Geometric mean: Difference between revisions

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: ''"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''1 and ''f''2 is

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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>

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

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>

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

=== To an equally spaced sequence ===

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>

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 ==

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* [[Mediant]]

[[Category:~~Theory~~]]

[[Category:Pythagorean means]]

[[Category:Terms]]

[[Category:Elementary math]]

@@ Line 1: / Line 1: @@
 : ''"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
@@ Line 25: / Line 25: @@
 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
-<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 ===
 This generalization connects the operation to [[equal tuning]]s.
@@ Line 43: / Line 42: @@
 <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 ==
@@ Line 54: / Line 53: @@
 * [[Mediant]]
-[[Category:Theory]]
+[[Category:Pythagorean means]]
 [[Category:Terms]]
 [[Category:Elementary math]]