Category Archives: Mathematics

Euclid’s molecules

One of these days, we will get around to studying the history of music, and you will understand why Wagner was the high fructose corn syrup of his century. We still need to work on constructing a foundation for the study of music theory. That requires us to return to number theory, and Euclid.

In an earlier post, we discussed Euclid’s atoms, the prime numbers:

The main thrust of that article was to prove what some call Euclid’s theorem: there are infinitely many prime numbers. To be more precise: any finite list of primes excludes some prime number. The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, and 23.

A prime number is defined as a natural number greater than one divisible only by itself and one. You might guess that prime numbers are very important in multiplication, and you would be right. Take the number 60, for example. Is it prime? No, it is composite, in multiple ways. 60 equals 2 times 30, 3 times 20, 4 times 15, 5 times 12, and 6 times 10.

In each of these factorizations, at least one of the factors is also composite, so it too may be factored as a product of smaller numbers. Keep factoring all the factors until you cannot factor any longer. Common sense tells us that the resulting irreducible numbers are prime numbers; otherwise we could continue the factoring process.

But will you and I always get the same list of factors? 60 = 2 x 30 = 2 x 10 x 3 = 2 x 2 x 5 x 3. However, it is also true that 60 = 6 x 10 = 2 x 3 x 2 x 5. The two lists of prime factors are not identical, unless – unless I am allowed to change the order. You can do that in multiplication, just as in addition. If we write the prime factors in order from smallest to largest, we must always get 2 x 2 x 3 x 5, or 2^2 x 3 x 5. (That first factor is 2 squared.)

Euclid’s Fundamental Theorem of Arithmetic states that every natural number can be factored as a product of prime numbers in only one way, assuming the factors are listed in order from smallest to largest. It is an incredibly important statement in number theory, that has major implications for musical scales.

But even though it seems obvious, as in our example, it is really not a triviality. In some of the more advanced number systems in higher mathematics, the Fundamental Theorem of Arithmetic is not true. Euclid’s genius lay in recognizing the importance of the proof, not to mention carrying out the proof.

Along the way, Euclid invented a technique known today as the Euclidean algorithm (a method) which has the reputation of being one of the most efficient algorithms in all of mathematics. It is so easy, however, a child could do it, because all it involves is long division of whole numbers.

For example, suppose we need to determine the greatest common divisor of 30 and 48 (the largest natural number dividing evenly into both 30 and 48). Euclid said: divide the smaller number into the larger – 30 into 48 – and keep track of the remainder. The number we seek is also the greatest common divisor of the remainder and the divisor.

In this case, 30 goes once into 48, with a remainder of 18. Now I seek the greatest common divisor of 18 and 30. I divide 18 into 30, which goes once with a remainder of 12. Next I seek the greatest common divisor of 12 and 18. I divide 12 into 18, which goes once with a remainder of 6. On we go – divide 6 into 12, which goes twice with a remainder of 0 – and that is the end. Euclid says that your last non-zero remainder, 6, is the number you sought: the greatest common divisor of 30 and 48. 6 divides both 30 and 48, and no larger number does both.

The Euclidean algorithm was pointless in this example, for anyone who knows the multiplication table. For much larger numbers, however, it is invaluable.

Next, Euclid challenges the person who might doubt the Fundamental Theorem of Arithmetic: find me the smallest natural number that can be factored in two irreconcilable ways, with different prime factors. That number must be composite, of course. Line up the two different rows of prime factors. All smaller factors of this number must have unique factorizations themselves. Euclid pulls out his famous algorithm to slowly match up, one pair at a time, the prime factors belonging to the two allegedly different factorizations. Voila! This number does not have two different factorizations after all. Silly goose!

Using this great theorem, we see that all the natural numbers are like molecules with unique chemical formulas. Just as water is always H2O, with two hydrogen atoms and one oxygen atom and never any other chemical elements, so too the number 60 can only be made up of two factors of 2, one factor of 3, and one factor of 5, and never any other prime factors.

The small diagram below shows a view of some of the smaller natural numbers, with 1 at the lower left. Every time you proceed up to the right along a maroon line, you are multiplying by 2. When you move up to the right along a blue line, you are multiplying by 3; green 5; lavender 7.

These are some of Euclid’s infinitely many molecules.

Euclid’s atoms

The Surak blog musical library has grown substantially in the past week, showcasing some of the core orchestral works of the major baroque and classical period composers. All of the recordings are available to the public on Youtube.

As a professor, I want more than merely to make pleasant background sounds available. Many of the works highlighted on this list are some of the top intellectual achievements of the human race. But in order for me to be able to explain why, I need to be able to discuss the details of structure with you. Mathematics may be, at some level, just the study of structure.

Synthesizing the philosophy of the ancient Greeks with that of Immanuel Kant, if that structure reveals itself in the course of time, we are studying arithmetic; if in space, we are studying geometry; and music is merely applied arithmetic.

Though music – even music similar to Western music – was developed across several civilizations, its foundation had its first rigorous theoretical grounding in Euclid’s Elements. Euclid himself summarized a substantial corpus of work by earlier mathematicians of Greece, along with Mesopotamia and Egypt, while adding critical chapters in arithmetic. The same discoveries often appeared in other civilizations, but without the proofs that Euclid first provided.

In order to discuss harmony intelligently, we must understand first the natural numbers: 1, 2, 3, and so on. The reason for this unexpected connection is the physical phenomenon of vibration in objects with fixed boundaries, such as the pipe of a wind instrument, or the string on a stringed instrument. Consider this diagram of the harmonics on a vibrating string, from Wikipedia.

Once the ends of the string are fixed, it seems obvious that the string will vibrate in such a fashion that the fixed points are evenly spaced. In reality, the string’s vibrations form a complex pattern combining these individual harmonics. The frequencies of the various harmonics are natural number multiples of the fundamental frequencies.

It seems to follow intuitively, and can be proven using trigonometry, that various frequencies will go well together if their ratio is a rational number: a fraction that is a quotient of two natural numbers, like 7/5. The smaller the numbers involved in the ratio, the more pleasant the harmony.

Thus as far as the study of harmony is concerned, the major operations are multiplication and division. Euclid discovered something important about these operations, which we can discuss in a future column. Suffice it to say for the time being this discovery has to do with the elements of multiplication, the prime numbers. These are the atoms out of which we will form the molecules of numbers.

Below, I present the periodic table of chemical elements from Wikipedia. Next is a mathematically meaningless whimsical rendition of the prime numbers.

Prime numbers are natural numbers greater than one that are divisible only by themselves and by one. They are immensely important in number theory. It is easy to see that the numbers above are prime. Are there any more prime numbers?

In fact, Euclid proved there are infinitely many prime numbers. Here is his argument. Imagine you only knew finitely many prime numbers: for example, only 2, 3, 5, and 7. Euclid said to do this: multiply together all of your known prime numbers, and either add one or subtract one. In our example, we would multiply 2 x 3 x 5 x 7 to get 210 (a number that is clearly a multiple of 2, 3, 5, and 7); and then either add one to get 211, or subtract one to get 209.

Euclid says that the new number you get is either a prime number itself, or else is at least divisible by some new unknown prime number. How can he be so sure? You can never have two consecutive natural numbers both being multiples of the same natural number greater than one. For example, 89 and 90 cannot both be multiples of 7.

Thus, since we agreed that 210 is, by construction, a multiple of 2, 3, 5, and 7, it follows that 209 and 211 cannot be multiples of 2, 3, 5, or 7. As it turns out, 211 is indeed a prime number, but 209 can be factored as 11 x 19. All these numbers (11, 19, and 211) are new prime numbers we did not know at the beginning of our example.

And of course, if you think you are now done listing all the prime numbers, Euclid says to you: just do my process all over again, and you will find even more prime numbers; the supply is endless.

Here’s looking at Euclid.