Inverse-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

2023-03-04T10:06:00Z

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

← Older revision		Revision as of 10:06, 4 March 2023
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	In tuning, the '''inverse-arithmetic mean''', '''utonal mean''', or '''length mean''' generates new pitch materials by taking the mean in the inverse-arithmetic scale of pitch i.e. the scale proportional to the length of a string to be divided. It can be said with respect to ~~frequencies or~~ frequency ratios on a certain common fundamental.		In tuning, the '''inverse-arithmetic mean''', '''utonal mean''', or '''length mean''' generates new pitch materials by taking the mean in the inverse-arithmetic scale of pitch i.e. the scale proportional to the length of a string to be divided. It can be said with respect to pitches in frequency as well as intervals in frequency ratios on a certain common fundamental.

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

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

2023-03-03T18:18:56Z

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

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

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

FloraC: Created page with "In tuning, the '''inverse-arithmetic mean''' or '''utonal mean''' generates new pitch materials by taking the mean in the inverse-arithmetic scale i.e. the scale of the length..."

2023-03-02T16:04:42Z

Created page with "In tuning, the '''inverse-arithmetic mean''' or '''utonal mean''' generates new pitch materials by taking the mean in the inverse-arithmetic scale i.e. the scale of the length..."

New page

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

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

<math>\displaystyle f = \frac{2}{1/f_1 + 1/f_2}</math>

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

<math>\displaystyle r = \frac{2}{1/r_1 + 1/r_2}</math>

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

== Examples ==
The inverse-arithmetic mean of [[1/1]] and [[3/2]] is [[6/5]]: 2/(1 + 2/3) = 2/(3/3 + 2/3) = 2/(5/3) = 6/5.

The inverse-arithmetic mean of [[5/4]] and [[6/5]] is [[60/49]]: 2/(4/5 + 5/6) = 2/(24/30 + 25/30) = 2/(49/30) = 60/49.

The inverse-arithmetic mean of [[9/8]] and [[10/9]] is [[180/161]]: 2/(8/9 + 9/10) = 2/(80/90 + 81/90) = 2/(161/90) = 180/161.

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

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

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

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

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

=== To an inverse-arithmetically spaced sequence ===

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

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

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

The ''m'' inverse-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 \frac{m}{(i/m)/r_1 + (1 - i/m)/r_2} \right\rbrace</math>

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

== Terminology ==
The term ''inverse-arithmetic mean'' signifies that it is the inverse of the [[arithmetic mean]]. The term ''utonal mean'' reflects the fact that it forms an utonal sequence by taking such a mean in JI. It would have made sense to call it ''subharmonic mean'' if not for an established usage of ''harmonic mean'' in math to mean the same thing, since the sense in math is hardcoded in terms of length instead of the more intuitive measurement of frequency.

== See also ==
* [[Pythagorean means]]
** [[Arithmetic mean]]
** [[Logarithmic mean]]
* [[Mediant]]

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