in the Western world there’s an agreement that a particular A has a frequency of 440Hz (the A above Middle C on a piano). But it’s just an agreement.
Before we standardised on “Concert A = 440Hz” a composer or conductor in Vienna might have that A sounding higher of lower than a composer in London. In fact there was an arms race where composers were shifting their tuning higher and higher so that their music would sound more lively. We know this because we still have their tuning forks and can plot the shift over time.
So the note which is half way between C and C# is a quarter-tone out from *standard* tuning. You could call it a sharpened C, or a flattened C#. You could even call it an even more flattened D if you like. Or if you don’t want to call it sharpened or flattened, you could call it a non-standard C.
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