Root mean square: Difference between revisions
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The root mean square of [[1/1]] (0{{cent}}) and [[3/2]] (≈701.955{{cent}}) is <math>RMS(\frac{1}{1}, \frac{3}{2}) = \sqrt{\frac{13}{8}}</math> (≈420.264{{cent}}). | The root mean square of [[1/1]] (0{{cent}}) and [[3/2]] (≈701.955{{cent}}) is <math>RMS(\frac{1}{1}, \frac{3}{2}) = \sqrt{\frac{13}{8}}</math> (≈420.264{{cent}}). | ||
The root mean square of [[5/4]] (≈386.314{{cent}}) and [[6/5]] (≈315.641{{cent}}) is <math>\sqrt{\frac{1201}{800}}</math> (≈351.699{{cent}}). | The root mean square of [[5/4]] (≈386.314{{cent}}) and [[6/5]] (≈315.641{{cent}}) is <math>RMS(\frac{5}{4}, \frac{6}{5}) = \sqrt{\frac{1201}{800}}</math> (≈351.699{{cent}}). | ||
The root mean square of [[9/8]] (≈203.910{{cent}}) and [[10/9]] (≈182.404{{cent}}) is <math>\sqrt{\frac{12961}{10368}}</math> (≈193.224{{cent}}). | The root mean square of [[9/8]] (≈203.910{{cent}}) and [[10/9]] (≈182.404{{cent}}) is <math>RMS(\frac{9}{8}, \frac{10}{9}) = \sqrt{\frac{12961}{10368}}</math> (≈193.224{{cent}}). | ||
== See also == | == See also == | ||
Revision as of 02:23, 21 March 2023
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{ 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{ 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{ RMS(\frac{9}{8}, \frac{10}{9}) = \sqrt{\frac{12961}{10368}} }[/math] (≈193.224 ¢).