: How do multiple sound waves interact physically to create music? What happens on a physical level when many notes/sound waves interact with each other at the same time? (For example, a big chord on the piano with all of its

The physics behind music can be explained by the concept "Fourier Series":
en.wikipedia.org/wiki/Fourier_series
Any sound can be decomposed as the sum of simple sine waves of different frequencies. The main frequencies found in this decomposition are the main harmonics of that sound.

Then, playing two sounds together within an harmonical context (i.e. a fundamental key), if their harmonics physically "fit" together, to our ears, the two sounds feel musically "right".

So, there is a physical explanation to what we perceive as "musical". Musical harmony theory explains the rules to what is sounding good to us and uses concepts like tonic key, dominant key, circle of fiths etc.

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You're asking about perception, so even though you're asking about "what happens on a physical level," it seems your question is ultimately about what's happening in our minds, not what's happening physically.

To that end, I'll point you to Bregman's Auditory Scene Analysis (see also the Wikipedia page), the leading theory on how auditory systems segregate different components.

As a very brief summary, he discusses a few types of listening processes:

One listening process is by involuntarily recognizing learned schemas. This is how, for example, you're able to hear your name in a crowded room.
Another process is voluntarily recognizing learned schemas. An example of this would be listening for your name in a list.

He also takes some time to look at other research. In one experiment (by someone else), a tone was held, and for some duration during that tone it was played louder. If that louder duration was short, it sounded like a second tone played in addition to the first. But if the louder duration was longer, it was perceived as a single adjusted tone.

With research like this in mind, Bregman incorporates a number of Gestalt principles to model how we perceive music. Some examples of these "preference rules":

Unrelated sounds seldom start or stop at exactly the same time.
A sound (or sequence of sounds) changes its properties slowly. (This links sounds with similar properties together.)
Changes will affect all components of the sound in the same way at the same time.

With these principles, we can start to make claims about why we hear some sounds as music and other sounds as noise. In short, if these preference rules are unable to help us clearly segregate the auditory streams, we are more likely to hear it as "noise."

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Adding to Todd Wilcox's and Elliot Svensson's answers, take a look at this video. It explains the relationship between the notes of a triad. This is a good piece of information that a lot of people studying and playing music for a long time don't even know about.

It shows that the sound waves of a natural major triad meet at regular periods (a.k.a. consonance), as seen in this image (extracted from the video):

The triad in the image is the D natural major (D?, F??, A?), and you can see that, in the time the D? wave goes through two complete periods, F?? goes through 2.5 periods and A? through 3 periods. After the same amount of time, they'll meet again and again.

Random notes/sounds played will probably have a lot of dissonance, which mean that their frequencies won't meet at regular intervals, so they interfere each other in a way that the resulting sound (compound wave) doesn't "sound good" to the ears and aren't identified as "music".

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What physical process happens when two sounds occur at the same time?

When two or more sounds happen at the same time, their sound waves (commonly depicted as a graph of increasing and decreasing air pressure) superimpose into one composite waveform, through "wave addition". For each tiny moment in time, the air pressure is the sum of what it would have been from each instrument alone.

Wave addition is how the speaker in your headphone can play bass and guitar at the same time (and drums, voice, etc). If the waves being added are identical, the signal doesn't change but just gets louder. Even more unusual, when the signals add up to zero they may cause silence. This is extremely unusual with real instruments, but it is what causes "beats" when two instruments play the same note just slightly out of tune with each other.

Why do some combinations become music but not others?

Well, if the combination has the ability to communicate to the listener a musical idea, then it's music. If there is too much noise for the listener to distinguish what's being played, or if there aren't really any ideas being sent, you won't hear music.

Some common musical ideas are melody, harmony, and rhythm, as well as "color", imitation, and mood.

What about harmonics and overtones?

When two frequencies sound good together we say they are "consonant", as opposed to "dissonant". Real instruments don't produce a single frequency sound but an assortment of frequencies ("overtones") which are all added together physically, and which are more or less related mathematically to the pitch (or fundamental frequency) the instrument is playing.

Math and physics have identified a sequence of frequencies called "harmonics" (or "modes of vibration") which generally sound good together. These are known to all brass (i.e. trumpet, trombone) players as the sequence of notes played at the same fingering going from low to high.

When an instrument or noisemaker has a composite sound that is mostly composed of the early harmonics in the sequence, we say that it has a "sweet" sound (or sweet timbre), like a bell. It's easy to communicate musical ideas such as melody and harmony using sweet sounds.

When superimposed tones (such as an out-of-tune ensemble or modes of an object's vibration) contain strong frequencies that are dissonant, we may call the sound "sour". Sourness obscures melody and harmony by making the intended fundamental frequency ambiguous to the listener and poisoning consonance.

Chords are primarily composed of notes in the harmonic sequence from some fundamental frequency, and many complex chords still do not stray from this rule, though they may contain higher harmonics. If a chord doesn't relate well enough to some fundamental frequency (or "tonic") via the harmonic series, it will sound sour or dissonant; but this is not necessarily bad: complex music frequently uses dissonance in producing tension or other desired qualities.

Instruments

With classical instruments, the fundamental frequency is a lot louder than the other frequencies. The next loudest frequencies are nearby harmonics, which sound consonant to the fundamental frequency, with the higher harmonics more or less quieter. You know that B and C are dissonant, but because the volume of each next overtone is less than the previous, a classical instrument playing C is not dissonant with one playing E, even though B is audible as a harmonic of E... the B is less loud.

Percussive instruments like cymbals produce so many different frequencies that a person doesn't perceive a fundamental frequency. These instruments can communicate any musical idea that doesn't depend on tone (i.e. melody or harmony).

Further reading on overtones

Voice

Depending on what letter a person is pronouncing, a person's voice may be sweet or sour. For instance, the short vowels A and O (as in, "awesome" and "oafish") are very sweet. But the consonants by themselves (B, C, D, F etc.) have no fundamental frequency, and would make a sour sound if they were sustained. This is why holding the R sound is never advised for singers.

Interestingly, U as in umbrella (called /??/ by linguists) has a strong overtone one octave higher than the strong overtone of E as in free (/i?/).

Further reading on voice

What is music?

No doubt this has been addressed elsewhere. But I would formulate it this way: Music is when a person uses sound to communicate ideas that are other than language.

Order and symmetry

Order and symmetry function like carrier waves so that listeners will perceive the ideas the musician is trying to communicate.

Pure symmetry and perfect order are boring but instantly recognizable in sound. The musician initiates the listener's recognition by order and symmetry; then whatever the musician wants to communicate is placed on top of the order and symmetry to be noticed instantly.

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What happens on a physical level when many notes/sound waves interact with each other at the same time?

This depends on where you're asking about. What happens in the piano is that a bunch of strings are set in motion which then cause a large piece of wood to vibrate which causes the air around the wood to vibrate which then propagates like a wave to your ears which causes your eardrum to vibrate which causes you cochlea to vibrate which stimulates tiny hairs which send nerve impulses to your brain which decodes the pulses and creates the sensation of music.

how there can be any type of order in so much complexity?

All of the frequencies generated by normal piano chords are related to each other in a fairly simple mathematical way. Each note is composed of frequencies that are integer multiples of the lowest frequency. Then the other notes of the chord are related by generally simple frequency ratios like 3/2 and 5/4. Most of the individual frequencies in a large complicated chord overlap, so even with a huge symphony orchestra all playing a major chord, there's generally only 20 - 30 different frequencies being played that are audible, and again, all of them are closely related to each other.

Our ears and brains are amazingly good at detecting and decoding these frequencies.

What are the physics behind sound waves combining to create music?

Sound waves combine just like ocean waves, but when they are related to each other like in music, they still create an overall periodic auditory signal. All periodic signals can be decoded again into their component pure frequencies, which is exactly what the cochlea and the brain do with musical sound.

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