Glossary
Lewin-Quinn FC-components
Glossary
Set Theory
David Lewin (1959) proposed a six-number representation (FC1 through FC6) of the interval content of a set class that measures how unbalanced the set class is relative to each of the six interval classes.
So a low value of 0.0 for FC1 means that the set class has no semitone content (or is perfectly balanced relative to semitones), and a high value means more semitone content (spread out more evenly), and so forth, for FC2 (whole-tone content) through FC6 (tritone content).
Computing each Fourier component involves summing vectors with certain angles for each pitch class present in the set class, then taking the length of the resulting vector. This length captures the amount of imbalance numerically. These values (say, FC1 again) can be compared across set classes that share cardinality, (see Sorting the Set Classes tables) and manifest quantitatively some interesting, intuitive music-theoretic trends.
Ian Quinn’s 2004 masters thesis (see References) compares many other set-class-to-set-class metrics proposed by modern theorists and shows how the Lewin FC-components generalize and capture all of the important aspects of those metrics without some of their limitations.
This representation has other advantages: it can be used to analyze other tuning systems besides 12-TET (Quinn uses 10-TET at points in his work on Q-space); complementary set classes share the same values for all FC-components FC1 through FC6, capturing that relationship in a robust way “for free;” and transforming a set class by the M-relation (trading semitones for fourths) simply trades values for FC1 and FC5 (preserving FC2, FC3, FC4, and FC6), something that makes sense intuitively (for 12-TET).
Vectors in 6-D Complex Space
Instead of turning a set class into a six-number vector (taking the length of the summed 2-D vectors mentioned above), if we take any of the 4,096 possible pitch class sets (OPC-equivalent) and compute the Fourier components as a vector in 6-D complex space, we have a meaningful metric space for comparing pitch class sets to one another, different to the voice-leading spaces of Tymoczko (2011) but related. The distance relationships between musical objects in this complex Fourier space correlate strongly to distances in the voice-leading spaces, but reveal new insights and allow for different analyses (see Tymoczko 2009). Pretty powerful for such a simple, easy-to-perform computation.
Set Theory
David Lewin (1959) proposed a six-number representation (FC1 through FC6) of the interval content of a set class that measures how unbalanced the set class is relative to each of the six interval classes.
So a low value of 0.0 for FC1 means that the set class has no semitone content (or is perfectly balanced relative to semitones), and a high value means more semitone content (spread out more evenly), and so forth, for FC2 (whole-tone content) through FC6 (tritone content).
Computing each Fourier component involves summing vectors with certain angles for each pitch class present in the set class, then taking the length of the resulting vector. This length captures the amount of imbalance numerically. These values (say, FC1 again) can be compared across set classes that share cardinality, (see Sorting the Set Classes tables) and manifest quantitatively some interesting, intuitive music-theoretic trends.
Ian Quinn’s 2004 masters thesis (see References) compares many other set-class-to-set-class metrics proposed by modern theorists and shows how the Lewin FC-components generalize and capture all of the important aspects of those metrics without some of their limitations.
This representation has other advantages: it can be used to analyze other tuning systems besides 12-TET (Quinn uses 10-TET at points in his work on Q-space); complementary set classes share the same values for all FC-components FC1 through FC6, capturing that relationship in a robust way “for free;” and transforming a set class by the M-relation (trading semitones for fourths) simply trades values for FC1 and FC5 (preserving FC2, FC3, FC4, and FC6), something that makes sense intuitively (for 12-TET).
Vectors in 6-D Complex Space
Instead of turning a set class into a six-number vector (taking the length of the summed 2-D vectors mentioned above), if we take any of the 4,096 possible pitch class sets (OPC-equivalent) and compute the Fourier components as a vector in 6-D complex space, we have a meaningful metric space for comparing pitch class sets to one another, different to the voice-leading spaces of Tymoczko (2011) but related. The distance relationships between musical objects in this complex Fourier space correlate strongly to distances in the voice-leading spaces, but reveal new insights and allow for different analyses (see Tymoczko 2009). Pretty powerful for such a simple, easy-to-perform computation.