Geometric mean - Revision history

ArrowHead294 at 19:52, 20 February 2026

2026-02-20T19:52:01Z

← Older revision		Revision as of 19:52, 20 February 2026
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	The geometric 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>		<math>\displaystyle f = \left(\prod_{i = 1}^{m} f_i\right)^{1/m}</math>

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

	=== To an equally spaced sequence ===		=== To an equally spaced sequence ===

	This generalization connects the operation to [[equal tuning]]s.		This generalization connects the operation to [[equal tuning]]s.

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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>		<math>\displaystyle \left\lbrace i \in \mathbb {Z} \mid r_1^{i/m} \cdot r_2^{1 - i/m} \right\rbrace</math>

	The geometric mean is found by setting ''i'' = 1 and ''m'' = 2.		The geometric mean is found by setting {{nowrap\|''i'' {{=}} 1}} and {{nowrap\|''m'' {{=}} 2}}.

	== Terminology ==		== Terminology ==

Fredg999 category edits: Moving from Category:Theory to Category:Pythagorean means using Cat-a-lot

2023-03-04T20:23:15Z

Moving from Category:Theory to Category:Pythagorean means using Cat-a-lot

← Older revision		Revision as of 20:23, 4 March 2023
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	* [[Mediant]]		* [[Mediant]]

	[[Category:~~Theory~~]]		[[Category:Pythagorean means]]
	[[Category:Terms]]		[[Category:Terms]]
	[[Category:Elementary math]]		[[Category:Elementary math]]

FloraC: Clarify that the arguments in this context are pitches or intervals, rather than arbitrary numbers

2023-03-04T10:04:51Z

Clarify that the arguments in this context are pitches or intervals, rather than arbitrary numbers

← Older revision		Revision as of 10:04, 4 March 2023
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	: ''"Mean" redirects here. For other types, see [[Pythagorean means]].''		: ''"Mean" redirects here. For other types, see [[Pythagorean means]].''

	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, such as [[cent]]s. 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, such as [[cent]]s. It can be said with respect to pitches in frequency as well as intervals in 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 geometric 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

FloraC: Mention cents to ease beginners

2023-03-03T18:25:38Z

Mention cents to ease beginners

← Older revision		Revision as of 18:25, 3 March 2023
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	: ''"Mean" redirects here. For other types, see [[Pythagorean means]].''		: ''"Mean" redirects here. For other types, see [[Pythagorean means]].''

	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.		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, such as [[cent]]s. 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 geometric 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

FloraC: FloraC moved page Logarithmic mean to Geometric mean without leaving a redirect: "Logarithmic mean" is in conflict

2023-03-03T18:22:13Z

FloraC moved page Logarithmic mean to Geometric mean without leaving a redirect: "Logarithmic mean" is in conflict

← Older revision	Revision as of 18:22, 3 March 2023
(No difference)

FloraC: Eliminate "logarithmic mean" since it's in conflict

2023-03-03T18:13:42Z

Eliminate "logarithmic mean" since it's in conflict

FloraC at 15:42, 2 March 2023

2023-03-02T15:42:38Z

← Older revision		Revision as of 15:42, 2 March 2023
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	: ''"Mean" redirects here. For other types, see [[Pythagorean ~~mean~~]].''		: ''"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
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	== See also ==		== See also ==
	* [[Arithmetic mean]]		* [[Pythagorean means]]
	* [[Inverse-arithmetic mean]]		** [[Arithmetic mean]]
			** [[Inverse-arithmetic mean]]
	* [[Mediant]]		* [[Mediant]]

FloraC: 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..."

2023-03-02T12:03:31Z

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..."

New page

: ''"Mean" redirects here. For other types, see [[Pythagorean mean]].''

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.

The logarithmic 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

<math>\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 ''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

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

=== To an equally spaced sequence ===

This generalization connects the operation to [[equal tuning]]s.

The ''m'' equal sequence of two frequencies ''f''1 and ''f''2 is

<math>\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 ''r''1 and ''r''2 on a common fundamental is

<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.

== Terminology ==
The term ''logarithmic mean'' was coined by analogy to ''arithmetic mean''. The term ''geometric mean'' comes from math. See [[Wikipedia: Geometric mean]].

== See also ==
* [[Arithmetic mean]]
* [[Inverse-arithmetic mean]]
* [[Mediant]]

[[Category:Theory]]
[[Category:Terms]]
[[Category:Elementary math]]

← Older revision		Revision as of 18:13, 3 March 2023
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	: ''"Mean" redirects here. For other types, see [[Pythagorean means]].''		: ''"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>		<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>		<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 ==		== 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 ==		== Generalizations ==
	=== To more frequencies or frequency ratios ===		=== 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>		<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>		<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>		<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 ==		== 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 ==		== See also ==