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As well as the mathematics, the pitch of a note on a practical instrument is affected by several factors. The recorder is close to your "theoretical instrument" and is one I am very familiar with, so I will use it as an example. Recorders have a 2 octave and a tone range. The bottom octave is the fundamental notes and the upper octave notes are "overblown" ...


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There are some pages that go into this in some detail. Clarinets are here: https://newt.phys.unsw.edu.au/jw/clarinetacoustics.html That page has links to others.


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The main reason is a physiological one. It's all in how our inner ear percieves sounds. Play an octave. Our ear percieves them as virtually the same sound (by the reasons exposed in other answers above). Musically, nothing is added by playing an octave above the root note you're playing. Yet, in physics, an octave is represented as a proportion of ...


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It was laid out that way, because there was no choice due to the nature of sound waves, thw workings of our ears/brain combination and the intercultural agreement, that octaves are the basis on which to operate. You somehow seem to think, that geometric is complicated and artifical, which is wrong in most of contexts related to physiology and perception: A ...


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The fundamental reason we define intervals as pairs of pitches with a certain ratio (e.g., 3/2), rather than a certain fixed frequency difference (e.g, 100Hz), is that pitched instruments create overtones. In a simplified picture, if you pluck a guitar string, you will get both the fundamental frequency (let's say, 220 Hz), which corresponds the entire ...


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There are probably more versions of this than I am aware of but I'll cite 2 or 3. In the just tuning system the other intervals were chosen to be a rational fraction of the lowest note in the scale (Do). For example the frequency of a fifth (Sol) is 3/2*(frequency of Do), and a second is 9/8*(Do), etc. You can look up the full chart on wikipedia. The ...


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