# Variance

Variance is a property of vectors which indicates how their entries vary with respect to the system's units: covectors vary in the same direction as the units, hence the co- prefix meaning "with", while (contra)vectors vary in the opposite direction to the units, hence the contra- prefix meaning "against".

In other words, let's say we divide octaves (our units) into 1200 parts to get cents, so that they're now 1/1200th the size they used to be. If we wanted our system to represent the same information, then, we have a couple options for recalibrating with respect to the new units.

1. We could choose to change our vectors. If we choose this, we must multiply the numbers in each of our vectors by 1200.
2. We could choose to change our covectors. If we choose this, we must divide the numbers in each of our covectors by 1200.

So we say that the vectors vary in the opposite direction to the units, but the covectors vary in the same direction as the units. In other words the vectors are "contravariant" with the units and the covectors are "covariant" with the units.

## Footnotes

1. This notion of variance is not to be confused with the unrelated notion of variance in statistics.
2. Interestingly, regular temperament theoreticians are more similar to physicists than mathematicians in a certain respect: we have units of measurement associated with our vectors, such as octaves or cents.