2

For one octave, we double the frequency, so 400Hz to 800Hz is one octave higher. To get the tone that is perceived to be in the middle of those two frequencies is however not 600Hz but 400*1.059463⁶=~565Hz. However if I have arbitrary frequencies like 400 and 760Hz, how do I get the percived "middle" tone?

5

It depends on what precisely you mean by "middle tone," but from your example, it seems you mean to divide an octave in half to get the tritone in the center.

Mathematically, the quickest way to do this for any two frequencies is the geometric mean, rather than the arithmetic mean. The mathematical reason you need to do this (rather than use the arithmetic mean, i.e., simply adding and dividing by 2) is because frequencies map to pitches on a logarithmic scale, rather than a linear scale.

In any case, to find the geometric mean between two numbers, simply multiply them together and then take the square root. For your examples:

  • For 400 and 800 Hz: square root of (400*800) = square root of 320000 = ~566 Hz
  • For 400 and 760 Hz: square root of (400*760) = square root of 304000 = ~551 Hz

Note that for arbitrary frequencies, the result of this calculation will not necessarily fall on a "standard" musical pitch or make "normal" musical intervals within the 12-semitone per octave chromatic scale. But it will give you the frequency that is acoustically sort of the "middle" between any two frequencies.

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  • 2
    It seems to me that rather than the geometric mean being the quickest way to find the frequency of the pitch midway between two frequencies, it is the only way to do so. Any method you might employ that is slower than multiplying and taking the square root is just a slower method for finding the geometric mean. Similarly, I would remove "sort of" in the last sentence. – phoog Dec 9 '19 at 17:56
  • 1
    @phoog - Just because something reduces to doing the geometric mean algebraically doesn't mean it is understood by someone as equivalent. Obviously the OP here calculated the tritone distance by multiplying by the 12th root of 2 (the semitone ratio) raised to the 6th power. Yes, I suppose you can reduce that to "finding the geometric mean," but those are very different ways of understanding the process. Also, I said "sort of" because that's not the only way of finding the "middle." For example, to the ancient Greeks, the harmonic mean was also a useful "middle" note between two pitches. – Athanasius Dec 9 '19 at 18:54
  • and for standard tuning (440 = A), D# is 622 Hz, in compliance with this formula. – Carl Witthoft Dec 10 '19 at 15:44
1

Find the geometric mean, not the arithmetic.

√(800*400) ≈ 565.7

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