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 .

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 (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 ... As we’ll see shortly, this 7-number sequence has a natural starting point: the number 4.

To derive the heptagram on CBS 1766 we begin by giving each fifth a number indicating the number of times we multiplied the fundamental by 2/3 to create it. For example, if the fundamental gives the note C, then G is the 1st fifth, generated by multiplying 2/3 by itself 1 time. Therefore, we write the number 1 radially outward from the note G (Figure 6). Similarly, we write the number Figure 5 12 outward from the note C, because if the fundamental is C then the 12 th fifth is another C, generated by multiplying 2/3 by itself 12 times.

Now, to continue, we must make one final assumption: that the Babylonians understood that each fifth can be moved into the same octave as the fundamental by doubling its string length a certain number of times. (This is not a new concept, but simply an application of the octave rule previously discussed.) As the twelve fifths are reduced into a single octave, they become arranged in the order that we now associate with an ascending : C#, D, D#, E, F, F#, G, G#, A, A#, B and C. As we will now see, it is the process of reducing the fifths that generates the sequence 4,1,5,2,6,3,7.

5 Dumbrill, R., Is the Heptagram in CBS1766 a Dial? (www.iconea.org) Figure 6 5 To begin, let’s list the fifths in the ascending chromatic order just described Table 1 (Table 1). Beside each fifth, in the column Fifth # , the numbers shown on Note Fifth # Figure 6 are written. C# 7 D 2 Now let’s explore the visual relationship of the ascending chromatic order shown in D# 9 Table 1 to the circle of fifths. Figure 7 duplicates Figure 6, adding a 12-pointed star. E 4 Following the diagonals of this star generates the ascending chromatic order. For F 11 example, moving along the purple line from C, the fundamental, we arrive at the next F# 6 note in the ascending order: C#. To arrive at the next note, D, we follow the green G 1 line. Continuing along the diagonals, we generate the ascending chromatic order G# 8 given in Table 1. A 3 A# 10 If we were to play the fifths, as we have arranged them, in ascending order, we B 5 would hear notes that jump up and down through seven octaves. This is illustrated by C 12 Figure 8, where the fifths are shown as yellow dots on the keyboard. (Note that the use of the keyboard does not imply , but simply allows us to view seven octaves.) Instead of having the fifths jump up and down in this way, let’s reduce the twelve fifths into a single octave to create an ascending 12-tone scale.

To accomplish this, we double the string length of each fifth, repeatedly when necessary, until we find all twelve fifths in the same octave. To keep track of how many times the string length of each fifth is doubled, the octaves need to be numbered.

The octaves in Figure 8 are numbered using scientific pitch notation: octave 0 extends from C0 (the fundamental, coloured red) up to but not including the next C. This next C is called C1, indicating that it is positioned at the bottom of octave 1. Figure 8 shows seven octaves (octave 0 to octave 6, inclusive) and one note in octave 7: C7. This note, C7, is both the 12 th fifth and the 7 th higher octave of the fundamental. Figure 7

Figure 8 Figure We now give each fifth a second number, indicating in which octave it is located: G, the 1 st fifth, is in octave 0; D and A, Table 2 the next two fifths, are in octave 1; E and B are in octave 2; Fifth Octave # F# is in octave 3; C# and G# are in octave 4; D# and A# are C# 4 in octave 5 and F is in octave 6. Table 2 lists these octave D 1 numbers. Notice that the first six numbers in the column D# 5 Octave # , coloured in red, are the first six numbers of the E 2 sequence 4,1,5,2,6,3,7, the same sequence that is found by F 6 following the diagonals of the heptagram on CBS 1766. F# 3 The first four numbers of the sequence are then found later G 0 in the column, coloured in green. This gives us an idea: can G# 4 the sequence 4,1,5,2,6,3,7 be formatted as a visual guide that A 1 can be used, in conjunction with the circle of fifths, to give A# 5 the octave numbers of the fifths? And, in fact, it can. B 2

6 Figure 9 duplicates Figure 7, highlighting the first six diagonals in red. As we highlight these lines, we add the octave number of each fifth, as given in Table 2, writing it radially inward from the note in question. For example, in Figure 9 the number 4 is written radially inward from C#, indicating that C# is found as a fifth in octave 4. Continuing with the octave numbers, we write the number 1 radially inward from the note D, the number 5 inward from the note D#, the number 2 inward from the note E, 6 inward from F, 3 inward from F# and 0 inward from G.

We can now use the Fifth numbers and the Octave numbers to create a 12-tone scale. To do this, we must simply understand that the string length of any note in the scale is determined by a combination of two processes. First, the fundamental is multiplied by 2/3 a certain number of times to generate a specific fifth. Then, the string length of that fifth is doubled as many times as necessary, to find that same note in a lower octave. This operation can be described f n Figure 9 by the formula 2/3 x 2 , where f is the number of the fifth that a particular note is derived from and n is the octave in which that fifth is found, relative to the octave of the scale being constructed.

Table 3 Table 3 combines Tables 1 and 2 from the previous page Note Fifth# Octave# String Length and adds the column String Length , which gives the string C# 7 4 0.9364 lengths of the notes in the ascending scale by inserting the D 2 1 0.8889 Fifth# and Octave# values for each note into the formula D# 9 5 0.8324 2/3 f x 2 n. The string length values in Table 3 are relative E 4 2 0.7901 to a fundamental, C, whose string length is assumed to be 1. F 11 6 0.7399 F# 6 3 0.7023 As an example, let’s calculate the string length of the note C#. G 1 0 0.6666 In Figure 9, the f value of C# (7) is written radially outward from G# 8 4 0.6243 the note C# and the n value of C# (4) is written radially inward A 3 1 0.5926 from the note C#. To calculate the string length of C# we insert A# 10 5 0.5549 these numbers into the formula 2/3 f x 2 n and arrive at the value B 5 2 0.5267 of 2/3 7 x 2 4 = .9364. To find the next note in the scale we move along the red diagonal from C# to D. The f value of D is 2 (it is the 2 nd fifth) and is written radially outward from the note D. The n value of D is 1 (as a fifth, it is located in the 1 st octave) and is written radially inward from the note D. Inserting these values into our formula, we calculate the string length of D as 2/3 2 x 2 1 = .8889. Continuing up the scale (and along the red diagonals), D# has a value of 2/3 9 x 2 5 = .8324; E has a value of 2/3 4 x 2 2 = .7901; F has a value of 2/3 11 x 2 6 = .7399 and F# has a value of 2/3 6 x 2 3 = .7023. We have now come to the note G, with an octave number of 0. Because we are aware of the sequence 4,1,5,2,6,3,7, it makes sense to ask ourselves if the number 0 can be replaced with the number 7. And, in fact, it can.

Because a line of twelve fifths is a repeating pattern, the 13 th fifth is another G. This G is in the 7 th octave. Therefore, if we consider G to be the 13 th fifth, rather than the 1 st fifth, we are correct in giving it an octave number of 7. But how will changing the Fifth # and Octave # of G affect the value of its string length? In fact, the difference caused is very slight: 2/3 1 x 2 0 = .6666 while 2/3 13 x 2 7 = .6577. This slight difference is, of course, the Pythagorean Comma and represents the difference in the string length of a note when calculated either as the 1 st fifth or as the 13 th fifth, reduced seven octaves. 7 Table 4: Ascending Scale In essence, what we are doing by substituting the number 0 for the number 7 is calculating the string length of the note in question by Note Fifth # Octave # String Length reducing a fifth that is located seven octaves above the octave that C# 7 4 0.9364 we are working in. In other words, we don’t use the note as it D 2 1 0.8889 appears as a fifth, located in the octave in which we are making the D# 9 5 0.8324 scale. Instead, we use the 12 th higher fifth, which is, in essence, the E 4 2 0.7901 same note, seven octaves higher. Therefore, when substituting an F 11 6 0.7399 octave number of 7 for an octave number of 0, the associated fifth F# 6 3 0.7023 G 13 7 0.6577 number must be increased by a value of twelve. For example, the G# 8 4 0.6243 fifth number of G must be increased from 1 to 13 (1+12=13), as A 3 1 0.5926 shown in Table 4. A# 10 5 0.5549

B 5 2 0.5267 Now let’s continue up the scale. Looking back at Table 3, we see C 12 6 that the next four notes after G have Octave #’s of 4,1,5 and 2, 0.4933 respectively. This repetition of the sequence 4,1,5,2,6,3,7 gives C# 7 3 0.4682 us an idea: is there some way to re-use the red lines and numbers D 14 7 0.4385 D# 9 4 0.4162 that we drew in Figure 9 to pair the upcoming Fifth numbers with E 4 1 0.3951 their Octave numbers? And, in fact, there is. F 11 5 0.37

F# 6 2 0.3512 Figure 10 duplicates the red shape shown in Figure 9, but replaces G 13 6 0.3288 the number 0 with the number 7. Also, another red line has been G# 8 3 0.3122 drawn to lead us from the number 7 back to the number 4. Notice A 15 7 0.2923 that this shape is now a heptagram. We can use this heptagram to calculate the string lengths of the next notes in our ascending scale. We do this by noting that G#, the next note in the scale, has an octave number of 4. So we turn the heptagram one position clockwise so that point 4 indicates G# (Figure 11). Now, just as before, we use the Fifth number of G# (8) and the Octave number of G# (4) to calculate its string length: 2/3 8 x 2 4 = .6243.

We can now continue up the scale, calculating the string lengths of the notes by inserting the Fifth numbers (given on the outer circle) and the Octave numbers (given by the points of the heptagram) into the formula 2/3 f x 2 n. These values are given in Table 4.

Figure 10

For example, the note A has a Fifth# of 3 and an Octave# of 1 (Figure 11), so the string length of A is 2/3 3 x 2 1 = .5926. Continuing up the scale and along the red diagonals, A# has a string length of 2/3 10 x 2 5 = .5549; B has a string length of 2/3 5 x 2 2 Figure 15 = .5267; C has a string length of 2/3 12 x 2 6 = .4933 and C# has a string length of 2/3 7 x 2 3 = .4682.

Figure 11 8 We have now come to the note D, which has an Figure 13 Octave number of 7. Because this note has an Octave number of 7, we are calculating its string length not as the fifth located in the octave we are working in, but by using the 12 th higher fifth and reducing it seven octaves. Therefore, we must increase the Fifth number of D by twelve. Consequently, D has a Fifth number of 14 (2+12 =14), as shown in Table 4. Here, then, is a general rule: every time we use an Octave number of 7 we add 12 to the related Fifth number . We then turn the heptagram and begin again at 4.

Having turned the heptagram (Figure 12), we continue to use the Octave numbers in conjunction with the Fifth numbers to calculate the string lengths in our Figure 12 ascending scale (Table 4). We see, therefore that a heptagram whose diagonals generate the sequence 4,1,5,2,6,3,7 provides a visual guide for calculating the string lengths of the notes in an ascending scale made by reducing perfect fifths.

There is another way to represent the f and n values visually – a way that illustrates their relationship to music very clearly. If we take a strip of paper the length of a guitar string and fold it repeatedly by 2/3 we portray a series of fifths (Figure 13). st Whenever a fifth fold falls between the 1 octave (the red line) and the end of the paper labelled “bridge,” we double its length, finding the same note in octave 0. Keeping a tally of the number of 2/3 folds on the right hand side of the paper we generate the f values in Table 4. Keeping a Figure 14 tally of the doubling folds on the left hand side of the paper we generate the n values in Table 4: Table 5: Descending Scale the repeating sequence 4,1,5,2,6,3,7.

Note Fifth # Octave # String Length If we assume the strip of paper to sound the note C,

C# 7 4 0.9364 then the distance from the end of the paper labelled

C 12 7 0.9865 “bridge” to each fold line is the length of string that

B 5 3 1.0535 sounds each note in our 12-tone scale. For example,

A# 10 6 1.1099 if we assume the whole strip to have a length of 1,

.5267 = A 3 2 1.1852 B is sounded by plucking .5267 of the string, as 2

G# 8 5 1.2486 2 x shown by the green arrow. If we compare our paper 5

G 1 1 1.3333 with the fret board of the guitar, we see a marked

F# 6 4 1.4047 similarity between the fold lines and the frets. F 11 7 1.4798 The discrepancy between the two illustrates the 2/3 = B

E 4 3 1.5802 difference between a scale made by reducing perfect

D# 9 6 1.6648 fifths and the equal tempered scale. This difference

D 2 2 1.7778 is observable, for example, between the fold line C

C# 7 5 1.8729 (12, 6) that has a length of .4933 and the true octave C 0 1 2 line that has a length of .5 (coloured red).

9 Just as rotating the heptagram in a clockwise direction generates an ascending scale, so rotating the heptagram counter clockwise will generate a descending scale. Figure 14 shows the heptagram from Figure 10 rotated one position counter clockwise. Table 5 projects the scale downward.

We have extended our scale 1½ octaves upward and 1 octave downward. How far can we continue? Since something can be divided in half indefinitely, a single string contains an infinite number of octaves. Therefore, we can generate fifths ad infinitum . As we generate these fifths, the sequence 4,1,5,2,6,3,7 repeats indefinitely, giving the Octave number of each fifth relative to the octave of the scale being constructed. Of course, the 12-tone scale created in this way is not practical, because none of the octaves remain pure. Nevertheless, I believe that its creation was understood and was an important theoretical exercise.

Let’s compare the heptagram we just generated, when creating a 12-tone scale made by reducing perfect fifths, with the one on CBS 1766. We saw earlier that R. Dumbrill 6 reproduces the heptagram on CBS 1766 as shown in Figure 15 – as a regular heptagram with the point at the top of the tablet labelled 1.

Figure 16

Figure 15

Notice however, that the heptagram on CBS 1766 is not regular. Notice also that this irregular heptagram has an axis of symmetry running through the point labelled 2 (Figure 16). Figure 17 shows the heptagram on CBS 1766, rotated so that the point labelled 2 is positioned at 12 o’clock. Figure 18 shows the heptagram that we derived earlier, when reducing perfect fifths to create a 12-tone sale. Notice that the two heptagrams bear a striking resemblance to each other. Firstly, the placement of the numbers is the same. Secondly, the shapes of the two heptagrams are almost identical.

Figure 17 Figure 18

6 Dumbrill, R., Is the Heptagram in CBS1766 a Dial? (www.iconea.org) 10 The question arises, therefore: Is the figure on CBS 1766 a depiction of a rotating dial – an analog device, perhaps the first ever invented – that was used to generate a 12-tone scale made by reducing perfect fifths?

To date, experts in the field believe that cuneiform tablet CBS 1766 describes the construction of the diatonic scale. And, in fact, the heptagram on CBS 1766 does have this secondary application. It must be argued, however, that the diatonic scale has no natural basis – why choose seven notes to create a scale, rather than six or five? Therefore, although the heptagram on CBS 1766 can be used to describe the construction of the diatonic scale, I believe that its discovery originated with the 12-tone scale – as a natural pattern generated by the ratios 1/2 and 2/3 – and that this discovery predated and inspired the creation of the diatonic scale.

Let’s look, now, at how the heptagram on CBS 1766 relates to the diatonic scale. To do this, we will continue with the assumption that the Babylonians were aware of the process of generating a 12-tone scale by reducing perfect fifths. In so doing, they would have re-arranged the twelve fifths in chromatic order: C#, D, D#, E, F, F#, G, G#, A, A#, B, C. I believe that the Babylonians then drew inspiration from the circular pattern generated by the fifths, arranging these chromatic notes in a circle. Moreover, I believe that they then superimposed upon this circle, the heptagram from CBS 1766 (Figure 19).

Why do I believe that they did this? Because the relationship of the points of the heptagram to the chromatic circle explains the construction of the diatonic scale. For example, if we follow the points of the heptagram around clockwise, starting at point 1, we select the notes in the scale of C major: C, D, E, F, G, A and B. If we then rotate the heptagram to any of the other 11 positions around the circle, its points will indicate the notes in the named by the note at point 1. Figure 19

Postulating the diatonic scale to have this origin resolves some fundamental questions. It explains, for example, why the mode beginning with the note indicated by point 2 (and then containing the notes at points 3, 4, 5, 6, 7, and 1, in that order) is symmetrical: because it begins on the note located at the point through which the axis of symmetry of the heptagram passes. Figure 20 I realize that what I have proposed does not correspond exactly with what is known about the tonal system in place in Babylon at the time of the creation of CBS 1766. But that is not the issue here. What is important to consider is the possibility that the diatonic scale was originally created in the image of the 12-tone scale – the heptagram generated by creating a 12-tone scale being simply re-applied to the chromatic circle to select the seven notes of the diatonic scale. If this is the case, then the Babylonian tuning system, as it developed over time, would have sought to preserve this original form, although the details of this form – the exact temperament of the notes within the scale – would have evolved. And this is, in fact, what we see: the heptagram on CBS 1766 describes the construction of the diatonic scale using a heptagram whose diagonals generate the sequence 4,1,5,2,6,3,7. Perhaps this duo function of the heptagram (that it can be used on both the circle of fifths and the chromatic circle) explains the reason for the two concentric circles that surround the heptagram on CBS 1766 (Figure 20): perhaps they are meant to indicate this double application. 11 * We have seen that the sequence 4,1,5,2,6,3,7 is generated when creating a 12-tone scale made by reducing perfect fifths. We have also seen that the same sequence explains the construction of the diatonic scale. It seems, therefore, that there was an intentional link or correspondence made between the two scales. Why was this done? I believe that the making of this correspondence or analogy illustrates a way of thinking that was inherent in the ancient world.

The English word analogy comes from the Greek word analogos , which translates as “proportionate.” It is made up of two smaller words, ana (“upon”) and logos . Logos is an important term in philosophy, psychology, and religion that translates as “reason,” but also “word,” and “ratio.” The deeper, implied meaning is that of a guiding principle that underlies all creation. This formative force, logos , is the divine “reason” that creates order in the universe: it is the holy “word,” spoken in the language of “ratio” or number – and this number was imagined, specifically, as emanating from the proportions ( analogia ) of music.

The fact that analogos means “proportionate” illustrates a basic mathematical principle: that the concept of ratio comes logically before the concept of proportion, for proportion is built “upon” ( ana ) “ratio” ( logos ). This fact also allows us to understand the origin of the concept of analogy, for an analogy was, originally, a comparison built upon number . And that is exactly how the sequence 4,1,5,2,6,3,7 is used: as a numerical correspondence, an analogy, that links the major scale to the 12-tone scale. Yet the major scale is just one of a series of analogies based on this sequence.

For example, the sequence 4,1,5,2,6,3,7 was also used to order the days of the week. To understand this, we must familiarize ourselves with the ancient model of the universe (Figure 21). This geocentric model placed the seven ancient planets (the sun, the moon and the five planets visible to the naked eye) in seven concentric spheres, arranging them according to their relative speeds, as seen from earth. For example, the moon, which appears to move most quickly through the constellations, was placed in the 1 st sphere while Saturn, the slowest moving of the visible planets, was placed in the 7 th sphere. Because the earth seems stationary, the sun appears to move at the relative speed of the earth in its orbit: faster than Venus but slower than Mars. Consequently, th Figure 21 the sun was placed in the 4 sphere. The Ancient Model of the Universe

The days of the week are named after the seven ancient planets: Sunday after the sun, Monday after the moon, Tuesday (in Latin: dies Martis) after Mars, Wednesday (dies Mercurii) after Mercury, Thursday (dies Iovis) after Jupiter, Friday (dies Veneris) after Venus and Saturday after Saturn. To understand how this order is generated we use a heptagram like the one on CBS 1766, writing the name of each body beside its sphere number (Figure 22). For example, we write “Sun” beside the number 4 because the sun was thought to occupy the 4 th sphere. Following the diagonals of the heptagram, we generate the week-day order: Sunday (the first day of the week, according to traditional Hebrew and Christian calendars), Monday, Tuesday, Figure 22 Wednesday, Thursday, Friday, Saturday: 4,1,5,2,6,3,7. 12 Is there proof that the ordering of the days of the week was, in fact, intentional, and based on the principles of music? The following quote from the 3 rd century Roman historian Cassius Dio confirms this:

“The custom...of referring the days to the seven stars called planets was instituted by the Egyptians... For if you apply the so-called "principle of the tetrachord" (which is believed to constitute the basis of music) to these stars, by which the whole universe of heaven is divided into regular intervals, in the order in which each of them revolves...you will find all the days to be in a kind of musical connection with the arrangement of the heavens.” 7

We also see the sequence 4,1,5,2,6,3,7 encoded into a ritual of the Mithraic Mysteries – a mystery religion that was practiced in the Roman Empire from the 1 st to 4th centuries, but that evolved from an older tradition. In this ritual, a ladder was used to symbolize a journey of ascent through the planetary spheres. The following is a description of this ritual from the writings of Origen, a Christian theologian of the 2nd century AD.

“In the mysteries of Mithras...there is a representation of...the planets, and of the passage of the soul through these. The representation is of the following nature: There is a ladder with lofty gates, and on the top of it an eighth gate. The first gate consists of lead, the second of tin, the third of copper, the fourth of iron, the fifth of a mixture of metals, the sixth of silver, and the seventh of gold. The first gate they assign to Saturn, indicating by the ‘lead’ the slowness of this star; the second to Venus, comparing her to the splendour and softness of tin; the third to Jupiter, being firm and solid; the fourth to Mercury, for both Mercury and iron are fit to endure all things, and are money-making and laborious; the fifth to Mars, because, being composed of a mixture of metals, it is varied and unequal; the sixth, of silver, to the Moon; the seventh, of 8 gold, to the Sun—thus imitating the different colours of the two latter.” Figure 23

Origen extracted the above quote from the writings of his contemporary, Celsus (whose works are now lost). Origen then comments on the quote, explaining that Celsus examines “the reason of the stars being arranged in this order” and gives ...“musical reasons...quoted by the Persian theology; and to these, again, he strives to add a second explanation, connected also with musical considerations.” 9

The most obvious connection to music is the fact that the ritual is accomplished by climbing a ladder. For the Latin word for ladder, scala , is the root of the musical term scale . If we then look at the arrangement of the rungs of the ladder we see another musical connection: the rungs of the ladder arrange the planets so that their sphere numbers generate the sequence 4,1,5,2,6,3,7 (Figure 23). We see, therefore, that the initiates who performed this ritual made a symbolic ascent through the heavens by climbing a musical scale.

According to Origen, Celsus identifies the idea that the soul ascends through the spheres as originating with Plato (428-348 BC). It shouldn’t surprise us, therefore, that when Plato recounts the tale of Er, whose soul journeys to the heavens, his description of the spheres generates a series of numbers that closely resembles the sequence 4,1,5,2,6,3,7.

7 Cassius Dio, Roman History, Book XXXVII 8 Origen, Contra Celsum , Book VI, Chapter 22 9 ibid. 13 Plato lists not seven but eight spheres (or whorls), identifying “the first and outermost whorl” as the sphere of the constellations inside which the other seven spheres revolve. He then describes the widths of these whorls:

“The first and outermost whorl has the broadest rim, and the seven inner whorls are narrower, in the following proportions – the sixth is next to the first in size, the fourth next to the sixth; then comes the eighth; the seventh is fifth, the fifth is sixth, the third is seventh, last and eighth comes the second.” 10

Notice the width of the whorls: 4,5,2,6,3,7,8,1 (Table 6). This arrangement of Ta ble 6 numbers is close enough to the sequence 4,1,5,2,6,3,7 to give us pause for thought, Whorl Width 1st 1 especially since Plato’s works survive only as copies of copies. Moreover, Plato nd describes his cosmological model as having a musical dimension: “...and on the 2 8 3rd 7 upper surface of each circle is a siren, who goes round with them, hymning a single th 11 4 3 tone or note. The eight together form one .” th 5 6 th 6 2 In describing the width of the wholrs using music, Plato echoes the teachings th 7 5 of Pythagoras, who is credited with saying: “There is geometry in the humming th 12 8 4 of the strings, there is music in the spacings of the spheres.”

Let’s return, now, to the previous description of the Mithraic ritual. According to Celsus, the rungs of the ladder were made of seven metals, each one corresponding to one of the seven ancient planets. These comparisons arose naturally from the fact that there were only seven metals known in the ancient world: gold, sliver, copper, iron, lead, tin and mercury. (This is the generally accepted chronological order of discovery.) The manufacture of bronze, an alloy of copper and tin, was also known and predated the discovery of mercury. On the Mithraic ladder, bronze (what Celsus refers to as a “mixture of metals”) is used instead of mercury.

Not all traditions linked the metals and the planets in exactly the same way: some of the correspondences varied according to time and place. For example, Celsus identifies copper as associated with Jupiter but according to most other sources it was associated with Venus. Nevertheless, three of the correspondences were universally accepted: gold was paired with the sun, silver with the moon and lead with Saturn. Table 7 lists the planets in their sphere order and gives their associated metals, according to Celsus.

Table 7 Table 8 Sphere Order Planet Metal Week-day Order Planet Metal 1 Moon silver 4 Sun gold 2 Mercury iron 1 Moon silver 3 Venus tin 5 Mars bronze 4 Sun gold 2 Mercury iron 5 Mars bronze 6 Jupiter copper 6 Jupiter copper 3 Venus tin 7 Saturn lead 7 Saturn lead

The arrangement of the metals in Table 7 is of no recognizable order. However, when we rearrange the planets as they are on the Mithraic ladder – using the sequence 4,1,5,2,6,3,7 (Table 8) – the metals take on a familiar pattern: they become arranged according to their traditional values. Gold, the most valued metal is at the top of the ladder with silver and bronze directly below. Lead, the least valued, is at the bottom. In other words, the traditional value system of the metals corresponds with the musical sequence 4,1,5,2,6,3,7.

10 Plato, The Republic, Book X 11 ibid. 12 The Houghton Mifflin Dictionary of Biography (2003), p 1250 14 The relationship of the metals to the sequence 4,1,5,2,6,3,7 also explains the origin of the alchemical tradition of turning lead into gold. As we have seen, when a heptagram whose diagonals generate the sequence 4,1,5,2,6,3,7 is used to create a scale by reducing fifths, the heptagram must be turned after the number seven: 4,1,5,2,6,3, 7 turn 4,1,5,2,6,3,7. In other words, when moving from one cycle of the sequence to the next, the number 7 (Saturn/lead) is followed by the number 4 (Sun/gold): lead becomes gold through the turning of the heptagram. And, in fact, there is evidence that this symbolism was used in alchemical initiation:

“A ladder with seven rungs was also preserved in alchemical tradition. A codex [in the Royal Library at Modena] represents alchemical initiation by a seven-runged ladder up which climb blindfolded men; on the seventh rung stands a man with the blindfold removed from his eyes, facing a closed door.” 13

It appears. therefore, that the sequence 4,1,5,2,6,3,7 was used to symbolize a process of transformation. This is true not only with respect to the metals but in the passage of the week-day cycle as well. We can see this clearly in the writings of Gregory of Nyssa, a 4th century Christian theologian, who believed that “the nature of time is circumscribed in the week of days.” 14

For Nyssa, the week-day cycle has a spiritual function: to provide a means for the soul to ascend toward God. And not only does the individual undergo this transformative passage; the entire history of the human race is a progression toward the divine, a process that Nyssa terms akolouthia , a Greek word that means “sequence.” According to Christian tradition, this journey will end only when the week-day cycle ceases, an end that will come with the return of Christ, on what Christians call the “eighth day.”

“Thus we accept the law concerning the octave which cleanses and circumcises because once time represented by the number seven comes to a close, the octave succeeds it. This day is called the eighth because it follows the seventh... and is no longer subject to numerical succession. Another sun makes this day, the true sun which enlightens...” 15

Let’s summarize. We have seen that the sequence 4,1,5,2,6,3,7 originates in the creation of a 12-tone scale and was used, analogically, to create the diatonic scale, to order the days of the week, to generate the traditional value system of the metals, to describe the alchemical tradition of turning lead into gold and to represent a ladder of ascension. Let’s look at some other possible analogies based on this sequence:

• According to several traditions, the world was believed to cycle through different ages. For example, the Denkard , a 10 th century Zoroastrian text, refers to ages of gold, silver, steel and “mixed with iron.” Each age was believed to be more degenerate than the previous one, until, at the end of the final age, the world was destroyed and the cycle began anew, with another Golden Age.

• The ancient Greeks believed every seventh year to be of critical importance. Such a year was called a klimacteric , a Greek word meaning “ladder.” According to Aulus Gellius, a Roman historian of the 2 nd century, this belief originated in with the Chaldeans (Babylonians). 16

• A similar cycle of seven years is described in the Torah. On the seventh year of the cycle, the Shmita year, Jews are instructed to let their land lie fallow and to forgive all debts. After seven cycles of seven years a Jubilee year is celebrated, during which the mercies of God are believed to be particularly manifest.

13 Eliade, Mircea, Shamanism: Archaic Techniques of Ecstasy, p 490 14 Gregory of Nyssa, On the Sixth Psalm, Concerning the Octave 15 ibid. 16 Aulus Gellius, Attic Nights 15 • Every week-day cycle closes with a day of rest – the Sabbath, which literally means “to rest.” For the Jews, the Sabbath is Saturday (7) and its observation is the edict of the 4th Commandment: “Remember the Sabbath day, to keep it holy.” The Hebrew word sabbat (probably from the Sumerian sa-bat meaning “mid-rest”) is the root of the word sabbatical , the name still given to the year of leave granted to a professor once every seven years.

• A definition of the word octave still found in most dictionaries is “a period of eight days beginning with the day of a church festival.” In the Christian Church, annual feasts are emphasized by extending their celebration over the period of an octave. Easter, the most important festival, is followed by a period of seven octaves – Eastertide – during which the Ascension of Christ is commemorated. A similar period of forty-nine days follows the celebration of Passover, the Jewish festival from which Easter evolved. Interestingly, the word week comes from the Old English word wice , of Germanic origin, that originally meant “series” or “sequence.”

• The term akolouthia (Greek: “sequence”) came to refer, in the Church, to the arrangement of the Divine Services, prayers that are said in sequence each day. Originally such prayers may have, sub-divided the day into seven: “Seven times a day I praise you for your righteous laws.” 17

Most of the traditions mentioned above are punctuated, at the end of each cycle, by a period of a transformation: the destruction of the world, the annulment of debts, the rest period of the sabbatical, etc. Here we see an echo of the pattern generated by the motion (i.e. transformation) of the heptagram, when creating a 12-tone scale: after the number 7, the heptagram is turned (i.e. a transformation occurs), and the cycle repeats again with 4.

To summarize, I believe that the sequence 4,1,5,2,6,3,7 was used analogically in the ancient world to link various traditions (diatonic scale = week = value system of the metals = ladder of ascension), creating an all- encompassing system. But is there proof that such a system existed? According to Aristotle, there was such a scheme: one that applied the numbers of music to describe the structure of the cosmos – a scheme that originated with the Pythagoreans.

“[T]he so-called Pythagoreans, who were the first to take up mathematics...saw that the modifications and the ratios of the musical scales were expressible in numbers;-since, then, all other things seemed in their whole nature to be modelled on numbers, and numbers seemed to be the first things in the whole of nature, they supposed the elements of numbers to be the elements of all things, and the whole heaven to be a musical scale and a number. And all the properties of numbers and scales which they could show to agree with the attributes and parts and the whole arrangement of the heavens, they collected and fitted into their scheme; and if there was a gap anywhere, they readily made additions so as to make their whole theory coherent.” 18

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It is a fascinating truth that the heptagram on CBS 1766 (and specifically the sequence 4,1,5,2,6,3,7) can not only help us understand the origin of various systems and traditions that we have inherited from the distant past, but can also be used, today, to explain the rudiments of music theory.

We saw earlier that the heptagram on CBS 1766, when superimposed on the chromatic circle, selects the notes in the diatonic scale (Figure 19). When selected from the circle of fifths, however, the notes in the diatonic scale appear, instead, as a continuous group (the yellow shape in Figure 24).

17 The Bible , Psalm 119 18 Aristotle, Metaphysics, Book 1 16 The fact that the notes in the diatonic scale form a continuous group on the circle of fifths lends itself to the teaching of music theory. For here is a visual representation of the diatonic key. It is now a simple matter to illustrate the notes within a key (they are those that lie within the yellow shape – what we’ll call the Keyfinder ) and those that are outside the key (they are those that lie outside the Keyfinder ).

Moreover, if the holes of the Keyfinder are labelled with the sequence 4,1,5,2,6,3,7, selecting the notes shown in consecutively numbered holes gives the notes in the seven different modes. For example, when we place hole 1 over the note C, the notes in the scale of C major (C ) are found in holes 1,2,3,4,5,6 and 7, in that order. To construct the six modes related to the scale of C major, we simply select the notes in the consecutively number holes, but start with a note in a hole other than hole 1. Moreover, rotating the Keyfinder to different positions allows us to construct all the other major scales and their related modes.

Because there is a direct relationship between the sharp/flat naming system and the circle of fifths, the Keyfinder can also be used to name the notes in any major scale (and its related modes). This is accomplished by adding an outer rim to the circle of fifths, divided into four sections – Naturals; Naturals, then Sharps; Naturals, then Flats; Sharps, then Naturals or Flats, then Naturals . If we then add a pointer that extends from hole 1 of the Keyfinder , as the pointer rotates on the outer rim it indicates how to name the notes in a particular key, scale or mode.

For example, Figure 24 shows the scale of B major, because hole 1 indicates the note B . Although the name A# also appears in hole 1, the pointer on the outer rim tells us to choose, whenever possible, natural names. If this is not possible (because no natural name is present in a specific hole) we choose the flat name. Consequently, the notes in the scale of B major, selected from holes 1 to 7, in that order, are named using Naturals, then Flats : B , C, D, E , F G and A.

To construct chords, we lay on top of the Keyfinder another rotating shape, made by fusing a triangle and a square together (what we’ll call the Chordfinder ). Different combinations of the points of the Chordfinder give the notes in different types of chords. For example, the notes in a major triad are given by yellow points M, 2 and 3, in that order. Similarly, the notes in a minor triad are given by blue points m, 2 and 3.

In order to name the notes in a chord, we simply observe the visual relationship between the Chordfinder and the Keyfinder : Figure 24: http//musicircle.net/ If all the points that indicate the notes in the chord can be made to lie on top of the Keyfinder, then that chord is “in” the key named by the note in hole 1 and its notes are named according to the pointer at hole 1.

In this way, the teaching tool in Figure 24 not only describes how to construct and name scales, modes and chords, but also illustrates which chords to use in diatonic composition (those that lie with in the key) and which chords can be added to create a chromatic composition (those that fall out side the key).

We see, therefore, that the sequence 4,1,5,2,6,3,7, found on cuneiform tablet CBS 1766, is not only of interest to archeomusicologists but also has a very real application in modern-day music theory pedagogy. 17

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In a previous section, examples were given that illustrate the analogical use of the sequence 4,1,5,2,6,3,7 in the ancient world. In most of these cases, the reason cited for making an analogy was musical. For example, Cassius Dio explained the week-day order as having a “musical connection with the arrangement of the heavens.” Similarly, Celsus gave “musical reasons” to account for the arrangement of the rungs on the Mithraic ladder. Aristotle went so far as to suggest that there was a “coherent” scheme based on “the properties of numbers and scales.” Where did this concept originate?

There is evidence that Pythagoras acquired much of his knowledge from his journeys to the Near East. For example, Iamblichus (245-325 AD) recounts that Pythagoras spent time in Babylon where “the Magi instructed him in their venerable knowledge and he arrived at the summit of arithmetic, music and other disciplines.” 19 It is possible, therefore, that the Pythagorean “scheme” to which Aristotle refers had its beginnings in Mesopotamia.

The construction of the diatonic scale would seem to confirm this. For the diatonic scale is, in fact, an analogy, in the original, mathematical sense of the word, in that it is based on a geometric/numeric pattern – a pattern that arises when creating a 12-tone scale. This pattern is, of course, the heptagram on CBS 1766.

Although, to date, no textual evidence has been found that proves that the Babylonians did, in fact, create a 12-tone scale by reducing perfect fifths, I believe that the heptagram on CBS 1766 provides visual proof of this . It is my belief, therefore, that when we encounter historical documentation of the use of the diatonic scale, we can be certain that enquiries into the ratios of music have been made and that the result has been the discovery of the 12-tone scale made by reducing perfect fifths and the consequent derivation of the heptagram found on CBS 1766. For how else can we account for the appearance of this very specific figure?

It might be proposed that another way to explain the appearance of the heptagram on CBS 1766 is to assume that the diatonic scale is innate (i.e. inscribed in the human subconscious) and then to postulate that, because the Babylonians had a sexagesimal number system, a visual depiction of this innate scale resulted in a non- regular heptagram, the points of which indicate seven of twelve equally spaced lines passing through the centre of a circle (the resulting shape being the heptagram on CBS 1766). But if the purpose of such a proposal is to support the belief that the diatonic scale is innate and that its use predated the discovery the circle (or spiral) of fifths, then perhaps we need to question these assumptions.

For as we have seen, the sequence 4,1,5,2,6,3,7 was used to symbolize a ladder (scala) spanning the heavens. So why, then, could not the circle of fifths (from which the sequence 4,1,5,2,6,3,7, is derived) also have been applied, analogically, to conceptualize the cosmos? This may be the case, for notice that when we superimpose the sequence 4,1,5,2,6,3,7 on the circle of fifths we see, in fact, a model of the ancient universe: when viewed from earth (the centre), the seven ancient planets (in their week-day order) appear to move around the outer circle. With respect to the ancient model of the universe, this outer circle represents the zodiac – the division of the ecliptic (the circular band of the sky through which the planets appear to move) into twelve constellations (Figure 25).

19 Iamblichus, The life of Pythagoras, p. 49 Figure 25: The Sequence 4,1,5,2,6,3,7 Superimposed on the Zodiac 18 According to historians, the zodiac originated in Babylon around the 7 th century BC. Yet, as we have seen, the circle of fifths must also have been known in Babylon around this time, because the sequence 4,1,5,2,6,3,7, which is derived from the circle of fifths, appears on tablet CBS 1766. One wonders, therefore, if the division of the zodiac into twelve constellations (and the related division, by the Babylonians, of the day into twelve hours) was inspired by the circle of fifths. If this is the case then the circle of twelve fifths may also have been the inspiration behind the creation of the sexagesimal base counting system used by the Babylonians, whereby the circle was divided into 360º.

Perhaps we will never know the answer to this question. But what will now be considered, I hope, is that the figure on CBS 1766 could be, in fact, the earliest known depiction of a tone-circle. For as we have seen, the heptagram contained in this figure is generated when calculating the string lengths of the notes in a 12- tone scale made by reducing perfect fifths. Moreover, when re-applied to a chromatic circle, it explains the construction of the diatonic scale. In both these applications it is not only the irregular shape of the heptagram that plays a part, but also the sequence of numbers generated by its diagonals: 4,1,5,2,6,3,7. The importance of this sequence cannot be over emphasized. For not only does it provide insight into the profound role played by the physics of music in the spiritual traditions of the ancient world, it also has a very real application, today, in music education.

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Bibliography

Aristotle, Metaphysics Aulus Gellius, Attic Nights Cassius Dio, Roman History Crickmore L., New Light on the Babylonian Tonal System Dumbrill, R., Is the Heptagram in CBS1766 a Dial? (www.iconea.org) Dumbrill, R., The Alacahoyuk ‘Guitar’ (www. iconea.org) Dumbrill, R.J., The Archaeomusicology of the Ancient Near East (2005) Eliade, Mircea, Shamanism: Archaic Techniques of Ecstasy Gregory of Nyssa, On the Sixth Psalm, Concerning the Octave Iamblichus, The life of Pythagoras Origen, Contra Celsum Plato, The Republic The Houghton Mifflin Dictionary of Biography (2003)