Category Archives: Research

Articles about Harmonics

Frequencies or Wavelengths, Pitch or String Lengths?


Ernest G. McClain’s Holy Mountains represent an ascending octave integer set as ascending pitch classes (frequencies) yet Greek scales were described as notes descending in pitch class with increasing string length. Such mountains were often “reciprocated” and visually inverted, pivoted on the octaval D, and if frequencies were being inverted, then the inverted mountain would then represent string lengths. It is these contradictions which need to be resolved, probably by accepting two different views with the familiar as McClain’s notation.


In Myth of Invariance (McClain, 1976)[1] the Samkhya term Purusha (cosmic principle or self) is used to describe an inverted mountain pivoted about D and Prakriti (material principle) is compared to the holy mountain itself, as it stands “upon the earth”. It seems clear that the material principle behind tuning is the instrumentality, such as a set of string lengths, whilst the more abstract principle is our modern term frequency, known only recently as the reciprocal of string length. But the modern notion of frequency seems conflated with how antiquity must have managed its own musical tuning theory, if and when it had one.

[1] (see “The Holy Mountain” page 50-53)

I remember being told (in Physics) that Pythagoras discovered musical ratios through experimenting with string lengths, this to introduce the (then) modern schoolboy to the inverse proportionality of wavelength (a.k.a. string length) and frequency. But how did the ancients understand the lower pitch produced by a lengthened string? – except as being directly proportional, the opposite of inversely proportional.

Our term frequency is only inversely proportional to wavelength because of the ascending directionality eventually chosen for frequency. String length and wavelength are more or less synonymous, and the modern interest in frequency only came to the fore after the logarithmic relationship of frequency to wavelength was discovered, a discovery related to the discovery of logarithms, the theory of functions and modern experimental equipment.


Figure 1 Part of Chart 11 in Myth of Invariance illustrates how the scale order of the holy mountains was interpreted in the sense of increasing frequency rather than string length.


If the ancient world thought only of string length and descending pitch, then an inverted holy mountain only appears to contain frequencies via the inverse law relating wavelength and frequency. When the inverted string numbers overlay the non-inverted string lengths, Jay Kapraff calls the mirroring of wavelength and frequency within an octave, a Tetrapolarity (Kapraff, 2002): where symmetry in string lengths across the axis of symmetry of an octave is matched by a symmetry in the corresponding string frequencies across the same axis of symmetry.

The earlier use made by McClain of an inverted holy mountain pivoted about D, was in fact to visually mask out tones which are not found in both the holy mountain and its pivoted inverse. One can then, through this visualisation, identify those tones symmetrically connected to both low and high D, as Plato’s “ambidextrous pairs of fighting men” which, of necessity, are those tones whose frequencies happen to be doubly connected to low and high D. The inverted mask, not of frequency but of symmetrical tones within a limit, is only a simple way to visually present these symmetrical twin tones as string lengths, so that,

  1. One must forget about “frequency” in an ancient context.
  2. the integer matrix in an ancient context should be read as of string lengths and
  3. higher numbers should generate a lower octave classes (note letters).

McClain showed the increasing numerical values (unfortunately called tone numbers) as higher in pitch so that the Pythagorean “tuning order” of FCGDAEB is a view based on ascension whilst the string length view would tune 3/2 from high D as G (rather than 3/2 from low D as A). If right, this means that the holy mountains have been notated incorrectly. Figure 2 shows the re-notated mountain for a limit of 720, indicating the significant changes when tone classes are allocated according to string length.



Figure 2 The pivoted mountain for limit D = 720, shown with tone classes allocated according to string lengths rather than frequency


It seems the holy mountain should not have been confounded with frequencies for the purposes of ancient musicology, or its attendant symbolism, as the mountain stands upon the earth and the idea of frequency has only been abstracted recently by modern science, as an ascending ordinality. The inverted mountain is actually showing the special relationship within an octave of one realised string length, across an octave’s axis of symmetry to another “twin”, a string whose length is similarly and “ambidextrously” connected, by intervals summing to two, to both low and high D.

A trial version of Harmonic Explorer can be tried prior to a “string length/pitch” checkbox being added later in the year to the standard version.

Works Cited

Kapraff, Jay. 2002. Beyond Measure. s.l. : World Scientific, 2002.

McClain, Ernest G. 1976. Myth of Invariance. New York : Nicolas Hays, 1976.

Mantles of Radiance and the 118th Root of Two

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

12 equal tones in tone circleA) 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

  1. A^2 approximately equals 81/80 but
  2. A^7 = 25/24
  3. A^9 = 254/243
  4. A^11 = 16/15
  5. A^13 = 27/25
  6. A^18 = 10/9
  7. A^20 = 9/8 (2 x 3 above)
  8. A^31 = 6/5
  9. A^38 = 5/4
  10. A^49 = 4/3
  11. 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.