FM modulation index ($\beta =\Delta f/f_m$)

For a message consisting of a single sinusoid with frequency $f_m>0$, the FM modulation index is $\beta =\Delta f/f_m$. It is dimensionless and equals the peak wave phase excursion in radians in the chosen sine-phase convention. Keeping $\Delta f=5\mathrm{Hz}$ while changing $f_m$ from $1$ to $2\mathrm{Hz}$ reduces $\beta$ from $5$ to $2.5$: the same frequency deviation acts for a shorter portion of each message cycle. A general message containing many frequencies does not have a single $f_m$ with this meaning.