Abstract: It is significant to create an appropriate cochleaogram (frequency-place map of a cochlea) to help hearing-impaired patients to achieve high level of speech and music recognition. A cochleaogram is also meaningful and useful in audiology, music therapy and music theory. Here, we originally propose a mathematical model of a logarithmic spiral function with a base 2, in a polar coordinates system, to plot a cochleaogram. The model is, fc (θ) = fms 2θ/θo, where, fc (θ) is a characteristic frequency of a cochlear basilar membrane, fms (2048 Hz) is the most sensitive frequency, θ is a spiral angle from the cochlear base to the cochlear apex (vice versa) around the origin of the system, θo (90°) is a spiral period constant of octaves. The origin is assumed at the spiral center of the cochlea. We estimate the fitness between our modeling results and the published data. The averaged relative deviation is smaller than 33%. A cochlea works as an informative decoder that decodes the sound information from a frequency domain to a spiral spatial domain: θ(fc) = θo log2(fc/fms). We also obtain perfect frequency ratios based on the semitone 21/12 for all 12 tones logarithmically and spirally with our model.