contraharmonic proportion

Three positive numbers  x, m, y  are in contraharmonic proportion, if the ratio of the difference of the second and the first number to the difference of the third and the second number is equal the ratio of the third and the first number, i.e. if

m-xy-m=yx. (1)

The middle number m is then called the contraharmonic mean (sometimes antiharmonic mean) of the first and the last number.

The contraharmonic proportion has very probably been known in the proportion doctrine of the Pythagoreans, since they have in a manner to (1) described the classical Babylonian means:

m-xy-m=mm  (arithmetic mean m)
m-xy-m=my  (geometric mean m)
m-xy-m=xy  (harmonic mean m)

The contraharmonic mean m is between x and y. Indeed, if we solve it from (1), we get

m=x2+y2x+y, (2)

and if we assume that  xy, we see that


The contraharmonic mean c is the greatest of all the mentioned means,


where a is the arithmetic meanMathworldPlanetmath, g the geometric meanMathworldPlanetmath and h the harmonic meanMathworldPlanetmath. It is easy to see that


Example.  The integer 5 is the contraharmonic mean of 2 and 6, as well as of 3 and 6, i.e.   2, 5, 6,  are in contraharmonic proportion, similarly are  3, 5, 6:

22+622+6=408= 5,32+623+6=459= 5

Note 1.  The graph of (2) is a quadratic cone surface  x2+y2-xz-yz=0, as one may infer of its level curvesMathworldPlanetmath


which are circles.

Note 2.  Generalising (2) one defines the contraharmonic mean of several positive numbers:


There is also a more general Lehmer meanMathworldPlanetmath:



  • 1 Diderot & d’Alembert: Encyclopédie. Paris (1751–1777).  (Electronic version
  • 2 Horst Hischer: “ Jahre Mittelwertbildung”.  — mathematica didactica 25 (2002). See also
  • 3 J. Pahikkala: “On contraharmonic mean and Pythagorean triplesMathworldPlanetmath”.  – Elemente der Mathematik 65:2 (2010).
Title contraharmonic proportion
Canonical name ContraharmonicProportion
Date of creation 2015-09-06 19:48:08
Last modified on 2015-09-06 19:48:08
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 24
Author pahio (2872)
Entry type Definition
Classification msc 26E60
Classification msc 11-00
Classification msc 01A17
Classification msc 01A20
Related topic ProportionEquation
Related topic Mean3
Related topic PythagoreanHypotenusesAsContraharmonicMeans
Related topic HarmonicMean
Related topic ConstructionOfContraharmonicMeanOfTwoSegments
Related topic ContrageometricProportion
Defines contraharmonic mean
Defines antiharmonic mean