# Chordal space

In music, chordal space is a mathematical model of relationships between chords in some musical system. These models are often graphs or tilings. Closely related to chordal space is modulatory space, which represents pitch classes; the chords of a chordal space are chosen from the pitch classes of a modulatory space.

## History of chordal space

One of the earliest graphical models of chord-relationships was devised by Johann David Heinichen in 1728; he proposed placing the major and minor chords in a circular arrangement of twenty-four chords arranged according to the circle of fifths; reading clockwise, ... F, d, C, a, G, ... (Lerdahl, 2001). The currently more popular major on the outside relative minor on the inside format was proposed by David Kellner (1737).

Gottfried Weber and F. G. Vial suggested a grid graph or square lattice model of chordal space; Weber's lattice centered on C major, is:

 d♯ F♯ f♯ A a C c g♯ B b D d F f c♯ E e G g B♭ b♭ f♯ A a C c E♭ e♭ b D d F f A♭ a♭ e G g B♭ b♭ D♭ d♭ a C c E♭ e♭ G♭ g♭

Lower case letters indicate minor key, uppercase major. This was first proposed by Vial (1767) (later Weber, Riemann, Schoenberg), the advantage over the circle of fifths being that it represents both relative and parallel major. (Lerdahl, 2001)

## Principles of chordal space

The Vial/Weber chordal space depicts two different sorts of relationships: shared common tones and efficient voice leading. For example, the proximity of the C major and e minor chords reflects the fact that the two chords share two common tones, E and G. Moreover, one chord can be transformed into another by moving a single note by just one semitone: to transform a C major chord into an E minor chord, one need only move C to B. Furthermore, the Vial/Weber chordal space is closely related to the two-dimensional lattices described in the article on pitch space: every chord on the Vial/Weber chordal space can be associated with a triangle on the Tonnetz or two-dimensional pitch class lattice.

The close correspondence between these properties — shared common tones, efficient voice leading, and the two-dimensional pitch lattices — is dependent on the fact that meantone temperament uses a consonant interval, the fifth, as generator, and may not apply in other cases.

However, when constructing a chordal space, several general principles are useful. The first is that chordal space should be or define a regular graph, whose regularity is linked to the regularity of the corresponding modulatory space. The second is that to start out with at least, only the most basic chords of the tonal system in question should be considered. The third is that two chords should be linked if and only if they share a common interval. Chords sharing a common note can then be reached via these closer connections.

## Cyclic chordal space

In the circular arrangement F - d - C - a ..., the chords F and d share two common tones, and can be linked by efficient voice leading. However, the chords d and C do not share any common tones, and cannot be linked by very efficient voice leading. By contrast in the series d - F - a - C - e - G ..., every chord shares two notes with its neighbors and can be transformed into them by moving one note by one or two semitones. The resulting pattern of chords can be generated in the Vial/Weber space, by moving upward along adjacent columns in the space.

We can now take the twenty-four major and minor triads of equal temperament and place them on the vertices of a regular 24-gon. We then draw lines from triads separated by one step, and also from each major triad to its parallel minor triad, and obtain a geometric picture of the regular graph in question, which satisfactorily models the triadic relationships in 12 equal temperament.

We may obtain very similar pictures for any of the other equal temperaments supporting the use of meantone temperament, including in particular 19edo and 31edo, by drawing a 38-gon or a 62-gon respectively, and linking all chords separated by one step, and the correct half separated by seven steps.

It might be considered that the presence of cycles, or closed looping paths, in these cyclic chordal spaces is a defect. However, removing them is neither necessary nor desirable; in fact, they result from a fundamental characteristic of temperament, namely comma pumps. These are cycles of chords which if traversed in just intonation would result in modulation by a small interval, or comma, but which in the temperament tuning simply returns to its starting point. The cycle C-a-F-d-D-b-G-e-C is characteristic of meantone, and so its presence in these cyclic chordal spaces should be expected. The closure of the 24-gon of 12 equal temperament tells us that the Pythagorean comma is also tempered out in this system; it is in fact a comma pump for the Pythagorean comma. The 12 equal temperament can be described as the temperament with both the syntonic and Pythagorean commas tempered out, and so we ought to find and do find both kinds of comma pumps. In other words, this isn't a bug, it's a feature.

Cyclic chordal spaces are by no means limited to equal temperaments supporting meantone. An examination of the diagram for 12 equal chordal space will show that it could also be described as a circle of fifths, with major triads represented by even numbers and minor triads by odd numbers, such that for each even number we add 17, and for each odd number we subtract 17, both modulo 24, and then draw a line between the two nodes. This is because -7 modulo 24 is the same as 17 modulo 24. We can do the same thing in other appropriately selected equal temperaments, for example 53edo, and obtain a chord space corresponding to schismatic temperament. However, this is not the only way of arranging the 106 triads of 53 equal. We may equally well began with the chain -A-a-C-c-Eb-eb-..., and draw a line between the node i and the node i+11 whenever i is even, and between i and i-11 whenever i is odd. This produces picture of 53 equal arranged in terms of hanson temperament, which tempers out the kleisma, or 15625/15552. The two graphs are in fact isomorphic, and so on a more abstract level the two pictures are identical, yet as depicted they are quite different.

## Linear chordal space

If we take a chain major and minor chords arranged as we arranged the chords in a circle of fifths, namely ... -d-F-a-C-e-G-b-D-..., and then draw loops from the minor triads ahead seven steps to the major triads, we obtain a picture of chordal space in generic, or logical, meantone. This is a meantone which is not tuned to an equal temperament, so that the circle does not close, as for example quarter-comma meantone. If we do the same sort of thing for the two pictures of 53 equal temperament discussed in the previous section, we now obtain two linear chordal spaces, one based on a chain of fifths, and the other a chain of minor thirds. These two chordal spaces are now not isomorphic, and represent chordal space for the schismatic and hanson temperaments respectively.

We may also produce linear and cyclic chordal spaces for higher limit harmony, such as septimal harmony. For example, we can take the 144 septimal tetrads of 72edo, arranged so that two steps from an otonal tetrad brings us to another otonal tetrad with root 7 steps of 72 equal higher, and likewise for utonal tetrads. We then may draw lines between chords whenever they are one, eleven, or fifteen steps away. Unlike in the case of triads, we do not need to worry about different kinds of chords for even and odd numbers of steps. We may unroll this picture of 72 equal into a linear chordal space for miracle temperament.

## Five-limit chordal space

The regional chart of Gottfried Weber represents the first attempt at defining the two-dimensional relationships of the chordal space of five-limit just intonation. It was, however, not intended as a system for representing just intonation, and consequently has the same chord name appearing in different locations. In pure just intonation, these would be different chords.

By beginning with the hexagonal lattice picture of five-limit modulatory space, we can arrive at a more theoretically satisfying picture of five-limit chordal space, which Lerdahl characterizes as "neo-Riemannian". The equilateral triangular regions of this correspond to triads, with the difference between major and minor triads represented by two different orientations. By taking the midpoints of these triangles to be verticies, and drawing an edge between them if the triangles share a side (which means the corresponding triads share an interval) we can create a tiling of the plane, called the dual tiling to the hexagonal lattice tiling, which is known as the hexagonal tiling. Each vertex of this tiling represents a triad, and from each vertex there are three edges each 120 apart from the others, connecting to the three triads which share an interval with the given triad. The three differing directions for the edges correspond to the three "neo-Riemannian transformations": R, which connects major to relative minor, for example C to a; L which connects a major chord to a chord containing a leading tone, for example, C to e; and P which connects a major triad to its parallel minor, for example C to c.

## Seven-limit chordal space

If we take the otonal and utonal tetrads of seven-limit just intonation,

and draw an edge between them if they share a common interval, we obtain the basic chordal space of seven-limit harmony. It has the remarkable property of forming a lattice; the tetrads are associated to triples of integers (a, b, c), which defines the cubic lattice, or three dimensional integer lattice. If

a+b+c is an even number, the corresponding tetrad is major with root

3^(-a+b+c)/2 5^(a-b+c)/2 7^(a+b-c)/2. If a+b+c is odd, it is minor, or utonal, with root

3^(-1-a+b+c)/2 5^(1+a-b+c)/2 7^(1+a+b-c)/2. This very regular structure is to some extent inherited by the cyclic or linear seven-limit chordal spaces, making them in some ways more regular than the corresponding five-limit spaces.