# Inverse-arithmetic mean

In tuning, the **inverse-arithmetic mean**, **utonal mean**, or **length mean** generates new pitch materials by taking the mean in the inverse-arithmetic scale of pitch i.e. the scale proportional to the length of a string to be divided. It can be said with respect to pitches in frequency as well as intervals in frequency ratios on a certain common fundamental.

The inverse-arithmetic mean *f* of two frequencies *f*_{1} and *f*_{2} is

[math]\displaystyle f = \frac{2}{1/f_1 + 1/f_2}[/math]

Similarly, the inverse-arithmetic mean *r* of two frequency ratios *r*_{1} and *r*_{2} on a common fundamental is

[math]\displaystyle r = \frac{2}{1/r_1 + 1/r_2}[/math]

Unlike mediant, how the ratios are written out has no effect on their inverse-arithmetic mean.

## Examples

The inverse-arithmetic mean of 1/1 and 3/2 is 6/5: 2/(1 + 2/3) = 2/(3/3 + 2/3) = 2/(5/3) = 6/5.

The inverse-arithmetic mean of 5/4 and 6/5 is 60/49: 2/(4/5 + 5/6) = 2/(24/30 + 25/30) = 2/(49/30) = 60/49.

The inverse-arithmetic mean of 9/8 and 10/9 is 180/161: 2/(8/9 + 9/10) = 2/(80/90 + 81/90) = 2/(161/90) = 180/161.

## Generalizations

### To more frequencies or frequency ratios

The inverse-arithmetic mean *f* of *m* frequencies *f*_{1}, *f*_{2}, …, *f*_{m} is

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

The inverse-arithmetic mean *r* of *m* frequency ratios *r*_{1}, *r*_{2}, …, *r*_{m} on a common fundamental is

[math]\displaystyle r = \frac{m}{\sum_{i = 1}^{m} (1/r_i)}[/math]

### To an inverse-arithmetically spaced sequence

This generalization connects the operation to inverse-arithmetic divisions.

The *m* inverse-arithmetic sequence of two frequencies *f*_{1} and *f*_{2} is

[math]\displaystyle \left\lbrace i \in \mathbb {Z} \mid \frac{m}{(i/m)/f_1 + (1 - i/m)/f_2} \right\rbrace[/math]

The *m* inverse-arithmetic sequence of two frequency ratios *r*_{1} and *r*_{2} on a common fundamental is

[math]\displaystyle \left\lbrace i \in \mathbb {Z} \mid \frac{m}{(i/m)/r_1 + (1 - i/m)/r_2} \right\rbrace[/math]

The inverse-arithmetic mean is found by setting *i* = 1 and *m* = 2.

## Terminology

The term *inverse-arithmetic mean* signifies that it is the inverse of the arithmetic mean. The term *utonal mean* reflects the fact that it forms an utonal sequence by taking such a mean in JI. It would have made sense to call it *subharmonic mean* if not for an established usage of *harmonic mean* in math to mean the same thing, since the sense in math is hardcoded in terms of length instead of the more intuitive measurement of frequency.