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) 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.
 (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,
- One must forget about “frequency” in an ancient context.
- the integer matrix in an ancient context should be read as of string lengths and
- 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.
Kapraff, Jay. 2002. Beyond Measure. s.l. : World Scientific, 2002.
McClain, Ernest G. 1976. Myth of Invariance. New York : Nicolas Hays, 1976.