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.

This scale divides the whole of the octave into twelve equally increasing semi tones and, in the outer rotational world, it is manifested as the division of the Sun’s path within the year. Planets pass through twelve equal areas called the Zodiac or circle of twelve animals. The origin of the Zodiac appears purely abstract since the only physical division of the year by twelve is the twelve lunar months which actually divide the solar year into 12 and 7/19ths, as one see’s in the Lunation Triangle. However, as noted by Hindu astronomy, Jupiter passes through 1/12th of the ecliptic in 19 squared or 361 days, [the Barhaspatya samvantara see Sacred Number and the Origins of Civilization, p 198-202]. Jupiter also resonates with the Moon at an exact 8:9 whole tone ratio, Saturn resonating similarly but with a 15:16 half tone [see especially Harmonic Origins of the World, Part 1]. Thus there is pure tone order underlying the organisation of the year into twelve signs.

Here I will show how the megalithic geometers could have generated equal tempered string lengths .

To divide the octave into twelve equal semitones one needs to find the twelfth root of two, two being the octave doubling. Using a calculator, the value of this is 1.059 but such calculators belong to recent times. The problem of achieving its value is not the only challenge since one needs to develop the full scale of twelve notes, as is done with the numerical yantras of McClain [see The Myth of Invariance] using integers, not just one interval.

Figure 2. The Babylonian Astrolabe
(after B. L. van der Waerden)

A geometrical solution is available through the N:N+1 triangle, where N is integer.  Such triangles can simulate growth based upon any logarithmic base and the logarithm, base 2, of the twelfth root of two is 1/12. This means that in the equal tempered zodiac of tones, a tone circle is divided equally into a twelve-spoked wheel.

If the baseline of an N:N+1 triangle is given the conceptual value 1, then there has to be a value for the actual N such that the conceptual hypotenuse equals the twelfth root of two, 1.059. Furthermore, once this triangle is constructed then the hypotenuse length can be arced to the base, to be raised as a perpendicular which then strikes an extended hypotenuse at the point (N+1) times (N+1) that is, an equal tempered whole tone.

We can then see that the successive applications of the same procedure results in dividing up the range 1:2 into twelve equally growing intervals. What is now required is to see what value of N can give access to the equal tempered scale.

The triangle is close to 1/17 different [1/16.817] between hypotenuse and base. However, this can be seen to be close to the 16/15 pure half tone (of course) but a little further from the septenary half tone of 15/14. However, 15/14 = 1.07142857 and its difference from 12√2 is 1.0112939 which is one part in 177/2 which is very close to the square of one part in 175, one of Neal and Michell’s metrological grid “tweaks”. If 15/14 is reduced by (175/176)2 then the hypotenuse becomes 1.059287 which is within one part in 6044 of 12√2 of 1.059463.

I was alerted to this possibility by another problem of the ancient world, namely “how to double the volume of the cubit altar of Apollo”. A perfect solution has been proved impossible but the answer in rational terms is to apply a rational approximation to four equal tempered semitones. This can be achieved by multiplying the existing side length by 5/4, the major third, and then to grow this by 1/125th. This correcting ratio is also known to ancient metrology via the combined grid adjustments (176/175) times (441/440) equals (126/125) which is one part in 125. The accuracy of the result is one part in 16000, beyond the accuracy of practical metrology.

Returning to the N:N+1 triangle that can generate the 12th root of two, what is the height of its third side? It is closely 7/20 in size. If we scale the triangle to have a base of 20 units in length, then the height is 7 units and this gives us a very simple way to set such a triangle up in practice, by simply constructing a 20 long base and raising a 7 long perpendicular. The resulting hypotenuse is 20 times 12√2 long. Alternatively (and conceptually) if a Royal step of 2.5 royal feet is used for the base, 5/2 * 8/7 feet = 20/7 feet, whereupon the perpendicular side is 7/7 = one English foot in height.

The relationship between the pure tonal ratios and Equal Temperament

We can then see a highly reasonable application of both metrology and of its shortcuts and adjustments, to make the equal tempered scale or any other logarithmic transform available long before the modern scheme of logarithms came to hand. The N:N+1 triangle not only enabled the capture of essentially logarithmic ratios in celestial time periods, but also provided logarithmic calculation when required since the iteration of such a triangle leads to logarithmic growth measures. Whilst such logarithms might not seem very accurate, higher accuracies are available to the builders of larger triangular calculators whilst relatively small triangles will deliver the accuracy necessary for acoustic comparisons with the pure tone scales of antiquity.

In a following post, I shall investigate ancient clues to equal temperament in the Egyptian iconography of Thoth.

Background to post

This post was written as a pdf for Ernest McClain in 2008, on the Bibal Yahoo Group. This was my second attempt to relate my own work to his, and the dialogue went as follows:

Dear Ernest, I tried to send this to yahoo but met with resistance perhaps due to forgetting id, lack of use etc, so send this direct. I am only very occasionally looking at the group but read this very interesting fragment:

Any early scribe might have noticed that three consecutive steps of 4:5 fail to double the octave at 64:128, falling short by 3 units at 125, and intuited a correction of one part in 125, witnessed in the size of Uruk in the Gilgamesh epic (12,600 area units) and in his reign of 126 years. This is one of the best kept secrets of ancient mathematics and leads to a “cube root” temperament within the necessary tolerances of modern equal temperament.

earlier Bibal post

I will take the risk and mention the Ancient Metrology of Michell and
Neal, purely because the ratio 126/125 is used to vary different types
of rational feet (called modules). This ratio is the result of 441/440
times 176/175, the former being the ratio of the mean to polar Earth
radii and the latter, at least, a difference in lengths of degrees of
latitude between 10 and 51 degrees. There is plenty of historical
evidence for the varied measures.

The two constituent ratios contain 11/7 and have been found in use in
circular structures. 441/440 is employed within the Great Pyramid. And
so on…

However, the uses of these ratios is still quite mysterious and you have
given one here, in the ability to correct within the tonal applications
of number. 126/125 is very good for removing the seven in the
denominator of the royal measures such as the royal cubit, root value
12/7 feet. For this reason such a varied measure is called Canonical
because it then becomes 1.728 feet (non-recurring in base 10), the cubit
found at Jerusalem.

I now see metrology as a necessary science for building the numerical structures
which preceded much of our own notational systems. The challenge of
*building” your yantras is something I hadn’t considered but it would
reduce the notational burden, metrology already having all the rational
fractions of a foot available through super-particular triangular construction.

Dear Richard: This is interesting, but I really don’t know what to make of it. I’m sending this also to a friend who may have some thoughts about it. I personally find some of Michell’s geometry historically anachronistic. I work within more restricted arithmetical limits. Ernest

I tried to write some response to reading your Quran book here (on March 24th):


This was a 2008 posting at SacredNumber.co.uk, a defunct Squarespace website.

Towards the end:

Taking up the “doubling of volume” for a 12 side cube, the result is a cube 12 times the cube root of two in side length. The latter is very nearly 1.26 [actually 1.25992105, one part in 16,000 different] and this raises some interesting “ancient metrological” considerations, for John Neal’s Standard Canonical transform is 126/125 or 1.008 of a foot. We can then see that 1.25 is 5/4 [the major third] and that this multiplied by 126/125 yields a highly accurate ability to create a cube that has doubled in volume. 5/4 times 126/125 is 1.26 which then times a side length of 12 gives 15.12 for the double volume cube’s sides.

It appears then that this metrological ratio happens to be a practical means to the “doubling of the cubic altar” problem.

from the above linked text

Which relates to your “This is one of the best kept secrets of ancient mathematics and leads to a “cube root” temperament within the necessary tolerances of modern equal temperament.”? Richard

Richard, It seems to me that we both know what we are doing with these numbers. If so, then the question becomes, “What do others think?” Ernest

Ernest, Spot on. Ouch. It is hard to find out what people think and the word remains multi-valued. I believe most people think it is irrelevant and it is also hard for the content not to exceed people’s interest. My interest is highly variable since my main objective is finding out why numbers are a  significant story.

I am working on a plot-line, a hypothesis, that would make the numerical content of the Megalithic and Mesopotamian an actual arrival of mind on earth (I have some evidence). This in order to make it relevant and also to then imagine how that process has progressed. It may then be that since the individuation of the mind, we are dealing with how we make meaning rather than the archetypes of the cosmic. That would explain our crisis. I attach a ring composition for this. It will be a minimalist trajectory. The Babylonian phase (in this story) rationalises the cyclic astronomical knowledge of the megalithic into the ideal causal structures of harmony. By the way, how are you getting along telling people that their scriptures are generated by numbers?

My blog functions as a place to put some material that I can work up into some reasonable form and not forget it. The concept of an audience (the other) is quite powerful even if not actual. I would never have written the second book without a strong sense of audience and need for a story to explain. Now I have created this mad pattern! How to make it relevant. Richard

Richard:  You ask: <<“By the way, how are you getting along telling people that their scriptures are generated by numbers?>>Not very well, naturally. But more fun lies ahead. Only a few have caught on. The evidence slowly becomes clearer. Their religions more more sense understood correctly. The ascent of man was by using what we were given, not by trashing it. And the authors were keenly aware of it. Irony is permanent part of the story. Ernest