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Appendices

Figure A1.  The first voice leading moves one note by two semitones, while the second moves three notes by one semitone each.

Figure C2. (a) On the Tonnetz, F major (triangle 3) is closer to C major (triangle 1) than F minor (triangle 4) is.  Consequently, the voice-leading (C, E, G)→(C, F, A) is represented a two-step motion, while it takes at least three steps to represent (C, E, G)→(C, F, Af). (b) In actual music, however, F minor frequently appears as a passing chord between F major and C major.

Figure D1.  The rows of this matrix correspond to the voice leadings in (b).  All of the strongly crossing-free (four-voice) voice-leadings from half-diminished to dominant seventh can be derived by adding a constant number to some row of this matrix.  For example, to combine interscalar transposition up by two steps with chromatic transposition downward by seven semitones, subtract 7 from the values in the third row of the matrix, as illustrated by (c).

Figure D3. Two equivalent interscalar interval matrices.  Each of the first six rows in the right matrix can be obtained by subtracting 12 from the next–lowest row in the left.  The last row of the right matrix is identical to the first row of the left.

Figure D4. The four voice-leadings corresponding to the four rows of the scalar interval matrix that takes (C, E, G, Bf) to itself.

Figure D5. The cross section of three-note chord space contains three line segments that connect 047 to a minor triad without touching the boundary (a).  Each corresponds to a different interscalar transposition between major and minor triads (b).

Figure E2a.  Pentatonic scales playing a prominent melodic role at the conclusion of Chopin’s Op. 10, No. 5 (a), in Dvorák’s Op. 96 “American” Quartet (b), and at the opening of McCoy Tyner’s solo on “Pursuance” (c).

Figure E2b.  Pentatonic scales playing a prominent melodic role at the conclusion of Chopin’s Op. 10, No. 5 (a), in Dvorák’s Op. 96 “American” Quartet (b), and at the opening of McCoy Tyner’s solo on “Pursuance” (c).  

Figure E2c.  Pentatonic scales playing a prominent melodic role at the conclusion of Chopin’s Op. 10, No. 5 (a), in Dvorák’s Op. 96 “American” Quartet (b), and at the opening of McCoy Tyner’s solo on “Pursuance” (c).   

Figure E3. The opening of Chick Corea’s solo on “Matrix,” from Now he sings, now he sobs (transcribed by Bill Dobbins in Corea 1988).  In jazz, the global harmonic stability of notes need not correspond to their local stability over tonic chords: Corea often plays F-Bf-Ef  for the F7 chord, but this does not mean that Bf is globally more stable than A.

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