User:Overthink/Tuning ranges of just ratios

Temperaments are often used 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 (266.9 ¢) and 6/5 (315.6 ¢) 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) (291.3 ¢). 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. Amazingly, there is a simple method to draw a line that does exactly that: the mediant. The mediant of two ratios [math]\displaystyle{ \tfrac {a}{b} }[/math] and [math]\displaystyle{ \tfrac {c}{d} }[/math] is [math]\displaystyle{ \tfrac {a+b}{c+d} }[/math]. The mediant is weighted toward the more complex ratio, so the tuning range of the more complex ratio is stricter than that of the simpler one. In this case, the mediant of 7/6 and 6/5 is (7+6)/(6+5) = 13/11 (289.2 ¢), which is slightly closer to 7/6 than to 6/5.

However, an interval near 13/11 will likely be very ambiguous between 7/6 and 6/5 or percieved as neither, so the actual range where a just interval sounds like 7/6 or 6/5 is narrower. Thus we may want to tighten the ranges by a factor [math]\displaystyle{ k>1 }[/math]. For example, if we let [math]\displaystyle{ k = 1.5 }[/math], then the upper range of 7/6 is 281.8 ¢, and the lower range of 6/5 is 298.0 ¢.

We use similar logic to find the boundary of 6/5 and 5/4: Their mediant is 11/9, an interval of 347.4 ¢. Letting [math]\displaystyle{ k = 1.5 }[/math], we find that the upper range of 6/5 is 336.8 ¢, and the lower range of 5/4 is 360.4 ¢. Considering 7/6, 6/5, and 5/4 as target intervals, we find the range for 6/5 is from 298.0 ¢ (17.6 ¢ flat) to 336.8 ¢ (21.2 ¢ sharp). We see that the range is wider sharpwards than flatwards, so our formula is not perfect. It also depends heavily on the target intervals; for example, if 13/11 and 11/9 were target intervals, then the range for 6/5 would be much narrower.