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Note nomenclature is well-defined for 12-tone equal tempered tunings, e.g. English notation vs German notation, etc.
My question is: how is it possible to reliably name notes when using different tunings, e.g. 24-TET or Pythagorean? Does one use the same note names as above, plus some tuning-specific accidental to indicate the difference?
The western standard terminology of note names has very little to do with 12-edo tuning, in fact the plentitude of enharmonic clashes demonstrates that these naming schemes are actually a lot finer than 12-edo. If you apply it to 12-edo it is, in fact, not well defined anymore!
Those note names, like most of the otherwise body of theory, are all about diatonic scales. Since Pythagorean is on of the two classical diatonic tunings (though the other one is arguably more relevant), it does naturally describe this scale just fine:
1/1 9/8 81/64 4/3 3/2 27/16 243/128 260.7 293⅓ 330 347.7 391.1 440 495 C D E F G A B
The names can by extension be used in any temperament that approximates diatonic scales. This includes, sure enough, 12-edo – like any other meantone temperament. In particular, it works for more accurate 5-limit equal temperaments like 34-edo, and also 31-edo (though that is rather more interesting as a 7-limit tuning). You will in each case get a slightly different frequency assigned to most notes, but they're still “conceptually the same notes”.
You won't necessarily get a name for every note in the temperament this way. 24-edo can also approximate diatonic scales... because it contains 12-edo as a subset. So, the quarter-tones don't get names this way. The “microtonal notes” do get names assigned in e.g. 31-edo though: unlike in 12-edo, enharmonicity doesn't “wrap around” the circle of fifths already after six sharp signs, but only after 16 of them, or 17 in 34-edo. You
https://en.wikipedia.org/w/index.php?title=31_equal_temperament&oldid=740902751#Scale_diagram
Of course, there's a lot of ambiguity if you consider multiple different temperaments and stray through many different diatonic keys. In this case, the western naming scheme just doesn't work very well anymore. I'm afraid there is no single universally accepted convention that adresses this problem. You can of course always just specify the scale and then enumerate the scale degrees, but that's not particularly insightful – essentially you're writing guitar tabs at this point.
More useful are naming schemes like Ben Johnston uses, based on higher-limit just intonation. You can again adapt these to any temperament that approximates the just limit.
