A fiber interferometer for absolute distance measurements is presented whereby wavelength variation is achieved via a sinusoidal strain modulation of a fiber Bragg grating to generate a series of beat wavelengths. The interferometer employs fiber laser sources where the design is based on the use of narrow-bandwidth fiber Bragg gratings. The accuracy of the beat wavelengths is improved compared to the use of multiple wavelengths measured with conventional optical spectrum analyzers or available wavemeters. Initial measurements are presented for beat wavelengths of $254.74\mathrm{mm}$ and $27.4\mathrm{m}$ over an optical path difference of $200\mathrm{mm}$ and $3.8\mathrm{m}$ , respectively. Combined with a two (or three) wavelength interferometer, this technique has the potential for ultrahigh dynamic range metrology ranging over several meters while preserving subfringe resolution and a low system complexity.