![]() This is why two on-paper-identical notes can end up audibly different, depending on what key you're starting with (and hence how they are approached). ![]() "12" means there's twelve tones between f and 2f the ratio between adjacent tones is defined to be 2^f = ~1.6818f, which is already 10% of the way to the next note. ![]() Modern tuning (C-f "12-TET" in the article) almost, approximately satisfies (a) and (b) simultaneously. Small-integer ratios are naturally occurring and very recognizable. You want (b) because small-integer ratios are pleasant sounding - partly culturally-acquired taste, partly because physics gives musical instruments acoustic spectra in integral multiples of a fundamental frequency: f, 2f, 3f. You want (a) because it you gives you nice algebraic properties (the music structure is invariant under frequency shifts). The paradox is that you can't create a theory of music whose notes are both (a) evenly spaced and (b) contain the integer ratios.
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