Divine 9 revisited

I’ve received a few questions about the details of the mathematics in my Divine 9 post, and why the Divine 9 theory qualifies as pseudo-science. In this post I’ll skip the musical part of the Divine 9 tuning, and will focus on the arguments about the divinity of the number 9.

Let’s start with the pseudo-science part. The usual scientific principle is to start with the facts, and then trying to draw a conclusion. Scientists try to find counter-examples to disprove their own theory. Pseudo-science just starts with a conclusion and tries to find facts supporting this conclusion, ignoring facts that don’t fit with the result. Now let me try to explain why the Divine 9 theory qualifies as pseudo-science.

If you take a look at the Divine 9 music site, you’ll read about the importance of the number 9 and how it should show up everywhere in nature. In fact, this connection with nature is based on spirals, the Fibonacci sequence, and the golden ratio. So far, so good. Both the Fibonacci sequence and the related golden ratio (I’ll get into the details), really do show up in nature. Interesting, but not that spectacular, since both nature and the Fibonacci sequence follow simple rules. Where it goes wrong is where the number 9 is lifted to a divine status. On the Divine 9 site, Gert Kramer gives examples of how the number 9 keeps showing up. Musical examples are John Lennon’s ‘Revolution 9’ and the fact that Beethoven wrote 9 symphonies. Conveniently, wikipedia maintains a list of song titles with a number in the title. Counting titles I don’t see a preference for 9. And though pi shows up, the golden ratio is missing. Wikipedia also features a list of symphony composers. Again, there are many composers who didn’t write exactly 9 symphonies.

The problem is that if you look for the number 9 you’ll find it, but the same holds for other numbers. In my previous Divine 9 post I mentioned Gert Kramer’s example of the Chinese dragon having 117 scales. Although there are 81 and 36 scales (both squares), in fact there is some relationship with 9 here. But that’s just because 9 turns out to be a lucky number for the Chinese, so they had a preference for it. If it had been western dragons, the number of scales might have been a multiple of 7. In fact, claims based on ancient knowledge are one of the characteristics of pseudo-science.

I noticed another mention of 117 on one of Kramer’s facebook posts: “Why did the original map of Lima, Peru created by Pizarro, consist of 9 × 13 roads resulting in 117 houseblocks?” He doesn’t give the answer, but my first thought was that this was just a coincidence. My second, that a grid of 9 × 13 roads in fact results in 8 × 12 = 106 house blocks. Googling for an old map of Lima (though not by Pizarro), it turns out that although there might have been 9 × 13 roads, the map doesn’t show a complete rectangular grid, so the number of blocks is much lower.

On the Divine 9 website there’s not much evidence for the importance of 9, but check any list of characteristics of pseudo-science and you can find most of them in the Divine 9 arguments.

Fibonacci sequence and Golden Ratio

The Fibonacci sequence is the sequence of numbers starting with 0 and 1, where each next number is the sum of the previous two:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …

For example, 89 = 34 + 55.

The Golden Ratio is the ratio A:B such that A:B = (A+B):A, and is close to 1.618. Due to the simple equality, the Golden Ratio has very nice properties and shows up both in art and nature. The relationship with the Fibonacci sequence is that the ratio between two consecutive numbers converges to the Golden Ratio. So, for example, 144/89 (=1.618…) is a good approximation for the Golden Ratio.

Due to the simple rules it shouldn’t be too surprising that both the Fibonacci sequence and the Golden Ratio show up in nature. For example, rows of sunflower seeds often follow the Fibonacci numbers. There’s also a relationship with spirals, such as again the pattern of sunflower seeds or nautilus shells. The Fibonacci sequence can be turned into a Fibonacci spiral, which also visualizes the Golden Ratio.

Mathematically, the nice thing about both the Fibonacci sequence and the Golden Ratio is that it’s (like e.g. primes) a property of numbers, not of our decimal (base-10) number system. Our decimal number system originates in the fact that we have 10 fingers, but the Fibonacci sequence and the Golden Ratio exist, whatever number system you use. Given that it’s an inherent and simple property of numbers, it shouldn’t be too surprising that you can find these numbers in nature.

Number systems

Our decimal number system uses 10 different digits (0 through 9). For example 321 means 3×10×10 + 2×10 + 1. Other number systems have been in use: the Maya used a base-20 system, but base-5, base-6, base-8 and base-12 have also been used. Interestingly, Kramer mentions the importance of the base-9 system, though I haven’t found any evidence that a base-9 system has ever been in use.

If you don’t know what a base-8 system would look like, it consists of 8 different digits (say 0 through 7), and the number positions indicate powers of 8 (instead of 10). So the number 321 in base-8 would be 3×8×8 + 2×8 + 1, or 209 in decimal.

The reason number 9 is so popular in numerology, is our base-10 system. In base-10, adding up the digits of a number results in the remainder after division by 9. So for 321 we have 3+2+1=6, and 321 = 35×9 + 6. This has nothing to do with 9 being special, except that it’s the highest digit in a decimal system. For example, 1000 = 999 + 1, or 9×111 + 1. So the remainder of 1000 divided by 9 is 1. This holds for any power of 10. Of course 2000 = 2×9×111+2, so the remainder is 2. Without giving a complete proof, I hope I’ve convinced you that the individual digits give you exactly the remainder after division by 9.

The interesting thing is that you can compute with these remainders, in what is called computation modulo 9. Formally you are computing with equivalence classes of numbers with the same remainder, but the easy way to think about this is clocks. On an clock, hours after 12 o’clock starts all over again at 1 o’clock. This is computing modulo 12. For example if it’s 3 hours after 11 o’clock, it must be 2 o’clock. Modulo 12 we have 11 + 3 = 14 = 2 (the remainder of 14/12 is 2). It’s nothing more than computing with remainders after division by 12.

The nice thing is that you can do the modulo-reduction any time in your computations. So if you compute 41+32 (modulo 9), you can either first do the sum (73) and then take the remainder (1), or you could first do the reduction: 41+32 = 5+5 = 10 (or 1) modulo 9. The same holds for multiplication.

Now unlike the Fibonacci sequence and the Golden Ratio, this adding of digits is specific for the number system you are using. In base-8 the sum of the digits gives the remainder after division by 7. The Maya would get the remainder after division by 19. This doesn’t mean that you can’t do computations modulo 19 in a decimal system, it’s just that modulo 9 is easier because you can just add individual digits.

In the Divine 9 tuning the modulo 9 computations are cleverly hidden. The frequencies are given in decimal notation, not as fractions of integer numbers. All frequencies turn out to have a rather arbitrary factor 9 put in the nominator (which has no musical meaning whatsoever). This implies that the fraction is by definition a multiple of 9. Yet, the results are presented as if by some divine intervention the sum of the digits is a multiple of 9 each time.

The (ir)relevance of 9 in the Fibonacci sequence

Finally, let’s have a look at the only connection given on the Divine 9 site between the number 9 and the Fibonacci sequence. Kramer computes the Fibonacci numbers modulo 9 (by the numerology trick of adding the individual digits). The resulting sequence is:

1 1 2 3 5 8 4 3 7 1 8 9 8 8 7 6 4 1 5 6 2 8 1 9 (Kramer writes ‘9’ for remainder 0)

He then splits the sequence into two sequences of 12 number:

1 1 2 3 5 8 4 3 7 1 8 9

8 8 7 6 4 1 5 6 2 8 1 9

Now the sums of corresponding digits is a multiple of 9 each time. This might seem like a small miracle, but in fact it is not that special. The only lucky thing is that the Fibonacci sequence contains two 8’s in a row. If you split before those eights, the first two sums are of course both 1+8, which is a multiple of 9. The rest of the sums are just a result of the definition of the Fibonacci sequence. For example, the next two numbers are 2 and 7. The 2 is the result of 1+1 (the previous two numbers), the 7 is the sum of 8+8 = 16 = 7 (modulo 9). Is it a coincidence that 2+7 is a multiple of 9? No, because 2+7 = (1+1) + (8+8) = (1+8) + (1+8) = 9 + 9 = 0 + 0. The first two sums are both 1+8=9=0. Each next sum is the sum of the previous two terms, which are both 0. Nothing special going on, and of course this keeps going on for the following sums. The sequence of sums is nothing more than a kind of Fibonacci sequence starting with two zeroes, and hence consists of zeroes only.

In fact, you can try the same trick in any number system. It doesn’t always work, but there are plenty examples apart from 9 that do. I’ve already shown you the base-6 equivalent. In the Maya base-20 system you would look for the Fibonacci sequence modulo 19, and look for two consecutive numbers 18. Unfortunately the sequence repeats before these two 18’s show up. However, for example base-8 (that is, modulo 7) does work:

1 1 2 3 5 1 6 0

6 6 5 4 2 6 1 0

Add the corresponding numbers, and each sum equals 7.

Interestingly, using the base-9 system that is so important according to Kramer, does not work.

As with the selective examples of Lennon and Beethoven, the Divine 9 site mentions this Fibonacci trick only for the number 9, ignoring all other numbers for which the same property holds. Their conclusion is that this should explain the numbers 999 (God), 9 and 117 (Chinese dragons), 72 (Adonai), 99 (Allah) and strangely enough 52 (Thoth), which isn’t a multiple of 9. Of course many other numbers from mythology which aren’t multiples of 9 are ignored again.

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