Kaaba’s Numerical Picture of Harmonic Origins

A recent plan of the Kaaba[1] indicates that its walls, by their odd-number proportionality, symbolised the numerical origins of musical harmony through the first six numbers, 1, 2, 3, 4, 5 & 6, sometimes called the Senarius meaning “existence out of six”.

According to Plato, the world was created using the rules of musical harmony in a scheme involving perfect fifths of 3/2, fourths of 4/3 and tones of 9/8, leaving “leftover” semitones of 256/243: a rudimentary musical scale. This only used prime numbers 2 and 3 and multiplications with themselves and each other; a system called Pythagorean tuning. The Kaaba incorporates another prime number 5, called the human number, this enabling two more large intervals called thirds, the major third of 5/4 and minor third of 6/5. Using 5 enables more and better scales to be formed and fills in the gap between 4 and 6 to show all the large intervals in the first six numbers, 1:2:3:4:5:6. This fuller tuning system has been found in the ancient Near East as long ago as the Sumerians, in their tuning texts on cuneiform (c. 3000 BC onwards).

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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.

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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
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