Geometric mean
- "Mean" redirects here. For other types, see Pythagorean means.
In tuning, the geometric mean, pitch mean, or simply mean generates new pitch materials by taking the mean in the logarithmic scale of pitch i.e. the scale proportional to the logarithm of frequency, such as cents. It can be said with respect to pitches in frequency as well as intervals in frequency ratios on a certain common fundamental. The idea of treating quarter-comma meantone as the "strict" meantone is backed by this type of mean.
The geometric mean f of two frequencies f1 and f2 is
[math]\displaystyle f = \sqrt {f_1 f_2}[/math]
Similarly, the geometric mean r of two frequency ratios r1 and r2 on a common fundamental is
[math]\displaystyle r = \sqrt {r_1 r_2}[/math]
Unlike mediant, how the ratios are written out has no effect on their geometric mean.
Examples
The geometric mean of 1/1 and 3/2 is sqrt (3/2): sqrt (1 × 3/2) = sqrt (3/2).
The geometric mean of 5/4 and 6/5 is sqrt (3/2): sqrt ((5/4)(6/5)) = sqrt (6/4) = sqrt (3/2).
The geometric 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 geometric mean f of m frequencies f1, f2, …, fm is
[math]\displaystyle f = (\prod_{i = 1}^{m} f_i)^{1/m}[/math]
The geometric mean r of m frequency ratios r1, r2, …, rm on a common fundamental is
[math]\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 \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 \left\lbrace i \in \mathbb {Z} \mid r_1^{i/m} \cdot r_2^{1 - i/m} \right\rbrace[/math]
The geometric mean is found by setting i = 1 and m = 2.
Terminology
The term geometric mean comes from math. See Wikipedia: Geometric mean. It would have made sense to call it logarithmic mean but for its established usage in math to mean something else. See Wikipedia: Logarithmic mean.