# 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 *f*_{1} and *f*_{2} is

[math]\displaystyle f = \sqrt {f_1 f_2}[/math]

Similarly, the geometric mean *r* of two frequency ratios *r*_{1} and *r*_{2} 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 *f*_{1}, *f*_{2}, …, *f*_{m} is

[math]\displaystyle f = (\prod_{i = 1}^{m} f_i)^{1/m}[/math]

The geometric mean *r* of *m* frequency ratios *r*_{1}, *r*_{2}, …, *r*_{m} 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 *f*_{1} and *f*_{2} 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 *r*_{1} and *r*_{2} 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.