For example, if I have the data

$$ \begin{array}{l|l|l|l|l|l|l} A & low & & medium & & high & \\ \hline B & standard & new & standard & new & standard & new \\ \hline means & m1 & m2 & m3 & m4 & m5 & m6 \end{array} $$

And I want to create contrasts, then apparently there are 5 that can be created.

$$ \begin{align} C_1 &: \text{New v Standard} \\ C_2 &: \text{Low v High} \\ C_3 &: \text{(Low and High) v Medium} \\ C_4 &: \text{Interaction between $C_1$ and $C_2$} \\ C_5 &: \text{Interaction between $C_1$ and $C_3$} \\ \end{align} $$

If I create the additional contrast

$$ C_6 : \text{Medium v High} $$

What has been violated?

I don't really understand why this isn't an option.

For example, if I have the data

$$ \begin{array}{l|l|l|l|l|l|l} A & low & & medium & & high & \\ \hline B & standard & new & standard & new & standard & new \\ \hline means & m1 & m2 & m3 & m4 & m5 & m6 \end{array} $$

And I want to create contrasts, then apparently there are 5 that can be created.

$$ \begin{align} C_1 &: \text{New v Standard} \\ C_2 &: \text{Low v High} \\ C_3 &: \text{(Low and High) v Medium} \\ C_4 &: \text{Interaction between $C_1$ and $C_2$} \\ C_5 &: \text{Interaction between $C_1$ and $C_3$} \\ \end{align} $$

If I create the additional contrast

$$ C_6 : \text{Medium v High} $$

What has been violated?

I don't really understand why this isn't an option.

Why there are only five

From my notes, with respect to the table provided, all I have to explain this is the following

Since the treatments occupy a six-dimensional vector space, it is only possible to find six orthogonal vectors. One of these dimensions is occupied by the overall mean:

$$ C_0 = \left[ 1 \quad 1 \quad 1 \quad 1 \quad 1 \quad 1 \right] $$

Hence there will only be 5 orthogonal contrasts - one for each treatment degree of freedom.