REVIEW of The Harmonic Origins of the World

During the latter part of the twentieth century, three divergent speculative perspectives opened up on the ancients’ cosmology: astronomical, musical, and metrological. The astronomical perspective found its classic expression in von Dechend and de Santillana’s Hamlet’s Mill. The musical perspective was spelled out, almost single-handedly, by Ernest McClain, beginning with his work The Myth of Invariance. The metrological perspective diverged into the practical (descending from Alexander Thom’s surveying in the nineteenth century), and the more theoretical work associated perhaps most famously with John Michell’s View Over Atlantis.  

These three perspectives shared an awareness that number was an indispensable guide. Number is invariant; three is always three, and always one plus two. Mathematics is a realm of order, and recurrent patterns like the seasons or the harmonic scale call for mathematical descriptions precisely because such descriptions find stability in change. 

As scoffers and skeptics like to point out, however, where there is pattern-finding, there is also often unconscious wishful human ingenuity. Moreover, because the astronomical, musical, and metrological perspectives were carried on sometimes in isolation from one another, their results diverged, and an apparent incommensurability emerged: how could they all be true? This gave scoffers an argument that was, on the face of it, difficult to answer: why not none of them instead? Perhaps the real answer was the skeptical shrug: the ancient myth-tellers and builders of stone circles were acting more or less haphazardly or moved by very terrestrial, local, and historical concerns. Was this not the simplest explanation?

Richard Heath for a quarter of a century has been building towards a case diametrically opposed to this.  From the beginning he worked with Thom’s practical metrological results, bringing them into dialogue with Michell and John Neal; then later with a further expansion of astronomical results that far outpaced von Dechend and de Santillana’s speculations on the precession of equinoxes. In The Harmonic Origins of the World, Heath goes a further step, bringing McClain’s results into dialogue with his previous work. Heath provides ample demonstration that the results of these various perspectives can clearly be seen to not diverge from one another. Suddenly it is very plausible that they might indeed “all be true,” because they were never, for the ancients, separate at all. 

According to Heath, there exists in our solar system a harmony of extraordinary beauty among planetary cycles. This harmony was observed by ancient astronomers, and enshrined in megalithic monuments; it was transmitted in oral and literary culture via a musical grammar of proportion, easily reproducible across various cultures, which informs scripture and speculation (in McClain’s phrase) “from the Rg Veda to Plato.”  These assertions are of course controversial and deserve scrutiny. But they give the lie to any facile dismissal of ancient cosmological sophistication on the grounds that reconstructions are inconsistent. Astronomy, metrology (practical and theoretical), and music are all comprehensible under a single analogical system. They hang together in a coherent, living dialogue.

This book is the most recent chapter and the most comprehensive introduction to a vital adventure in ideas. It is a detailed account of how human beings on the ground could make sense of the sky by way of the octave. In it, rigor and common sense meet wonder and awe. 

Harmonic Genesis of the Sumerians

Here I start by publishing an important diagram that shows how the earliest known references to musical tuning (early 4th millennium BC) on clay cuneiform tablets, using “regular numbers” whose factors are products of only the numbers 2, 3 and 5, led to the cosmological vision of their gods, the primary god, Anu, being a balanced mix of all three numbers as 60 but also called ONE. This is the source of their Sexagesimal  or base-60, still employed in measuring angles and time called minutes and seconds. All comes from ONE.

The emergence of 2, 3, 5 from ONE then combining as ANU and leading to the differentiation of the World along various paths. The creation proceeds through three prime number dimensions, Ea (as in Earth) through 2, Enki through 3 , Enlil through 5. Anu remains the fountainhead associated with all three and with the Zodiac, which emerged in later Babylonian as a seasonally relevant calendar.
Continue reading “Harmonic Genesis of the Sumerians”

Natural Evolution of our Modern Tuning System

The diatonic or natural scale, consisting of five whole tones and two opposed semitones, is most familiar today in the white notes of the piano [Apel. see Diatonic]. On the piano this would be called C-major, which imposes the sequence of tones (T) and semitones (S) as T-T-S-T-T-T-S in which the initial and final tetrachords are identically T-T-S, leaving a tone between F and G, the two fixed tones of the Greek tetrachordal system

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.


Figure 1 Tone circle and tetrachords for C-Major also called the modal scale of Ionian

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.

Continue reading “Natural Evolution of our Modern Tuning System”

Distribution of Prime Numbers in the Tone Circle

first published 13 February 2018

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.

The role of odd primes within octaves

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.


Figure 1 Viewing the harmonic primes 3 and 5 as a mountain of their products, seen as integer numers or as to these harmonic primes
Continue reading “Distribution of Prime Numbers in the Tone Circle”

Equal Temperament through Geometry and Metrology

The form of musical scale we use today is the (apparently modern) equal tempered scale. Its capabilities express well the new mind’s freedom of movement in that it allows us to change key to play compositions that move between alternative frameworks. This possibility was known to ancient tuning theory, could be approximated within Just intonation’s chromatic notes and was discussed by Plato as forming the constitution of one of his harmonic city states called Magnesia.

Relationship of the Equal Temperament Keyboard to the (logarithmic base-2) tone circle of an octave. We choose D (the Dorian scale) because it is symmetrical on both keyboard and tone circle. Equal Temperament supplies tones which enable any scale to be played starting from any note, though it is the Ionian (C-major) which defines modern key signatures.
Continue reading “Equal Temperament through Geometry and Metrology”