The Pythagorean Octave

The musical octave has been used as a way of understanding the cosmos for thousands of years. The ancient Greeks used it to represent the harmonies of the universe, and in the renaissance it was used to show hidden relationships of harmony in the music of the spheres. More recently the philosopher G I Gurdjieff used the Law of Seven to understand how processes work at all levels of existence, explaining how things can go wrong in our simplest plans, and what we can do to stop this happening. The octave can be found in the Kabbalistic Tree of Life, and Saros philosophy shows how it arises as a simple abstract pattern.

In the beginning

The story of the octave, as far as we know it, begins with Pythagoras the Greek mathematician and philosopher, who in the 6th Century BC came up with two key ideas:

  • That musical harmony is based on mathematical ratio. He found that strings of different length vibrate with different sounds, and that they sound most harmonious together when the lengths are in a simple numerical ratio: 2 to 1, or 3 to 2 for example.
  • That the universe is harmonious, with the planets and stars moving according to mathematical rules.

He reasoned that the motion of the planets should correspond to musical harmony, laying the foundation for the harmony of the spheres, which was used as a way of understanding our world for thousands of years.


Engraving from Renaissance Italy (Gafurius’s Practica musice, 1496) showing Apollo, the Muses, the planetary spheres and musical ratios. (From Wikipedia)

Plato’s Timaeus

An interesting description of the octave is contained in Plato’s dialogue on cosmology called the Timaeus. This dialogue, written around 360 BC, was Plato’s attempt at a ‘theory of everything’ describing how the world and everything in it was made [1]. The crucial part for us is where he talks about the way ‘the divine craftsman’ created the world-soul, using a mathematical division to lay out the circles which describe the apparent motion of the fixed stars and the planets around us.

The mathematical division is based on simple numerical ratios, which define a Pythagorean octave. He starts with the simplest numerical ratios, formed from whole numbers:

2 to 1   do to do (an octave)
3 to 2   do to sol (a perfect 5th)
4 to 3   do to fa (a perfect 4th)

This gives a framework of notes: do … fa sol … do

Plato then fills out the framework, taking the ratio between fa and sol as the basic tone. The ratio of fa to sol is 4/3 to 3/2 which simplifies to 9/8, so Plato adds two notes re and mi spaced at this ratio from do, and then la and si spaced by the same ratio. He ends up with this picture:


The octave in Timaeus

Do-re-mi and fa-sol-la-si are all spaced by a full tone, 9/8 ratio, but there are two intervals (mi-fa) and (si-do) which are smaller, and filled by a ratio of 256/243 which is roughly a semi-tone. The white notes on a piano, CDEFGABC correspond closely to the notes of the octave. [2]

You can play around with the sounds of the notes to listen to the musical ratios using a musical instrument, or using an online keyboard. A good place to begin an appreciation of the musical intervals is this video.

The structure of the musical octave

Before moving on, it is worth summarising some of the structural elements of the octave:

  • The octave is made up of seven notes (do-re-mi-fa-sol-la-si).
  • Most of the notes are separated by a full tone, but the notes mi-fa and si-do are separated by semi tones. What does this mean? What is different about these intervals and why?
  • Considering each tone as two semitones, we can see there is a total of 12 semitones in the octave.
  • The perfect 4th and perfect 5th ratios are also present between other notes in the octave.

Next: Gurdieff’s Law of Seven (to come).


[1] Timaeus:
See: “Music and Mathematics in Plato’s Timaeus” which describes the octave process.
See: for an interesting description of how Plato went on to describe the universe in terms of the five platonic solids.

[2] Plato’s semitone ratio (256/243) is approximately half of a full tone ratio (9/8), because (256/243)(256/243) ≈ 9/8. The equivalence is only approximate however. In tuning a modern piano, a slightly different set of ratios are used, with a semitone defined so that 12 equal semitones give an octave (s12 = 2), and a tone is exactly two semitones (t = s2).

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Cailleach’s bed by the hearth

There’s an interesting article on the Old European Culture blog about alcoves found in old houses in parts of Ireland. The alcoves, built next to the warm hearth, were apparently called cailleach, which of course reminds us of the name of the old veiled one, the goddess.

Whether this is coincidence or not, it reminds me of the little house in Scotland made to keep the Cailleach and her family warm through the winter, and I imagine a time when each house would have a place for the Cailleach near to the fire.


The source for the article is a record of a house plan in Galway from the Irish Folklore Photograph Collection. The note reads:

“The bed outshot was a common feature of houses in north-west Ulster and north-west Connaught, consisting of an alcove in the back wall of the house beside the hearth. The purpose of this small extension to the house was to provide additional sleeping accommodation and it was often occupied by elderly people. A common name for the outshot was the cailleach, which according to folk etymology derives from “cúil theach” (the back of the house).”




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Winter Solstice

In the dark mid-winter,
warm underground,
light pulses,

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The Living Mountain

Nan Shepherd was a Scottish writer and hill-walker, who described her experience of the Cairngorms as a kind of meditative participation in the landscape.

Last night I watched a beautiful and moving BBC TV documentary about her, and one of the quotations seemed to me to go to the heart of walking-as-meditation. She wrote, in a letter to a friend in 1940:

“To apprehend things – walking on a hill, seeing the light change, the mist, the dark, being aware, using the whole of one’s body to instruct the spirit… it dissolves one’s being. I am no longer myself, but a part of a life beyond myself.”


Beinn a’Chaorainn by Mark Knapton at English Wikipedia

The BBC TV Documentary, in the Secret Knowledge Series is called The Living Mountain: A Cairngorms Journey and there is an interesting article about the documentary and Nan on the BBC News website here.


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The Cailleach at Sligo

This summer I visited Sligo on the Northwest coast of Ireland, looking for traces of the Cailleach, the crone of Irish legend. What I found there was a prehistoric ritual landscape of breathtaking scale, and finally an intimation of the Cailleach herself. At the centre of the landscape is Carrowmore, a collection of 30 neolithic passage tombs. To the northwest the great hill of Knocknarea dominates the landscape.

From the top of Knocknarea a huge prehistoric cairn, the tomb of Queen Mebd, looks down on the monuments of Carrowmore.


Looking up to the cairn on Knocknarea from the centre of Carrowmore

In the opposite direction lie the Ballygawley mountains. The soft curves of these hills suggest to some people the shape of a giant woman lying down. In legend, the hills are associated with the Cailleach Bhérra – the old woman of Beare.


The Ballygawley mountains from the centre of Carrowmore

The Cailleach

The Cailleach is an important figure in Irish and Scottish folklore. As the old woman, the veiled one, or the hag, she is associated with many landscape features and prehistoric sites. In folklore she is of great age, remembering previous generations and landscape changes on a geological timescale. For example in one story, she remembers when the ocean was a forest, full of trees. [1]

As well as great age, she also has giant stature and supernatural strength. Although she is old, she renews her youth every generation by bathing in a deep pool before sunrise on beltane. [1]

She is also a shaper of the landscape, and there is a story of her throwing large rocks around in a contest with another cailleach. The contest takes place at dawn, while the people of the place are all still asleep. The story explains the landscape: anyone who comes to that district today can still see a pile of large rocks on the hilltop, with no other rocks anywhere near them.

Gearóid Ó Crualaoich suggests that this story is “a metaphor for the terrifying cosmic energies that have gone into the rendering of the physical landscape of human times into its shape and form, energies beyond the tolerance of humans who live their lives in a smaller and, hopefully, safer compass. The physical features of the landscape today, however, and legends about its formation, remind the human community of the otherworld dimension of their environment, a dimension available to them in the legends and other imaginatively creative narrations of cultural tradition.” [2]

As mentioned above, the Ballygawley mountains southeast of Carrowmore are associated in legend with the Cailleach. One of the peaks is named after the Cailleach Bhérra and there is a passage tomb there called the Cailleach’s house. When viewed from the centre of Carrowmore, an interesting alignment comes into focus.

The centre: Listoghil

At the centre of Carrowmore is a large passage tomb called Listoghil. Originally the monument would have appeared as a large cairn or mound of stones. There would have been a passage giving access to the centre of the cairn, where six large upright stones supported a massive roof-slab.  The monument has now been partially restored, so that the central chamber can be seen surrounded by walls of cairn-stones held back by wire fencing. In effect a visitor walks into a cut-away version of the original cairn to find the central chamber.


The central monument of Listoghil, with the partially restored cairn in the background (originally the monument would have been completely covered by stones with just a passage leading in from the left).

Researchers at Listoghil have recently discovered a fascinating seasonal alignment. The passage into the centre of the cairn, now represented by the entry path to the monument, is directed towards a distant point called the saddle on the Ballygawley mountains.


Looking out from the centre of Listoghil along the passage towards the saddle on the Ballygawley mountains.

Sunrise Alignments

At the autumn equinox, the sun rises exactly in the east, but then each morning the sun rises a little bit more towards the south, until it reaches the most southerly sunrise on the mid-winter solstice. It then begins to swing back towards the east. From the centre of Listoghil, this movement of the sunrise takes place over the Ballygawley mountains, and the sun rises over the saddle around Samhain, so that the rising sun shines in to the centre of Listoghil. By mid-Winter the sunrise has moved along the mountains to the Cailleach’s House, before turning around for the journey back. By winter’s end, as the sunrise comes back towards the east, the light once again shines into Listoghil around Imbolc.


Ballygawley mountains showing the progression of the winter sunrise

In his book Listoghil: A Seasonal Alignment? [3]  Pádraig Meehan describes a moment when he saw the Samhain sunrise:

31 October 2008 was a beautiful crisp morning, one of the year’s first frosts. In the moments before sunrise the Ballygawley Mountains appeared to burn, and fingers of sunlight threaded the sky. The four peaks of the Ballygawley Mountains huddled on the horizon, the passage tombs at each of their summits clearly discernible; Aghamore Far to the north, Sliabh Dhá Éan, Sliabh Dargan and Cailleach a Bhérra. As the light intensified before sunrise the sense of expectation increased. Suddenly, a brilliant sliver of sun-disk appeared, close and immediate. The land darkened under the fire. On October 31, the sun rose just off-centre left in the saddle and shone directly down the ‘passage’ of Listoghil, and through the chamber. Queen Meadbh’s cairn on Knocknarea was illuminated by sunlight approximately one minute before the sun reached the central tomb in Carrowmore.

Meehan relates that there is a local story describing the form of the rounded hillocks of the Ballygawley mountains as the hag lying on her back, so that the sunrise travels up her belly and breasts to rest at her head at midwinter. Seen from the slopes of Knocknarea, the hills look to me more like a figure lying on her side with her legs drawn up.


Ballygawley mountains from Knocknarea. If the figure is lying on its side, then the hills from left to right would be hips, elbows, shoulder and head.

In either case, there seems to be a relationship painted by the landscape of the sun rising over the Cailleach’s body during the winter months, and a polarity between Knocknarea and the Cailleach’s body, with everything centred on Listoghil. In fact the progression of the sunrise along the Cailleach’s body is mirrored by a progression of sun-sets behind Knocknarea during late spring and summer.

Interestingly, the poet and esotericist W B Yeats seemed to be aware of this polarity. In the opening of his poem The Hosting of the Sidhe, he has the faerie host riding from Knocknarea and over the grave of Clooth-na-Bare (the name he used for the Cailleach a Bhérra).

In the central chamber of Listoghil, the passage is partially blocked by a stone with a blunted peak, which forms a triangular spear of shadow in the chamber when the sunrise shines down the passage. Pádraig Meehan describes lying on his back and watching the shadow cross the roof slab. He wondered “Did a ritual specialist once lie in this way to observe light and shadow in the chamber? Or was the entire event, or sequence of events, arranged not for the living, but for the dead?”


The blocking stone in the central chamber

The view from Knocknarea

The climb up to the summit of Knocknarea is steep, but worth the effort. On a clear day there are views of the sea on three sides, with the Ballygawley mountains in the south-east. Looking down the hill the monuments of Carrowmore seem tiny and insignificant.


The view from Knocknarea towards the north

The cairn at the top is huge, built from large stones.  It is the largest unopened cairn in Ireland and is thought to conceal a neolithic passage tomb dating back to 3,000 BC. Queen Medb of Connaught is supposed to be buried in the cairn, standing up with her spear and shield. According to Wikipedia, Mebd was originally a sovereignty goddess, whom a king would ritually marry as part of his inauguration. Her name, cognate with the English word ‘mead’, means ‘intoxicating one’. This perhaps then was the summer face of the goddess, with the Cailleach as her winter person.


Medb’s Cairn on Knocknarea

Michael Quirke, a local woodcarver and folklorist introduces a third person to this local form of the goddess: Findabair as the maiden. In this video he also mentions another local name for the Cailleach: Garavogue which is the name of the river running through Sligo.

The Stones of the Cailleach

I just want to finish by talking a little bit about an impression I got of the Cailleach from this visit. The rocks that I saw both at Carrowmore and on the top of Knocknarea had an organic quality. For example, look at this boulder, part of one of the Carrowmore circles:


It’s made of gneiss – a rock formed under extreme temperature and pressure. Its shape flows, in curves and bands suggestive of half-formed living shapes. This is the same rock from which the ‘Cailleach’s body’ – the Ballygawley mountains – is formed.

Here is another rock, this time from the top of Knocknarea:


I think this is fossilised coral, and there are many fossil rocks like this in on Knocknarea, again suggesting an idea of half-formed living things in the rock. The third type of rock is more about touch than appearance. This is the type of rock sometimes found in caves, with a layer of small crystals one one side, somewhere between sandpaper and velvet to the touch. This type of rock was also on Mebd’s cairn on Knocknarea.

Letting this all percolate in the back of the mind brought about a picture of the Cailleach as a shaper of forms, working very deep, with the earth and with the living creatures of the earth. The shaper of the mammoth and the mountain. She whom the cave painters addressed.



[1] The Cailleach Bheurr and Loch Bà, pp. 113-120 of Ó Crualaoich, Gearóid. The Book of the Cailleach: Stories of the Wise-Woman healer. Cork: Cork UP, 2003.

[2] The story of Cailleach Bhéarra’s Shower of Stones, pp. 104-105 of Ó Crualaoich, Gearóid. The Book of the Cailleach: Stories of the Wise-Woman healer. Cork: Cork UP, 2003.

[3] Listoghil: A Seasonal Alignment? by Pádraig Meehan is available at the Carrowmore visitor centre, but is seemingly out of stock on online retailers. Much of the same information from the book is available online at


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Loch dá Ghedh

I recently came across a fascinating Irish folklore site called dú, which includes digitized images of thousands of pages of folklore written down by schoolchildren in the 1930s:

Approximately 740,000 pages (288,000 pages in the pupils’ original exercise books; 451,000 pages in bound volumes) of folklore and local tradition were compiled by pupils from 5,000 primary schools in the Irish Free State between 1937 and 1939.

Volunteers are welcome to help transcribe the folklore they find interesting. It’s quite easy to do, so I transcribed a story about Loch dá Ghedh:

Loch dá Ghedh

On the top of Slieve da En facing towards the town of Sligo and situated between two of the mountain peaks, is a lake called Loch Ghedh. This lake has the reputation of being the deepest lake in the world. It was supposed by the old people to be enchanted and the devil’s mother was said to be drowned in it. There is a legend told of it and it was of a man who went out this mountain one fine day cutting rods or sticks with a bell hook. When evening came the man did not return to his home. Of course a big search was made to find the missing man.

The search proved in vain and the people formed the opinion that in returning to his home in the evening with the bundle of sticks on his back, the man in passing by Loch Ghedh slipped off the slopes on the side of it and fell in and was drowned. This man had a daughter in America and in some years after this sad happening she decided to come home to her native Sligo and help to console her people on their great loss. When coming in on the boat to Sligo bay she observed in the sea a bundle of rods and at once recognised her father’s bell-hook stuck in them. The old people say that the lake is connected with the sea by an underground passage and that this accounted for the bundle of rods being found in the Ocean.

Loch dá Ghedh is identified with the ‘Lough Ia’ mentioned by the poet W B Yeats as the grave of Clooth-na-Bare, who  “went all over the world, seeking a lake deep enough to drown her faery life, of which she had grown weary, leaping from hill to hill, and setting up a cairn of stones wherever her feet lighted, until, at last, she found the deepest water in the world in little Lough Ia, on the top of the bird mountain, in Sligo.”

As discussed in my previous postClooth-na-Bare was a corrupted form of Cailleach Bhéara – the old woman of Beare, an important figure in Irish and Scottish folklore. Presumably it is she who is being described as the devil’s mother in this story.

In looking for a photo of the lake, I came across another interesting story about it in a comment by swoop on the Mountain Views website. In this story, the lake is called Lough Dhá Géanna (the name seem to mutate quite a bit – I’ve also seen Lough Dagee and Lough Dha Ghe):

Under Sliabh Dá Éan (‘mountain of two birds) lies a small but beautiful hidden lake. Legend has it that got its name Lough Dhá Géanna from the story of king Sweeney. Sweeny was a king condemned by an angry cleric to wander, naked and nervous as a bird throughout Ireland. Any sharp sound, like the ringing of a bell, would send Sweeney into madness. Because of the curse he began to levitate and move about like a bird. Being bird-like he could never trust humans but fled from place to place, naked and hungry. Near Lough Gill, in the eastern Ox mountains, Sweeney fought with a Cailleach, a wise woman, and both of them in their struggle transformed into geese and dived into the lake. Never to be seen again.


Lake of the two geese, image via swoop

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The House of the Old Man

Many years ago I read about an ancient shrine to the goddess hidden away in a remote highland glen. It was written about in Twilight of the Celtic Gods [1]:

“This rocky shrine is in all probability connected with the pagan Celtic cult of the Mother Goddess. It may be the only surviving example of its kind in the whole of the British Isles. But this is no lifeless pile of stones, for the shrine is part of a living Celtic tradition which has been continued into recent years by a guardian – a lone shepherd – who has performed a vital ritual at the little shrine, as his father and grandfather had done before him. At the door of the little stone house, from May to October, sit three strange stones, keeping watch over the glen. The tallest, 46 centimetres (18 inches) in height, is known as the Cailleach, Old Woman or Hag. Her partner is the Bodach or Old Man, and there is a third, the smaller Nighean, or daughter.”

The shine is known as Tigh na Cailliche, ‘The Hag’s House’ or the house of the goddess, or alternatively – on the map – as Tigh nam Bodach, the old man’s house. Earlier this year a friend and I were inspired by a post from William A. Young’s blog Feral Words to make a pilgrimage to the house, and this is the story of that visit.


Tigh nam Bodach with the strange stones sitting outside

William had visited the shrine during the winter, when snow was on the ground. He and his companion took a challenging route up to the glen from Bridge of Orchy, camping out overnight. When he arrived he found that the stones representing the old woman and her family were snugly tucked away inside the little house, for that is the ritual of the place: every year at Samhain the family of stones is moved inside the house for the winter months, and then at Beltane they are taken out again, to sit in front of the little stone house.

I’m not too good at long walks, so I looked for an easier path to the shrine, and found one mentioned here, accessible by car and then by foot along the shore of Loch Lyon. I was also hoping that going in mid-summer would make things easier. It did, but the journey still took me near to the limits of my strength!

Meeting ‘the Old Man’

On this kind of magical trip, signs and omens become important. Everything seen and done becomes part of an unspoken dialogue with the shining ones. The afternoon before our walk we stayed near Loch Tay, and went for a stroll in the warm summer sunshine, looking for some cup-marked rocks that were shown on the map. As we wandered around a field, an white-bearded sheep farmer came by on a quad bike, his collie sitting on the back. I asked him about the cup-marked rocks, but he wasn’t sure where they might be. We talked more. He spoke quietly and thought a long time about each thing he said. I felt myself slow down. It was like he was full of the countryside and the light. “I was born in the Western Isles,” he said, “but my wife is from here.” Somehow this encounter stayed with me, and cheered me on our walk. I felt I had been checked out by ‘the old man’.

Thunder and Lightning

During the night the fine weather broke and we were woken by thunder and lightning. In the morning it was still raining, but we had waterproofs and after some hesitation we decided it was now or never. Driving over the mountain pass from Loch Tay into Glen Lyon, we were surrounded by mist and low clouds as the rain fell, with lightning flickering above us. After splashing twenty miles along single-track roads, we finally arrived at the start of our walk by the hydroelectric dam at the east end of Loch Lyon.

As we began to walk the weather relented, and although the clouds continued to lour over the hills, the weather remained dry as we walked by the dam and followed the track alongside Loch Lyon.



Track alongside Loch Lyon

The River Ford

During the walk we were reminded that nothing worthwhile is achieved without difficulty. In so many stories the heroes and heroines must perform tasks and overcome obstacles in order to reach their goal. It felt like the morning’s rain had been an obstacle like this, and we encountered others. For example, the map optimistically described the crossing of Allt Meurain as a ‘ford’, but it turned out to be quite deep. We ended up wading knee-deep through rapidly flowing water. The pain of walking barefoot on the sometimes sharp stones in the rocky riverbed had to be endured, for the body’s natural reaction of aversion risked falling headlong into the icy water.

Luckily we did not see the Washer at the Ford, which I took as a good sign!

In Glen Cailliche

After the ‘ford’ we began walking up Glen Cailliche, the valley of the Cailleach, and just when I was beginning to despair of finding it, we came in sight of the Tigh nam Bodach, down below the track on the northern side of the river. There were more than just three stones sitting outside – all with the same characteristic rounded river-worn shape. As you can see in the picture below there was one big one with a clearly defined head. Behind it was a dumpy, almost acorn-shaped stone. To the left were two smaller stones with defined heads, lying on their sides. To the front there were three other quite small stones which seemed part of the pattern. In Twilight of the Celtic Gods [1] it says that there is a local legend that every hundred years or so, the Cailleach gives birth to another stone, and “though the baby of the family is still very small, people swear it is growing and will one day be just as big as the others”.



The Legend

Anne Ross, in her Folklore of the Scottish Highlands [2], retells fragments of the cult legend surrounding the house which have survived orally:

“Many years ago, in a fierce snowstorm, an unnaturally large man and woman were seen coming down the mountain-side of the upper glen. They asked the people who were still settled there, for hospitality and shelter. These were willingly given to them. This pleased the supernatural pair well and they took up residence in the glen when the inhabitants had built a thatched house large enough to accommodate them. The woman was pregnant and in due course gave birth to a daughter. The weather was always favourable when they dwelt there. The stock flourished and the crops were always of the best.

Then one day the time came when they decided they must go. Before doing so, they promised that as long as they were remembered and their house kept in order, and everything done as they themselves had done it, they would bring it about that winters would be mild, the summers warm, and peace and prosperity would always be with the people who had been so generous to them.  In memory of this event of long ago a small shrine in the form of the house was constructed and every May Day the three stones representing three deities would be taken out of the house and placed facing down the glen, there they remained until the house was re-thatched and made warm and comfortable for the winter and they were returned to the miniature house on the eve of 1 November, Hallowe’en. When the upper glen was flooded and the people moved away, it became the shepherd’s duty to continue this ritual and this was faithfully carried out.”


Looking over the shrine up Glen Cailliche


The feeling of the shrine seemed to me very pleasant and warm. I had half expected a rather more fierce encounter with the Cailleach, but if I had to put my feelings into words, I would say that I found her a mother, one who can comfort and provide strength in need. I suspect though that she needs to be approached properly. After a while resting at the shrine, the mist began to descend down the Glen and we began our walk back to the dam.




May she be with you in your hour of need


[1] Twilight of the Celtic Gods: An exploration of Britain’s Hidden Pagan Traditions, David Clarke with Andy Roberts, Blandford 1996, pp. 66-67

[2] Folklore of the Scottish Highlands, Anne Ross, The History Press; 2nd edition (1 Nov. 2000)

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