certain of these insights which relate music, number and mythology, to the Castlerigg Stone Circle, 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 N 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 Alexander Thom’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, Cumbria, England. 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-f etc6. 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 irrational number. 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 Pythagoras had Hippasus 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 henges 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.