Convexity and the well-formedness of musical objects

Abstract

It is well known that subsets of the two-dimensional space ℤ² can represent prominent musical and music-theoretical objects such as scales, chords and chord vocabularies. It has been noted that the major and minor diatonic scale form convex subsets in this space. This triggers the question whether convexity is a more widespread concept in music. This article systematically investigates the convexity for a number of musical phenomena including scales, chords and (harmonic) reduction. It is hypothesised that the notion of convexity may be a covering concept of musical phenomena and possibly reflects other mathematical properties of these musical structures. Furthermore, convexity can be used in a pitch-spelling model.

Acknowledgements

This article benefited from discussions with many people. We wish to thank in particular Dion Gijswijt and Yoav Seginer for discussion on some mathematical topics. Furthermore, we are grateful to Henkjan Honing, Jelle Zuidema and two anonymous reviewers for helpful comments.

Notes

¹The symbol ℤ is here used in its standard mathematical meaning of the set of integers. ℤ² is therefore a two-dimensional lattice of points aligning with integers on each axis. ℤ _n is the set of integers modulo n, so ℤ₁₂ is the set {0, 1, … 11}, like the hours of the clock. ℝ is the set of real numbers.

²Convexity has also been observed in rhythm space (Desain & Honing, 2003).

³We can write: with k = u + v + 2w, l = v + w, m = v. Or, in vector notation:

This map is invertible if T is invertible; T is invertible if Det(T) ≠ 0. It is easy to calculate that Det(T) = 1 so T is invertible.

⁴A basis-transformation is by definition a bijective map (one-to-one correspondence). For all possible basis-transformations, the determinant of transformation matrix should equal 1. For the proof, see Honingh (2003).

⁵From here on, when we write convex sex in the note name space or pitch number space, we mean the set that is obtained by projecting the corresponding (convex) set in the interval space to the note name or pitch number space.

⁶E.g., the dominant seventh chord can (in 5-limit JI) be tuned as 36 : 45 : 54 : 64 : = 4 : 5 : 6|27 : 32 or as 20: 25 : 30 : 36 = 4 : 5 : 6|5 : 6 (and there are more possibilities), but there is no theory that tells us which tuning is preferred. See, e.g., “The Alternate Tunings Mailing List” for an ongoing discussion about tuning (http://launch.groups.yahoo.com/group/tuning/).

⁷Since we are using octave equivalence, we are not taking into account the different inversions of a chord. Schenker (1906) and Salzer (1962) state that it depends on the bass whether a chord progression is a harmonic progression. Here we treat all triads having an harmonic function.

⁸The dotted lines indicate how the scales proceed to both sides, so that there is more than one possibility of choosing the region (convex or not convex) of two adjacent chords. E.g., for triads IV and V, it is not directly clear that they can form a convex set, but if V is chosen as indicated by the dotted lines, they do. Again, it depends on the intonation of the triad which one is preferred.

⁹Leaving out the progressions involving III(harmonic) and VI(ascending melodic), since these are rarely used according to Piston and DeVoto (1989).