# Category Theory Package

# What is symmetry breaking?

To understand symmetry breaking we must first understand what a symmetry is. For example, a wall painted a uniform color exhibits maximum symmetry. This is because no patch on the wall is any different than any other patch. The moment that you hang a picture on that wall the symmetry is broken since there is a portion of the wall that is no longer the same as all the others.

But notice that even in the case of symmetry breaking, there are good brakes and bad breaks, consonant symmetry breaking and dissonant symmetry breaking.

For example, hanging a picture in the exact center of the wall preserves as much of the symmetry as possible from the wall. This is because even though there is an area of the wall that is different than the others, it is equally distant from the borders, and in all regards is its most symmetric placement. Similarly, if I hang the picture closer to the right side but halfway between the roof and the ceiling, I’ve once again broken the symmetry, but not as evenly as in the previous case. And finally, if I hang the picture in a random non-central location, then I have maximally broken the symmetry since the patch of wall that is different is no longer equally distant to anything.

In CGT, I apply the principle of symmetry breaking in a similar fashion with the nature of the symmetry breaking correlating with musical concepts of consonance and dissonance, as well as the structures of triads and chords. Thus the central mathematical thread that unites the theory starts with the notion of an infinite line of positive frequency as reflecting maximal symmetry with the selection of pitch and their subsequent organization into pitch-classes as two symmetry breaking events which define most if not all of the structures in contemporary music theory.

Due to my personal background with symmetry breaking [ref] and its usage within physics theory[ref], in this site will provide a series of tutorials which will showcase a physicist’s view on symmetry and symmetry breaking. This will set up the distinction between spontaneous symmetry breaking event and a explicit symmetry breaking event as well as providing insight, analogies, and visualizations which will ultimately help us understand how symmetry breaking works within this one-dimensional frequency model and eventually music.