Notation. Write $\mathbb{R}_\mathbb{Z}$ for the set of all real numbers for which there exists a unique nearest integer. Explicitly:
$$\mathbb{R}_\mathbb{Z} = \mathbb{R} \setminus(0.5 + \mathbb{Z}),$$ where the backslash denotes relative complementation.
Given a real number $x$ not of the form $n/2$ where $n$ is an integer, write $\lfloor x \rceil$ for the nearest integer (I've read that this is standard notation in cryptography, which is the only reason I included cryptography in the tags). In this notation, the signed distance to the nearest integer is the function defined below. $$f : \mathbb{R}_\mathbb{Z} \rightarrow (-0.5, 0.5), \quad f(x) = x - \lfloor x \rceil.$$
Question. Is there a standard name for this function (and/or the number it produces) that's reasonably succinct? E.g. the deviation of $x$, the residue of $x$, the deficiency of $x$, etc.
Motivation. This function shows up whenever we try to fit microtonal music into a semitonal context. For example, suppose I have tune $C$ to its standard 12TET value. Suppose I want $E$ to be a JI major third above that. The question arises of how far to retune $E$ from its standard 12TET value in order to achieve this goal. The JI ratio for the major third is $5/4$, and consequently its semitonal size is $12 \log_2(5/4).$ Hence the amount we have to retune $E$ is $f(12 \log_2(5/4))$. Since this is approximately $-0.14$, this tells us that we need to flatten $E$ by about $14\%$ of a semitone to achieve our goal. In the hope of communicating with other musicians more easily, I'd therefore like to have a succinct name for this function.