User:Currywurst44/Consistency Rewrite
An edo represents the q-odd-limit consistently if the closest approximations of the odd harmonics of the q-odd-limit in that edo also give the closest approximations of all the differences between these odd harmonics; for example, if the difference between the closest 7/4 and the closest 5/4 is also the closest 7/5. An edo is distinctly consistent (or uniquely consistent) in the q-odd-limit if every interval in that odd limit is consistent and mapped to a distinct edostep. For example, an edo cannot be distinctly consistent in the 7-odd-limit if it maps 7/5 and 10/7 to the same step (in this case, the semi-octave of 2edo, tempering out 50/49).
The page Minimal consistent edos shows the smallest edo that is consistent or distinctly consistent in a given odd limit while the page Consistency limits of small edos shows the largest odd limit that a given edo is consistent or distinctly consistent in.
Mathematical definition
S shall be a set of intervals and M a tuning's pitch mapping of these intervals. r1 and r2 shall be in S with r1 * r2 = r3 also in S. A tuning is consistent to distance d when the error of all M(r1 * r2) < 1/(2d) and the interval mapping is linear with M(r1 * r2) = M(r1) + M(r2).
A tuning is distinctly consistent when all M(r3) are different.