Geometric mean: Difference between revisions

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: ''"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. 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.

In tuning, the '''geometric mean''', '''pitch mean''', or simply '''mean''' generates new pitch materials by taking the mean in the [[Wikipedia: Logarithmic scale|logarithmic scale]] of pitch i.e. the scale proportional to the logarithm of frequency. 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''1 and ''f''2 is

The geometric mean ''f'' of two frequencies ''f''1 and ''f''2 is

<math>\displaystyle f = \sqrt {f_1 f_2}</math>

Similarly, the ~~logarithmic~~ mean ''r'' of two frequency ratios ''r''1 and ''r''2 on a common fundamental is

Similarly, the geometric mean ''r'' of two frequency ratios ''r''1 and ''r''2 on a common fundamental is

<math>\displaystyle r = \sqrt {r_1 r_2}</math>

Unlike [[mediant]], how the ratios are written out has no effect on their ~~logarithmic~~ mean.

Unlike [[mediant]], how the ratios are written out has no effect on their geometric mean.

== Examples ==

The ~~logarithmic~~ mean of [[1/1]] and [[3/2]] is sqrt (3/2): sqrt (1 × 3/2) = sqrt (3/2).

The geometric 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 geometric 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).

The geometric 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 ''f''1, ''f''2, …, ''f''''m'' is

The geometric mean ''f'' of ''m'' frequencies ''f''1, ''f''2, …, ''f''''m'' is

<math>\displaystyle f = (\prod_{i = 1}^{m} f_i)^{1/m}</math>

The ~~logarithmic~~ mean ''r'' of ''m'' frequency ratios ''r''1, ''r''2, …, ''r''''m'' on a common fundamental is

The geometric mean ''r'' of ''m'' frequency ratios ''r''1, ''r''2, …, ''r''''m'' on a common fundamental is

<math>\displaystyle r = (\prod_{i = 1}^{m} r_i)^{1/m}</math>

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<math>\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.

The geometric 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]].

The term ''geometric mean'' comes from math. See [[Wikipedia: Geometric mean]]. It would have made sense to call it ''logarithmic mean'' but for its established usage in math to mean something else. See [[Wikipedia: Logarithmic mean]].

== See also ==

@@ Line 1: / Line 1: @@
 : ''"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. 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.
+In tuning, the '''geometric mean''', '''pitch mean''', or simply '''mean''' generates new pitch materials by taking the mean in the [[Wikipedia: Logarithmic scale|logarithmic scale]] of pitch i.e. the scale proportional to the logarithm of frequency. 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 geometric mean ''f'' of two frequencies ''f''<sub>1</sub> and ''f''<sub>2</sub> is
 <math>\displaystyle f = \sqrt {f_1 f_2}</math>
-Similarly, the logarithmic mean ''r'' of two frequency ratios ''r''<sub>1</sub> and ''r''<sub>2</sub> on a common fundamental is
+Similarly, the geometric mean ''r'' of two frequency ratios ''r''<sub>1</sub> and ''r''<sub>2</sub> on a common fundamental is
 <math>\displaystyle r = \sqrt {r_1 r_2}</math>
-Unlike [[mediant]], how the ratios are written out has no effect on their logarithmic mean.
+Unlike [[mediant]], how the ratios are written out has no effect on their geometric mean.
 == Examples ==
-The logarithmic mean of [[1/1]] and [[3/2]] is sqrt (3/2): sqrt (1 × 3/2) = sqrt (3/2).
+The geometric 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 geometric 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).
+The geometric 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 ''f''<sub>1</sub>, ''f''<sub>2</sub>, …, ''f''<sub>''m''</sub> is
+The geometric mean ''f'' of ''m'' frequencies ''f''<sub>1</sub>, ''f''<sub>2</sub>, …, ''f''<sub>''m''</sub> is
 <math>\displaystyle f = (\prod_{i = 1}^{m} f_i)^{1/m}</math>
-The logarithmic mean ''r'' of ''m'' frequency ratios ''r''<sub>1</sub>, ''r''<sub>2</sub>, …, ''r''<sub>''m''</sub> on a common fundamental is
+The geometric mean ''r'' of ''m'' frequency ratios ''r''<sub>1</sub>, ''r''<sub>2</sub>, …, ''r''<sub>''m''</sub> on a common fundamental is
 <math>\displaystyle r = (\prod_{i = 1}^{m} r_i)^{1/m}</math>
@@ Line 43: / Line 43: @@
 <math>\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.
+The geometric 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]].
+The term ''geometric mean'' comes from math. See [[Wikipedia: Geometric mean]]. It would have made sense to call it ''logarithmic mean'' but for its established usage in math to mean something else. See [[Wikipedia: Logarithmic mean]].
 == See also ==