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Suppose that 
$0<|\unicode[STIX]{x1D70C}|<1$ and 
$m\geqslant 2$ is an integer. Let 
$\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D70C},m}$ be the self-similar measure defined by 
$\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D70C},m}(\cdot )=\frac{1}{m}\sum _{j=0}^{m-1}\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D70C},m}(\unicode[STIX]{x1D70C}^{-1}(\cdot )-j)$. Assume that 
$\unicode[STIX]{x1D70C}=\pm (q/p)^{1/r}$ for some 
$p,q,r\in \mathbb{N}^{+}$ with 
$(p,q)=1$ and 
$(p,m)=1$. We prove that if 
$(q,m)=1$, then there are at most 
$m$ mutually orthogonal exponential functions in 
$L^{2}(\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D70C},m})$ and 
$m$ is the best possible. If 
$(q,m)>1$, then there are any number of orthogonal exponential functions in 
$L^{2}(\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D70C},m})$.
