Tuning Theory

Explores how musical intervals come into existence through limiting number made up of prime numbers two, three, five and occasionally seven, populating the octave boundary provided by doubling with tones in which the higher primes can divide the pitch continuum within the octave. The methodology used here, proposed as ancient and recently reconstructed by Ernest G. McClain, is the perfect framework for studying the harmonic time periods of planets.

The ancient notion of holy mountains, intuited by Ernest G. McClain in the 1970s, was based on the cross-multiples of the powers of the prime numbers three and five, placed in an table where the two primes defined 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 and, furthermore, the prime number two would have been seen as not instrumental in placing where, on such holy mountains, each number must appear. Thus, an inherent duality was recognised 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.

Primes3and5 asnumberandpowers

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 

What has come down as Jesus' teaching were largely oral "sayings", some then integrated into later hagiographies called gospels. The fundamental gospel was that ascribed to Mark, since Luke and Matthew both drew stories from it and from sayings in a Q document. Mark was soon edited and this in a detectably different mood, more critical to disciples and including additional miracles, perhaps to match those of Elija and Elisha. Translators and church doctrines later affected what appears in the canonical versions we now use.

This article discusses the feeding of a large crowd of five thousand men in the original Mark and, in the following inserted section of the editor, the feeding of four thousand. These "feedings" follow the same motif: Late in the day, people need feeding and the small amount of food found available, five loaves and two fishes or seven loaves, is distributed by Jesus and the "fragments" left are gathered into twelve or seven baskets, leaving more after all the eating than could possibly have existed to start with.

John Dart notes in Decoding Mark, with others, how weak the second feeding story is since it appears to merely repeat the previous miracle, in a lesser fashion. However, a day or two after the second feeding miracle, of the 4000, the disciples are worried when they have forgotten to bring any bread for their journey by boat. Jesus says they should "beware of the leaven of the Pharisees, and of the leaven of Herod … Why reason ye, because ye have no bread? perceive ye not yet, neither understand? have ye your heart yet hardened?:

8:19 When I brake the five loaves among five thousand, how many baskets full of fragments took ye up? They say unto him, Twelve.

8:20 And when the seven among four thousand, how many baskets full of fragments took ye up? And they said, Seven.

On the surface, Jesus is saying the disciples should know by now that food can come from being with Jesus, but on another level is this incident an example of his teachings use harmonic allusion, a technique found quite widely within both Biblical writings and those of the near east, in which numbers could refer to the harmonic world.

We don’t know exactly how many people there were, only that "they that did eat of the loaves were about five thousand men." The numbers 4000 and 5000 seem related and centuries earlier, Plato had used vague references to exact numbers in Dialogues written in Greek - a lingua franca in Alexandria and the middle east. If Jesus was the esoteric master he appears to have been, then the harmonic allusion of the ancient near east and eastern Mediterranean would have been an "open book" to him.

One can start by looking at 4000, which has three factors of 5 (32 x 125) and 5000 which has four (8 x 625). Harmonic numbers are best studied using mountians formed of the products of prime number two, three and five - a technique pioneered by Ernest McClain to identify harmonic allusion in ancient texts.

HE 6250

Figure 1 A Mountain Index of Harmonic Numbers between 3125 and 6250, useful for studying numbers around 4000 and "about" 5000, these numbers to be seen top left under the index number, 6250.

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).

development of the squareFigure 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.

Marduk and his horned dragon

Marduk is a character like Indra who was associated with a very large number and antediluvian floods, though Noah seems adapted not to mention 8,640,000,000. The reality of these floods is based upon an idea revealed in this number. The head number 864 is the earliest which can form the seven note heptatonic octave whose intervals are far from perfect. Winds can be viewed as blowing into the octave's tone circle, and the heptatonic has seven but having defined seven tones (including the tonic here to be called D) the gaps remaining between B and C and between E and F, that we call semitones, are unplanned so as to be numerically and acoustically imperfect as equal the ratio 256/243, This imperfection was called by Pythagoreans the leftovers (leimmas), and they highlight the fact that when a cycle of fifths and fourths emanate from D (clockwise and anticlockwise within in the octave's tone circle), powers of three can only be eliminated to achieve the exact 2 required of octave doubling, by a semitone fraction involving three to the power of five = 243 (figure 1).

HE sevenWinds

Figure 1 The Serpentine cycle of ascending and descending fifths leading to the Pythagorean Heptatonic using imagery generated by Harmonic Explorer.