At the suggestion of a colleague I’ve added a works in progress page to give folk an idea of where I’m at on a number of fronts. Herewith is an extract from one of them: Music in the Key of D: Harmonic Explorations of John Dee’s Monas Hieroglyphica.

For those unfamiliar with the Monas Hieroglyphica, a bit of background first. Dee considered Monas Hieroglyphica (1564) one of his most important works. Despite being written over twelve days in something of a mystical state it had obviously had a lengthy gestation period (the graphic device appears on the titlepage of his Propaedeumata Aphoristica of 1558). A short work, it is a series of 24 ‘theorems’ chiefly expounding the alchemical significance of his graphical ‘monad’, which contains within it the sigils of the seven planets, four elements and zodiacal exaltations of the sun and moon:

Although biased towards an alchemical interpretation of the sign – a strategic move on Dee’s part to win himself patronage by demonstrating that that he (theoretically) had the key to perfecting metals – Dee is keen to point out in his prefatory letter to Maxmillian II that the Monad had applications to all arts and sciences. It was the mention of music in particular that piqued my interest and has led me to use principles derived from the Monad in a large number of my compositions. These musical strategies, along with a historical discussion of Dee’s musical education and interests are examined at length in the full work. However, here I would like to present an extract concerning the Monad and its relation to the Pythagorean diatonic scale. Due to the limitations of WordPress, footnotes and references have been omitted. The entirety of work is still very much a draft and thrown out here for interest and comment.

Part 3. A Harmonic Exploration of the Hieroglyphic Monad

I. The Monad as Monochord

In his letter to Maximilian, Dee wrote of his Monad:

How justly may the musician be struck with wonder when here he will perceive inexplicable, celestial harmonies without any movement and sound.

Given that the study of music with which Dee was most accustomed was primarily the theoretical method of harmonic investigation I consider that it may be useful to begin by exploring the Monad on these terms. The penultimate theorem gives instructions for the geometrical construction of the Monad based on a straight line CK:

It would be well to notice, you who know the distances of our mechanism, that the whole of the line CK is composed of nine parts, of which one is our fundamental, and which in another fashion is able to contribute towards the perfection of our work . . .

The notion of deriving a meaningful figure from the division of a line can be seen as analogous to deriving meaningful intervals from the division of a string, such as the monochord: the instrument for investigating harmonic ratios which as we have seen would have been well known to Dee. Along with the importance of the tetrakys, the fact that he suggests constructing the Monad using a line divided into nine portions may also have been what prompted him to describe the figure in terms of harmonic science: the line divided being analogous to the string of the monochord.

There are many ways to sound out, or construct, a seven tone Pythagorean scale on the monochord. The ancient Greeks favoured the ‘descending’ method in which the string was lengthened by a certain proportion for each tone, thus creating a scale that descended from its initial note as the string became longer. In the Middle Ages a system in which the string was both lengthened and shortened was in use, while by the 1500s an ascending method of division was most commonly practised. Here the string was progressively shortened, or stopped, to create a scale of tones increasing in pitch from the fundamental note.

The construction of scales in the ascending form depends on the division of the string into proportions of nine and four parts. Dee’s Monad is, of course, nine units in height and four in width. To create the second interval of a scale, that is, one whole tone up from the fundamental note (indicated by the numeral I), the string is divided into nine. On the monochord the movable bridge would then be placed after the first portion, resulting in a string length that is 8/9 of the original:

For the successive whole tones in the scale the string is divided once more by nine and stopped at the proportion 8/9. . If we were to consider this interval in relation to the entire length of the string (rather than from the previous interval) we would find that it had the more complicated ratio of 64/81 (e.g. 8/9 * 8/9):

Considered in relation to the previous interval, the semitone between E and F has the seemingly complicated ratio of 243/256, but can actually be found by stopping the string at 1/4 of the original length, allowing the remaining 3/4 to vibrate:

A complete octave can then be constructed by repeating these simple divisions:

If this process is continued we can note down the divisions as they occur in relation to the entire length of the string we can construct the following series of values, here plotted up to the interval of 1/9 of the entire string length:

In the above table, the note names and corresponding octaves are shown for a string with the fundamental C3 along with the frequencies according to the method of Pythagorean tuning. Those intervals which correspond to divisions of the string into ninths are highlighted and shown below alongside the Hieroglyphic Monad itself – the Monad as monochord:

This diagram of the Monad shows where stops along a monochord divided into ninths coincide with Pythagorean tuning. As can be seen above the intervals that correspond to the nine-fold division of the Monad are I, II and V in mixed octaves.  It is worth noting the symbolism in that the upper (celestial) portion of the monad has three intervals (I-II-V) perhaps suggesting a symbolic analogy with the holy (or alchemical) trinity, while the lower portion has four intervals (II’-V’-II”-III”’) analogous to the four elements.

Viewed as a tonic progression, I-II-V opens in interesting avenue for further speculation since three movements of John Dowland’s celebrated Passionate Pavans (1604) are based on the progression I-I/II-V/V-I (the intervals of the upper half of the Monad). Peter Holman notes that the progressions used in these pieces Lachrimae Tristes, Coactae and Amantis “recall the ‘strange and informal’ modulations to the supertonic advocated by Campion.” This is an interesting coincidence considering the speculation in recent years about the possibility of Dowland’s inspiration deriving from a peculiar Elizabethan notion of transcendental and Hermetic melancholy. Quite what Dee would have made of Dowland’s progressive music with its expressive and ‘painful’ use of chromatic inflections is hard to know, although it is almost too tempting to suggest that when Dee described the harmonies of the Monad as ‘inexplicable’ he may have had something like this in mind.

I-I/II-V/V-I progression in the bass part of Dowland’s Lachrimae Coacte.

It is worth considering here the significance of the number nine as it relates to the concepts surrounding the Hieroglyphic Monad. While the Monad, or one, is indivisible, nine also has a curiously incorruptible quality in that all multiples of nine add up to nine and every number that adds up to nine is divisible by the same. The number nine has a further connection with the history of Hermetic music, as found in the writings of Marsillo Ficino who asserted that “the ancient theologians maintained that the nine Muses were the musical songs of the eight spheres, and in addition the one great harmony arising from all the others.”