Geometric mean: Difference between revisions
Clarify that the arguments in this context are pitches or intervals, rather than arbitrary numbers |
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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 == | ||
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* [[Mediant]] | * [[Mediant]] | ||
[[Category: | [[Category:Pythagorean means]] | ||
[[Category:Terms]] | [[Category:Terms]] | ||
[[Category:Elementary math]] | [[Category:Elementary math]] | ||