Let \(\mu _{p,q}\) be a self-similar spectral measure with consecutive digits generated by an iterated function system \(\{f_i(x)=\frac{x}{p}+\frac{i}{q}\}_{i=0}^{q-1}\), where \(2\le q\in {{\mathbb {Z}}}\) and q|p. It is known that for each \(w=w_1w_2\cdots \in \{-1,1\}^\infty :=\{i_1i_2\cdots :~\text {all}~i_k\in \{-1,1\}\}\), the set

is a spectrum of \(\mu _{p,q}\). In this paper, we study the possible real number t such that the set \(t\Lambda _w\) are also spectra of \(\mu _{p,q}\) for all \(w\in \{-1,1\}^\infty \).

令\（\ mu _ {p，q} \）为自相似频谱度量，具有由迭代函数系统\（\ {f_i（x）= \ frac {x} {p} + \ frac { i} {q} \} _ {i = 0} ^ {q-1} \），其中\（2 \ le q \ in {{\ mathbb {Z}}} \）和q | p。已知对于\ {-1,1 \} ^ \ infty中的每个\（w = w_1w_2 \ cdots：= \ {i_1i_2 \ cdots：〜\ text {all}〜i_k \ in \ {-1,1 \} \} \），集合

是\（\ mu _ {p，q} \）的频谱。在本文中，我们研究了可能的实数t，使得集合\（t \ Lambda _w \）也是所有\（w \ in \ {-1 ）的\（\ mu _ {p，q} \）的谱，1 \} ^ \ infty \）。