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THE UBIQUITY of the DIATONIC SCALE Leon Crickmore

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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 octave. 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 definiturbestia (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 Phrygian Mode 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.
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