Arithmetic mean: Difference between revisions

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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''', '''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

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== 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 [[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 ==

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== 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 ==

* [[~~Mean~~]]

* [[Pythagorean means]]

* [[Inverse-arithmetic mean]]

** [[Geometric mean]]

** [[Inverse-arithmetic mean]]

* [[Mediant]]

[[Category:~~Theory~~]]

[[Category:Pythagorean means]]

[[Category:Terms]]

[[Category:Elementary math]]

@@ Line 1: / Line 1: @@
-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''', '''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
@@ Line 12: / Line 12: @@
 == 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 [[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 ==
@@ Line 44: / Line 44: @@
 == 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 ==
-* [[Mean]]
+* [[Pythagorean means]]
-* [[Inverse-arithmetic mean]]
+** [[Geometric mean]]
+** [[Inverse-arithmetic mean]]
 * [[Mediant]]
-[[Category:Theory]]
+[[Category:Pythagorean means]]
 [[Category:Terms]]
 [[Category:Elementary math]]