Geometric mean
- "Mean" redirects here. For other types, see Pythagorean mean.
In tuning, the logarithmic mean, geometric mean, or simply mean generates new pitch materials by taking the mean in the logarithmic scale i.e. pitch. It can be said with respect to frequencies or frequency ratios on a certain common fundamental.
The logarithmic mean f of two frequencies f1 and f2 is
[math]\displaystyle{ \displaystyle f = \sqrt {f_1 f_2} }[/math]
Similarly, the logarithmic mean r of two frequency ratios r1 and r2 on a common fundamental is
[math]\displaystyle{ \displaystyle r = \sqrt {r_1 r_2} }[/math]
Unlike mediant, how the ratios are written out has no effect on their logarithmic mean.
Examples
The logarithmic mean of 1/1 and 3/2 is sqrt (3/2): sqrt (1 × 3/2) = sqrt (3/2).
The logarithmic mean of 5/4 and 6/5 is sqrt (3/2): sqrt ((5/4)(6/5)) = sqrt (6/4) = sqrt (3/2).
The logarithmic mean of 9/8 and 10/9 is sqrt (5/4): sqrt ((9/8)(10/9)) = sqrt (10/8) = sqrt (5/4).
Generalizations
To more frequencies or frequency ratios
The logarithmic mean f of m frequencies f1, f2, …, fm is
[math]\displaystyle{ \displaystyle f = (\prod_{i = 1}^{m} f_i)^{1/m} }[/math]
The logarithmic mean r of m frequency ratios r1, r2, …, rm on a common fundamental is
[math]\displaystyle{ \displaystyle r = (\prod_{i = 1}^{m} r_i)^{1/m} }[/math]
To an equally spaced sequence
This generalization connects the operation to equal tunings.
The m equal sequence of two frequencies f1 and f2 is
[math]\displaystyle{ \displaystyle \left\lbrace i \in \mathbb {Z} \mid f_1^{i/m} \cdot f_2^{1 - i/m} \right\rbrace }[/math]
The m equal sequence of two frequency ratios r1 and r2 on a common fundamental is
[math]\displaystyle{ \displaystyle \left\lbrace i \in \mathbb {Z} \mid r_1^{i/m} \cdot r_2^{1 - i/m} \right\rbrace }[/math]
The logarithmic mean is found by setting i = 1 and m = 2.
Terminology
The term logarithmic mean was coined by analogy to arithmetic mean. The term geometric mean comes from math. See Wikipedia: Geometric mean.