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Surprisingly nobody has referred to Unicode’s Musical Symbols block (PDF). It contains a surprisingly thorough assortment of symbols and even some mechanisms for connections/grouping. Unfortunately most operating systems don’t come with fonts that will show those glyphs, so it ends up looking like ��mojibake��.

The Unicode Standard (PDF) does a great job of explaining/justifying the scope of what these symbols are for (emphasis mine):

Because of the complexities of layout and of pitch representation in general, the encoding of musical pitch is intentionally outside the scope of the Unicode Standard. The Musical Symbols block provides a common set of elements for interchange and processing. Encoding of pitch, and layout of the resulting musical structure, involves specifications not only for the vertical relationship between multiple notes simultaneously, but also in multiple staves, between instrumental parts, and so forth. These musical features are expected to be handled entirely in higher-level protocols making use of the graphical elements provided. Lack of pitch encoding is not a shortcoming, but rather is a necessary feature of the encoding.