69.1 Pythagoras differentiated three types of means: 

4 is the arithmetical mean in the progression 2 : 4 : 6;
4 is the geometrical mean in the progression 2 : 4 : 8;
4 is the harmonic (or musical) mean in the progression 3 : 4 : 6.

Touches of Sweet Harmony.

69.2 Diagram. The Three Kinds of Proportion 

"This illustration exemplifies the three kinds of mathematical proportion. There is an arithmetical proportion 10 : 25 : 40, where each term varies from the preceding term by a constant number, 15. There is a geometrical proportion 10 : 20 : 40, where each term varies from the preceding term by a constant ratio, 2, so that the differences between adjacent terms also reflect this ratio, such as 20 : 10. And there is a musical or harmonic proportion 10 : 16 : 40, where the third term, 40, has the same ratio to the first term, 10, as the difference between the third and second term, 24, has to the difference between the second and first term, 6, so that 40 : 10 = 24 : 6".

From Scientific American is a definition for mean proportional, (those lying between the standard ratios as arithmetic and harmonic means). Mean proportions are found everywhere in Renaissance architectural dimensions.


Scientific American.
Dec. 1967.

69.4 Harmonic Mean 

Expressed in algebraic terms, in order to find the harmonic mean, b, within the terms a b c , the formula is applied:

b = 2ac / a+c

For the arithmetic mean:

b = a + c / 2


69.6 Pythagoras's Harmonic Mean 

"The harmonic mean expresses a pitch ratio between neighboring notes that is a good deal more subtile than the arithmetic mean, which only averages them or the geometric mean which equally tempers their proportion".


Harmony of the Spheres

69.8 Definition of Mean 

"The middle term...the three most important... arithmetic, harmonic and geometric..In the following equations the two extremes are A + C, and the mean term is B":

Arithmetic mean: B = (A + C) / 2
Harmonic mean: B = 2AC / A + C
Geometric mean: B = Sq rt.A x C


Pythagorean Sourcebook and Library

69.10 Mean / Square Root 

"The interpolation of the arithmetic means leads directly to approximative values for the square root."

Pythagorean Musical Theory

69.11 Mean, Harmonic / Arithmetic / Geometric 

Harmonic mean = The reciprocal of the arithmetic mean of the reciprocals of two or more quantities.

Arithmetic mean = A quantity formed by adding quantities together in any order and dividing by their number.

Geometric mean = The 'n'th root of the product of 'n' statistical variables.

Views from the Real World
By J. D. Salzmann
196. G9615v

Impossible Correspondence Index

Copyright. Robert Grace. 1999