Okay, so let's start at the start. Firstly, sound is basically the conscious experience of vibrations of the air that have been picked up by the ear. Secondly, a pure tone like 440Hz is very rare to find in the environment around us, unless people are playing electronic music near you; most sounds are complex, made up of the air vibrations caused by an event, but also by the way those air vibrations interact with the environment around us. As a result, a sound - the sound of a footstep, or a piano - with a fundamental frequency of, say, 440Hz typically has harmonic partials - other associated sounds that go along with it. Some of those harmonics are mathematically related to the original fundamental frequency of the sound - 880Hz (double the original sound), and 1760Hz (double the first harmonic), and 3520Hz (double the second harmonic) are all very common frequencies to also be hearing when you hear a frequency of 440Hz. These harmonics are, as you say, logarithmic (like the rate of new coronavirus diagnoses if left unchecked). Not all of the frequencies associated with a sound are harmonic partials - there's also inharmonic partials, but that's not important for our story.

It seems to be a basic property of the mammalian nervous system that our nervous systems associate such mathematically(/logarithmically)-related sounds together so that they're categorised as basically the same sound, or at least related sounds - just because this is typically what regular sounds in the environment do.

Musical notes in the Pythagorean scale (that the equal temperament tuning scale on a modern piano is a variation of) correspond with, essentially, sounds that have a relatively simple mathematical relationship between the original tone (which we might call the tonic): the fifth has a 3:2 mathematical relation to the tonic (e.g., if it's the tonic - an A note in this case - at 440Hz, it's the the fifth - the E note in this case - at 660Hz), a fourth has a 4:3 mathematical relationship to the tonic (if it's the tonic at 440Hz, it's the fourth - a D note in this case - at 586.667Hz).

In terms of the history of the understanding of these kinds of relationships in Western music, this goes back to the Pythagoreans and other Ancient Greek philosophers, who could see the relationships between the notes in the resonances of those notes in glasses of water. They thus could mathematically work out those basic relationships (as the Pythagoreans were wont to do). This information was not lost in the medieval era; this area of knowledge was typically taught as one of the liberal arts in medieval universities.

One problem with instruments based around Pythagorean tuning was that they needed to be retuned every time there was a new piece to play in a new key, as the logarithmic ratios that work for one key aren't necessarily relevant in another - if you try this, you get 'wolf' tones that sound particularly discordant. By the modern era in Europe, you get various attempts at attempting to create instruments that could change keys by carefully working out a series of 12 tones within an octave (i.e., within that range of 440Hz and 880Hz, where 440Hz and 880 Hz are both regarded as an A note, the octave is that distance between the two closest A notes) that are close enough to those logarithmic ratios that listeners consider them to be mathematically related, but where the relationships are equivalent across the notes - e.g., that the ratio between A and B is the same as the relationship between B and C# on a logarithmic scale, which isn't necessarily the case with Pythagorean tuning. This equal temperament between keys are basically what modern Western instruments like the piano, the saxophone and the guitar are designed to use.

Given this long history of focusing on these kinds of relationships in music from a Western academic tradition, a lot of music theory for a long time has been very focused on the complexities of relationships in this equal temperament scale. There is certainly a tendency in some research in the Western academic tradition to assume that music has to be in this tradition, and that other ways of looking at pitch are impossible.

However, this 12 tone equal tempered scale, specifically, is a cultural artifact of a particular European tradition.

That said, almost all cultures that have been studied by ethnomusicologists, as far as I'm aware, do appear to associate the same notes at different octaves. Most have notes in the scale that correspond to the simpler ratios - fourths and fifths - because these are relatively simple ratios mathematically, there may be some neural mechanism by which they are likely to sound good. But outside of the simpler ratios, there's no particular reason why you have to have a minor sixth ratio (128:81 in Pythagorean tuning) in the scale - with a complex ratio like that, well, there's plenty of other complex ratios that are possible.

And just because the auditory system associates some natural sounds doesn't mean that the art and meaning we create out of sound has to follow those associations of natural sounds - witness the various forms of 20th century music that reject the 12 tones, like musique concrete (which creates music out of found sounds) or electroacoustic music, which sounds more like bursts of noise rather than music to the uninitiated.

It's also not the case that the 12 tones on a piano are the true and natural way to assort sound - there's complex tuning systems associated with non-European musical cultures, like the classical Indian tradition, which makes use of microtones (i.e., notes that are between the European 12 tones - why is 12 tones the best way to divide an octave?). The 'blue note' of jazz has its origins in a note used in some of the West African cultures where the African-Americans who influenced jazz were enslaved. This is pitched between the major and minor third - there's no reason why that's a wrong note, it's just a different ratio to the octave to the notes used in the Western scale. Its survival in America to become part of jazz is a testament to its ability to create meaning regardless of what the specific logarithmic ratio actually is.

But also, the role of melody and pitch within a musical culture can vary widely; some cultures have much simpler harmonic and melodic structures than in the Western musical tradition, and some have much more complex rhythmic or timbral structures, to the extent that it's hard for Western audiences to hear the complexity, because they didn't grow up listening to it. The way that pitch is used in one culture in terms of what role it plays might be quite different to the next; when pitch isn't used in the same way as it is in the Western tradition, it might also follow different principles.

Developmental psychologists looking at music find that infants and children grow to understand the specific parameters of music in their particular cultural tradition, to respond to sounds and notes that fit that tradition, and to be confused by others - whether that tradition is the standard Western one with the 'set logarithmic scale' or not. Ultimately humans are pretty good to adapting to whatever information they're exposed to, if they're presented with that information enough - and that goes for the Western musical tradition as much as it goes for the Pitjantjatjara vocal music of Central Australia which (according to a 1996 paper by Will & Ellis) appears to avoid logarithmic scales, instead appearing to use a linear scale.