How To Derive Note Frequencies From The Harmonic Series? [closed]

Question

The harmonic series give us the (natural or just) frequencies of the notes in an octave. To achieve that we need to transpose the higher-order harmonics to the same octave as the fundamental tone. But how?

The third overtone of a fundamental tone with frequency=440 Hz has a frequency of 1320 Hz. It is obviously an octave above 440. How do we determine its frequency in the proper octave?

• divide by a power of two until we reach the destination range (440-880)? If we trust that each tone has a frequency double that of the same tone an octave below, then that should work
• we find the ratio of 1320 to the tonal centre of its octave (880), and then, using the same ratio, find the tone on the octave starting from 440.

Which one? Or both? Or another.

Answer 1

As far as your two bullets go, they are both the same.

I usually tend to think of the process in terms of your first bullet.

Answer 2

I'll admit I'm not entirely sure what you are asking, but I'll take a shot at it anyway.

First, 1320 Hz is roughly an E. I can find that by assuming that 880 Hz is A and noting that the ratio of 1320Hz to 880 Hz is 3:2, which is the just interval of a perfect fifth.

Every tone does have a frequency double the same tone an octave below, since that's what an octave is.

A handy visualization to the ratios and frequencies involved in a pure harmonic series (from Wikipedia):

The numbers above the staff are cents above or below the equal-tempered pitch displayed on the staff.