Let μ be a probability measure with compact support in R . The measure μ is called a spectral measure if there exists a countable set Λ ⊆ R , called a spectrum of μ , such that the family of exponential functions { e − 2 π i λ x : λ ∈ Λ } forms an orthonormal basis for L 2 ( μ ) . In this paper we study the structure of spectra and the real number t such that both Λ and t Λ are spectra for a class of self-similar spectral measures, which have symmetric spectra. Some examples are given to explain our theory.