In your question, you point out a number of different requirements that a system for representing music could fulfil. I think you'll find that there are systems that each satisfy some of your requirements; I don't think you will find a 'Swiss army knife' that unifies them all.
One requirement you mention is wanting to see relationships more easily, and being bothered by the same interval being represented in different ways. This problem might be a job for a chromatic staff - http://musicnotation.org/ has a lot of information about this approach. Sagittal might be of interest if you are mostly interested in microtonality.
In many ways, these systems are still very similar to standard notation in that they are concerned with representing music as a set of notes with pitch on a vertical axis - but for a more numerical approach, you might prefer to use pitch class sets.
If analysis is the main goal, the tonnetz provides a visual and conceptual space in which the harmonic movement of a piece of music may be analysed.
When it comes to a form of notation that allows reproduction by computers, you might consider that the Music-N family of languages are a kind of imperative notation oriented around synthesis (although perhaps not so good for analysis).
Computer-generated music might also simply be represented as an equation that produces a waveform, or the implementation of this equation in some programming language.
All of these mentioned so far are abstractions of musical sound. At the more concrete end of the spectrum, it might be worth observing that high-fidelity recording has in some areas of musical activity (I'm talking pop!) replaced musical notation as the way of storing and transmitting 'reference' versions of musical works. Recordings are also easily transformed into spectrograms, which are another way of viewing any audio (including music).