Is there some inherent quality in the frequencies that make them notes, or is it just an arbitrary human construct?
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It is sort of arbitrary but there is a quality of the frequencies that make them sound good together. In general notes that share a lot of the same harmonics sound good together. We have after lots of negotiations over hundreds of years agreed that 440Hz is a note. And any notes that is double or half frequencies of a note is also considered a note. This is because they share half the harmonics, we even say they are the same notes just in different octaves. So for example 880Hz is a note. Then we have decided that a frequency that have a ratio of 3:2 to a note is also a note. Again they share a lot of harmonics. Lastly we also consider frequencies of ratios 5:4 and 6:5 to other notes to be notes. This covers the four most basic harmonic ratios used in western music.
It turns out that a lot of these ratios land pretty close to each other, even if they are not exactly right. This is because multiplying ratios can give different numbers depending on which ratios you multiply but the actual numbers get close. So we group these close ratios together. We therefore end up with 12 different frequencies in each octave that we consider to be notes. The frequencies that lie between these do not share many harmonics with any of the other notes and is therefore not considered notes.
We could use different ratios and get different notes. This is sometimes the case in non-western music. And even some western music use ratios not established in certain songs. So it is kind of arbitrary which note we started with and exactly which ratios we use. You can make good music with frequencies that is not considered notes in the western music scale.
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