User:Overthink/Tuning ranges of just ratios

Musicians often use temperaments to approximate ratios in just intonation. However, it is not always clear whether or not a tempered interval is close enough to represent that just ratio. This page explores some systems to determine what approximations of a just ratio are "acceptable".

(Avoiding) Ambiguity

Lets say that a minor third is tuned between the just 7/6 (267 ¢) and 6/5 (316 ¢) ratios. Where is the line such that a minor third above this line is more likely heard as 6/5, and a minor third below this line is more likely heard as 7/6? A simple answer is halfway between them, or sqrt(7/5) (291 ¢). However, this does not account for the fact that simple ratios are more likely to be registered than complex ones, so the line should be lower to expand the range of 6/5 and shrink the range of 7/6.