Root mean square
In mathematics and tuning, the root mean square (RMS) of two frequencies [math]f_1[/math] and [math]f_2[/math] is equal to [math]\sqrt{\frac{f_1^{2} + f_2^{2}}{2}}[/math]. The RMS is also known as the quadratic mean.
In regular temperament theory, it is used in the context of tuning schemes such as TOP-RMS and miniRMS, as well as RMS temperament measures.
Examples
The root mean square of 1/1 (0 ¢) and 3/2 (≈ 701.955 ¢) is [math]\operatorname {RMS}(\frac{1}{1}, \frac{3}{2}) = \sqrt{\frac{13}{8}}[/math] (≈ 420.264 ¢).
The root mean square of 5/4 (≈ 386.314 ¢) and 6/5 (≈ 315.641 ¢) is [math]\operatorname {RMS}(\frac{5}{4}, \frac{6}{5}) = \sqrt{\frac{1201}{800}}[/math] (≈ 351.699 ¢).
The root mean square of 9/8 (≈ 203.910 ¢) and 10/9 (≈ 182.404 ¢) is [math]\operatorname {RMS}(\frac{9}{8}, \frac{10}{9}) = \sqrt{\frac{12961}{10368}}[/math] (≈ 193.224 ¢).