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Fig. 7. Plots of Ph, versus pitch-class sum for 100,000 randomly chosen trichords, tetrachords, and penta- chords Fourier component is closely correlated with the transposition of FE’, the perfectly even n-note chord that is “nearest” to X, with distance measured by the size of the smallest voice leading from X to some subset of E’s notes. (These subsets are represented by unisons in the nth Fourier space.) Figure 7 plots Fourier phase against the transposition of E for 100,000 randomly chosen trichords, tetrachords, and pentachords in continuous space. The correlation and its approximate nature are both clear: while the two quantities are generally related, it is possible for them to diverge substantially, particularly in the case of chords with small Fourier magnitude.

Figure 7 Plots of Ph, versus pitch-class sum for 100,000 randomly chosen trichords, tetrachords, and penta- chords Fourier component is closely correlated with the transposition of FE’, the perfectly even n-note chord that is “nearest” to X, with distance measured by the size of the smallest voice leading from X to some subset of E’s notes. (These subsets are represented by unisons in the nth Fourier space.) Figure 7 plots Fourier phase against the transposition of E for 100,000 randomly chosen trichords, tetrachords, and pentachords in continuous space. The correlation and its approximate nature are both clear: while the two quantities are generally related, it is possible for them to diverge substantially, particularly in the case of chords with small Fourier magnitude.

Related Figures (7)

Fig. 1. The third Fourier component in complex space for pitch classes and major triads, and the fifth component for pitch classes and minor ninth chords.
Fig. 1. The third Fourier component in complex space for pitch classes and major triads, and the fifth component for pitch classes and minor ninth chords.
Fig. 2. Major triads in Phg and the circular dimension of voice-leading space
Fig. 2. Major triads in Phg and the circular dimension of voice-leading space
(~ 45054) )——> ( Fig. 4. The voice leading (C, E, G B, E, Gf) in F3
(~ 45054) )——> ( Fig. 4. The voice leading (C, E, G B, E, Gf) in F3
Fig. 3. The voice leading (C, D, E)“—""s(G#, E, F#) in F3
Fig. 3. The voice leading (C, D, E)“—""s(G#, E, F#) in F3
Fig. 5. Diatonic scales (labeled by number of accidentals) in Phs
Fig. 5. Diatonic scales (labeled by number of accidentals) in Phs
Fig. 6. Positions of {CEG} and {C#EF¢} in F3 and circular voice-leading space.
Fig. 6. Positions of {CEG} and {C#EF¢} in F3 and circular voice-leading space.
Fig. 8. Calculation of Ph3 and the nearest subset of a perfectly even chord for {CFBb} the pitch classes of the chord; if we adopt the Euclidean metric, the resulting vector is one of the n vectors that can serve as the “average” of the n points on the circle. Figure 8 illustrates in the case of the “fourth chord” (C, F, Bb), where the two methods coincide (compare Figure 1).
Fig. 8. Calculation of Ph3 and the nearest subset of a perfectly even chord for {CFBb} the pitch classes of the chord; if we adopt the Euclidean metric, the resulting vector is one of the n vectors that can serve as the “average” of the n points on the circle. Figure 8 illustrates in the case of the “fourth chord” (C, F, Bb), where the two methods coincide (compare Figure 1).
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