The Constant-Q Harmonic Coefficients: A timbre feature designed for music signals [Lecture Notes

Abstract

Timbre is the attribute of sound that makes, for example, two musical instruments playing the same note sound different. It is typically associated with the spectral (but also the temporal) envelope and assumed to be independent from the pitch (but also the loudness) of the sound <xref ref-type="bibr" rid="ref1" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[1]</xref> . This article shows how to design a simple but effective pitch-independent timbre feature, well adapted to musical data, by deriving it from the constant-Q transform (CQT), a log-frequency transform that matches the typical Western musical scale <xref ref-type="bibr" rid="ref2" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[2]</xref> , <xref ref-type="bibr" rid="ref3" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[3]</xref> . The decomposition of the CQT spectrum into an energy-normalized pitch component and a pitch-normalized spectral component is demonstrated, the latter from which a number of harmonic coefficients are extracted. The discriminative powers of these constant-Q harmonic coefficients ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">CQHCs)</i> are then evaluated on the NSynth data set <xref ref-type="bibr" rid="ref4" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[4]</xref> , a publicly available, large-scale data set of musical notes, where they are compared with the mel-frequency cepstral coefficients (MFCCs) <xref ref-type="bibr" rid="ref5" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[5]</xref> , a feature originally designed for speech recognition but commonly used to characterize timbre in music.