Arithmetic mean

In tuning, the arithmetic mean or otonal mean generates new pitch materials by taking the mean in the arithmetic scale i.e. frequency. It can be said with respect to frequencies or frequency ratios on a certain common fundamental.

The arithmetic mean f of two frequencies f₁ and f₂ is

[math]\displaystyle{ \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{ \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) = 161/144.

Generalizations

To more frequencies or frequency ratios

The arithmetic mean f of m frequencies f₁, f₂, …, f_m is

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

The arithmetic mean r of m frequency ratios r₁, r₂, …, r_m on a common fundamental is

[math]\displaystyle{ \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{ \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{ \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 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.

See also

• Pythagorean means
  • Logarithmic mean
  • Inverse-arithmetic mean
• Mediant