The Geometry of Musical Chords — Science

Claims

1. consonance-efficiency
   Because acoustically consonant chords divide the octave nearly evenly, they lie near the centre of the orbifolds and can be connected by efficient, independent voice leading. Tymoczko argues that this central feature of Western counterpoint is made possible by composers' interest in acoustic consonance.

   "It follows that the acoustically consonant chords of traditional Western music can be connected by efficient voice leading. ... These chords are therefore clustered near the center of the orbifolds (Table 1) and can be linked by efficient, independent voice leadings. Traditional tonal music exploits this possibility (Fig. 1, A to C, and movie S4). This central feature of Western counterpoint is made possible by composers' interest in the harmonic property of acoustic consonance."

   Locator: page: 73-74 · Quote language: en
2. harmony-counterpoint
   Western music lies at the intersection of harmony, which delimits acceptable chords and chord sequences, and counterpoint or voice leading, the technique of connecting the notes of successive chords to form simultaneous melodies. Chords are usually connected so that the voices move independently, efficiently (by short distances) and without crossing, which facilitates performance, engages explicit aesthetic norms and lets listeners distinguish simultaneous melodies.

   "Western music lies at the intersection of two seemingly independent disciplines: harmony and counterpoint. Harmony delimits the acceptable chords (simultaneously occurring notes) and chord sequences. Counterpoint (or voice leading) is the technique of connecting the individual notes in a series of chords so as to form simultaneous melodies. Chords are usually connected so that these lines (or voices) move independently (not all in the same direction by the same amount), efficiently (by short distances), and without voice crossings (along nonintersecting paths) (Fig. 1, A to C). These features facilitate musical performance, engage explicit aesthetic norms (1, 2), and enable listeners to distinguish multiple simultaneous melodies (3)."

   Locator: page: 72 · Quote language: en
3. history-diagrams
   Musicians have investigated efficient voice leading for almost three centuries. The circle of fifths, first published in 1728, depicts efficient voice leadings among the 12 major scales, and the Tonnetz, originating with Euler in 1739, represents efficient voice leadings among the 24 major and minor triads.

   "Musicians have been investigating these questions for almost three centuries. The circle of fifths (fig. S1), first published in 1728 (4), depicts efficient voice leadings among the 12 major scales. The Tonnetz (fig. S2), originating with Euler in 1739, represents efficient voice leadings among the 24 major and minor triads (2, 5)."

   Locator: page: 72 · Quote language: en
4. orbifold
   Tymoczko represents a chord as a point in a geometrical space called an orbifold, in which line segments represent voice leadings between chords. The space of unordered pairs of pitch classes is a Möbius strip. Efficient voice leading between structurally similar chords is possible only when the chords are nearly symmetrical under transposition, reflection or permutation.

   "A musical chord can be represented as a point in a geometrical space called an orbifold. Line segments represent mappings from the notes of one chord to those of another. Composers in a wide range of styles have exploited the non-Euclidean geometry of these spaces, typically by using short line segments between structurally similar chords. Such line segments exist only when chords are nearly symmetrical under translation, reflection, or permutation."

   Locator: page: 72-73 · Quote language: en