube itself, as archetype of space, is merely divided by the side length of the unit cornerstone, which is 1/8th the volume and therefore reciprocal to the volume of 8 leaving the cube singular.

This may not seem important but, by dividing a whole cube, one is releasing more and more of the very real behavior that exists between numbers, within the cube. For example the number 8 gives relations between numbers 1 to 8, such as the powers of two {1,2,4,8} and the harmonic ratios {2/1, 3/2, 4/3,5/4,6/5}. These can give an important spine of {4/3, 5/4,6/5} which equals 6/3 = 2 of the yet to (numerically) be octave of eight note classes. Moving to side length 3, the cube of three is twenty seven (27), as seen in figure above, top right. To obtain it, the corner stone must be side length 1/3, and volume 1/27 so that, in these units, the volume of the cube is 27.

If one were to reciprocally double the 1/3 side length, each cornerstone unit would have 8 subunits, so that the volume of 27 would be times 8 which equals Plato’s number of 216. Another view is then that the cornerstone side length has divided the bottom right cube into six units which number, 6, cubed, is 216 a perfect number for Plato.

By accepting the cube of one as the whole, this form of thinking reciprocally divides that whole side length to generate an inner structure within the whole cube of one, equal to the denominator of the reciprocation. The role of the whole is then to be the arithmetic mean between a number and its reciprocal. This procedure maintains balance between what is smaller than the whole (the reciprocal) and what is larger than the whole (in this case the volume).

In ancient tuning theory this was expressed by the two hexchords descending and ascending from the tonic (we might call do), expressed by the two hands. The octave of eight and the cube are both wholes to be broken into by numbers greater than one by means of reciprocation.

Ernest G. McClain revealed the scale of such thinking was massive, whilst also but secretly reciprocal, so that a limiting number could express how different wholes will behave due to their inner diversity of numbers at work within them.

In this example, a musical code for planetary resonance is revealed within the metrology of the Parthenon (above), implied tone set (right) and octave mountain of numbers below 1440 (left bottom). In this case the number 24 has been multiplied by 60 to give a limiting number of 1440. The cornerstone in this case is bottom left of the mountain = 1024, a pure power of 210.

By simply quoting a limiting number, in passing, ancient texts could, in the hands of an initiate, create an enormous world of tonal and impied religious meaning – through a kind of harmonic allusion.

It is only by using the conceptual approach of the ancients, that their intellectual life can be recovered – just by adding the waters of number and some powers of imagination.

Music of the Olmec Heads

Seventeen colossal carved heads are known, each made out of large basalt boulders. The heads shown here, from the city of San Lorenzo [1200-900 BCE], are a distinctive feature of the Olmec civilization of ancient Mesoamerica. In the absence of any evidence, they are thought to be portraits of individual Olmec rulers but here I propose the heads represented musical ratios connected to the ancient Dorian heptachord, natural to tuning by perfect fifths and fourths. In the small Olmec city of Chalcatzingo [900-500BCE] , Olmec knowledge of tuning theory is made clear in Monument 1, of La Reina the Queen (though called El Rey, the King, despite female attire), whose symbolism portrays musical harmony and its relationship to the geocentric planetary world *(see picture at end).

* These mysteries were visible using the ancient tuning theories of Ernest G. McClain, who believed the Maya had received many things from the ancient near east. Chapter Eight of Harmonic Origins of the World was devoted to harmonic culture of the Olmec, the parent culture of later Toltec, Maya, and Aztec civilizations of Mexico.

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Starcut Diagram: geometry to define tuning

This is a re-posting of an article thought lost, deriving in part from Malcolm Stewart’s Starcut Diagram. The long awaited 2nd edition Sacred Geometry of the Starcut Diagram has now been published by Inner Traditions. Before this, Ernest McClain had been working on tuning via Gothic master Honnecourt’s Diagram of a Man (fig. 2), which is effectively a double square version of the starcut diagram.

The square is the simplest of two dimensional structures to draw, giving access to many fundamental values; for example the unit square has the diagonal length equal to the square root of two which, compared to the unit side length, forms the perfect tritone of 1.414 in our decimal fractional notation (figure 1 left). If the diagonal is brought down to overlay a side then one has the beginning of an ancient series of root derivations usually viewed within the context of a double square, a context often found in Egyptian sacred art where “the stretching of the rope” was used to layout temples and square grids were used to express complex relationships, a technique Schwaller de Lubitz termed Canevas (1998). Harmonically the double square expresses octave doubling (figure 1 right).

Figure 1 left: The doubling of the square side equal 360 units and right: The double square as naturally expressing the ordinal square roots of early integers.

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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).

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Introduction to my book Harmonic Origins of the World

Over the last seven thousand years, hunter-gathering humans have been transformed into the “modern” norms of citizens (city dwellers) through a series of metamorphoses during which the intellect developed ever-larger descriptions of the world. Past civilizations and even some tribal groups have left wonders in their wake, a result of uncanny skills – mental and physical – which, being hard to repeat today, cannot be considered primitive. Buildings such as Stonehenge and the Great Pyramid of Giza are felt anomalous, because of the mathematics implied by their construction. Our notational mathematics only arose much later and so, a different maths must have preceded ours.

We have also inherited texts from ancient times. Spoken language evolved before there was any writing with which to create texts. Writing developed in three main ways: (1) Pictographic writing evolved into hieroglyphs, like those of Egyptian texts, carved on stone or inked onto papyrus, (2) the Sumerians used cross-hatched lines on clay tablets, to make symbols representing the syllables within speech. Cuneiform allowed the many languages of the ancient Near East to be recorded, since all spoken language is made of syllables, (3) the Phoenicians developed the alphabet, which was perfected in Iron Age Greece through identifying more phonemes, including the vowels. The Greek language enabled individual writers to think new thoughts through writing down their ideas; a new habit that competed with information passed down through the oral tradition. Ironically though, writing down oral stories allowed their survival, as the oral tradition became more-or-less extinct. And surviving oral texts give otherwise missing insights into the intellectual life behind prehistoric monuments.

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