The Octave in Saros Philosophy

This post is part of a series about the octave as a way of understanding the world.
The previous post was about The Octave and the Tree of Life

Saros Philosophy presents a different approach to the octave or Law of Seven, showing how its structure can arise from very simple laws. The full process is described in The Book of Jubilee [1] and is outlined below:

Three

The root of the octave is in the law of three. In any interaction, three forces act: an impulse, a reaction, and a result. Here the three are labelled Affirming, Denying and Unifying (A, D and U).

Six

The three forces can act in different orders, giving six combinations called constructors: ADU, AUD, DUA, DAU, UAD, UDA. The constructors can be represented as shown below.

cons6

Twelve

The six constructors can link together, one following on from another in a tail-eating sequence. For example UAD ends with the denying force D, and so it could lead on to a constructor which begins with D, such as DUA. This would make the pair UAD-DUA, or written more simply, UADUA, the D in the middle being both the last part of UAD and the first part of DUA.

The six can combine in different patterns, but if we require that each of the six should appear once and only once, there are only three possibilities. One of these is the following:

UAD-DAU-UDA-AUD-DUA-ADU-

This sequence of six constructors repeats again and again, because the last one leads back to the first one. One representation of this sequence is shown below:

cosmos5

Rewriting the sequence without duplicating forces gives a repeating sequence of 12 forces:

U A D A U D A U D U A D –

Seven

It is in this sequence of 12 forces that the octave is found. The diagram below shows the sequence U A D A U D A U D U A D drawn out in a circle, starting at 10 o’clock and moving clockwise.

7octave

The full tones do-re, re-mi, fa-so, and la-si are mediated by the Affirming force. The sequence of forces in each transition is UAD or DAU.

At the first semi-tone interval or shock point mi-fa, the transition is from U to D with no mediating third force. At the other shock point, si-do, again the transition is unmediated, this time from D to U.

The disharmonised point (Gurdieff’s Harnel Aoot [2]) so-la, is a full tone, made of three forces, but this time two of the forces are the same – UDU, so the transition seems like UD followed by DU.

The glyph in the middle of the diagram shows the common forces of the notes within the octave. The three Denying notes (si, re, fa) and the four Unifying ones (la, do, mi, sol).

The following diagram shows the same structure related to an octave on a piano keyboard. The seven white notes are the notes of the octave, all U and D forces. The 5 black notes are the semitones – four Affirming forces, and one Denying.

kboctave

Conclusions

I think that the key contribution of Saros Philosophy to the theory of the octave is showing how a very simple method of linking three elements can lead to the structure of the octave with its patterns of tones and semitones. The method of producing the octave structure is entirely independent of the mathematical ratios and musical interpretation used by Pythagoras, and in addition to the two semi-tone intervals, it also highlights the so-la disharmonisation noted by Gurdjieff.

The relations and patterns that lead to the octave can be appreciated and explored in different ways. For example, the four denying forces are equally spaced, whereas the spacing of affirming and denying forces change: as one ‘breathes in’, the other ‘breathes out’. Contemplating these patterns can be worthwhile.

The tail-eating pattern of six constructors used above is not the only possibility. Nor is the octave structure found within it. Other possibilities can be found, and can throw further light on the octave itself.

Good hunting!


Notes:

[1] The Book of Jubilee, a Primer of Saros Philosophy. https://thebookofjubilee.wordpress.com/

[2] Chapter 39 of “Beelzebub’s Tales To His Grandson”. An electronic version of Chapter 39 is here.

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Basajaun – the Lord of the Forest

I recently wrote this for the Sareoso Website.

Sareoso

On my first visit to the Basque Country, I stayed in Zubieta, a small village in Navarre. The town is famous for the yearly carnival when the Joaldunak march to the neighbouring village, swinging huge cowbells to wake the world up from winter.  Zubieta is in a quiet river valley, and is surrounded by hills and mountains. On the slopes there are woods and pastures where sheep and cows graze.

CIMG0001 Zubieta street scene

Bells in the Night

At night all is quiet, and very dark except for the occasional street light in the village itself. Every now and then you can hear the bells worn by the grazing animals on the slopes surrounding the village. Most of the time, the bells ring just occasionally, but sometimes there is an outburst of bells ringing. Something is happening up on the hill! Nothing can be seen in the dark, but the bells…

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The Centre is Everywhere

The [limitless light] has been called a circle whose
centre is everywhere and whose circumference is nowhere.
– Dion Fortune, The Mystical Qabalah

This paradoxical saying by Dion Fortune can be traced back to The Book of the Twenty-Four Philosophers, a medieval work thought by some to have been composed in the 4th
century AD:

God is an infinite sphere whose centre is
everywhere and whose circumference is nowhere.

The saying inspired the following exercise, which I have been doing most days for the last few months.

I take a few minutes each day, standing still or sitting, usually eyes open, indoors or outdoors, and I try to remember the infinite universe around me. I imagine the six directions extending from my heart: in front of me, behind me, to my left and right, above and below, extending without limit, for where is the circumference?

At this specific location, at this moment in time, the universe is aware of itself through me. I’m breathing in and out, just holding that awareness.

Sometimes there is a hint of something in the breathing. I imagine breathing in the limitless light, the Ain Soph Aur of Kabbalah, or if you prefer physics, whatever is on the other side of the big bang, and then breathing it out into our world.

 

 

 

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Part 2: Geometrical Models in the Sepher Yetzirah and Bahir

This post follows on from Part 1.

Second Model

An alternative way of representing the same geometry is to use a model constructed from spheres (eg. table-tennis balls) instead of lines and corners.  To do this we start by making a 5-layer tetrahedron, and then add a group of four balls to each face, thus making two intersecting tetrahedra – a ‘star’ shape.

balltet1

5-layer tetrahedron

 

balltet2

Adding to each face to make the star shape.

Looking at the star shape from a corner shows the intersecting tetrahedra – one ‘pointing up’ towards the viewer and the other ‘pointing down’. Looking at the star shape from the side shows the cube outline:

balltet3

Although less obvious, the octahedron is still embedded within the star shape, with the six directions at the intersections of the edges of the tetrahedra (the centers of the cubic faces). For example, the ball marked ‘E’ in the diagram above represents one of the six directions. By cutting off the 8 corners of the tetrahedra, the octahedral shape is exposed:

octahedron

Also concealed within the star shape (and indeed within the octahedron) is the dymaxion or cuboctahedron shape. This shape, where one central ball is surrounded by 12 others, is an aspect not seen in the ‘line-and-corner’ model.

balltet5

The dark coloured balls make the dymaxion

dymaxion

Dymaxion

One way of thinking about the star shape is that there is:

  • One ball at the very centre
  • Twelve balls that surround it and make the dymaxion (13 balls)
  • Add a ball for each of the six directions to make the octahedron (19 balls)
  • Add four groups of 4 balls to make a 5-layer tetrahedron (35 balls)
  • Add four groups of 4 balls to make the star shape (51 balls).

threeshapes.jpg

The introductions to the Yetzirah and Bahir include other numerical interests in the star shape:

  • The star shape has seven levels.
  • The surface contains 50 balls (all but the central ball is on the surface). The letter Nun equals 50 in numerical value.
  • Each tetrahedron contains six edges (Vav) and the number of spheres along these edges adds up to 22.
  • When it is on its base, the number of visible spheres in a tetrahedron is thirty-one (all except the one in the middle and the three on the inner part of the bottom face). The total of all spheres in the tetrahedron is 35. when it is remembered that it was customary to include in the numbering the whole that is formed, we have the combination of the 32 and the 36 upon which much of the Bahir’s explanation is built.

One of the comments in the introduction to the Yetzirah is a little unclear: “Also each of the subsidiary tetrahedra contain 10 spheres which answer to the four worlds of logos, creation, fashioning and making.” A three-level tetrahedron contains 10 spheres, and there are several ways that three-level tetrahedra can be seen in the star shape. Precisely which four of these are meant to correspond to the four worlds is not clear.

 

A PDF version of Part 1 and Part 2 is available here.

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Part 1: Geometrical Models in the Sepher Yetzirah and Bahir

In the introduction to their translations of the Sepher Yetzirah and Sepher ha-Bahir[1],  W.G. Davies and G. Zur suggest that the structure of the Sephirot described is consistent with that of a cube containing a double tetrahedron and octahedron. The aim of this note is to try and clarify this geometry.

Making the model

We start with a cube, shown in red. By joining alternate corners of the cube, we make a tetrahedron (a triangular pyramid), coloured green and labelled 1,2,3,4:

cube cubetet

We make a second tetrahedron, joining the other four corners of the cube (shown below in black). This makes two intersecting tetrahedra as illustrated below.

cubetet2 

The edges of the two tetrahedra cross at the middle of each face of the cube, giving six points. Joining these points together makes an octahedron, shown in blue and labelled 5,6,7,8,9,10:

cubeoct

(For the sake of clarity, only one of the tetrahedra is shown).
Another view, without the tetrahedron:

Image: Birgit Lachner, Wikimedia

 

The figure below shows the same structure, viewed from a different direction lining up with point 4. Considering Verse 8 of Chapter 1 in Sepher Yetzirah, we interpret the four corners of the green tetrahedron as representing: 1-Spirit, 2-Spirit (or Air) from Spirit, 3-Water from Spirit, and 4-Fire from Water. The corners of the octahedron are interpreted as aligning with the six directions (Height, Depth, East, West, South, North).[2] Thus these points are the 10 Sephirot.

Hexmap

The four unlabelled points on the intersecting tetrahedron (black line) perhaps correspond to the four measurements in Chapter 1, Verse 4: boundless good, boundless evil, boundless beginning and boundless end[3]. The directions again correspond to the octahedron corners.

In Chapter 5, Verse 2 of the Sepher Yetzirah, the twelve simple letters are linked to the 12 diagonal directions: East-Depth, South-Height, South-East and so on. These correspond to the edges of the octahedron. Each corner of the octahedron corresponds to one of six directions, and so the lines joining them correspond to the 12 diagonal directions (as illustrated below, showing the signs of the zodiac corresponding to each of the 12 simple letters).

12simples

Visualisation

As a result of this understanding, it might be possible to imagine oneself standing in the centre of the octahedron. Each corner of the octahedron is one of the six directions, sealed with the different combinations of the letters yod, vav and heh. The twelve edges of the octahedron represent the twelve simple letters, or the signs of the zodiac. Through each triangular face of the octahedron, one looks out to a corner of one of the two tetrahedra, or (equivalently) a corner of the cube. Spirit, air, water, fire, beginning, end, good and evil.

End of Part 1.
A PDF version of Part 1 and Part 2 is available here.

 


[1] Electronic copies available at http://www.bizworld.co.uk/saros/pubs.html

[2] Note that the numbering of the points in the diagram of the octahedron do not correspond to the numbering of the directions in verse 8.

[3] In the introduction to Sepher Ha-Bahir, where this diagram originates, the three visible corners are labelled Good, Height and Depth. I suspect that Height and Depth were labelled in error and should have been Beginning and End.

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Kabbalah and the Big Bang

In medieval times the Kabbalistic Tree of Life incorporated a model of the physical universe. In this model the earth was at the centre, surrounded by spheres of the known planets: the Moon, Mercury, Venus, the Sun, Mars, Jupiter, and the sphere of fixed stars.  We can see this model in the planetary attributes of the sephiroth. (See my previous post for more on the Tree and the planetary attributes).

flammarion

Nowadays we have a different model of the physical universe: starting with an infinitely small singularity out of nowhere which suddenly expands and inflates in a big bang, creating space and time as it goes, and then there’s some cooling and clumping together to begin the formation of matter and eventually the universe we experience today.

cmb_timeline300_no_wmap

Chronology of the Universe

I love reading about the big bang theory because it is so awesome – and it stretches my mind in the same way that working with the Tree of Life does. When I think about Kether breaking forth from the veils of negative existence, I also think about the singularity bursting, and what there was before that moment. And then perhaps we can make other correspondences. Cosmic Inflation = Hokhmah and Cooling and Clumping = Binah for example?

Here’s a nice video on inflation, with some nice animations from about halfway through.

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The Company of Nine – An ancient template for women’s magic

Guest Post by Cherry Gilchrist

We, who come among the dead as far
as to the very Goddess, nine girls,
maidens, lovely in our dancing,
in bright loveliness of folded
woven-work, with fine-sawn necklaces
of ivory, shine, brilliant
to the dead eye as forgotten daylight[i]

Recent research has revealed the presence of an age-old template for women’s votive groups: it’s a company of nine women, often known as the nine sisters, maidens or ladies. Here, it’s embedded in an ancient Greek poem, describing how nine female spirits pass through the realms of the dead to the Goddess. Fast forward to the British myth of nine sisters conveying the dying King Arthur to the otherworld of Avalon, add in the classical Nine Muses and we may begin to see that there is a story here – a story of nine women engaged on sacred tasks.

I knew about this to some extent when I first wrote The Circle of Nine, back in the 1980s. The book was based on nine feminine archetypes that certain groups of women were using to understand the principles of the female psyche, and how they worked as archetypal forces in our own lives. We called it Nine Ladies, taking the name from the stone circle in Derbyshire which we had visited on one auspicious occasion. As I wrote the book, I came across a few more instances of the nine, but with no internet resources to draw on, I couldn’t investigate much further. Now, in the re-writing of the book, I’ve been able to research the Company of Nine more fully. And my findings show that it is more than an occasional occurrence – it seems to be, rather, the fundamental template for groups of nine women engaged in a votive or sacred task. It’s widespread in time and space, found from Africa to Russia, and from around 10,000 BCE until just about the present day.

The Circle of Nine coverThis has been an exciting discovery – there’s a wide range of types of these companies, and I can only describe them briefly here.[ii] However, you’ll find more in my book, where I’ve dedicated the opening chapter to the Company of Nine. (The Circle of Nine: An Archetypal Journey to Awaken the Divine Feminine Within – Red Wheel Weiser, Sep 2018).  Each ‘company’ has its own identity, and ranges too from groups of women with a historical presence, to those who exist in myth, folklore, or within the symbolic contours of the landscape. Each group seems to have a specific task or function; their briefs range widely, from divination to healing, serving a saint or goddess, or even simply dancing for joy. Their job is often to help others, work magic or see into the realms of the future.

The Nine Priestesses of Sena

A very fine example was recorded by the Roman geographer Pomponius Mela. He reported that a group of nine priestesses lived on an island called Sena, lying off the coast of Brittany – possibly the one known today as the Ile de Sein in the Bay of the Dead, thought of as a portal to the ‘otherworld’. Here, the women tended the oracle of a Gallic god, and were ‘endowed with singular powers’. Navigators visited the priestesses asking for guidance, wanting to know their future, and to gain assistance in charming the winds and seas. The women were also renowned for their ability to shape-shift into different animal forms, and to heal serious wounds and diseases.

These priestesses seem to have had a historical basis, and later historical accounts relate that in medieval times, companies of nine women were known to travel around Scandinavia, acting as seeresses. One such ‘volva’ or spae-queen is recorded in an Icelandic saga from Greenland, and described as she visits a remote village, dressed in special robes, and pronouncing her predictions from a throne erected by the locals for the occasion. These forecasts related to weather, health, and future marriages within the community.[iii] Legends abound too; in Brittany, nine witches or spirits are said to inhabit the mountain of Dol. These myths spread out geographically –  St Samson, a dragon-slaying saint who is also associated with Mount Dol as well as with Wales, is said to have had an encounter with a magical woman – or ‘crazed hag’, depending on the source – a wild woman from a company of nine sisters living in a wood. She claims to be the last of her kind. [iv] Indeed, it’s often the case that such wise women, oracles, witches and healers are said to be ‘the last of their kind’. This perhaps embodies the memory that the company of nine comes from an ancient source, and hints at the slipping away of this way of expressing feminine wisdom, as societies and religions evolved.

Nine Stones in the Landscape

dav

Visiting the Nine Maidens Circle at Belstone, Dartmoor

The landscape too bears many traces of the nine, coming back to my first association with the Nine Ladies stone circle in Derbyshire. Nine Maidens, Sisters or Ladies appear many times over in the UK as stone circles, stone rows, and wells. According to one study, for example, there are at least six circles in Cornwall called Nine Maidens or Nine Stones.[v] Sometimes, although the names don’t carry female connotations, examining local folklore shows that they’re related to tales of nine women. Many of the stone circles bearing the name of ‘nine’ also carry a legend about nine girls being turned into stone for dancing there  at a prohibited time, such as the Sabbath. In the Nine Maidens circle at Belstone, Dartmoor, a site I often visit, it’s said that if you enter the circle at midday, and let the world go still around you, the nine maidens will start their dance again…

One curious proof of the importance of nine as the number of the maidens in a stone circle, is that this is often in defiance of the actual number of stones standing. Nine is stubbornly adhered to, even when scholars assert that it should be ‘men’, as the Cornish word for stones, or ‘noon’ – as we’ve seen,  the time of day at which standing stones are often said to move – or even ‘nineteen’ as the true number of stones. Of the six Cornish circles named for the Nine Maidens, apparently only one has nine stones and it’s doubtful that this was the case originally.[vi] Plainly, none of these explanations holds good for all the different geographical examples of nine maidens in the landscape, and it’s a far more natural and plausible conclusion to accept that ‘nine’ is what folk intended to call them! William Bottrell, the nineteenth century Cornish folklorist, who denied the veracity of ‘nine’ in the names of the megaliths nevertheless said: ‘You know everybody hereabouts uses nine in all their charms and many other matters.’ Nine does indeed have magical connotations, and it’s likely that the template for the ‘company of nine’ derives from the idea of the triple goddess, a feminine trinity of cosmic forces. This has often been related to the phases of the moon, and to the phases of a woman’s life as maiden, mother and crone.

Nine Maidens, nr Zennor, Cornwall

The Nine Maidens stone circle at Zennor, Cornwall. Painting in Oils by Robert Lee-Wade

Ancient Tradition

I believe that our original call to form the Nine Ladies groups was prompted by touching on a very ancient tradition, which was associated with that particular stone circle in Derbyshire. But although the tradition of the company of nine women may be ancient, it’s important to emphasise that it was not necessarily associated with these stone alignments when they were first built, some 4000 years ago. It’s more likely, in my view, that the nine were attributed to these places later on, after the original purpose of the megaliths was forgotten, at which point the stone circles and rows could well have served as ritual places for a different mythology. But that’s not to say that this mythology itself may not have very ancient origins.There is one really early image of the nine, which could push the timeline right back to between 10,000 and 7,000 BCE. This is a cave painting from Catalonia, which depicts nine women dancing round a male with an erect phallus. Perhaps the dance of the nine maidens was one of the very earliest of rituals.

Spain Moros cave

Nine Young Women dancing around a male figure, from a prehistoric cave painting in Catalonia

There is much more of the story still to uncover, but my guess is that the nine maidens played a part in an early magical and perhaps shamanic type of religion, allied to the elements and the spirits of nature. It probably arose in pre-historic times, spreading to different parts of the globe and remaining in folk memory. Traces of it still abound. Here’s a shaman’s song from Mongolia, describing the Nine Sisters as they dance between heaven and earth:

We play on the rays of the sun
We ride on the rays of the moon
We rise into the heavens
We descend onto the hills

…Nine young ones danced
They met a glowing mother
Three times in the ritual
We will dance the ancient dance
All nine will dance together! [vii]


P1010772Cherry Gilchrist is the author of books on alchemy, Russian mythology, women’s spirituality and traditional Tarot. She’s travelled to far-flung places such as Easter Island, Siberia and the Silk Road in her quest for ancient traditions. Her own background is in Tree of Life Kabbala and western meditation. Cherry currently lives in Topsham, Devon, with her husband, artist Robert Lee-Wade.

The Circle of Nine: An Archetypal Journey to Awaken the Divine Feminine Within
is published by Red Wheel Weiser in Sep 2018.

Pre-publication, you can access a sample e-reader text at https://www.calameo.com/read/0055709746d8b1d726d86


[i] From a 7th century BCE Papyrus.Translated by Brooks Haxton from the ancient Greek, Dances for Flute and Thunder , (Viking 1999) Quoted by kind permission of the author.

[ii] The only other specific study I’ve found is The Quest for the Nine Maidens, Stuart McHardy, (Luath Press Ltd, 2003) . This has many useful references and focuses primarily on Scotland.

[iii] The Saga of Erik the Red posted at http://sagadb.org/eiriks_saga_rauda.en, translated J. Sefton, 1880

[iv] See Celtic culture: a historical encyclopedia. Vol. 1-, Volume 3 edited by John T. Koch, p1558 and also Stuart McHardy, p65

[v] “The Stone Circles of Cornwall”, B. C. Spooner, Folkore (Vol 64, Dec 1953), pp. 484-487

[vi] ibid

[vii] Siberian Shamanism: The Shanar Ritual of the Buryats (Inner Traditions, 2015) pps 18 & 19 Translated and reproduced by courtesy of the authors  Sayan Zhambalov, Virlana Tkacz and Wanda Phipps

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Day of the Cailleach

Today is the traditional day of the Cailleach in Scotland, called Là na Caillich. On March 25th, the winter reign of the Cailleach comes to an end. Here’s a little video all about it with some beautiful images.

You can read about Là na Caillich on the Tairis Gaelic Polytheist Website.

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The Otherworld in the Celtic tradition

Interesting article by Sharon Blackie about the Otherworld and Shamanism.

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The Red Spear of Lyra

A little bit of space amazingness: it seems that our recent interstellar visitor has an unusual shape – long and thin as in this artist’s reconstruction. The interstellar object, named ‘Oumuamua (Hawaiian for scout) came from the direction of the constellation Lyra, passing through the solar system earlier this year, swinging past the sun in early September at a speed of nearly 200,000 miles per hour. It’s now on it’s way out again.

Eso1737a

‘Oumuamua

By ESO/M. Kornmesser (http://www.eso.org/public/images/eso1737a/)
[CC BY 4.0 (http://creativecommons.org/licenses/by/4.0)%5D,
via Wikimedia Commons

 ‘Oumuamua is not icy like a comet, but has a reddish colour like some outer solar system objects. A bit more reading suggests that the colour could be down to Tholin or ‘space tar’, a complex mix of organic chemicals formed in space from simple molecules by radiation. Some researchers have speculated that Earth may have been seeded by organic compounds early in its development by tholin-rich comets, providing the raw material necessary for life to develop.

It all seems like a science fiction film – but it’s not.

What a time to be alive!

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