Why do we recognize certain frequencies as notes?
We don't. Frequencies can change based on how a musical instrument is tuned.
However since quite some time it became common to tune them according to equal temperament.
Our ears enjoy notes played together that conform to simple ratios, like 2/3. The more complicated a ratio is, the more dissonant it sounds to us. There are different ways to tune instruments because it's basically impossible to have all 12 notes in an octave conform to a perfect ratio, some notes will always be slightly off. Equal temperament distributes the dissonance on all 12 notes equally, which makes it much easier for composers and players.
And if a ratio is doubled or halved, our ears perceive it as the same "note". That's why notes keep repeating on the Piano.
It has to do with the way our brain analyses sound. Every human has a slightly different pitch. Women are higher pitched than men. Instead of analyzing every frequency on it's own, our brain sort of doubles and halves everything such that in the end it only has to analyze a limited amount of input. (It "normalizes" the input). That's why middle C and high C somehow sound different yet totally the same.
Additionally, have you wondered why the very some note, say middle C, sounds different on a Piano and on a Guitar? How can they sound different to us if they are the same frequency?
That's because only a sine wave would produce a pure frequency. Notes played on instruments have "Overtones". The frequency repeats a certain pattern in higher frequencies that varies slightly by instruments. Our brain is really good at analyzing those frequencies, such that if a speaker only plays the overtones, you start to imagine the corresponding lower frequency. Earbuds use this to make you imagine bass sounds, as the speakers are in fact to small to actually play low frequencies. That's also why lower quality speakers sounds "Metallic". They can only play part of the overtones and the illusion starts to fade.
Our brain is incredibly good at analyzing those overtones and it appears that there is a relationship between music rules and those overtunes. Basically, when you hear music certain parts of your brain that analyze sound according to some math are overstimulated in a way they'd never be stimulated by nature. Humans and animals have different musical taste however (birds have their own "music theory" that is different from ours), because there is also a big relationship between the human voice and how we emotionally perceive music.
If you want to learn this stuff, I recommend specifically searching for 'scientific' music theory. It will not teach you how to compose. But regular music theory is to a large extend outdated and has many myths. If you wish to compose rock, edm or any modern genre anticipate to break a lot of music theory rules.
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