Damage is a measurement of how much an interval deviates from another interval it is approximating. In the simplest case it is the absolute value of error, and variations are defined where the error is also weighted, typically by a complexity measure.
The use of the term "damage" in the modern sense was popularized by Paul Erlich through his paper A Middle Path, published in August of 2004. Leading up to the publication of this paper, Erlich had begun using it on the Yahoo Groups tuning lists, where it had already started to gain some traction with other members of the community.
In Middle Path, Erlich used the word "damage" to refer to the absolute value of error weighted by harmonic distance. As the term damage grew in popularity and began to be generalized to be weighted by other complexities (such as TE), the original measurement eventually came to assume the more specific and technical name of "TOP damage". The "TOP" part of the name is due to the fact that when Erlich first described it, he described it in the context TOP tuning, a tuning which is designed to minimize this particular type of damage. This sets a historical precedent, then, that other types of damage may be named either after a tuning that minimizes it, or the complexity used as a weight in its calculation (sometimes these are the same thing).
- Dave Keenan & Douglas Blumeyer's guide to RTT: tuning fundamentals#Damage: for a step-by-step textbook tutorial style introduction to this topic
- Erlich referred to error as "mistuning" in this paper.