In "The Geometry of Music", Dmitri Tymockzo discusses voice crossings in measures of voice leading sizes:
"I consider a measure of voice-leading size to be “reasonable” if it satisfies two basic requirements: first, it should depend only on how far the individual voices move, with larger motion leading to larger voice leadings; and second, it should judge voice leadings with voice crossings to be larger (or at the very least, no smaller) than their natural uncrossed alternatives."
"I propose that any method of measuring voice leading is acceptable, as long as it does not have the counterintuitive consequence that “voice crossings” make a voice leading smaller. It is actually quite easy to formulate this principle mathematically: if a metric is not to favor voice crossings, then it must treat the collection of distances {x1 + c, x2, …, xn} as being at least as large as {x1, x2 + c, …, xn}, whenever x1 > x2 and c ≥ 0. Intuitively, this means that the metric should not prefer an uneven distribution of distances, such as {4, 0, 0} over a more even distribution, such as {1, 1, 2}"
Why should voice leadings with voice crossings be judgded as larger than their uncrossed alternatives? And why does such metric does not prefer an uneven distribution of distances over a more even one?