The Connection of Chords over Zn

by Daniel Schell, Ola Rinta-Koski an d Thomas R.R. Pintelon

Abstract

Chords are optimally connected if their voicing sums to a minimum of steps. The basic theory of harmony shows how to connect chords with small intervals, leading-tone half-step lines and, where possible, held common notes. This is particularly in use in the conduct of voices in choral works.

Piano players, specially the improvising ones, know how to minimise the motion of their fingers when they go from one chord to the other. This could be formulated as a search for minimal distances. Similarly, we could consider that, at some moments, intervals in the individual voices of choral works should be minimised. If we accept that the energy spent by a singer performing a leap is proportional to the step interval value, then the optimal connection of chords becomes equivalent to a problem of sparing resources under constraints.

Composers usually connect 3-chords (triads) and 4-chords (tetrads) over Z12, but it could be interesting to generalise this procedure to chords of any size – p-chords – over any system of integer octave division Zn. This is also an attempt to generalise the description of traditional harmony. Considering that a formula of minimally connected chords such as I7-IIm7-V7-I7 is commonly used on Z12, we might ask us why, and if there are other similar formulas to be found, with different chord sizes and on different Zn.

The process of connecting chord sequences is divided in two:

Connecting equal sized chords in pairs, crossing-free and bijective.
Finding the shortest path between the sequences of pair-connected chords.

For connecting chords in pairs we use and compare three different methods: (i) exhaustive enumeration, (ii) transition matrix search by assignment method (the Hungarian algorithm) (iii) Depth-first search.

For the shortest path between sequences of chords we use Travelling Salesman type of algorithms.

For small chord and scale sizes, both the exhaustive enumeration (i) and the two search methods (ii) and (iii) have been used and compared, allowing us to verify the results of the search algorithms. The latter are then used for larger chords on a Z24 note scale, which would be too computationally expensive with exhaustive methods.

Key words: Musical Harmony, Combinatorial Optimisation, Optimality, Connection of sets, Applications in Music Theory, Enumeration of Partition, Patterns, Tiles

The list of karo-s

A karo is a partitioning tile of Z12. It contains four tri-chords (or triads) . We have named this structure a ‘karo’ , from the French Carreau, because when represented on a circle it looks like a ‘Beau Carreau’ (Carreaux de Delft, for instance).

Hereunder is the list of 1306 existing karos, in other words tiles of four 3-chords partitioning Z12, ordered by their name, in ascending lexicographic order.

First column: Structure of the tile. Each chord is described by its name, the Interval Vector with the smallest ambitus,
Second column: Number of instances, depending on its symmetry. If a tile has no or one axe symmetry, then there will be 12 different instances or rotations. If there is two axes of symmetry, thus a symmetry of order 2, then it will have 6 instances and so on.
Third column: TAS, Total Absolute Step interval, a measure of its connecting ability.

Karo_List_10 by Daniel Schell, Ola Rinta-Koski, Thomas Pintelon