Ernest McClain wrote a short note called Mantles of Radiance as assumed, computed, and interpreted. In it he compares the three situations found in the logarithmic tone circle for equal temperament, the cycle of pure fifths and Just intonation. The tone circle as shown in Harmonic Explorer by default beside the holy mountain (aka matrix) for each limit, covers the invariant tones generated by the limit, when these are symmetrically paired, left and right – these also shown on the mountain as “wetted” bricks.
Mantles of Radiance as assumed, computed, and interpreted
A) The cycle (circle) as 12 in both circumference and area, not by computation but by cultural assumption. To Plato this meant that our “Equal Temperament” model was anticipated as convenient “boundary markers” for the cultural norm.
B) The serpentine “spiral” of musical fifths and fourths naturally approximates the cultural assumption. Only on the 13th pitch class is a cumulative excess in musical fifths discovered, with a consequent defect in fourths, by the overlap or gap of a “comma”.
C) An alternate “triadic” Just tuning correlates consonant major thirds of 5:4 with resulting semitones of three different sizes, leaving a defect of a comma. This tuning correlates directly with clock and calendar within the limit of 720 “days plus nights.”
If you were able to detect it, each Just tone is 81/80 different with each of the cyclic fifths, leading to double lines in Ernest’s Holy mountains, generated using a single limiting number as high D for an octave.
D) Correlation of the two systems displays commas at nine different locations surrounding the assumed perfection. They are Plato’s metaphorical “no man’s land” and locus of eternal conflict, resolved only by moderation in demanding “exactly what is owed.”
McClain goes on to say;
Sumerian affection for “mantles of radiance” assigned to the gods assured close scribal attention to music as an easy manipulable model for the heavens, algebraically and geometrically. This level of computing accuracy is displayed in the Shuruppak “grain-constant” of 1,152,000 units surviving on tablets from two different houses. When the city was burned c. 2600 BC the soft clay tablets of this “problem text” were baked for posterity. Working with symbols only for 1, 10, 60, 600, 3600, and 36000 students were required to discover the whole from its division into 164,571 thousand portions of 7 each, with a remainder of only 3. The answer, correct on one of the tablets, required 32 symbols for 36,000. Platonic decomposition of 1,152,000 exposes a brilliant assembly of harmonic formulas, “sealed” forever.
Analysis of the “Mantles of Radiance”
There is more to these mantles of radiance than meets the eye for in larger limits one can see that the mantles all fall according to some inherent granularity of spacing within this logarithmic world. I was intrigued, whilst Ernest was alive, to add a graticule to Harmonic Explorer which could match the granularity observed between the tones we call harmonic.
The solution at a simple level is to give each graticule an angle so as to divide the circle into 118 parts. One then sees that the tones all fall upon the graticule and hence fall into discrete units of the 118th root of two.
Dividing the logarithmic world of the octave tone circle into 118 equal parts means that every tone lands upon one of the divisions
Dividing the logarithmic world of the octave tone circle into 118 equal parts
means that every tone lands upon one of the divisions
The 118th root of two is 1.005891415 which equals 171/170 = (9 x 19) / (17 x 10) to high accuracy. If we call this ratio A then
- A^2 approximately equals 81/80 but
- A^7 = 25/24
- A^9 = 254/243
- A^11 = 16/15
- A^13 = 27/25
- A^18 = 10/9
- A^20 = 9/8 (2 x 3 above)
- A^31 = 6/5
- A^38 = 5/4
- A^49 = 4/3
- V^69 = 3/2 where 49 + 69 = 118
and so on. The explanation in the world of number arises from the nature of the ratio 171/170 in that it can be decomposed into two ratios, 9/10 and 19/17 which are very close to each other and when multiplied yield the 118th root two. So 171/170 = 9/10 times 19/17 = the ratio for the 118th root of two which divides the tone circle into 118 parts. The resulting octave is slightly short at 1.997874 rather than 2.
This cross product is interesting because the first ratio different by two instead of one (numerator to denominator), equal to 10/9, has been found as a near ratio – which need not have given the 118th root of two. In practical terms, a right triangle with hypotenuse 10 and base 9 can be extended at that invariant angle until, at hypotenuse 19 and base 17 the near approximation to whole number lengths, with the same angle between, is 19/17.
To see the graticule of 118 in action, use this link to a development version of Harmonic Explorer.