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 f1 and f2 is
[math]\displaystyle f = \frac{2}{1/f_1 + 1/f_2}[/math]
Similarly, the inverse-arithmetic mean r of two frequency ratios r1 and r2 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 f1, f2, …, fm is
[math]\displaystyle f = \frac{m}{\sum_{i = 1}^{m} (1/f_i)}[/math]
The inverse-arithmetic mean r of m frequency ratios r1, r2, …, rm 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 f1 and f2 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 r1 and r2 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.