Dynamic mechanical spectroscopy (DMS), which allows measuring frequency-dependent viscoelastic properties, is important to study soft materials, tissues, biomaterials, polymers. However, the existing DMS techniques (nanoindentation) have limited resolution when used on soft materials, preventing them from being used to study mechanics at the nanoscale. The nanoindenters are not capable of measuring cells, nanointerfaces of composite materials. Here we present a highly accurate DMS modality, which is a combination of three different methods: quantitative nanoindentation (nanoDMA), gentle force and fast response of atomic force microscopy (AFM), and Fourier transform (FT) spectroscopy. This new spectroscopy (which we suggest to call FT-nanoDMA) is fast and sensitive enough to allow DMS imaging of nanointerfaces, single cells, while attaining about 100x improvements on polymers in both spatial (to 10-70 nm) and temporal resolution (to 0.7 s/pixel) compared to the current art. Multiple frequencies are measured simultaneously. The use of 10 frequencies are demonstrated here (up to 300 Hz which is a rather relevant range for biological materials and polymers, in both ambient conditions and liquid). The method is quantitatively verified on known polymers and demonstrated on cells and polymers blends. Analysis shows that FT-nanoDMA is highly quantitative. The FT-nanoDMA spectroscopy can easily be implemented in the existing AFMs.