Is CBS 1766 a Tone-Circle?

1 Is CBS 1766 a Tone-Circle? by Sara de Rose Abstract This paper illustrates how the heptagram on tablet CBS 1766 can be derived as a geometric/numeric pattern when creating a 12-tone scale by reducing perfect fifths. It also shows that the sequence 4,1,5,2,6,3,7, generated by following the diagonals of the heptagram on CBS 1766, explains the construction of the diatonic scale. The paper suggests that the sequence 4,1,5,2,6,3,7 was originally derived from the 12-tone scale and then later re-applied, intentionally, to create the diatonic scale. It is suggested that this re-application of the sequence 4,1,5,2,6,3,7 to create the diatonic scale was, in fact, an analogy, in the original sense of the word (i.e. a comparison based on number) and that a series of further analogies using the sequence 4,1,5,2,6,3,7 were also created – analogies that extended beyond the practical applications of music. Finally, the paper shows that the sequence 4,1,5,2,6,3,7 has a role, today, in teaching music theory, for it is, quite literally, what musicians call the key . 2 Is CBS 1766 a Tone-Circle? 1 In New Light on the Babylonian Tonal System , Figure 1: CBS 1766 L. Crickmore suggests that musicologists pose the question “could CBS 1766 be the earliest known example of a tone-circle?” This paper explores that possibility. There are two assumptions that I will make before exploring the diagram on CBS 1766 as relating to a tone-circle. The first is that, at the time of the creation of CBS 1766 (circa 500BC?), the Babylonians were aware of the ratio 1/2 and its relationship to the interval of the octave. The second is that, at this time, they were also aware of the ratio 2/3 and its relationship to the interval of the fifth. But how realistic are these assumptions? In The Archeomusicology of the Ancient Near East , R. Dumbrill discusses a cylinder seal (Figure 2: BM WA 1996-10-2, 1), acquired by Dr Dominique Collon on behalf of the British Museum. The seal, which dates from 3500-3200 BC, depicts, among others, the figure of a seated lute player. Figure 2: BM WA 1996-10-2, 1 Dumbrill writes that “The importance for a lute represented at such an early period implies that the Sumerians were aware of the usage of ratios in the division of the strings since the frets, or position of the strings on the neck of the instrument implies that knowledge.” 2 According to Dumbrill, “There is suggestion that the neck of the lute was divided into 60 units of length, (fingers) with a first fret position at 50/60th of the length, the second fret at 40/60th and a third fret at 30/60th of the length. Should the free string produce a fundamental ‘c’, then the first fret would give an ‘e’ flat, the second fret a ‘g’ and the third fret the octave ‘c’.” 3 If the Sumerians did, in fact, divide the neck of their lute in this way, then at the time of the making of CBS 1766, the Babylonians, who inherited much of their musical knowledge from the Sumerians, could well have understood that the interval of the octave is produced by plucking 1/2 (30/60) of the fundamental and that the interval of the fifth is produced by plucking 2/3 (40/60) of the fundamental. I will make these assumptions. Before I discuss the relationship of the diagram on CBS 1766 to a tone-circle, I must extend these two assumptions to include the further hypothesis that the Babylonians of the time were also capable of conceptualizing that a single string contains multiple octaves and multiple fifths, although, in each case, only the first few are actually playable. Let’s start with the idea of multiple octaves. 1 Crickmore L., New Light on the Babylonian Tonal System 2 Dumbrill, R.J., (2005) The Archaeomusicology of the Ancient Near East , p.321 3 ibid. 3 When a string is divided in half, the half string sounds the octave of the fundamental. In plucking half of the string, we have essentially birthed a new entity – for we see before us a vibrating string that is an exact replica of its parent, but half the size. Upon reflection, it becomes apparent that this new string can also be split in two, just as its parent was, to create another octave, another generation. Theoretically, this process can be repeated indefinitely. I believe that knowledge of this fundamental musical concept – that a single string contains multiple octaves – logically pre-dates the playing of standardized intervals, such as the ones suggested by R. Dumbrill to have been sounded by the Sumerian lute. The simplest way to divide something is in half. To generate multiple octaves, we re-apply this ratio, plucking 1/2 of 1/2 of 1/2, etc. The next simplest way to divide something is in thirds. It is by plucking 2/3 of the fundamental that we hear the interval of the fifth. As we saw earlier, there is suggestion that the neck of the Sumerian lute had a fret that shortened the length of the whole string to 40/60 or 2/3 of its length, thereby sounding the interval of the fifth. But were the Sumerians and the Babylonians capable of conceptualizing the generation of multiple fifths, sounded by plucking 2/3 of 2/3 of 2/3, etc? There is a depiction of a fretted instrument from the region of the Ancient Near East that pre-dates the making of CBS 1766 and that suggests an awareness of this concept. This is the relief popularly known as the “Hittite Guitar” (Figure 3), unearthed at Alaca Hoyuk, in modern day Turkey and dating from 1400 BC. The term “guitar” is used to describe the depicted instrument because its neck shows definite fret marks, similar to those on a modern-day guitar. Upon analysis, Dumbrill 4 found seven frets, placed at varying positions on the neck. He calculated that the fret closest to the bridge of the “guitar” would have shortened the whole string to 4/9 of its length. This fraction, 4/9, is equal to 2/3 x 2/3. In other words, the interval that would be sounded by shortening the whole string at this fret would be the fifth of the fifth of the fundamental. Here, then, is evidence of knowledge of generating the fifth of the fifth. With this in mind, let us imagine that the Babylonians, at the time of the making of CBS 1766, were able to conceptualize continuing this process, generating the fifth of the fifth of the fifth, Figure 3: “Hittite Guitar” etc... I am aware that the consensus among most archeomusicologists is that the musicians of Sumer and Babylon did not consciously relate ratios to musical intervals or manipulate these ratios to develop a musical system based on a mathematical understanding of music. However, I believe that making the assumptions that the Babylonians did, in fact, understand that a single string contains multiple octaves and multiple fifths to be an important exercise, for as we will see, these assumptions may lead us to understand the origin of the diagram on CBS 1766. We will now create a tone-circle using only the intervals of the octave and the fifth (i.e. the ratios 1/2 and 2/3). In so doing we will derive the heptagram on CBS 1766, a process that may cause us to reconsider our understanding of how the Sumerians and Babylonians conceptualized the relationship between mathematics and music. 4 Dumbrill, R., The Alacahoyuk ‘Guitar’ , (www. iconea.org) 4 If we generate a series of eleven fifths from a fundamental that sounds the note C, we will hear these pitches: G, D, A, E, B, F#, C#, G#, D#, A# and F. (Note: This is a series of perfect fifths. Western nomenclature is used simply as a means of identifying the fifths, not to suggest that the fifths have been tempered.) If we then create a 12 th fifth, we generate a note that is almost, but not quite, the 7 th higher octave of the fundamental (2/3 12 ≈ 1/27). The fact that twelve fifths almost exactly equal seven octaves creates a repeating pattern. This can be depicted, visually, by arranging the twelve fifths in a circle. Today, we call this arrangement the circle of fifths (Figure 4). I realize that, because of the discrepancy between the 12 th fifth and the 7 th octave of the fundamental, it would be more correct to represent the twelve fifths not as a closed, repeating system (i.e. a circle), but as a spiral. As we will now see, however, it is possible to represent the spiralling nature Figure 4: Circle of Fifths of the fifths using a circle, when the circle is used in conjunction with a heptagram like the one on CBS 1766. Let’s look more closely at the heptagram on CBS 1766. Figure 5 shows the heptagram on CBS 766, as deciphered by R. Dumbrill in Is the Heptagramon CBS 1766 a Dial? 5 The numbers 1 to 7 are written in a clockwise direction around the outer circle, each number positioned at one of the points of the heptagram. Following the diaginals of the heptagram generates the numbers 1,5,2,6,3,7, 4,1,5,2,6,3,7 ..

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