in an earlier post I said that the arithmetic mean of two frequencies an octave apart is an interval of a perfect fifth, and the geometric mean gives a tritone. This post will look at a few other means.

Intervals

The harmonic mean (HM) gives exactly a perfect fourth.

The arithmetic-geometric mean (AGM) gives a pitch about midway between a tritone and a fifth, a tritone plus 50 cents.

The arithmetic mean gives a perfect fifth.

The contraharmonic mean gives approximately an interval of a major sixth.

If we take the means of A 440 and A 880, the AGM is an E half-flat (hence the backward flat sign above).

Equations

Here are the equations for the various means:

begin{align*} HM(a, b) &= frac{2ab}{a + b} \ GM(a, b) &= sqrt{ab} \ AM(a, b) &= (a + b)/2 \ CHM(a, b) &= frac{a^2 + b^2}{a + b} end{align*}

The AGM is defined iteratively: Take the GM and AM of the pair of numbers, then take the GM and AM of the result, and so on, taking the limit. More detail here.

Frequencies

Here are the frequencies of the means.

    |------+-----|
    | Mean |  Hz |
    |------+-----| 
    | HM   | 586 |
    | GM   | 622 |
    | AGM  | 641 |
    | AM   | 660 |
    | CHM  | 733 |
    |------+-----|

Lilypond

Here’s the Lilypond code that was used to create the music notaton above.

begin{lilypond}

new Staff with { omit TimeSignature} {
  relative c''{
     <a d>1 <a ees'>1 <a eeh'>1 <a e'>1 <a fis'>1 |
  }
  addlyrics{"HM" "GM" "AGM" "AM" "CHM" }
}

end{lilypond}

The post More ways of splitting the octave first appeared on John D. Cook.

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