What happens if you go down by the same steps:
- 1 step down : 403.33Hz
- 2 steps down : 366.67Hz
- 3 steps down : 330.Hz
- 11 steps down : 36.67Hz
- 12 steps down : 0Hz
- 13 steps down : -36.67Hz
So, using your "equally divided" logic, we are at zero Hz after 12 steps, and the next step beyond that is minus 37 Hz! What does that even mean? But ok, let's follow your logic a little bit ... what's the frequency exactly in the middle of the octave 440 - 880 Hz, that would be 660 Hz. What's an octave above that? That would be 2 * 660 Hz = 1320 Hz. What would be the steps in that octave - 660 Hz / 12 = 55 Hz? Ok, then let's take one step up from 660 Hz, that's 660 Hz + 55 Hz = 715 Hz. But wait ... the step was supposed to be 37 Hz, not 55 Hz??? Does your step size depend on the start and end points of the octave? Or does it take a sudden jump at 880 Hz - steps below 880 would be 440 / 12, but above 880 they would be 880 / 12? Where does such a divider come from, is it embedded in nature? I thought A = 440 Hz was only an agree-upon convention, not a law of nature.
Where did you get the 880Hz? By multiplying by 2, i.e. one octave higher. I guess the same has to apply for any frequency, not only 440Hz? For example, one octave higher from 880Hz has to be 880Hz * 2? And any other frequency like 1000Hz... one octave above that must be 2000Hz. If the interval of an octave is calculated with multiplication, how could other intervals be calculated with addition?
So, ask yourself: if F1 and F2 are the frequencies of two consecutive semitones, what is the relationship between F1 and F2, if (F1 * 2) and (F2 * 2) has to have the same relationship?
You're looking for a function f(F) such that f
applied 12 times gives 2*F.
f(f(f(f(f(f(f(f(f(f(f(f(F)))))))))))) = 2 * F
If you step up one semitone from F, you get a frequency f(F). The frequency one octave higher from that is 2 * f(F).
If you first step up an octave, you get F*2. And if you step up one semitone from that, you get f(F*2), which should be the same frequency, so:
2 * f(F) = f(2 * F)
What might function f be like?
From the subject line "why hertz differences are not the same but element 12th of two?" I assume that you already know that consecutive semitones have a ratio of 2^(1/12).