Root mean square
In mathematics and tuning, the root mean square (RMS) of two frequencies [math]\displaystyle{ f_1 }[/math] and [math]\displaystyle{ f_2 }[/math] is equal to [math]\displaystyle{ \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 RMS tuning and RMS temperament measures.
Examples
The root mean square of 1/1 (0 ¢) and 3/2 (≈ 701.955 ¢) is [math]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ \operatorname {RMS}(\frac{9}{8}, \frac{10}{9}) = \sqrt{\frac{12961}{10368}} }[/math] (≈ 193.224 ¢).