Relative interval error
This article is about the error of intervals measured in relative cents. For the relative error of temperaments, see Tenney-Euclidean temperament measures #TE simple badness.
The relative error of an interval in an edo is the interval's error in cents divided by the cents of an edostep, or equivalently stated, the error in relative cents.
For example, in 24-edo, 3/2 has an absolute error of about -2¢, meaning that the nearest interval in the edo is about 2¢ flat of 3/2. One edostep is 50¢, and -2 / 50 = -0.04, therefore the relative error is about -4% or -4 relative cents. In contrast, 12-edo has the same absolute error, but a smaller relative error of -2%. (In fact, 12-edo's absolute and relative errors are always identical.)
Formula
The formula for closest mapping of any JI interval is
[math]\displaystyle{ e(n, r) = (\text{round} (n \log_2 r) - n \log_2 r) \times 100\% }[/math]
where n is the edo number and r is the targeted frequency ratio.
The unit of relative error is relative cent or percent.
With a direct mapping via the ratio's cents, the relative error ranges from -50% to +50%. With an indirect mapping via patent val or other val, it can be farther from zero. To obtain the relative error in patent val mapping, first find the relative errors of each prime, and then find the dot product of this vector with the ratio's monzo.
Additivity
There are two additivities of relative errors.
First, for the same edo, a ratio which is the product of some other ratios have their relative errors additive, that is, if r3 = r1r2 for n, then e (n, r3) = e (n, r1) + e (n, r2).
If the error exceeds the range -50% to +50%, it indicates that an inconsistency occurs, and there is a discrepancy in patent val mapping and closest mapping, so is the error. The patent val mapping error is unchanged, and that of closest mapping is the previous result reduced by an integer to fit it into the range.
For example, the errors of 2/1, 3/1 and 5/1 in 19-edo are 0, -11.43% and -11.66%, respectively. Since 6/5 = (2/1)(3/1) / (5/1), its error is 0 + (-11.43%) - (-11.66%) = 0.23%. That shows 19-edo has fairly flat fifths and major thirds, yet they cancel out when it comes to minor thirds and results in a very accurate approximation.
Second, for the same ratio, an edo which is the sum of some other edos have their relative errors additive, that is, if n3 = n1 + n2 for r, then e (n3, r) = e (n1, r) + e (n2, r). This also needs to be reduced by an integer to fit into the range. In special, if an edo duplicates itself, and if the mappings do not change, then the error also duplicates.
For example, the errors of 3/1 for 26-edo and 27-edo are -20.90% and +20.60%, repectively, and their sum -0.30% is the error of 3/1 for 53-edo.