Can you see the pattern here? Anytime you want to use intervals based on third harmonics, you’re multiplying and dividing by 3, but anytime you want to use intervals based on fifth harmonics, you’re multiplying and dividing by 5. (Notice that the conflicting notes always conflict by the same amount, too, a ratio of 81/80.) Starting from C, it’s possible to produce any note if you multiply or divide your frequencies by 3 enough times, but those notes won’t be in tune with the notes you’d get multiplying or dividing your frequencies by 5, because 3 and 5 don’t mutually divide evenly. This is not just an abstract mathematical issue. It’s the reason that it’s impossible to have a guitar be in tune with itself.
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