certain of these insights which relate music, number and mythology, to the Castlerigg , it will attempt to complement Thom’s astronomical interpretation with a subsidiary musical one. With regard to the ancient quadrivium of arithmetic, 9-17 music, geometry and astronomy, McClain believed that he had found evidence of the use of similar CASTLERIGG: numbers and mathematical constructs in Sumer, Babylon, India, Egypt, Palestine and Greece. As a Stone or Tone Circle? result, he postulated the evolution and existence of a single global mathematical, musical, geometric, Leon CRICKMORE Most Northely CASTLERIGG (L 1/1) Setting Moon Midsummer This paper is intended as a tribute to the musicologist Setting Sun Ernest McClain (1918-2014) and his seminal insights, on the basis of which it will attempt to complement ’s astronomical interpretation of the Castlerigg Stone Circle with a subsidiary musical one. Altar

Castlerigg is a Stone Circle in the Lake District Eq. Sun National Park, near Keswick, , . 2 MY Impressively encircled by mountains and fells, it is 4 thought to have been built about 3200 BC. Outlier 196 ft. 6 Candlemass Rising Sun Diameter 107.1 ft. Midsummer Rising Sun To North and Most Southerly Setting Moon Most Southerly Rising Moon 10 0 10 20 30 40 50

Figure 2. Alexander Thom’s archeo-astronomical plan of Castlerigg. After Thom (1967). Figure 1. Castlerigg Stone Circle. astronomical and spiritual tradition. In particular, Probably the best available interpretation of he has certainly uncovered much of the missing the purpose of the Castlerigg Circle is that of arithmetic underlying the concept of what is Professor Alexander Thom1, who considered it usually described as the ‘Harmony of the Spheres’. to have been an ancient astronomical observatory At the beginning of The Myth of Invariance, (see figure 2). McClain points out how the poets of the Rg Veda This paper celebrates the work of the seem obsessed by a concern about ‘the exact musicologist Ernest McClain (1918-2014) and number of everything they encounter’. Later, he the seminal insights contained in his three books: claims that ‘the central geometrical image of the Rg The Myth of Invariance; The Pythagorean Plato2, Veda is the mandala of the “single-wheeled chariot and Meditations through the Quran3. By applying of the Sun”, harmonizing moon months with 1 Thom, A. (1967), Megalithic Sites in Britain, Fig.12.10, solar years and the signs of the zodiac’, quoting, in Oxford University Press. support, from the Rg Veda 1.164.11 & 48: 2 As Plato recognised (Republic, 530 c-e), astronomy and music have long been viewed as sister disciplines. our understanding of the universe. Knowledge has thereby 3 Since the seventeenth century, however, modern tended to be reduced to evidence within the limits of statistical science has gradually ‘untuned the heavens’ and demythologized mathematics. 10 CASTLERIGG: Stone or Tone Circle?

needs to be said. ‘Formed with twelve spokes, by length of Time, unweakened, rolls round the Heaven this wheel of during order, Twelve are the fellies, and the wheel is single.’ In his book Beyond Measure4, the mathe- matician, Jay Kappraff generously devotes the whole of chapters 3 and 4 to McClain’s speculations and the mathematics of music in general. On page 60, he presents a diagram of McClain’s ‘Mandala of the Single-heeled Chariot of the Sun’. For figure 3, this has been superimposed onto a later analysis of the site, supportive of Thom’s conclusions, by D. P. Gregg5. Since, strictly speaking, Castlerigg is shaped like a flattened egg rather than a circle, the alignments are approximate. Anachronistically, the signs of the zodiac and the pitches of the equal-tempered scale are also included. Kappraff comments: ‘This hypothetical tonal zodiac shows how Figure. 3. McClain’s mandala superimposed on Professor a twelve-spoked mandala harmonizes music Gregg’s diagram. and astronomy at an abstract geometrical Equal temperament is a system of tuning level although in ancient times neither the constellations nor the intervals of the in which the octave is divided into twelve equal chromatic scale divided the cycle equally’. semitones. Pitch is calculated as a logarithm Notice how the co-ordinates of each diameter of frequency. The whole system comprises a 7 – and these include the important astronomical logarithmic spiral , which has to be flattened to line between the Midsummer Setting Sun and form McClain’s mandala. The ancients, however, the Candlemas Rising Sun (line A-B in figure 3) had no means of measuring frequency accurately. – correspond to musical tritones (augmented They had no logarithms. Nevertheless, they fourthsdiminished fifths): c-f sharp; b-f6 etc . skilfully calculated musical pitch by means of ratios Kappraff also points out that, when the twelve of string or pipe length, having observed that two points of this diagram are joined in various ways: pitches an octave apart are in a ratio of 1:2. But the the sides any equilateral triangle define major tritone, at the mid-point of the octave (2 in equal thirds (e.g. C-E-G sharpA flat); the sides of any temperament) is an . As such, it hexagon define a whole tone scale (e.g. C--E-F presented a serious problem to the ancients. There sharp- G sharp-A sharpB flat) ; the sides of any is a story that had thrown square define minor thirds (e.g. C-E flat-G flatF into the sea for making such an alarming truth sharp- A); while a connected twelve-pointed star public. In mediaeval times, the tritone was still displays a circle of perfect fifths - a tuning method known as diabolus in musica. However, knowledge dating back before recorded history, but still in use of logarithmic mathematics would not have been 8 by modern piano-tuners today. Yet in the context necessary to construct Castlerigg. John Hill has of the Castlerigg Stone Circle, much more still demonstrated, through actual practice, how stone circles and could have been designed by 4 Kappraff, J., (2002) Beyond Measure: a Guided Tour through Nature, Myth and Number, World cientific. See 7 By a curious coincidence, a large spiral has been also Ch. 11.7, ‘The Stone Circles’. discovered incised onto one of the stones at Castlerigg. See 5 Gregg, D.P., (2013) Castle Rigg Circle Analysis, figure Beckensall, S., (2002) Prehistoric Rock Art in Cumbria, Tempus A9E in private communication. Publishing, pp. 70-8. otice that there is no stone for E flat. Some 8 Hill, J., Video: Ancient Knowledge – The sacred commentators believe there were originally forty stones in all. geometry behind British Stone Circles, (Web reference). Leon CRICKMORE 11 means of rope-folding; and McClain9 has provided eighteenth and nineteenth century scribal schools a number of analogous paper-folding exercises, by of Larsa, Ur and Nippur12. which a monochord can be tuned in various ways - The tuning system of ancient Greece, even including equal temperament. attributed to Pythagoras and found explicitly in We must now return to the ancient system the writings of Plato is based on numbers in the of tuning by means of perfect fifths - that is, in form 2p3q. It produces what modern musicologists accordance with the twelve ‘fellies’ of ‘the single- call ‘Pythagorean tuning’. The Babylonian tuning wheeled chariot of the sun’, drawn as a continuous system, on the other hand, additionally uses the line so as to form a twelve – pointed star. The prime number five, and is thus based on numbers tuning ratio for a perfect fifth is 2:3. The Egyptians, in the form 2p3q5r13. It thereby produces ‘Just’ whose arithmetic was usually carried out by means rather than ‘Pythagorean’ tuning. Every tone- of reciprocals, known as Egyptian fractions10, had number has to be an integer hence the difficulty a special hieroglyph for two-thirds. In Gilgamesh, over the tritone, mentioned earlier. In order to dating from the second millennium BC, Urshanabi define a tritonal pitch an approximation has to be (Old Babylonian: Sursunabi), the boatman who employed. Tone numbers can be freely multiplied sailed across the Waters of Death, is described as or divided by two (a procedure to which modern the ‘Servant of Two-Thirds’ – a reference to his musicologists refer as ‘octave equivalence’). Such a service to Ea, patron of music, whose symbolic multiplication or division leaves the ‘pitch-class’ of number was 40, which is two-thirds of that of any tone unchanged. the head of the pantheon, Anu, whose number There is evidence from the seventh century was 60. Mesopotamian mathematicians, of course, BC14 for the existence of a simple rule of thumb to counted sexagesimally. enable tuning by a series of perfect fifths, namely: Hilprecht has published four tables of divisors ‘add or subtract one-third’ from successive string of 604, dating from about 2200BC Commenting or pipe lengths. Since in the Babylonian and ancient on an anomaly in the first line of each of them, Greek tuning systems, the addition or subtraction he writes: of ratios called for multiplication or division, ‘They all read: ‘’1, 8,640,000 A-AN’’. The this rule can also be expressed as ‘a tone-number quotient being 2/3 of 12,960,000, we should multiplied by 4/3 and 2/3, successively’. To rather expect 1½ instead of 1 as its divisor, for calculate the tone numbers for a series of thirteen 12,960,000 divided by 3/2 is = 12,960,000 x 2/3.’ perfect fifths, which McClain suggests might be the Hilprecht11 suggests that we should probably origin of various dragon and serpent myths, one 12 regard this anomaly as ‘an abbreviated expression must start with a reference value of 3 = 531,441 19 well understood by the Babylonians’ – especially, and apply the rule until one reaches 524,288 = 2 one might add, by musicians. For each of the – or, to be more strictly accurate – 262,144, which numbers in these four tables is appropriate for use then has to be doubled in octave equivalence. as a tone-number, defining a musical pitch in terms 531,441: 524,288 constitutes the Pythagorean of hypothetical string or pipe length. This is also Comma, the difference in pitch between seven true of the numbers in the related standard Tables octaves in the ratio 2: and 2 perfect fifths in the of Reciprocals, to be found so frequently in the ratio 3:2. It is therefore the amount by which the thirteenth tone to be generated will be out of tune. 9 McClain, E., (1978) The Pythagorean Plato, Appendix This calculation may perhaps be the origin of our IV, ‘Introduction to the Monochord’, Nicolas-Hays, York Beach, Maine, pp. 169-75. 10 Crickmore, L., ‘Egyptian Fractions and the Ancient 12 Robson, E., (2002), ‘Words and Pictures: New Light Science of Harmonics’, ICONEA 2009-2010, eds. Richard on Plimpton 322’, American Mathematical Monthly, 109: 105-20. Dumbrill & Irving Finkel, Co-Published by Gorgias Press 13 For further details see Crickmore, L., (2010) LCC, pp. 1-8. New Light on the Babylonian Tonal System, ICONEA 11 Hilprecht, H.V., (1906) The Babylonian Expedition of 2008, eds. Richard Dumbrill & Irving Finkel, ICONEA the University of Pennsylvania, Series A: Cuneiform Texts, Volume PUBLICATIONS, London, pp. 11-22. XX, Part 1, published by the Department of Archaeology, 14 Nakaseko, K., (1957) ‘Symbolism in ancient Chinese University of Pennsylvania, 25. music theory’, Journal of Music Theory 1 (2), pp. 147- 80. 12 CASTLERIGG: Stone or Tone Circle? modern superstition that thirteen is an unlucky and Akkadian in UET VII 126, proposed an number. alternative musical interpretation of the text as In ancient Greece, musical scales comprised a visual aid for the tuning of seven heptachords seven octave species or eight-note modes. on a seven-stringed instrument. The names of In Babylon, the corresponding scales were the seven strings also appear in CBS 10996. The heptachords, the octave-note being analogous to present author has subsequently published his the first day of a new seven-day week. In Egyptian own musical and mathematical interpretation of mythology, Thoth was known as ‘the Eighth which the seven-pointed star (reference in footnote 15). completes the Seven’. The best available evidence While the date and provenance of CBS 1766 still for the Just tuning of the seven tones of each of remain uncertain, Joran Friberg, who possesses a the seven Babylonian heptachords is probably deep understanding of the entire Old Babylonian the remarkable cuneiform text CBS 1766, which mathematical corpus, has pointed out that its is headed by a seven-pointed star within two unusual format is similar to that of Plimpton 322. concentric circles15. Figure 4 shows the cuneiform Robson19 reports that the on Plimpton tablet CBS 1766. 322 ‘is laid out in a landscape-orientated table with the final heading MU.BI.IM (‘its name’) for the numerical data’. Importantly, she adds that these were ‘formal features of administrative tables from Larsa during the period of rigorous standardisation in 1790-80 B.C.E.’. In the light of all this, Friberg concludes (reference in footnote 15) that ‘CBS 1766, like Plimpton 322, in all probability is an Old Babylonian text from Larsa, dating from the period 1790-1780 B.C.’. He also acknowledges that it is ‘a text with a mixed topic’ – that is, both a description of how to draw a heptagon by means of a continuous Figure 4. Cuneiform tablet CBS 1766. line, analogous to the continuous line forming the CBS 1766 originally photographed by twelve-pointed star mentioned by Kappraff and 16 Hilprecht and published in 1906 with the corresponding to a cycle of perfect fifths, and also description ‘an astronomical tablet from the a description of how to tune heptachords, though Temple Library’, was saved from oblivion and the tuning he suggests is Pythagorean rather than republished with a mathematical interpretation by Just. My musical and mathematical interpretation 17 Horowitz in 2006. A year later, Waezeggers and of CBS is shown in figure 5. 18 Siebes , noticing that around the seven-pointed The seven-pointed star, originally within two star on this tablet, there stand the names of seven concentric circles in CBS 1766 is shown at the of the nine musical strings listed in both Sumerian centre in red, though the drawing on the tablet is rougher. On the tablet, the points of the star are 15 Crickmore, L., (2008) ‘A Musical and Mathematical numbered from 1-7 and are labelled as follows: Context for CBS 1766’, Music Theory Spectrum,Volume 30, no. 2 : 327- 338, and Friberg, J., (2011) ‘Seven-Sided Star Figures 1 the foremost; 2 the next; 3 the third thin; 4 and Tuning Algorithms in Mesopotamian, Greek and Islamic the one made by (the god) Ea; 5 the fifth; the Texts’, Archiv fur Orientforschung 52, pp. 121-55. 16 Hilprecht, V., (1903)Explorations in Bible Lands During fourth behind (that is, from the other end of the the 19th Century, Philadelphia, Molman, A.J. instrument); and 7 the third behind. With two 17 Horowitz, W., (2006) ‘A Late Babylonian Tablet with Concentric Circles from the University Museum (CBS additions, these names also appear in the list of 1766)’ Journal of the Ancient Near East Society, 30, pp. 37-53. 18 Waerzeggers, C., and Siebes, R., (2007), ‘An 19 Robson, E., (2003) ‘Tables and tabular formatting Alternative Interpretation of the Seven-Pointed Star in CBS in Sumer, Babylonia, and Assyria, 2500 B.C.E. – 50 C.E.’ in 1766’, Nouvelles Assyriologiques Breves et Utilitaires (N.A.B.U.) 2, Campbell Kelly, M. et al (eds. The History of Mathematical Tables, pp. 43-5. Oxford University Press, pp. 18-47. Leon CRICKMORE 13 nine strings in UET VII 126 /Nabnitu XXXII the second column produces a regular heptagon. a bilingual lexical text – the earliest evidence CBS 10996 also contains two columns of the we have about string/pitch names - in which numbers 1-7, as follows: (1) 1-5; 2-6; 3-7; 4-1; 5-2; the fourth string is described as ‘four-small’ in 6-3;7-4 and (2) 7-5; 1-6; 2-7; 1-3; 2-4; 3-5; 4-6. All the pairs in the first column from CBS 1766 appear in CBS 10996, though in a different order. In CBS 10996, the fourteen pairs of integers are preceded by the logogram SA, meaning ‘string’. In her pioneering work on Babylonian music, Kilmer20 interprets the even numbered lines between 11 and 24 - my column (1) – as a means of tuning by perfect fourth and perfect fifths each of the seven Babylonian heptachords. Smith21 and Kilmer have interpreted the numbers in the second column as a method of further fine tuning, so as to sweeten the intervals of a third and bring the original Pythagorean tuning closer to Just tuning. This process is indicated by the pairs of green lines in figure 5. When transcribing the seven heptachordal scales into modern letter notation, Kilmer presents them chromatically, within a single octave, in a manner described by musicologists as ‘thetically’. There is also another cuneiform tuning – or, more strictly speaking – modulating text: UET VII 74. When it was originally published by Gurney22 in , everyone assumed that the scales defined by the text were rising. However, later, when the musicologist Vitale23 argued that the scales should Figure 5. The cuneiform tablet CBS 1766. be falling, the Assyriologist, Krispijn24, proposed Sumerian, but as ‘Ea-created’ in Akkadian. CBS an amendment to line twelve, where a damaged 1766 also displays two columns of the numbers sign, previously read as NU SU (‘end of sequence’) 1-7 arranged in pairs: (1) 2-6; 6-3; 3-7; 7-4; 4-1; ought to have been read as nusuĥ(u-um), meaning 1-5; 5-2, and (2) 1-7; 5-4; 2-1; 6-5; 3-2; 7-6; 4-3. ‘to tighten’. The text now fell into two sections, the Its format then includes several empty columns, first involving the ‘tightening’ of certain strings, suggesting that the tablet might have been intended while the second involved their ‘loosening’. The 25 as an exercise to be completed by a student. There tablet therefore was describing a method for is also some further writing on the tablet which has modulating through all the seven heptachords, so far proved indecipherable. However, Friberg’s 20 Kilmer, A.D., (2001) ‘Mesopotamia’ The New Grove nd diagram (Figure 6.1 in op.cit. footnote 15) shows Dictionary of Music and Musicians, 2 edition, 16, pp. 480-7. 21 Smith, J.C., and Kilmer, A.D., (2000) ‘Continuity and the possible beginning of the heptachordal names Change in the Ancient Mesopotamian Terminology for Music išartum and kitmum, which would support my and Musical Instruments’, Orient Archaeologie 7, pp. 113-9. 22 Gurney, O.R., (1968) ‘An Old Babylonian Treatise reconstruction (op.cit. footnote 15). Friberg points on the Tuning of the Harp’, Iraq 30, pp. 229-33. out that a seven-pointed star figure can be drawn 23 Vitale, R., (1982) ‘La musique suméro-akkadienne, Gamme et notation musicale’, Ugarit-Forschungen 14, pp. by the use of an uninterrupted chain of straight 241-65. lines through the pairs of points listed in the first 24 Krispijn, Th.J.H., (1990) ‘Beitrage zur altorien- alischen Musikforschung I’, Akkadica 70, pp. 1-27. column; while joining each of the paired points in 25 For details of the method see Crickmore, L., op.cit. 14 CASTLERIGG: Stone or Tone Circle? by means of sharpening or flattening one of the Pythagorean tuning all tones are 9:8, in Just tuning components of the tritone in the heptachord to there are two forms of tone: 9:8 and 10:9, and the which a nine-stringed instrument was presently semitone is 15:16. tuned. Gurney published a revised transliteration in 199426. But all this created a problem for Kilmer, A further question now arises: how can the since if the scales were falling, it would appear that heptagonal geometry of CBS 1766 be related to she had been mistaken over her identification of stone circles and henges, and in particular to the the names of the heptachords. The present author27 geometry of Castlerigg and the ‘twelve-spoked has suggested a resolution of this problem: that wheel of the chariot of the sun’? Two books by the Babylonian heptachords – unlike the ancient John Michell can help: (1) The Dimensions of Greek octave species or our modern diatonic scales Paradise.29 with the subtitle ‘the proportions and – must have been modal patterns of tones and symbolic numbers of ancient cosmology’ and (2) semitones which remain unchanged, regardless of How the World is Made: the Story of Creation the direction of the scale. Greek modes and modern according to Sacred Geometry. The opening diatonic scales, on the other hand, are ladders sentence of the former reads: of pitches, the modal patterns of which differ ‘Ancient Science was based like that of according to whether the scales are rising or falling. today on number, but whereas number is In figure 5, the names of the falling heptachords now used in the quantitative sense for secular purposes, the ancients regarded numbers as are shown in black, and their scales are to be read symbols of the universe, finding parallels to the right; the rising heptachords are named in between the inherent structure of number and red, and the pitches of their respective scales have all types of form and motion’. to be read to the left – thus reversing the directions hen reflecting on the differences between shown in figure.328. Modern letter-name notation modern and ancient mathematics, one must has been used to define the relative pitches of the additionally bear in mind that while the fulcrum of notes of each heptachord. This procedure was modern mathematics is zero, with positive numbers chosen for the sake of simplicity, since this makes extending in one direction and negative numbers in it possible to define all seven heptachords using the the other, ancient mathematics knew no zero. Its white keys of a piano only. Musicologists call such fulcrum was the monad. From unity, numbers were a form of presentation ‘dynamic’, in contrast to conceived as an arithmetic progression upwards: 2, Kilmer’s ‘thetic’ or chromatic notation. 3, 4, 5, 6.... and a harmonic progression downwards: In adding the names of the seven Chaldean 1/2, 1/3, 1/4, 1/5, 1/6 ... Michell’s book is full planets to the diagram, they have been ordered of examples of how twelve combined with seven according to their diminishing orbital cycles to has been used to symbolize unions of differences match (though not proportionately) the shortening such as heaven and earth, solar and lunar years, and of the musical string in the falling scales. Finally, mind and body. the tone-numbers and the ratios of string-length Three, the first odd number unity was not defining each of the pitches in Just tuning have considered as a number in itself, but rather as been included. It should be noted that whereas in the monad from which all other numbers are generated plus four the first square number in footnote 3, pp. -5, figures 5 and , andop.cit. in footnote add up to seven, while three multiplied by four is 15, pp. 336-7, Appendix. twelve. The latter book, Michell’s last and probably 26 Gurney, O.R., (1994) ‘Babylonian Music Again’, Iraq 56, pp. 101-6. most beautiful, contains a section headed: ‘Twelve 27 Crickmore, L., (2010) op.cit. footnote 10. and Seven: The Supreme Numerical Marriage’ (pp. 28 In the modern world, musical pitch is defined by frequency and usually expressed in Herz (Hz.), signifying 219-231). vibrations per second. Scales are conceived as rising – a Musicians are familiar with this mystery convention adopted by both McClain and Kappraff. But since the ancients defined their scales in terms of ratios and reciprocals of string-length, it is sometimes preferable to 29 Michell, J., (1988) The Dimensions of Paradise Thames reverse this procedure. & Hudson. Leon CRICKMORE 15 through its expression in the seven white and five published a general book about them31. Figure 7 black keys of a modern keyboard. shows drawing number 291. Hancox interprets Amongst Michell’s illustrations is figure , it as the geometry underlying an engraving by in which a twelve-pointed figure is created by arcs, Theodore de Bry: The Theatre of Human Life while the eighty-four divisions of a 7-pointed star (Theatrum Vitae Humanae, 1596). Frances Yates are drawn by means of straight lines. At the end of has reproduced this engraving in her book Theatre this section, the caption to figure 2 reads: ‘Seven of the World, where she likens it to the speech and Twelve united and the Heavenly City (i.e. the of Jaques in Shakespeare’s As You Like It: ‘All New Jerusalem) revealed’. The earliest features the world’s a stage....’. The geometrical drawing of the Circle are the Aubrey Holes consists of two interlinked circles, the upper of (c.3000 BC). Figure 6 indicates how this set of which contains a seven-pointed star. In the lower fifty-six holes has been interpreted by .P. Gregg circle, there is a set of diameters measuring: 30, 32 as the basis for both a heptagon and its seven- 36, 40, (41, 42,) 45, 48, (51, 54 = 27 x 2, 56, and 60 pointed star. = 30 x 2 units32. The numbers in heavy type appear as tone-numbers in my interpretation of CBS 1766 (Fig. 4), where, as ratios of string-length, they define the Babylonian heptachordišartum (falling).

Figure 6. Gregg’s Aubrey Circle heptagon to match CBS 1766. The heptagon shown in figure is only one of a number of geometrical interpretations of the Aubrey Holes at Stonehenge by D P Gregg30. The last of my examples of a geometrical Figure 7. Byrom Collection 291: Reproduced with permission for purposes of scholarship marriage of seven and twelve comes from ©Joy Hancox, The Byrom Collection, Cape, the Byrom Collection. The Byrom Collection 1992. comprises 516 geometrical drawings of which The diameter numbers 42, 51 and 56 are John Byrom (1691-1763), a Jacobite iophantine approximations for 2, (the musical sympathiser, was the custodian. These drawings tritone) used geometrically33 to estimate the were discovered by Joy Hancox, while she was 31 Hancox, J., (1992), The Byrom Collection, Jonathan writing a biography of Byrom. She has since also Cape. 32 Further details can be found in Crickmore, L., (2009). Appendix: The Byrom Collection’, Harmonic Mythology: Nine Interdisciplinary Research Notes, Arane, Volume 1, 30 Gregg, D.P., (2014) The Stonehenge Codes: A New ICONEA Publications. Light on Ancient Science , Green Man Books, pp. 22. 33 , The Mathematics Useful for 16 CASTLERIGG: Stone or Tone Circle? approximate length of the diameters of squares of fashioned the World Soul’. Plato’s World Soul sides 30, 36 and 40 respectively. was, in essence, the arithmetic ratios of string- Finally, readers who are in sympathy with the length that define the ancient Greek orian octave mode of thinking underlying Ernest McClain’s tone species in Pythagorean tuning. A thousand years mandalas and tuning systems might care to try the earlier, the first seven tones of this mode, in Just following experiment. Take any pair of figures 2-. tuning, already defined the Babylonian heptachord Superimpose them in such a way that the centres nid qablim. Ernest McClain’s life’s work and of their circles coincide. Place a light behind them, contribution to the debate over the Harmony of and then rotate some point of symbolic significance the Spheres has been summed up as follows37: ‘He in the lower diagram into the 12 o’clock/pitch D/ hypothesized prime number harmonics as the key North position in the upper (or, in the case of driver and shaper of historical mythology’. Castlerigg to the first stone to the right of orth), in the manner that seems most appropriate. Using Conclusions these diagrams playfully in this way - as a kind of 1. As Alexander Thom has suggested, the Jungian mandala for contemplation - this exercise Castlerigg Stone Circle was in all probability an can sometimes highlight fresh correspondences ancient astronomical observatory. and connections for their viewer. Some of these 2. As Plato recognised (Republic, 530 c-e), new insights may even be contradictory, for as astronomy and music have long been viewed as Joscelyn Godwin34 admits, when writing about the sister disciplines. Harmony of the Spheres: 3. Since the seventeenth century, however, modern science has gradually ‘untuned the ‘the way of understanding lies through finding the appropriate standpoint from heavens’ and demythologized our understanding which to consider every theory. Without that of the universe. Knowledge has thereby tended readiness to shift attitudes and to keep one’s to be reduced to evidence within the limits of mind open to several different and sometimes contradictory levels of being, any study of statistical mathematics. planet-tone or zodiac-tone theories is limited 4. Nevertheless, as the title of Kappraff ’s to an academic approach which regards its book Beyond Measure suggests, some of the subject as historically instructive but devoid of most important dimensions of human life – intrinsic truth’. including love, beauty and subjectively significant For readers of a more reductionist frame of coincidences – are often ambiguous and rarely mind, however, insights of the kind generated susceptible to the level of precision required by by the exercise described are likely to be deemed scientific method. wishful thinking, at best, and at worst, sheer 5. It seems unlikely that Ernest McClain’s ideas madness. will ever be considered mainstream. But for those In another book35, Godwin has traced the of us who believe that at the core of his work development of the concept of the Harmony of there is some intrinsic truth, and who are minded the Spheres through texts from the time of Plato to continue further to explore his hypothesized up to the present day. He summarizes this process prime-number mythologies, a new name for such a developing body of interdisciplinary research will as ‘a many-faceted commentary on the passage in be required. Graham Fearnhead38 has suggested: Plato’s Timaeus36 that describes how the Demiurge ‘Harmonic Mythology’. Understanding Plato, (translated from Greek/French edition of J. Dupuis by R & D Lawlor), San Diego (Cal.), Wizard’s Bookshelf, (1979). 34 Godwin, J., (1987), Harmonies of Heaven and Earth, Thames & Hudson: 124-5. 37 Ernest Glenn McClain, Obituary, New York 35 Godwin, J., (1993), The harmony of the Spheres: a Times, May 6, 2014. See Crickmore, L., (2009 Research Note Sourcebook of the Pythagorean Tradition in Music, Inner Traditions IX,)’Harmonic Mythology: Nine Interdisciplinary Research International, Rochester, Vermont. Notes’, ARANE, Volume 1, pp. 63-4. 36 Timaeus 34-7. See also Crickmore, L., (2009), 38 See Crickmore, L., (2009 Research Note IX,) ‘A possible Mesopotamian origin for Plato’s World Soul’, ’Harmonic Mythology: Nine Interdisciplinary Research Hermathena 186, pp. 5-23. Notes’, ARANE, Volume 1, pp. 63-4. Leon CRICKMORE 17

Bibliography

Beckensall, S., (2002) Prehistoric Rock Art in Cumbria, Tempus Publishing Ltd. Godwin, J., (1987) Harmonies of Heaven and Earth, Thames and Hudson, London; (1993) The Harmony of the Spheres: a Sourcebook of the Pythagorean Tradition in Music, Inner Traditions International, Rochester, Vermont. Gregg, D.P., (2014) The Stonehenge Codes: A New Light on Ancient Science, Green Man Books; (2013) Castle Rigg Circle Analysis, private paper. Hancox, J., (1992) The Byrom Collection, Jonathan Cape, London. Hilprecht, H.V., (1906) The Babylonian Expedition of the University of Pennsylvania, Series A: Cuneiform Texts, Volume XX, Part 1, published by the Department of Archaeology, University of Pennsylvania. Kappraff, J., (2002) Beyond Measure: a Guided Tour through Nature, Myth and Number, World cientific Publishing Co. Lawlor, R. and D., (1979) Theon of Smyrna, The Mathematics Useful for Understanding Plato, (translated from Greek/French edition of J. Dupuis), Wizard’s Bookshelf, San Diego (Cal). McClain, E.G., (1976) The Myth of Invariance: the Origin of the Gods, Mathematics and Music from the Rg Veda to Plato, Nicolas-Hays, York Beach, Maine, (1978); The Pythagorean Plato: Prelude to the Song Itself, Nicolas-Hays, York Beach, Maine, (1981); Meditations through the Quran: Tonal Images in an Oral Leon Crickmore was born in 1932 and Culture, Nicolas-Hays, York Beach, Maine. educated at King’s College, Cambridge, and the Michell, J., (1988) The Dimensions of Paradise, University of Birmingham. After working in Thames and Hudson, London; (2009); How the further education, he became Dean of the Faculty World is Made: the Story of Creation according to Sacred of Arts at the North East London Polytechnic, and Geometry, Thames and Hudson, London. later HM Staff Inspector of Music. In the latter Thom, A., (1967) Megalithic Sites in Britain, Oxford role, he was involved in the establishment of music University Press. as a subject in the National Curriculum, and of degree courses in the Conservatoires. In 1997 he was awarded an honorary fellowship of the Royal Scottish Academy of Music and Drama. Since 2000, he has been engaged in interdisciplinary research into the ancient science of harmonics, with particular reference to Babylonian and Greek tonal systema.