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.