LEON CRICKMORE

ICONEA 2012-2015, XXX-XXX The picture in figure 1 shows the old ‘School of Music’ here in Oxford. Right up to the nineteenth century, Boethius’s treatise De Institutione Musica was still a set book for anyone studying music in this University. The definition of a ‘musician’ contained in it reads as follows1: Is vero est musicus qui ratione perpensa canendi scientiam non servitio operis sed imperio speculationis adsumpsit THE UBIQUITY (But a musician is one who has gained knowledge of making music by weighing with reason, not through the servitude of work, OF THE DIATONIC SCALE but through the sovereignty of speculation2.)

Such a hierarchical distinction between music Leon Crickmore theory and music practice was not finally overturned until the seventeenth century onward, by the sudden Abstract: increase in public musical performances on the There are an infinite number of possible one hand, and the development of experimental pitches within any . Why is it then, that in science on the other. But the origins of the so many different cultures and ages musicians distinction probably lie far in the past. Early in the have chosen to divide the octave into some kind of diatonic scale? This paper seeks to eleventh century, Guido D’Arezzo had expressed a demonstrate the ubiquity of the diatonic scale, similar sentiment, though rather more stridently, in and also to reflect on some possible reasons why a verse which, if that age had shared our modern this should be so. fashion for political correctness, would have been likely to have been censured: Musicorum et cantorum magna est distancia: Isti dicunt, illi sciunt quae componit musica. Nam qui facit quod non sapit definitur bestia (There is a great distinction between musicians and performers. The latter sing or play, while the former understand what constitutes music. And someone who does something without understanding its nature is by definition abeast 3!)

I like to imagine that this tradition which distinguishes philosophically between the theoretical and the practical arts is reflected in the well-known cartoon-like shell-inlay on one of the lyres from Ur illustrated below:

Fig. 1. Schola Musicae, Bodeleian library, University of Oxford Fig. 2. Ornamental shell plaque from the front of a lyre THE UBIQUITY OF THE DIATONIC SCALE In this paper, I hope to demonstrate the scales with which we are to be concerned, since ubiquity of the diatonic scale. To do this, I shall be there are a number of possibilities. For McClain’s borrowing an insightful idea from Ernest McClain, interpretation of the relevant tone-numbers, he by using the tone-numbers and as what 288 120 Tone-number Pitch Pitch Recirpocals he calls ‘alternative “indices” for the double-octave 1 30 C E 288 form of our basic diatonic scale4.’ McClain points s=16:15 s=15:16 out that for King David’s Tabernacle (c.1000 BC), 2 32 B F 270 there were 288 Levites with skill in divine chant5. While in King Solomon’s Temple (c.950 BC), t=9:8 t=8:9 there were 120 trumpeter-priests6. Treated as tone- 3 36 A G 240 numbers, these limits can accommodate a double- t=10:9 t=9:10 octave diatonic scale in reciprocal arithmetical 4 40 G A 216 forms. It seems possible that during their exile t=9:8 t=8:9 in Babylon, some educated Jews served as civil 5 45 F B 192 servants and absorbed sexagesimal mathematics s=16:15 s=15:16 and its symbolism into their consciousness. 6 48 E C 180 Returning to Jerusalem, they may then have t=9:8 t=8:9 incorporated some of these symbolic numbers 7 54 D D 160 into their own culture, by including them in the t=10:9 t=9:10 so-called P-Document, (c.450 BC), which is one of 8 60 C E 144 7 the four hypothetical sources of the Pentateuch . s=16:15 s=15:16 However, my presentation of the double-octave 9 64 B F 135 diatonic scale will differ from McClain’s in two s=9:8 s=8:9 respects. McClain interprets the basic diatonic scale 10 72 A G 120 as , following modern practice, where pitch rising t=10:9 t=9:10 is measured in frequencies. But the ancients were 11 80 G A 108 only able to measure string and pipe length with t=9:8 t=8:9 the necessary precision. Their scales were initially conceived as falling, and I have therefore chosen 12 90 F B 96 to follow their convention. Secondly, McClain s=16:15 s=15:16 has used modern chromatic notation, a form 13 96 E C 90 of presentation which the ancients might have t=9:8 t=8:9 called ‘thetic’. Whereas I have preferred ‘dynamic’ 14 108 D D 80 notation – that is using the white keys of a modern t=10:9 t=9:10 keyboard only – since I believe this enables the 15 120 C E 72 8 case to be presented more clearly . Table 1 shows Table 1. Author’s version of McClain’s table. my version of McClain’s own table in which the selects D, the central pitch in our modern tonal pitches printed in black (falling scales) are system, as his tonic. His rising diatonic scale in expressed in ratios and reciprocal ratios of string- increasing ratios of frequency is thus our modern length. The pitches printed in red (rising scales) D major; while its reciprocal falling form defines represent tone-numbers interpreted as reciprocal the ancient Greek Dorian octave species in a ratios and ratios of frequency. The scales can be chromatic presentation. In my table, the falling viewed in either direction: 30-120; 120-30; and diatonic scale, in increasing ratios of string- 288-72; 72-288, by using the appropriate form of length, is our modern C major scale. (Amusingly, ratio. and in parenthesis, Gerard Manley Hopkins once But before citing some examples that demon- commented in a letter9: ‘All keys are the same to strate the ubiquity of the diatonic scale, I need me and to everyone who thinks that music was to clarify further the particular diatonic scale or before instruments and angels before tortoises LEON CRICKMORE and cats’!). My corresponding reciprocal scale is Example Two: CBS 1766 the rising form of the ancient Greek Dorian octave Both the date and the provenance of the species. Different cultures and different ages pre- cuneiform tablet CBS 1766 remain uncertain. It fer different norms. Concerning this topic Joscelyn was originally photographed by Hilprecht12, and Godwin10 has wisely commented: later published with a suggested mathematical/ 13 As the Dorian or First Mode of the Middle Ages, astronomical interpretation by Horowitz . At the and the diatonic of Ancient Greece, this head of the tablet there stands a seven-pointed D-mode (i.e. D E F G A B C D or D C B A G F E D) with star with added text, some of which is still its perfect intervallic symmetry has a good claim to primacy. 14 The modern major, however, is the inversion of the Greek indecipherable. In 2007, Waerzeggers and Siebes , diatonic Dorian (E D C B A G F E), the mode preferred who had noticed that seven of the nine string- by the Demiurge of Plato’s Timaeus for the construction names listed in UET VII 126 were inscribed by of the world; so, allowing for a reversal up and down, it is also a strong contender. Such matters could be argued ad the points of the star, suggested that the diagram nauseam: every speculative musician has a favourite scale. could be interpreted as a visual aid for the tuning of a seven-stringed instrument. Finally, Friberg15 has Example One: UET VII, 126 = nabnitu xxxii interpreted the star both as a means of drawing a UET VII, 126 is a late Babylonian copy of an heptagram by the use of an uninterrupted chain of older lexical list, containing the earliest information straight lines through points which are numbered we have about Mesopotamian string/pitch names. in the text below it, and also as a method for It lists nine strings in both Sumerian and Akkadian: tuning the seven Babylonian heptachords on ‘foremost; next; third/thin; fourth small/ Ea – a seven-stringed instrument. Figure 3 shows a created (in Akkadian); fifth; fourth behind; third speculative interpretation of CBS 1766 by the behind; second behind; first behind – nine strings’. present author16, in which the following additions The strings are numbered palindromically: 1 2 3 4 to the information in the cuneiform text have 5 4 3 2 1. Dumbrill11 has suggested an ingenious been made: (1) the names of the seven falling interpretation of this information, as a means Babylonian heptachords in black, to be read to the of tuning a nine-stringed instrument. Assuming right; (2) the rising heptachords in red, to be read epicentric tuning by perfect fifths and fourths from to the left; (3) modern letter-name notation of the a central string (5-1; 1-4; 5-2; 4-3; 3-3), and selecting hypothetical relative pitches, using a ‘dynamic’ form D for the tuning of string 5, produces the scale G of presentation; (4) the simplest tone-numbers A B C D E F G A (rising) and its reciprocal A G F defining these pitches in numbers of the form E D C B A G (falling). The first seven tones of this 2p3q5r (i.e. in Just tuning), together with the ratios series of pitches, in either direction, define the and reciprocal ratios between them; (5) and finally Babylonian heptachord pitum, which incidentally – rather more speculatively – the corresponding matches the ancient Chinese scale Shang, and is names of the seven Chaldean planets, listed in also close to the Hindu diatonic scale and possibly order of their decreasing orbital cycles to match some mbira tunings from Africa. I postulate that (though not proportionately) the shortening of the the Babylonian heptachords were modal patterns musical string to sound the rising scales. Notice of tones and which remained the same that the sun and moon correspond to the tritone regardless of the direction of the scale. With the F-B. With these tones omitted (just as in ancient addition of an eighth note, the new octave scales Chinese astronomy the sun and moon were not became ladders of pitches, the modal patterns counted as planets) the scale sounded would be of which differ according to whether the scale is pentatonic: C, D, E, G, A. All the tone-numbers rising or falling. In table 1, the scale resulting from in figure 3 can be found in the standard tables of Dumbrill’s tuning system for UET VII,126 can be thirty reciprocals which have been unearthed in th found between lines 11-3 (pitches in black) and considerable numbers in the scribal schools of 18 th 3-11 (pitches in red). and 19 century Larsa, Ur and Nippur (e.g. MLC 1670). They also occur as ‘divisors’ in Hilprecht’s Four tables of Divisors of 604 (c.2200 BC). THE UBIQUITY OF THE DIATONIC SCALE

Nevertheless, while ample documentary evidence Example Four: Queen PU-ABI’s Harp (c. 2600 exists for the quantification of musical scales in BC) (See figure 5 overleaf.) ancient Greece, no such cuneiform evidence has The reconstruction of Queen Pu-abi’s Harp so far been discovered to confirm this practice in pictured above indicates that it had thirteen strings. Babylon. All the tone-numbers added to CBS 1766 If the tuning used for the Silver Lyre is extend- in figure 3 can also be found between lines 1-7 of ed again by a in each direction, this the 30-120 column in table 1, providing 54 (line 7) harp could have been tuned diatonically to: E F. The reconstruction of Queen Pu-abi’s Harp pictured overleaf indicates that it had thirteen strings. If the tuning used for the Silver Lyre is ex- tended again by a semitone in each direction, this harp could have been tuned diatonically to: E F G A B C D E F G A B C. Such a tuning would then permit a performer to play each of the seven Babylonian heptachords and its reciprocal – that is in either direction – without any adjustment to the tuning being necessary18. Interestingly, in Chapter 11 of his Manual of Harmonics19, Nico- machus mentions ‘a thirteen-stringed system is found,the seventh string being fixed diatonically from either end’. However, apart from another obscure reference to such a scale (perhaps bor- rowed from Nicomachus) by the Byzantine the- orist, Pachymeres, there is no other evidence in all the existing literature for the existence of such a scale. Musicologists and Greek scholars have there- fore concluded that Nicomachus’s testimony on CBS 1766 as a Tone Circle and Planets this matter is not to be taken seriously. Neverthe- Notes and Key: less, taking D as a seventh and central string, and The tritonic procedures of UET VII, 74 can be applied to the falling scales extending from there by tones and semitones: first t = Tone as far as the heptachord kitmum (A-G); secondly, s = Semitone c’ = Middle C to Dumbrill’s tuning for the nine strings of UET Figures in red indicate reciprocal (inverse) scales. VII 126 G-A); thirdly, to his proposed tuning for Dichords in CBS 10996 the Silver Lyre of Ur (F-B); and finally to my hy- Initial Tuning (5ths and 4ths) Fine Tuning (5ths and 4ths) pothetical tuning of the thirteen-stringed Queen’s Harp, suggests that Nicomachus may have been almost right after all. is considered as 27 doubled in ‘octave equivalence’.

Example Three: The Silver Lyre of Ur. (See figure 4 overleaf.) I believe that when Richard Dumbrill built his playable reconstruction of the Silver Lyre of Ur, he found that his hypothesized diatonic tuning for the nine strings named in UET VII 126, extended by a tone at either end – that is, F G A B C D E FG A B – best suited the known measurements of the instrument. LEON CRICKMORE

Figure 5. Queen PU-ABI’s harp (ca. 2600 BC)

Figure 4. Dumbrill’s replication of the silver lyre of Ur, BM 121199 (ca. 2600 BC) THE UBIQUITY OF THE DIATONIC SCALE

Example Five: Plato’s Timaeus Scale

Plato’s dialogue Timaeus (34-37) describes In table 1, the ancient Greek Dorian octave through myth how the Demiurge fashioned the species (falling) appears between lines 6-13 in the world21. For Plato the world was a living organism, column 30-120. The tone numbers which define it and like all living creatures had a ‘soul’. The World- are also shown in the upper line of the table above: Soul is conceived as a mathematical series of terms The numbers in the lower line of table 3 show which together correspond to a musical scale, hav- the initial tone-numbers multiplied by 8 (23). ing a compass of four and a major sixth. Its To bring about a complete match with the first nine tone-numbers define the ancient Greek tone-numbers of the Timaeus scale, the three Dorian and Phrygian octave species. Plato de- figures underlined (480, 640 and 720) have scribes the former of these in Laches (188d) as the to be further multiplied by 81/80, to pro- ‘only harmony that is genuinely Greek’. The latter duce 486, 648 and 729, respectively. 81/80 is the only octave species which is palindromic and is known to modern acousticians as the ‘syn- remains unchanged regardless of its direction. Fur- tonic comma’. It is the difference between a thermore, these are the only two scales which Plato will admit into his ideal republic (Republic 398- E D C B A G F E 399c). In reciprocation, the falling 48 54 60 64 72 80 90 96 (ttsttts) becomes our modern major scale (stttstt), 384 432 480 512 576 640 720 768 while the Phrygian remains unchanged. Unlike all previous examples, the ratios between the pitches Pythagorean ditonic third (81:64) and a pure Just- in these Greek tunings exclude multiplications by tuned major third (5:4). Moreover, 80 and 81 the prime number five. Such tuning, based solely are often the final two numbers which appear in on numbers in the form 2p3q, is referred to by mu- the Babylonian Standard Tables of Reciprocals. sicologists as ‘Pythagorean’. The earlier Babyloni- Friberg22 quotes an example of a mathematical an system of Just tuning has been replaced in the problem from the Seleucid period (AO 6484:7a) Timaeus scale by a standardized Greek one. For which involves a reciprocal pair of sexagesimal the Babylonian priest-musicians there were two numbers (igi and igi bi) such that their product types of tone: 9:8 and 10:9, and three kinds of is equal to 1 (any power of 60). His solution to the semitone: 16:15, 25:24 and 27:25. In the Timaeus problem is that igi =81/80 and igi bi = 80/81. scale, all tones are 9:8, and all semitones 256:243. LEON CRICKMORE Henceforth, in the West at any rate, after the tuning, is that diatonic tunings tend to generate time of Plato, Pythagorean tuning seems to have clearly recognisable pitches which match the become the musical orthodoxy: the norm, for ex- rational number ratios of the acoustical Harmonic ample, for the diatonic tunings in the De Institu- Series: tione Musica of Boethius; for the eight mediae- Harmonic Series Written as Notes val Church modes; and for Robert Fludd’s Great Monochord23 Just tuning generally disappeared, or maybe just went underground for use by Her- metic coteries. It does occasionally re-appear in Establishment circles, as, for example in Ptolemy’s ‘intense diatonic tuning’ (9:8; 10:9; 16:15)24, or in Kepler’s Songs of the Planetary Motions25. But All the items in the Harmonic Series which p q r overall, it is as if with the loss of knowledge of cu- have a sexagesimal number in the form 2 3 5 neiform Just tuning was lost too. One suspects that generate clear pitches (in particular numbers 1-6). evidence relating to the quantification of musical The others, such as those numbered 7, 11, and 13 pitches using sexagesimal numbers, and even per- generate sounds which to a keen musical ear will haps concerning the thirteen-note scale mentioned sound slightly flat or sharp. For example, the pitch later by Nicomachus, was possibly destroyed with shown as F# could as easily have been transcribed the Library in Alexandria. There has always been a as F natural. To obtain an F, tuned by perfect power-politics of knowledge. Thus the influence of fifths, one would have to postulate a hypothetical the Timaeus persisted until the early sixth century, reciprocal or inverse harmonic series, whose and was communicated after that time through the undertones may or may not exist. Latin translation of the earlier part of the text by The hypothetical reciprocal or inverse harmonic Calcidius. In the preface to his book The Harmony series: of the Spheres (op. cit. note 24), Godwin admits that the entire book is in effect ‘a many-faceted commentary on the passage in Plato’s Timaeus that 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 describes how the Demiurge fashioned the World- Soul’. Conclusions: How then would I answer the question Discussion: concerning numeracy and metrology formulated in This paper has tried to demonstrate how the our Conference title as: ‘Arithmetical Subjectivism diatonic scale has been in continuous currency for or Unconscious Knowledge’? My preference at least five-thousand years. There have, of course, would be for a ‘both-and’ rather than an ‘either- been some notable temporary lapses: such as the or’ approach. We must always aim to bridge C. dominance of the chromatic and enharmonic P. Snow’s ‘two cultures’ by means of some kind modal forms in the music of the fifth century of interdisciplinary analysis. Modern thinking is Athenian theatre, and, in atonal compositions understandably strongly influenced by modern of modern times. Even more recently, however, Science. But it is sometimes too easy to forget diatonicism has been considerably re- invigorated that Science can only deal with items which can through its use in jazz and popular music. So the be accurately measured. Science is therefore basic question which arises is why, when there intrinsically limited by a tendency to become are an infinite number of possible pitches within reductionist, or over evangelically eager to de- any octave, should musicians of various cultures mythologize. Besides, with regard to the concept and ages have so frequently chosen the diatonic of the ‘unconscious’, I tend to follow Jung rather division? One answer which springs quickly to than Freud. mind, and is also particularly appropriate as an A Just-tuned diatonic scale can be interpreted explanation for the ancient popularity of Just as a response of the human brain to the acoustical THE UBIQUITY OF THE DIATONIC SCALE phenomenon of the harmonic series. Four tables Tone-number Pitch Pitch Recirpocals 4 26 of divisors of 60 from Nippur (c.2200 BC) and 1 30 C E 288 numerous standard tables of reciprocals from s=16:15 s=15:16 nineteenth and eighteenth century Ur, Larsa and 2 32 B F 270 27 Nippur contain all the numbers needed to enable t=9:8 t=8:9 the Chief Musicians in the temples of these ancient 3 36 A G 240 cities to quantify their diatonic scales. Despite the t=10:9 t=9:10 absence, so far, of documentary evidence to this 4 40 G A 216 effect, some of us are prepared to believe that t=9:8 t=8:9 these sexagesimal numbers must have been used 5 45 F B 192 in ancient music theory. By tradition from the time s=16:15 s=15:16 of Pythagoras, and confirmed by documentation from the time of Plato, the story of how the 6 48 E C 180 Greeks reformed the Babylonian harmonic system t=9:8 t=8:9 and established their own version of it, and how 7 54 D D 160 this new orthodoxy was maintained through some t=10:9 t=9:10 kind of power politics of knowledge for about the 8 60 C E 144 next two millennia has been described earlier. s=16:15 s=15:16 9 64 B F 135 Let us now return to Table 1 to the right: s=9:8 s=8:9 We have here in the red column, two octaves 10 72 A G 120 of the Greek Dorian octave species defined in Just t=10:9 t=9:10 tuning, rising in reciprocal ratios of frequency, 11 80 G A 108 through the tone-numbers 288-72. Reading the t=9:8 t=8:9 scale from the foot of the column upwards, and 12 90 F B 96 inverting the ratios (e.g. 10:9, 9:8 etc.), the tone- s=16:15 s=15:16 numbers 72-288 also define the Dorian octave 13 96 E C 90 species, falling in ratios of string-length. On the t=9:8 t=8:9 other hand, in the black column, we have a falling 14 108 D D 80 scale which is the reciprocal of the Dorian octave t=10:9 t=9:10 species, namely the Lydian octave species, which 15 120 C E 72 matches our modern major scale. It is defined by the tone-numbers 30-120 in ratios of string-length. Table 1 Notice also that the first 11 integers in this column (30-80) can all be found in the Mesopotamian tables of divisors and reciprocals. Lines 1-13 match my hypothetical tuning for the Queen’s Harp in Example Four, and can accommodate all the seven heptachords illustrated in figure 3 (CBS 1766). The upper portion of table 3 extends the pat- tern provided by the reciprocally numbered strings in UET VII, 126 (shown in red) in both an upward and a downward direction. In the lower portion, hypothetical relative pitches are postulated for each of the string-sets. LEON CRICKMORE

1 1 3 1 2 1 5 1 2 3 2 1 7 1 2 3 4 3 2 1 9 1 2 3 4 5 4 3 2 1 11 1 2 3 4 5 6 5 4 3 2 1 13 1 2 3 4 5 6 7 6 5 4 3 2 1

1 D 3 E D C 5 F E D C B 7 G F E D C B A 9 A G F E D C B A G 11 B A G F E D C B A G F 13 C B A G F E D C B A G F E Table 3 Number of tuned strings Description of tone-series and scales

1 Keynote 3 Major 3rd, 5:4 (Just); 81:64 (Pythagorean) 5 Tritone: √2 in equal temperament 7 kitmum in CBS 1766: Hypophrygian octave species 9 pitum in UET VII, 126; Hypodorian octave species 11 nīš tuĥrim; Hypothetical tuning of the silver lytre of Ur 13 išartum; Hypothetical tuning for Queen PU-ABI’s harp The three missing heptachords (qablītum, nīd qablim and embūbum) can be found either by extending lines 1, 3 and 5 to seven pitches, or by reading tyhem from line 13. Table 4 The upper two lines in the table below show The lower two lines show the fifteen pitches an extension of the thirteen-string scale to fifteen of the ancient Greek Greater Perfect System, the strings. Like all the odd-numbered sets of strings tone-numbers of which would have been tradition- with D at their centre illustrated in the previous two ally expressed in Pythagorean tuning, a tables, the pattern of tones and semitones charac- above, or a below embubum – in terizing this two-octave scale is palindromic. This effect its ‘plagal’ form. It corresponds to the an- particular tuning is that of the Babylonian hepta- cient Greek falling Hypodorian octave species and chord embubum. If Plato were to have known the the Babylonian heptachord pitum. Such a scale can tone-numbers of this scale, I suspect that he would also be found between lines 3 & 10 of my version have considered them to be ‘a pattern laid up in of McClain’s ‘alternative indices’ diagram. In its heaven’ (Republic 592b). reverse direction, it matches Dumbrill’s postulated D C B A G F E D C B A G F E D tuning for the nine strings of UET VII, 126. t s t t t s t t s t t t s t All the diatonic scales contained in the table above can be defined using the tone-numbers of A t G t F s E t D t C s B t A t G t F s E t D t C s B t A either Pythagorean or Just tuning. The latter, however, are more closely related THE UBIQUITY OF THE DIATONIC SCALE to the natural harmonic series. But, with the 14 Waerzeggers, C. and Siebes, R. (2007) ‘An alternative historic change-over, first to Pythagorean and later interpretation of the seven-pointed star on CBS 1766’, NABU 2007/2: 43-45 to the more mathematically precise equal tempered 15 Friberg, J. (2011) ‘Seven-Sided Star Figures and Tuning tuning, Just tuning seems to have been driven Algorithms in Mesopotamian, Greek and Islamic Texts’, Ar- underground or even at times actively suppressed. chiv fur Orientforschung 52: 121-155 16 Crickmore, L. (2008) ‘A Musical and Mathematical Con- Nevertheless, even in contemporary music-making, text for CBS 1766’, Music Theory Spectrum 30 no.2: 327-338 performers – especially singers and string players 17 Hilprecht, H. V. (1906) The Babylonian Expedition of – still often prefer a Just-tuned major third to an the University of Pennsylvania – Series A: Cuneiform Texts (Hilprecht ed.) Volume XX, Part I, published by the Depart- equal-tempered one. As with our division of time ment of Archaeology, University of Pennsylvania: Pls 10, 11, into sixty minutes and sixty seconds, perhaps we 12 and 14: CBM 11340 + 11402; 11368; 11902 and 11097 have here in such a spontaneous musical practice 18 Crickmore, L. (2012). ‘A Musicological Interpretation of the Akkadian Term siĥpu’, Journal of Cuneiform Studies, another precious remnant in our modern world of 64:59-66 the rich early culture of now sadly ravaged Iraq. 19 Levin, F.R. (1994) The Manual of Harmonics of Nico- Finally, towards the end of his fascinating book machus the Pythagorean, Phanes Press: 154 and 159-160 20 Levin, op cit 17: 170, n.6 Beyond Measure: A Guided Tour through Nature, 21 See Crickmore, L. (2009) ‘A Possible Mesopotamian Or- Myth and Number, Jay Kappraff29 has raised igin for Plato’s World-Soul’, Hermathens 186: 5-23 a profound question that could, in the future, 22 Friberg, J. (2007) Amazing Traces of a Babylonian Ori- gin in Greek Mathematics, World Scientific: 67-68 become extremely pertinent to the subject of our 23 See Godwin, J.(1979) Robert Fludd, Thames and Hudson: 53 discussion. He asks: ‘Is it not possible that our 24 Solomon, J. (2000) Ptolemy Harmonics, Brill: 103 brains are wired to the tones of the diatonic scale, 25 Godwin, J. (1993) The Harmony of the Spheres, Inner Traditions International, Vermont: 232 an area for neuroscientists to explore?’ In raising 26 Hilprecht, H. V (1906) The Babylonian Expedition of this question, Kappraff has certainly highlighted a the University of Pennsylvania, Series A: Cuneiform Texts, potentially significant area for future neuroscientific Vol. XX, part I, published by the department of Archaeology, University of Pennsylvania: II, p.21 and Plates 10, 11,12, and research. However, to follow his lead, would be to 14 – CBM 11340 + 11402; 11368; 11902; and 11097 enter an area that lies well beyond the scope of a 27 Robson, E. (2002) ‘Words and Pictures: New Light on mere musicologist! Plimpton 322’, American Mathematical Monthly 109: 105-120 28 Tone-numbers: embubum: Just tuning 432, 480, 512, notes 576, 640, 720, 768, 864, 960, 1024, 1152, 1280, 1440, 1536, 1728; Pythagorean tuning 432, 486, 512, 576, 648, 729, 768, 864, 972, 1024, 1152, 1296, 1458, 1536, 1728. Greater Perfect 1Anicus Manlius Severinus Boethius (c.510 AD)De Insti- System: Pythagorean tuning 576, 648, 729, 768, 864, 972, tutione Musica, I,34 1024, 1152, 1296, 1458, 1536, 1728, 1944, 2048, 2304; 2 Translated by Calvin M. Bowe (1989) in Fundamentals of Just Music, Yale University Press: 50-51 tuning 576, 640, 720, 768, 864, 960, 1024, 1152, 1280, 1440, 3 Free translation by the present author 1536, 1728, 1920, 2048, 2034. 29 4 McClain, E.G. (1976) The Myth of Invariance, Nico- Kappraff, J. (2002) Beyond Measure: A Guided Tour las-Hays, York Beach, Maine: 124-125 Through Nature, Myth, and Number, World Scientific: 562. 5 I Paralipomena (Chronicles) 25: 7-8 6 II Paralipomena (Chronicles) 5: 12-13 7 See McClain, E.G. (1981) Meditations through the Quran, Nicolas-Hays, York Beach, Maine: 160 8 For the origins of this distinction, see Ptolemy, Harmon- ics, II, 5 (51.19) in Solomon, J. (2000) Brill: 73 9 Hopkins, G. M. Letter, I:289-90 10 Godwin, J. (1987) Harmonies of Heaven and Earth, Thames and Hudson: 181 11 Dumbrill, R. (2010 ‘Evidence and Inference in Texts of Theory in the Ancient Near East’, ICONEA 2008 (Proceed- ings), Iconea Publications/Lulu: 105-115 NB p.108 Fig. 6 12 Hilprecht, V. (1903) Explorations in Bible Lands During the 19th Century, A. J. Molman, Philadelphia: 530 13 Horowitz, W. (2006) ‘A Late Babylonian tablet with concentric circles from the University Museum (CBS 1766)’, JANES 30:37-53 LEON CRICKMORE Bibliography

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