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.
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
Impossible Correspondence Index
© Copyright. Robert Grace. 1999