Arithmetic mean
In tuning, the arithmetic mean, otonal mean, or frequency mean generates new pitch materials by taking the mean in the arithmetic scale of pitch i.e. the scale proportional to frequency. It can be said with respect to pitches in frequency as well as intervals in frequency ratios on a certain common fundamental.
The arithmetic mean f of two frequencies f1 and f2 is
[math]\displaystyle f = (f_1 + f_2)/2[/math]
Similarly, the arithmetic mean r of two frequency ratios r1 and r2 on a common fundamental is
[math]\displaystyle r = (r_1 + r_2)/2[/math]
Unlike mediant, how the ratios are written out has no effect on their arithmetic mean.
Examples
The arithmetic mean of 1/1 and 3/2 is 5/4: (1 + 3/2)/2 = (2/2 + 3/2)/2 = 5/4.
The arithmetic mean of 5/4 and 6/5 is 49/40: (5/4 + 6/5)/2 = (25/20 + 24/20)/2 = 49/40.
The arithmetic mean of 9/8 and 10/9 is 161/144: (9/8 + 10/9)/2 = (81/72 + 80/72)/2 = 161/144.
Generalizations
To more frequencies or frequency ratios
The arithmetic mean f of m frequencies f1, f2, …, fm is
[math]\displaystyle f = \sum_{i = 1}^{m} f_i/m[/math]
The arithmetic mean r of m frequency ratios r1, r2, …, rm on a common fundamental is
[math]\displaystyle r = \sum_{i = 1}^{m} r_i/m[/math]
To an arithmetically spaced sequence
This generalization connects the operation to arithmetic divisions.
The m arithmetic sequence of two frequencies f1 and f2 is
[math]\displaystyle \left\lbrace i \in \mathbb {Z} \mid (i/m)f_1 + (1 - i/m)f_2 \right\rbrace[/math]
The m arithmetic sequence of two frequency ratios r1 and r2 on a common fundamental is
[math]\displaystyle \left\lbrace i \in \mathbb {Z} \mid (i/m)r_1 + (1 - i/m)r_2 \right\rbrace[/math]
The arithmetic mean is found by setting i = 1 and m = 2.
Terminology
The term arithmetic mean comes from math. See Wikipedia: Arithmetic mean. The term otonal mean reflects the fact that it forms an otonal sequence by taking such a mean in JI. It would have made sense to call it harmonic mean if not for its established usage in math to mean the inverse-arithmetic mean since it is hardcoded in terms of length instead of the more intuitive measurement of frequency.