I worked at least five years on the theory presented in this article, but it was never published.

"Pornography," said a Supreme Court justice, "can't be defined, but I know what it is when I see it." Many people feel the same way about beauty. We can tell a beautiful face from an ugly one, and the same is true of music, paintings, and architecture. Most mathematicians and theoretical physicists will also tell you that they can tell a beautiful proof or theory from an ugly one, though they will not be able to define that beauty other than to note that it is related to simplicity and elegance.

Periodically my son and I argue about whether classical music is superior to heavy metal rock, where "superior" means, if not beauty, then at least quality. My position is that it is and that I can prove it.  His position is that it is merely a matter of opinion and that I can prove nothing. I will give you my proof that beauty (or quality) is not just subjective, but has a strong objective component, and let you decide who is correct.

To begin at the beginning, we must go back to Carnot's studies of heat engines. These engines, he found, are not reversible. That is, a certain amount of heat energy is always lost when one form of energy is converted into a different form of energy. This discovery is embodied in the second law of thermodynamics, which holds that energy can be changed from one form to another, but one cannot increase the amount of usable energy. Later, Boltzmann found that this idea was equivalent to saying that it is more probable for order to go to disorder than the reverse. Still later, Claude Shannon applied these ideas to the transmission of information and the disorder or noise that arose during transmission.

Let us keep that last idea in mind and consider two pieces of music that I hope we can all agree are totally ugly. The first piece consists of a note of a single pitch and duration repeated endlessly, once a second. In the second piece, the pitches, durations, and intervals of the notes are all random. Why are these songs ugly? I suggest it is because they tell us nothing. They convey no, or almost no, information. To escape ugliness and move towards beauty, information must be conveyed.  And the more information that one conveys for a given number of notes, the more beauty the music has. Beauty is information content per unit data.

Now consider a piece of music that is universally thought of as beautiful, "Danny Boy." Notice how the piece is built on four, four note upwardly moving phrases, each of which starts at a higher note and ends on a note in the dominant chord (C, E, G, and E in the key of C). Notice how the first two notes of the second phrase in the song repeat the last two notes of the preceding phrase.  The song is filled with these relationships and every part is tied to another part. There is no randomness and no exact repetition. In other words, it has a very high information content per unit data.

What is this information? It is not as explicit as "Go to the store." The information is the relationship between the bits of data. In music, a relationship is formed by two or more notes set apart from other notes. A single repetition of a relationship tells us that it is not random relationship, but is a relationship that has meaning. That meaning is the information conveyed. Of course, the endless repetition of the same relationship would use up a lot of data to convey no more information, and we would be back to ugliness again.

On the other hand, if a relationship was never repeated, we would have to conclude that it was a random relationship that conveyed no information and we would also be back to ugliness again. Thus, in order to move towards beauty, we are constrained in the following way: All succeeding
notes must create relationships that repeat some aspect of a prior relationship, but an exact repetition of a prior relationship should be avoided. Because a relationship between notes repeats some part of a previous relationship, we know that the notes have meaning, and that the meaning or information conveyed is the difference between the two relationships. That is why, after the "da-da-da-dum" in Beethoven's Fifth Symphony, the next "da-da-da-dum" is a fourth lower.

One can quickly see how this principle applies to areas other than music. A good novel does not consist of a single repeated word nor does it consist of random words. What happens later in the story ties in to what happened in the beginning of the story. Everything that is said in a good novel is important and relates to the overall story.   Architecture is ugly if one part of a building just repeats another part, or there is no relationship between the parts. Frank Lloyd Wright emphasized unity in his architecture - the entire building expressed a single concept and there was "balance" between the parts, which means that the parts are related in that they repeat part, but not all, of their form. In his most famous structure, Fallingwater, the cantilevered terrace is duplicated several times, but each terrace is a different size and projects from a different angle.

Faces are more difficult to analyze, due to the constraints of biology, but clearly scars,
infections, and other blemishes, which Shannon would regard as noise in the transmission of information, are ugly. Scientists have found that a highly symmetrical face (and body) implies fewer genetic defects since a large, very symmetrical body cannot be grown from a single cell when genetic defects are present. They have also found that this symmetry is seen as attractive to members of the opposite sex. Asymmetries are a random element and constitute a loss of information content since they convey no information but still use data.

As a person grows, his body becomes a more and more ordered system. After maximum growth, the body gradually becomes more and more disordered. If beauty is correlated with entropy, we would expect people to have maximum beauty at the age of minimum entropy, which would be at the end of their period of growth, the late teen years. You may judge for yourself if that is true.

Does this mean that beauty can be objectively analyzed and that my son is wrong because classical music contains more information per note than heavy metal? Not quite. The problem arises with the concept of relatedness. How do we know whether or not a relationship between notes repeats some aspect of a prior relationship between different notes? Surely, the second "da-da-da-dum" in Beethoven's Fifth is related to the first since the rhythm and pitch differences between the notes in each "da-da-da-dum" is identical. But how about Rachmaninoff's "Rhapsody on a Theme of Paganinni," where a fast melody is later played upside down to form a lyrical slow melody? To someone with great musical sensitivity, the two melodies are related, but to the rest of us they are not.

Thus, beauty is objective to the observer who appreciates the relationships, and is subjective to the observer who is too insensitive to appreciate them. Tell me the skill of the observer, and I will tell you if beauty is objective.

So is my son right after all? Well, yes and no. Since he cannot appreciate the relatedness in classical music, to him it contains no more information per note than heavy metal. His criticism of classical music - that it is the same collection of pieces while popular music constantly changes - is consistent with this thesis. One hearing of a popular song is enough for me because that is all it takes to grasp all it has to tell me. That is also nearly true of people who like popular music as well, which explains why popular music quickly becomes dated and abandoned. I can hear Beethoven many times without losing interest, however, because he has so much to say that I can never grasp it all. So there is a sense in which a work is not beautiful, regardless of its information content per unit data, unless the observer is capable of deciphering it.

But suppose the creator of the work intends it to contain all kinds of relationships, which he will cheerfully point out to anyone who asks, but the work still seems ugly because the relationships are just too remote. Is that possible? As an example, suppose he has two notes that move up a major third at the beginning of the piece and two notes that move down a minor third near the end of the piece. He claims they are related and wants "credit" for them in calculating the information content of his work. The relationship is not obvious, even to professional musicians. Do we give him "credit"? This raises an interesting question. Is it possible to objectively measure the amount of beauty - information per unit data - in different works? I think the answer is "no" because it is impossible to know if one set of data is related to another. For example, if it is a C to D that appears isolated at the beginning of a piece, is it related to an isolated C to D at the end? Maybe, if the piece isn't too long. Suppose it's a D to C at the end? Or the C to D isn't isolated or it's a C to E or F at the end?  There is no way to answer these questions.

Is it possible to construct an algorithm that will generate works of great beauty? I doubt it for the reasons given in the preceding paragraph, but there is no harm in trying. Is it "unromantic" to analyze beauty in these terms, like saying a magnificent sunset is just light of various frequencies? Maybe, but if the thesis is correct, wouldn't you rather know about it?  In the end, I would not worry about someone attaching "beauty numbers" to works of art or about the next great composer being a computer.