Arithmetic mean - Revision history

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

2023-03-04T10:02:21Z

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

← Older revision		Revision as of 10:02, 4 March 2023
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	In tuning, the '''arithmetic mean''', '''otonal mean''', or '''frequency mean''' generates new pitch materials by taking the mean in the arithmetic scale of pitch i.e. the scale proportional to frequency. It can be said with respect to ~~frequencies or~~ frequency ratios on a certain common fundamental.		In tuning, the '''arithmetic mean''', '''otonal mean''', or '''frequency mean''' generates new pitch materials by taking the mean in the arithmetic scale of pitch i.e. the scale proportional to frequency. It can be said with respect to pitches in frequency as well as intervals in frequency ratios on a certain common fundamental.

	The arithmetic mean ''f'' of two frequencies ''f''<sub>1</sub> and ''f''<sub>2</sub> is		The arithmetic mean ''f'' of two frequencies ''f''<sub>1</sub> and ''f''<sub>2</sub> is

FloraC: +"frequency mean"; "logarithmic mean" -> "geometric mean"

2023-03-03T18:17:10Z

+"frequency mean"; "logarithmic mean" -> "geometric mean"

← Older revision		Revision as of 18:17, 3 March 2023
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	In tuning, the '''arithmetic mean''' or '''~~otonal~~ mean''' generates new pitch materials by taking the mean in the arithmetic scale i.e. the scale of frequency. It can be said with respect to frequencies or frequency ratios on a certain common fundamental.		In tuning, the '''arithmetic mean''', '''otonal mean''', or '''frequency mean''' generates new pitch materials by taking the mean in the arithmetic scale of pitch i.e. the scale proportional to frequency. It can be said with respect to frequencies or frequency ratios on a certain common fundamental.

	The arithmetic mean ''f'' of two frequencies ''f''<sub>1</sub> and ''f''<sub>2</sub> is		The arithmetic mean ''f'' of two frequencies ''f''<sub>1</sub> and ''f''<sub>2</sub> is
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	== See also ==		== See also ==
	* [[Pythagorean means]]		* [[Pythagorean means]]
	** [[~~Logarithmic~~ mean]]		** [[Geometric mean]]
	** [[Inverse-arithmetic mean]]		** [[Inverse-arithmetic mean]]
	* [[Mediant]]		* [[Mediant]]

FloraC: Fix and polish

2023-03-02T15:48:45Z

Fix and polish

← Older revision		Revision as of 15:48, 2 March 2023
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	In tuning, the '''arithmetic mean''' or '''otonal mean''' generates new pitch materials by taking the mean in the arithmetic scale i.e. frequency. It can be said with respect to frequencies or frequency ratios on a certain common fundamental.		In tuning, the '''arithmetic mean''' or '''otonal mean''' generates new pitch materials by taking the mean in the arithmetic scale i.e. the scale of frequency. It can be said with respect to frequencies or frequency ratios on a certain common fundamental.

	The arithmetic mean ''f'' of two frequencies ''f''<sub>1</sub> and ''f''<sub>2</sub> is		The arithmetic mean ''f'' of two frequencies ''f''<sub>1</sub> and ''f''<sub>2</sub> is
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	The arithmetic mean of [[5/4]] and [[6/5]] is [[49/40]]: (5/4 + 6/5)/2 = (25/20 + 24/20)/2 = 49/40.		The arithmetic mean of [[5/4]] and [[6/5]] is [[49/40]]: (5/4 + 6/5)/2 = (25/20 + 24/20)/2 = 49/40.

	The arithmetic mean of [[9/8]] and [[10/9]] is [[161/144]]: (9/8 + 10/9)/2 = (81/72 + 80/72) = 161/144.		The arithmetic mean of [[9/8]] and [[10/9]] is [[161/144]]: (9/8 + 10/9)/2 = (81/72 + 80/72)/2 = 161/144.

	== Generalizations ==		== Generalizations ==
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	== Terminology ==		== Terminology ==
	The term ''arithmetic mean'' comes from math. See [[Wikipedia: Arithmetic mean]]. The term ''otonal mean'' reflects the fact that it forms an otonal sequence by taking such a mean in JI. It would have made sense to call it ''harmonic mean'' if not for its usage in math to mean the [[inverse-arithmetic mean]] since it is hardcoded in terms of length instead of the more intuitive measurement of frequency.		The term ''arithmetic mean'' comes from math. See [[Wikipedia: Arithmetic mean]]. The term ''otonal mean'' reflects the fact that it forms an otonal sequence by taking such a mean in JI. It would have made sense to call it ''harmonic mean'' if not for its established usage in math to mean the [[inverse-arithmetic mean]] since it is hardcoded in terms of length instead of the more intuitive measurement of frequency.

	== See also ==		== See also ==

FloraC at 15:38, 2 March 2023

2023-03-02T15:38:38Z

← Older revision		Revision as of 15:38, 2 March 2023
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	== Examples ==		== Examples ==
	The arithmetic mean of [[1/1]] and [[3/2]] is [[5/4]]: (1 + 3/2)/2 = (2/2 + 3/2)/2 = [[5/4]].		The arithmetic mean of [[1/1]] and [[3/2]] is [[5/4]]: (1 + 3/2)/2 = (2/2 + 3/2)/2 = 5/4.

	The arithmetic mean of [[5/4]] and [[6/5]] is [[49/40]]: (5/4 + 6/5)/2 = (25/20 + 24/20)/2 = 49/40.		The arithmetic mean of [[5/4]] and [[6/5]] is [[49/40]]: (5/4 + 6/5)/2 = (25/20 + 24/20)/2 = 49/40.
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	== See also ==		== See also ==
	* [[~~Mean~~]]		* [[Pythagorean means]]
	* [[Inverse-arithmetic mean]]		** [[Logarithmic mean]]
			** [[Inverse-arithmetic mean]]
	* [[Mediant]]		* [[Mediant]]

FloraC: Created page with "In tuning, the '''arithmetic mean''' or '''otonal mean''' generates new pitch materials by taking the mean in the arithmetic scale i.e. frequency. It can be said with respect..."

2023-03-02T11:37:55Z

Created page with "In tuning, the '''arithmetic mean''' or '''otonal mean''' generates new pitch materials by taking the mean in the arithmetic scale i.e. frequency. It can be said with respect..."

New page

In tuning, the '''arithmetic mean''' or '''otonal mean''' generates new pitch materials by taking the mean in the arithmetic scale i.e. frequency. It can be said with respect to frequencies or frequency ratios on a certain common fundamental.

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

<math>\displaystyle f = (f_1 + f_2)/2</math>

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

<math>\displaystyle r = (r_1 + r_2)/2</math>

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

== Examples ==
The arithmetic mean of [[1/1]] and [[3/2]] is [[5/4]]: (1 + 3/2)/2 = (2/2 + 3/2)/2 = [[5/4]].

The arithmetic mean of [[5/4]] and [[6/5]] is [[49/40]]: (5/4 + 6/5)/2 = (25/20 + 24/20)/2 = 49/40.

The arithmetic mean of [[9/8]] and [[10/9]] is [[161/144]]: (9/8 + 10/9)/2 = (81/72 + 80/72) = 161/144.

== Generalizations ==
=== To more frequencies or frequency ratios ===

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

<math>\displaystyle f = \sum_{i = 1}^{m} f_i/m</math>

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

<math>\displaystyle r = \sum_{i = 1}^{m} r_i/m</math>

=== To an arithmetically spaced sequence ===

This generalization connects the operation to [[arithmetic division]]s.

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

<math>\displaystyle \left\lbrace i \in \mathbb {Z} \mid (i/m)f_1 + (1 - i/m)f_2 \right\rbrace</math>

The ''m'' arithmetic sequence of two frequency ratios ''r''1 and ''r''2 on a common fundamental is

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

The arithmetic mean is found by setting ''i'' = 1 and ''m'' = 2.

== Terminology ==
The term ''arithmetic mean'' comes from math. See [[Wikipedia: Arithmetic mean]]. The term ''otonal mean'' reflects the fact that it forms an otonal sequence by taking such a mean in JI. It would have made sense to call it ''harmonic mean'' if not for its usage in math to mean the [[inverse-arithmetic mean]] since it is hardcoded in terms of length instead of the more intuitive measurement of frequency.

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

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