Upon first hearing the news about Ben Chasny’s Hexadic System and album, one of the aspects that most interested me was the connection between the Hexadic System and the area of ‘speculative music‘, which is one of the perennial concerns of this blog. This post is a companion to the previous post on the practical aspects of the Hexadic System, and will look more closely at the relationship between the Hexadic System, atonality and ‘speculative music’. It’s a long read, but hopefully worthwhile for those interested in such things!

I. Speculative Music, Past and Present

In its most exoteric (outer, superficial) form, speculative music was traditionally concerned with matters such as tuning, temperament and rhythm: it was the theoretical side of music, opposed to the practice of playing and composing. However, within the neoplatonic worldview that dominated the Western world prior to the Enlightenment, such matters – dealing as they did with ratio and number – were inextricably associated with esoteric doctrines, primarily that of the music of the spheres: some commentators suggesting that the planets, like the notes in a musical scale, were distributed according to similarly simple, harmonic ratios. Given the supposition by esotericists that the universe is relational – as Agrippa had it spanning three worlds: the sublunar, celestial and divine, or supra-celestial – the ratios inherent in the planetary music would have sympathies with that which flows from the ‘divine’ world of souls and spiritual essences beyond them, and that which happens on earth below them. As John Dowland wrote in the high-minded and loftily poetic style of the age: “the whole frame of Nature, is nothing but Harmonie, as wel in soules, as bodies.”

"... Nature is nothing but harmony", from John Dowland's dedication of his Second Booke of Songs or Ayers.

“… Nature is nothing but harmony”, from John Dowland’s dedication of his Second Booke of Songs or Ayers.

Despite reaching a somewhat belated zenith in the work of Robert Fludd, chiefly his Temple of Music, the speculative elements of music were generally distinct from the compositional and performative ones (notable exceptions to be made for Ficino and Kepler!). The presence of the speculative was felt at least philosophically and poetically in the way that, for example, certain modes were said to influence the humours and emotions – but perhaps because it was considered by many part of the ontological nature of music it was a ‘given’, without need of a specific articulation. JFW Hasler’s PhD thesis and compositional portfolio attempts to address this by defining a typology by which music could be considered ‘speculative’ (which I’ve summarised here). Hasler generally considers any composition on mystical or Hermetic themes to be considered speculative either on an aesthetic, symbolic or ‘truly’ speculative sense, where esoteric ideas influence the selection of musical materials or creation of new system.

Personally, it often feels to me that the development of a ‘truly’ speculative music was, counterintuitively, something that had to wait until developments in the early 20th century, although it was of something of a different order to Fludd’s Rosicrucian expositions on the theme. Here, a movement away from the established tonal systems allowed deeper drives to create new musical systems reflecting psychological and spiritual concerns, for example: Schoenberg’s atonality seemed to resonate with unease about Freud’s psychoanalytic approach and the exposure of humankind not as rational animals, but ones driven by unconscious powers; furthermore, his music was championed by Kandinsky, a vocal exponent of the spiritual value of abstract art; Josef Hauer considered twelve-tone equal temperament to be the perfection of the musical system and used I-Ching as one method of method of exploring it; Ivan Wychnegradsky‘s quest for microtonal music was influenced by his own visions and the ideas of Theosophy; Dane Rudhyar carried on Schoenberg’s notion that the musical system was one that evolved (from single rhythms, to pentatonics, to modes, to keys, to the ’emancipation of dissonance’), and made the links with spiritual evolution and speculative music even more explicit, declaring the composer of new music to be “an evoker, a magician. His material is his musical instrument, a living thing.”; the “arch-modernist” Stockhausen could similarly be called the arch-speculative musician; and obviously both Russolo and Cage’s relationship with sound beyond the traditionally notated has significance to a wider conception of speculative music and the idea of the musica mundana.

A 'chromatic drawing' by Ivan Wyschnegradsky.

A ‘chromatic drawing’ by Ivan Wyschnegradsky.

Following the Second World War, the cause of modernism became disenchanted: as a reaction to the ‘irrationality’ of the Third Reich, atonal music came to be not an expression of a transcendent integration of consonance and dissonance, but an approach that, according to Adorno “perceives is the untransfigured suffering of man”, possibly serving part of a revolutionary, Marxist, positivistic agenda. This was a completely necessary reaction, and symptomatic of larger cultural trends away from the gnostic and toward the material, although of course some composers such as Orff, Panufnik, Crumb, Stockhausen and so on had spiritual and speculative preoccupations that ran counter to this. In the 21st century, it would be naive to suggest that music relates to the world, man and greater universe in the sense that Fludd posited, or that the development of tonal systems reflect a sort of universal spiritual sea-change. However, the philosophical overcompensation of the arts toward reductive and political agendas may be tempered by acknowledging that there is a complex and undeniable relationship between music and the ‘speculative’, symbolic, imaginative and phenomenological, and that one of the tasks of the contemporary musician or composer could be said to be the mediation between the tensions of radical, traditionalist, and esoteric (- though some would say bourgeois and obscurantist -) aspects of modernism.

With these thoughts in mind, the Hexadic System can be seen to have both exoteric and esoteric aspect: the former of these being found in the compositional strategies, ‘games’ and so on upon which Ben’s book concentrates most upon; the latter being latent in the contexts surrounding the use of ‘ars combinatoria’ and ‘magic squares’. Mediating between them is the creative imagination, which could also be considered on a scale or exoteric and esoteric: from the reductive ‘that which allows us to solve problems’, to the Romantic, ‘imaginal’ notion of imagination as an organ of perception itself. Both the exoteric and esoteric have resonances for continuing the contemporary exploration of the ‘problems’ of tonality, atonality and the integration of the two – or at least for furthering the appreciation of the possibilities of an expanded tonal palette.

II. Ars Combinatoria

The ars combinatoria is most readily associated with the work of the 13th century Catalan mystic, philosopher and logician Ramon Llull, and is often also referred to as the Lulllian art. Integral to this ‘art’ are a series of nested wheels, each one divided into compartments, which can be rotated to yield different combination:

A contemporary Lullian device by Mnemosyne Arts.

A contemporary Lullian device by Mnemosyne Arts.

The wheels could be rotated to mechanically create an exhaustive series of philosophical or theological questions, for example: what is the duration of wisdom? what is the virtue of wisdom? what is the power of wisdom? and so on. The figures of the Llullian art became machines for exploring the nature of the medieval cosmos and a tool for rhetorically converting ‘infidels’ to Christianity. Hasler has suggested possibilities for using the ars combinatoria in a compositional context, although he does not pursue it in his portfolio or dissertation. Of course, the notion of combinatorial approaches to musical material can be found throughout the history of Western music, for example in the enumeration of possible rhythms in Arbeau’s Orchesography (ca. 1546 [pdf]), or in the cycling of material between each of the four voices in Josef Hauer’s ‘melic design‘, however, it is rare that all the possible combinations of materials are sounded out as the basis of a composition (- although I am sure that Stockhausen must have done something along these lines; and perhaps there is a musical offshoot of Oulipo out there?).

Some works referred to in this article: Chasny's The Hexadic System, Arbeau's Orchesography & Agrippa's Occult Philosophy.

Some works referred to in this article: Chasny’s The Hexadic System, Arbeau’s Orchesography & Agrippa’s Occult Philosophy.

The most apparent manifestation of this combinatorial thought in the Hexadic System is the suggestion that the Hexadic Figure itself can be considered as a similar set of nested wheels, which can be rotated to yield new, variant tonal fields, in what Ben calls a ‘combinatorial move’. In the spirit of musical speculation, I decided to write a short Hexadic study that moves through all 216 possible tonal fields arising from a Hexadic figure. This piece uses the same figure as the ‘Progressions‘ study in the last post. However, after each tonal field has been played, the inner ‘wheel’ rotates one ‘step’. After six steps, the central wheel will move one step, and the inner wheel will move through another six rotations, and so on, until all possible combinations of tonal fields have been played.

The hexadic figure for the 'progressions' study.

The hexadic figure for the ‘progressions’ study.

III. The Square of the Sun

One of the more explicitly esoteric organisational mechanisms explored in The Hexadic System is the ‘magic square of the sun’, and takes us a step closer to the astrological concerns that have previously underpinned speculative music theory and practice. The magical squares are mathematical objects, which have the property that every row, column and diagonal adds up to the same value. In Western esotericism the squares are often called kameas, a word also related to cameo and the Paracelsan gamaheu and Dee’s gamaaea, all words for charms or talismans: in the case of Paracelsus and Dee these words explicitly referred to the impression of celestial influences on material bodies (see the notes on my album, Gamaaea). It was claimed that, by association of certain numbers with celestial bodies, six for the sun, for example, that the kameas could be used as tools to capture planetary influences in talismans: the influence of the sun for health and riches, for example. Below is an example from Israel Hibner’s Mysterium Sigillorum (1651), a relatively late text on astrological medicine, but an interesting one in that the talismans often incorporate the latest astronomical observations alongside more traditional esoteric symbolism: a suitable example of the synthesis of the innovative and the traditional that has been concerning us here. The talisman of the sun, below, incorporates both the traditional kamea, or magic square, and a representation of the sun with sun spots:

Solar talisman from Hibner's Sigillorum Mysterium.

Solar talisman from Hibner’s Sigillorum Mysterium.

As expressions of a hidden, numerical order in synergy with the celestial powers, the kameas also became tools by which the names and numbers relating to the magical ‘intelligences’ and ‘spirits’ of each planet could be expressed by relating the Hebrew letter values of certain names to cells on the squares. Each planet also has a ‘seal’, a graphic design which accounts for each square on the table:

The square of the sun, as it appears in the second book of Agrippa's Occult Philosophy.

The square of the sun, as it appears in the second book of Agrippa’s Occult Philosophy.

The square of the sun consists of 36 (6×6) cells. 36 is obviously an important number in the Hexadic System: it relates to three chromatic octaves on the fretboard of a guitar, and the number of cells on the Hexadic figure. The square of the sun as presented in Ben’s book presents is quite a static construct: the cell containing number 1 relates to the lowest open string on the guitar (e.g. E), the cell containing number 2 relates to F, and so on.

Inspired by Peter Maxwell Davies’ use of magic squares, in which he ‘projects’ a tone-row derived form a melodic sketch ‘through’ a square to create a new series of tonal relations, I was intrigued by the possibility that each unique Hexadic figure could also be ‘projected’ through the square to create a complementary series of tonal relations. The paths, gestures or ‘sigils’ can then be traced on the resulting square to create melodies and chords. A similar process can be pursued with rhythm also. Below is an example of the process, along with 1975’s Ave Maris Stella, which projects the titular Gregorian plainchant through the square of the moon.

An example of Maxwell Davies 'projection' of Gregorian chant through the magic square of Mercury (from Cross, 'Composing with Numbers, Sets, Rows and Magic Squares', in Fauvel, Flood & Wilson (eds.), Music and Mathematics, 2003, OUP)

An example of Maxwell Davies ‘projection’ of Gregorian chant through the magic square of Mercury (from Cross, ‘Composing with Numbers, Sets, Rows and Magic Squares’, in Fauvel, Flood & Wilson (eds.), Music and Mathematics, 2003, OUP)

For my own experiment with the magic square of the sun, I decided to ‘project’ a Hexadic figure through the square of the sun, by mapping each cell of the figure to a cell in the sun square. I could then use the pre-existing sigils relating to the spirits, intelligences and seal of the sun to create complementary harmonic materials. The collected tonal materials for the piece are shown below:

Hexadic figure, squares and sigils for the 'Sorath' study.

Hexadic figure, squares and sigils for the ‘Sorath’ study.

In the resultant piece, entitled ‘Sorath’, after Agrippa’s ‘spirit of the sun’, the seal of the sun is broken apart to create three introductory and three concluding chords. Within the body of the piece harmony is created by alternating between the root tone of the current tonal field, two or more tones from the tones on the magic square that comprise the sigils of the intelligence (Nachiel) and spirit (Sorath) of the sun. It can be seen that it’s quite easy to extract the ‘Sorath’, ‘Nachiel’ and ‘Seal’ tones from the hexadic figure itself, by taking the following cells:

Sorath: (6, 2, 4)
Nachiel: (5, 20, 10, 1, 3)
Seal: (19, 32, 27, 28, 35, 24) (34, 30, 23, 17, 12, 4) (3, 7, 14, 20, 25, 3) (18, 5, 10, 9, 2, 13) (6, 11, 16, 21, 26, 31)

The material for the other voices in this piece comes from progressions around the tonal fields, as detailed in Ben’s book. I think that in terms of a short composition using the Hexadic System, the following piece stands pretty well on its own terms.

IV. Astrology and the Decans

As previously mentioned, the number 36 is important within the Hexadic System, but also has resonances in traditional astrology, which divides the ecliptic into partitions of ten degrees. Originally a time-keeping mechanism, the decans became associated with planetary sympathies and symbolic magical images during the medieval and Renaissance flourishing of astrological magic. Each zodacal sign therefore comprises three decans – the decans of Aries, from a 16th century German manuscript are shown below.

Astrological images of the 'decans' of Aries (from a 16th century German manuscript, University of Hiedelberg)

Astrological images of the ‘decans’ of Aries (from a 16th century German manuscript, University of Hiedelberg)

The resonance between the 36 cells of the Hexadic figure and the 36 decans suggests a multitude of ways to develop compositions or collections of tonal material based on astrological conditions. The ascending decan at the time of composing or interpreting the figure could also be used as an alternative to clock time as a way to choose the first interval tone in a series of Hexadic progressions (- see Ben’s book for more information).

The following piece uses the magical square of the sun along with a mapping of astrological data to the Hexadic figure. The figure itself is the same as used for the preceding study. The traditional ‘figure of the heavens’, generated with Morinus, for the time of composition was as follows:

Astrological figure for the study 'Sigils/Decans'.

Astrological figure for the study ‘Sigils/Decans’.

The decans which the ascendant (‘A’) and the seven planets occupy are then mapped onto the Hexadic figure:

Hexadic figure for 'Sigils/Decans'.

Hexadic figure for ‘Sigils/Decans’.

You will notice that there are also arrows associated with the moon and ascendant. The astrological figure was observed over 24 hours, during which the ascendant passed through all 36 decans and the moon moved from the first decan of Capricorn to the second. In this piece, one voice (the ascendant) plays through all the cells on the Hexadic figure, from the 9th cell onward. Chords gradually build up based on different entrances of the tones associated with the planets: when these chords have reached their full ‘swell’ the cycle starts again, the chords vanishing and coalescing five times over the first part of the piece. The first coalescence is an introduction, overlapping with the entrance of the ‘ascendant’ voice, playing through the cells of the Hexadic figure. The four subsequent coalescences are accompanied by bass tones based on plotting the Hebrew words (according to Crowley’s Sepher Sephiroth) for dawn, noon, evening and night on the magic square/Hexadic figure, they are: BQR (dawn), NGB (noon), a’aRB (evening) and LYL (night). Following the fifth coalescence, the piece becomes more subdued and minimal, and one of the words is given to each voice, which play all the possible combinations of the letters/cells, e.g.: BQR, BRQ, RBQ and so on.

The decans, fixed stars and lunar mansions are all aspects of astrology that have fascinated me for years. The decans were also a subject of an earlier composition of mine on Rosicrucian Enlightenment, a series of speculative experiments that I recorded under the pseudonym of Peter Cora (a play on the name ‘Pitagora’, or Pythagoras). However, the Hexadic figure presents a more consistent scheme of composing with decans than that used in my earlier piece, which was based on the random distribution of tonal material over 36 ‘fragments’.

V. Musical Steganography

The final part of the Hexadic System I’d like to look at is the area of ‘musical steganography’. Steganography derives from the Greek words for ‘concealed writing’ and, as a branch of cryptography, is concerned with hiding secret messages in plain sight. The foundation text of this practice is the Steganographia of the 15th century Abbot, cryptographer and esoteric teacher Johannes Trithemius. In Trithemius’ opus, secret messages are hidden within seemingly innocent messages, and furthermore, the secrets of decoding those messages are disguised as conjurations of spirits, fuelling centuries of debate about whether the Steganographia was wholly a work of cryptography, magic, or somewhere between the two.

The use of musical notation as a steganographic method is has a long lineage, and examples can be found in many of the cryptographic manuals of the preceding centuries, for example in Porta’s Occultam Litterarum Notis [pdf] (ca. 1593):

A historical musical cipher.

A historical musical cipher.

A similar cipher became the musical language of the moon-people the 17th century proto-science fiction of Godwin’s The Man in the Moone [pdf]:

The name 'Gonsales' in Godwin's lunar language.

The name ‘Gonsales’ in Godwin’s lunar language.

The use of musical steganography in Ben’s book is not quite so developed as some of the other material, but in the spirit of an ‘open system’ it is ripe for extension, and provides some very interesting possibilities through its use of the magic square of the sun, as well as the possibility of projecting such a square into a further accompanying Hexadic figure.

Screen Shot 2015-05-24 at 21.24.06

It was one of those beautiful coincidences that my copy of the Hexadic System arrived just after I’d made my own experiment with a form of musical steganography, which I will include here by way of contrast. I’d been thinking about music-to-text ideas as a way to begin generating material for the next Hawthonn project, and had been reading Elizabeth U. Harding’s KaliI was astounded by the devotional poetry of Ramprasad, in particularly the verses beginning:

Taking the name of Kali, dive deep down, O mind,
Into the heart’s fathomless depths
Where many a precious gem lies hid.
But never believe the bed of the ocean bare of gems
If in the first few dives you fail;
With firm resolve and self-control
Dive deep and make your way to Mother Kali’s realm.

There are a lot of interesting resonance here: the connection between the ocean and the imaginal realm of Kali, and the command to the ‘mind’ was reminiscent of the similar way in which the Hermetic magus commands his own soul to travel to other places in the Corpus Hermeticum 10.120-124 [pdf]. This seemed like some good material for a musical exploration. I decided to use the ‘crow’ scale (C#, D, F, Gb, A, Bb) that I had used to compose Angelystor, each word being assigned a tone and octave on a letter-by-letter basis, for example:

Taking = A5 C#4 Bb4 A4 D5 F#4
the =A5 A4 F4
name =D5 C#4 C#5 F4
of =D5 F#4
Kali = Bb4 C#4 C#5 A4

Each word was then combined into mixtures of chords and melodies, and further repeated in different transpositions dictated by the intervals within the ‘crow’ scale itself. The result is certainly a very interesting musical universe of its own, and upon reflection seems to have something about it suggestive of the depths of Kali’s imaginal ocean, replete with glittering gems and strange, quasi-crystalline flora:

VI. Closing Thoughts

It may be critically said that there is not much that is necessarily ‘new’ about the ideas above, yet that is to be expected since the lineage of the work can be posited within the esoteric strains of speculative music and early developments in atonal language. However, the central innovation of the Hexadic figure is a striking way to organise and navigate material; and the context in which the work has been presented relates more to practical (guitar-orientated) music-making and the world of underground/indie rock/folk/psychedelic music. It’s interesting to see the enthusiasm that the system has stirred up amongst musicians not necessarily embedded in formal composition and non-standard tonalities.

My own journey through the system over the last month has been interesting learning process. In my playing and composition, I had tended to work modally up until Angelystor, being deeply involved in the Hermetic lineage of the modes (e.g. harking back to the magical musics of Ficino and co). Outside of the modes, I tended to think ‘semiotically‘ when composing, rather than in terms of distributions of fixed notes. Taking an alternative approach has been an interesting challenge, since the system was obviously developed for guitar there are a few practical, but minor, issues in applying it as a wider compositional system due to large intervals that are often expected. However, concepts like Ben’s centre tones and taking the spirit, intelligence and seal, as discussed above, help to create more flexible collections of pitch material. Another way around this ‘problem’ could be found in laying out Hexadic figure using several pre-prepared decks, for example, laying out a deck stocked with lower registers along the inner ‘wheel’ of the figure first, to create more partitioned registers of material for composition with other instruments.

Yet, what is it about the strange and atonal that still holds appeal for those exploring the more esoteric byways of contemporary music making? Bachelard wrote that a stable and completely realised image clips the wings of the imagination: in atonality is an ambiguity, a semi-stable image that allows the imagination, whether considered reductively or Romantically to engage with the materials on a very personal level, rather than under the influence of established musical expectation. Of course, this generally yields music that does not follow the norms other musics, that requires time, empathy, imagination and patience to appreciate. The ‘esoteric’ resonances within the Hexadic System as well as the practical, compositional elements present a highly compelling key to the twelve-tone universe, and one that feels very much of the present in the light of the recent revival of interest in the spiritual roots of the modernist aesthetic.