User:FloraC/Hard problems of harmony and psychoacoustically supported optimization: Difference between revisions
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Chordal rootedness is said with respect to an individual chord. A chord with chordal rootedness is dubbed a rooted chord. Such a chord feels distinctly resolved and self-contained: it speaks of itself as a center of reference, a result from the alignment of the chord's formal root with its virtual fundamental. Let us clarify these terms before we proceed. | Chordal rootedness is said with respect to an individual chord. A chord with chordal rootedness is dubbed a rooted chord. Such a chord feels distinctly resolved and self-contained: it speaks of itself as a center of reference, a result from the alignment of the chord's formal root with its virtual fundamental. Let us clarify these terms before we proceed. | ||
A formal root is simply the pitch that is denoted in terms of ratios as unity. Determination of a chord's formal root is part of the traditional pedagogy of harmony, but it is really hard to speak of as it depends on many factors: the specific chord structure, the texture and perhaps the genre of the piece. So it is often found heuristically. The bass supported by the perfect fifth is typically the best candidate of a formal root in | A formal root is simply the pitch that is denoted in terms of ratios as unity. Determination of a chord's formal root is part of the traditional pedagogy of harmony, but it is really hard to speak of as it depends on many factors: the specific chord structure, the texture and perhaps the genre of the piece. So it is often found heuristically. The bass supported by the perfect fifth is typically the best candidate of a formal root in a positive, octave-equivalent context, but such a structure is lacking in many chords especially in nontertian harmony. Another reasonable strategy in the same context is to always place the formal root on the bass. Anyway, this root is considered to be the starting point of a chord on top of which other notes are built. | ||
By contrast, the chord's actual root is what we know as the virtual fundamental. This is the particular pitch on which a chord appears to be a single harmonic note after the phenomenon of timbral fusion, so analytically it emerges at the GCD of all the ratios of the chord. This root, being simultaneously "virtual" and "actual", presents an interesting case on how we think about it. Its virtuality is reflected by the fact that the pitch need not be present – neither as a note nor even as an energy stream in the spectrogram. Its actuality is evidenced by the fact that we literally hear it due to the gestalt of harmonic series. | By contrast, the chord's actual root is what we know as the virtual fundamental. This is the particular pitch on which a chord appears to be a single harmonic note after the phenomenon of timbral fusion, so analytically it emerges at the GCD of all the ratios of the chord. This root, being simultaneously "virtual" and "actual", presents an interesting case on how we think about it. Its virtuality is reflected by the fact that the pitch need not be present – neither as a note nor even as an energy stream in the spectrogram. Its actuality is evidenced by the fact that we literally hear it due to the gestalt of harmonic series. | ||