Traditionalists, he asserts, would consider each passage a separate dimension. Suppose that a unique rank ordering supports two different linearizations. These linearizations must maintain the rank order, but on their different equal-interval metrics.
Cubes can be rank-ordered in size by volume or by length of a side.
These always yield the same rank order. The crux of the matter is the definition of the latent variable. Clearly the volume scale, not the side-length scale. Interval scaling requires a concatenation rule. Then there is only one linear scale, which is defined by the concatenation.