Geometric mean: Difference between revisions
Created page with ": ''"Mean" redirects here. For other types, see Pythagorean mean.'' In tuning, the '''logarithmic mean''', '''geometric mean''', or simply '''mean''' generates new pitch..." |
mNo edit summary |
||
| Line 1: | Line 1: | ||
: ''"Mean" redirects here. For other types, see [[Pythagorean | : ''"Mean" redirects here. For other types, see [[Pythagorean means]].'' | ||
In tuning, the '''logarithmic mean''', '''geometric mean''', or simply '''mean''' generates new pitch materials by taking the mean in the [[Wikipedia: Logarithmic scale|logarithmic scale]] i.e. pitch. It can be said with respect to frequencies or frequency ratios on a certain common fundamental. | In tuning, the '''logarithmic mean''', '''geometric mean''', or simply '''mean''' generates new pitch materials by taking the mean in the [[Wikipedia: Logarithmic scale|logarithmic scale]] i.e. the scale of pitch. It can be said with respect to frequencies or 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 logarithmic mean ''f'' of two frequencies ''f''<sub>1</sub> and ''f''<sub>2</sub> is | The logarithmic mean ''f'' of two frequencies ''f''<sub>1</sub> and ''f''<sub>2</sub> is | ||
| Line 49: | Line 49: | ||
== See also == | == See also == | ||
* [[Arithmetic mean]] | * [[Pythagorean means]] | ||
* [[Inverse-arithmetic mean]] | ** [[Arithmetic mean]] | ||
** [[Inverse-arithmetic mean]] | |||
* [[Mediant]] | * [[Mediant]] | ||
Revision as of 15:42, 2 March 2023
- "Mean" redirects here. For other types, see Pythagorean means.
In tuning, the logarithmic mean, geometric mean, or simply mean generates new pitch materials by taking the mean in the logarithmic scale i.e. the scale of pitch. It can be said with respect to frequencies or 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 logarithmic mean f of two frequencies f1 and f2 is
[math]\displaystyle{ \displaystyle f = \sqrt {f_1 f_2} }[/math]
Similarly, the logarithmic mean r of two frequency ratios r1 and r2 on a common fundamental is
[math]\displaystyle{ \displaystyle r = \sqrt {r_1 r_2} }[/math]
Unlike mediant, how the ratios are written out has no effect on their logarithmic mean.
Examples
The logarithmic mean of 1/1 and 3/2 is sqrt (3/2): sqrt (1 × 3/2) = sqrt (3/2).
The logarithmic mean of 5/4 and 6/5 is sqrt (3/2): sqrt ((5/4)(6/5)) = sqrt (6/4) = sqrt (3/2).
The logarithmic 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 logarithmic mean f of m frequencies f1, f2, …, fm is
[math]\displaystyle{ \displaystyle f = (\prod_{i = 1}^{m} f_i)^{1/m} }[/math]
The logarithmic mean r of m frequency ratios r1, r2, …, rm on a common fundamental is
[math]\displaystyle{ \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 f1 and f2 is
[math]\displaystyle{ \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 r1 and r2 on a common fundamental is
[math]\displaystyle{ \displaystyle \left\lbrace i \in \mathbb {Z} \mid r_1^{i/m} \cdot r_2^{1 - i/m} \right\rbrace }[/math]
The logarithmic mean is found by setting i = 1 and m = 2.
Terminology
The term logarithmic mean was coined by analogy to arithmetic mean. The term geometric mean comes from math. See Wikipedia: Geometric mean.