Traditional Western music has 12 notes. Other music systems have different amounts of notes. However, I'm wondering if there are any really general systems that can account for all the systems, similar to how the binary number system (base 2) can represent all numbers, and is what computers use. Wondering if there is anything like that for music (I don't know what it would even look like or I would try to describe it better).
The short answer is no. The long answer starts by saying that there's a bit of confusion here between number systems and representations of data.
Binary numbers are relevant in practice because of how easily digital data can be converted to an electrical signal if you express it in a number system with two digits - each digit can be represented as a higher or lower voltage and that's it. However, the same digital data can be expressed in any other number system, too.
The real distinction that matters is digital vs. analog. In the real world, sound is analog: there is no artificial limitation of which frequencies you can use. Analog signals, however, can't be directly processed by computers, so that's why analog data frequently gets converted into digital data (and back when you need to bring it back into the real world).
Converting analog to digital works by sampling the data. Without getting into the theory too much, this process converts a real-world analog signal into something that can be digitally processed, up to a certain maximum frequency (higher frequencies in the original analog signal are not preserved in the transformation).
When we talk about notes, that's a slightly related situation: there's an unlimited number of frequencies (there's infinitely many different frequencies just between 440 Hz and 440.1 Hz) but in most music we don't care about all of them, at least when talking about notes. Most music around here centers around the 12-note systems and each note in each octave maps to a different frequency, but with many instruments all of the surrounding frequencies come into play, too, for more expressiveness (e.g. portamento).
In terms of theory, any note system takes away flexibility from the real deal of what (base) frequencies occur in some kind of sound. So, no matter which note system you come up with that doesn't have infinitely many notes, it will never be "completely universal".
You could base a unified theory on digital sound (up to the sampling limit I mentioned earlier), but digitally sampled data doesn't make it extremely easy to make out individual notes, so it wouldn't be a good choice for a note system.
I think there is no answer to this question, due to a misconception of bases in numeral systems.
We usually use decimal system for describing numbers, meaning we use 10 different symbols to describe the first 10 whole numbers, then we start to repeat them and queue them up to describe higher numbers. Binary numbers are base-2 numbers, meaning you would only have 2 different symbols, therefore can describe 2 numbers without repeating them. Binary system is not more general than the decimal system, or hexadecimal system, or any system, because all systems can describe all whole numbers!
Analogically, an octave would be this threshold, from which you need to repeat the note names. So a 12 note scale system has 12 different "symbols", or notes, and repeat them in the next octave. But unlike numeral systems, different note systems cannot describe all notes, because they all consider this octave to be the same size, and divide them in different ways!
I will try to make an analogy to clarify my thoughts:
Let's consider the first decimal numbers to be: 0 1 2 3 4 5 6 7 8 9 10
The same numbers in binary system would be: 0 1 10 11 100 101 110 111 1000 1001 1010. We just started repeating symbols earlier in the second example, but we are talking about the exact same numbers, with less symbols.
If 12 note system has the notes: C1 C#1 D1 D#1 E1 F1 F#1 G1 G#1 A1 A#1 B1 C2
A "binary" note system to describe the same notes with less symbols would be: C1 C#1 C2 C#2 C3 C#3 C4 C#4 C5 C#5 C6 C#6 C7 ... Which doesn't make any sense.
Now suppose an octave can be divided into 12 notes. If you would divide into a different number of notes, for simplicity 2, you could only describe a few of these 12 notes, or even none of them, because you can use any frequency you want! Let's create a 2 note scale with only C and G. Therefore this scale could only describe the notes: C1 G1 C2 G2 C3 G3 ...
An analogy for this in the numeral system would be a system that can only describe 2 numbers within a ten, for example: 1 6 11 16 21 26 ... Which doesn't make any sense!
Did I make sense???
I think your numeric example might have obscured rather than clarified, but I think I got the idea.
I believe it's good to start with the following question: what differentiates notes between each other? This excellent answer states that a note is a named pitch (that is, a particular pitch that was given a name), and a pitch is a particular frequency of sound. We can then differentiate notes by their pitch.
Since pitch is a frequency, we can measure it on
Hertz. On this context,
Hertz represent the amount of cicles of the generated vibrations in the air (sine wave or waveform) per second. We can then state that a system based on
Hertz can possibly represent all notes, and this is accurate but impractical since, as other people already commented, noone would try to remembers notes on chord
220Hz (omitting decimals), even if this is just a power chord with an octave.
So does a current practical system exist for all notes? Unfortunately no. Each region has different notations of each pitch (different notes) and instruments to play them. One might think to unify them, maybe considering all pitches inside an octave and determining notes within to satisfy all different musical notations, and this is plausible, but it is comparable to creating a universal language from all current languages, it's utopic.