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Explain how there’s any debate over the exact frequencies of notes like IAMA math nerd who thinks you’re all chumps if you’re not multiplying by the 12th root of 2.(r/musictheory)
This probably only comes up as an issue of temperament. You see, if you tried to tune everything in perfect Pythagorean temperament where 5ths are 3:2 and octaves are 2:1… they conflict.
So let’s run back over a classic experiment involving two monochords.
Let’s say you started with a C1 and tuned the string 7 octaves up using the 2:1 ratio. So now you have a C7 perfectly tuned to C1 in octaves.
Now take another monochord and tune it in 5ths starting on C1 at a ratio of 3:2. That would be C, G, D, A, E, B, F#, C#, G#, Eb, Bb, F, C. So you end up on C7 again. Tuned perfectly from C1 using fifths.
Play the two monochords together on that same C7 tuned perfectly using the two methods, and they aren’t in tune. The reason is basically your answer.
Try running it with actual numbers if you like as well.
If you want a more detailed answer of the battle of temperaments, read Stuart Isacoff’s Temperament. It’s honestly pretty interesting and not nearly as dry as it sounds.
We’ve settled with an equal temperament system that is close enough but things like 5ths and 4ths aren’t quite pure. A piano is essentially not perfectly in tune. However, ensembles of non-fixed pitch instruments as well as a capella groups can slightly adjust pitch as needed, so they can actually sing more pure intervals as need as the chords change. The E that is the 5ths of an A chord is several cents too high if you suddenly make that same E the 3rd of an upcoming C chord. If someone were holding that note across, they would make a subtle change (I don’t remember the exact numbers because my ear just does naturally now). They basically form a Just temperament system on the fly.
Therefore, the pitches aren’t quite exact. So, sure, you can square root for approximations of a given pitch, but in the context of other notes in a chord, or scale, or piece of music, the frequency may not be quite in tune.
There also hasn’t even been a super standard for tuning. We now think of A440 as more or less than standard and it is widely accepted, but there are still groups that tune a few cents higher or lower. Period groups will sometimes tune as much as 20 cents lower. Many period instruments (especially strings) physically cannot be tuned up to 440.