A Working Document

It was difficult for me to understand how singular "limiting" numbers could have generated a musical picture that inspired the mythic narratives of Sumeria, Egypt, Babylonia, the Bible, the Rig Veda, Homer and Plato. This idea, pioneered by Ernest G McClain, seems both far-fetched and numerically complicated, yet it soon reveals a simple but powerful framework for expressing musical truths. The web application, Harmonic Explorer, aims to turn limiting numbers into the musical information driving ancient myth making and, when powers of two, three and five are added using its buttons, we are exploring a world known formerly only to the elite priests and scribes of the ancient world. We will therefore concentrate on reading what happens between these musical quantum realities, referenced by a single limiting number, and what these harmonic worlds might have been thought to "say".

When we declare a limiting number, it might well be based upon another one that creates musical intervals within an octave, such as 12 (or 60 or 360 in later modules). It must have been seen, through tokenised arithmetic, that such numbers had the prime number three within them, and that this prime number could naturally populate an octave with fifths (the interval ratio of 3/2 to do¹) from below, to form a note that is also a perfect fourth from above (the ratio 2/3 from do² being 4/3 from do¹) because these two intervals equal an octave doubling. with a whole tone (9/8) between these two notes. At this point the limiting number 12 is implicit since it is 4 times 3, enabling the intermediate numbers to be 9:8 [this being shown as 6:8:9:12]. The UTILITY of the limiting number is exactly in its composition of having two twos and one three so as to form the ability of the 3/2, 2/3, 4/3, 3/4 and 9/8 INTERVALS to have the INTEGERS 8 and 9 that can "play host" to that most perfect harmony within the octave, the ascending fifth and descending fifth from do² (equalling an ascending fourth from do¹). All limiting numbers follow from this humble example.

ABOVE: The diminutive brick pile for limiting number equal to Twelve. The tone circle has D at the top, representing the start and end of an octave (low do = 6 and high do 12, respectively). A perfect fifth and perfect fourth equal the whole circle, so the symmetrical notes A and G generate "legs" separated by a whole tone interval of 9/8 (the DIFFERENCE between the interval ratios of fourth and fifth, as 3/2 divided by 4/3 = 9/8.) In this excellent example, A and G actually have the value of 9 and 8 found within the ratio 9/8.

However, if only the numbers two and three are involved then a particular type of musical scale will result with only five super safe notes, the Pentatonic scale, though the octave can be populated with twelve notes by bringing a series of fifths, otherwise spanning twelve octaves, down to a single octave to create a SPIRAL of FIFTHS. However, that practical technique was not a means to studying harmony through limiting numbers which required another prime number, FIVE, to break out of the world of FIFTHS and, to our linguistic confusion, create THIRDS. That is three creates fifths and five creates thirds. If we want to add visual confusion also, the limiting number 144 generates five pentatonic notes that Ernest McClain calls the FIVER, whilst 144 = 12 times 12 contains only four twos and two threes. These FIVE notes C G D A E are an ideal core that emanate from the world of limiting numbers and are shown on Harmonic Explorer as white note names (they are like white notes) expressing a good foundation.

And now I think it is time you were introduced to SIN. The problem with all prime numbers is that they do not divide into each other, they only divide into each other as the zeroth power which is UNITY, for all numbers to the power of zero equal one AS IF the universe of number came out of a single unit or impulse, a primordial atom. With the creation therefore, differences and incompatibilities arise and within the world only according to three, with its cycle of fifths, and there seems to be no audible problem until, on the THIRTEENTH such generation, an octave is sounded with the same note again. But alas, twelve fifths (3/2 to the twelfth) do not PERFECTLY equate to seven octaves (2 to the seven) and the difference is called the Pythagorean Comma, one of many commas that are counted as SINS that must be managed by any world of limiting numbers. This can be demonstrated within Harmonic Explorer using the limiting number 2239488 which, containing only powers of two and three, cannot rise up the "brick pile" above its base. It forms along the base a spiral of fifths that generate all our familiar notes but with a duplication of G sharp and A flat, opposite D above and where a SINGLE NOTE would be expected. This demonstrates (figure below) that D is taken to be "do" because tones are then symmetrically paired using the (to be revealed) technique of building these brick piles. Therefore, the SIN of the Pythagorean comma, separating G sharp and A flat is projected (in the TONE CIRCLE) into the location opposite perfect unison as D, the God of Octave Doubling, raised on high like Indra in the Rig Veda. (see 2239488 in Harmonic Explorer: you can show and hide the brick pile using a check box labelled BRICK PILE)

ABOVE: The brick pile and tone circle for limit 2239488 which has 13 darker tone bricks along its base. Note that the tones furthest apart bracket the point opposite D above because there is an accumulating difference between powers of 2 in octave doubling and the powers of 3 in forming fifths.
BELOW: The bottom bricks placed next to the musical notation and, below that, McClain's description of the Pentatonic Fiver (central to the spiral) bracketed by the  heptatonic scale and then the "cosmological" scale, functional in the world of Equal Temperament, having twelve equal semitones between notes.

This spiral of fifths is shown as darker bricks because these are the only tones in the pile that work to generate symmetrical tones about the centre. We will see this "damp patch" of bricks rise up as soon as one or more fives are added to the limiting number because the height in rows of D(=do) equals the powers of five in the limiting number. This also can tell us how many powers of three are in 2239488, the above limiting number. The bottom left brick, called a key stone, ALWAYS expresses no power of three but only powers of two. Therefore, since D is seven bricks along from the cornerstone, the limiting number 2239488 must contain just seven powers of three (D being the central seventh of the thirteen bricks). Three to the seven is 2187 and the limiting number 2239488 is that times two to the tenth power.

The two notes G sharp and A flat bracket the geometric mean of every octave which is the square root of octave doubling, that is the square root of two, a tone which cannot be perfectly simulated using powers of three or of five, seven being required to get closer to it. This comma is but the beginning of the problem of musical sins, as this mountain will demonstrate, for each note will come to spawn sound-alikes that are subtly different, as powers of five cause D to rise up the "mountain of god" for ancient tonal theorists, also known as scribes because they wrote what they saw but often in veiled terms that we inherited as stories based upon musical values.

Your toy dramatizes these very ancient insights more vividly that any visual aid accessible to Plato's students. Today any 10-year old in the fourth grade can see this happening over at the touch of your buttons,  and over and puzzle over meaning at his own leisure