Continuous Chromagrams and Pseudometric Spaces of Sound Spectra

Abstract

In this paper we extend the ubiquitous music information retrieval technology of the quantized chromagram (or chroma feature) to propose a continuous chromagram, an octave-reduced spectrogram that logarithmically reduces the frequencies of a sound spectrum onto a half-open interval [a, 2a) we call a chroma octave. We prove that for any real number $r>1$, any two logarithmically reduced spectrograms onto intervals of reduction $[a_1, ra_1)$ and $[a_2, ra_2)$ with $a_1 \ne a_2$ are equivalent up to logarithmic scaling and rotation. In the case $r=2$ this proof shows why all chroma octaves bounded by both the upper and lower frequencies of the sound spectrum in question yield essentially the same continuous chromagram. We then propose a family of pseudometrics on sound spectra and discuss potential applications to analysis and composition.