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. A chord is a geometric figure. ... In geometry, a tiling (also called tessellation, mosaic or dissection) of a given shape S consists of a collection of other shapes which precisely cover S. Often the shape S to be tiled is the Euclidean plane, but other shapes and three-dimensional objects are considered as well. ... In music, modulation is most commonly the act or process of changing from one key (tonic, or tonal center) to another, also known as a key change. ...

History of chordal space

Historically, chordal space goes back to 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). Johann David Heinichen (1683 - July 16, 1729) was a Baroque composer and theorist active in Dresden at the court of Augustus the Strong. ... In music theory, the circle of fifths is a model of pitch space. ...

Gottfried Weber and S. G. Vial suggested a grid graph or square lattice model of chordal space; Weber's "regional chart" centered on C major, is: In graph theory, a grid graph is a graph corresponding to the square lattice, so that it is isomorphic to the graph having a vertex corresponding to every pair of integers (a, b), and an edge conecting (a, b) to (a+1, b) and (a, b+1). ... Upright square tiling. ...

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)

In music theory, the key identifies the tonic triad, the chord, major or minor, which represents the final point of rest for a piece, or the focal point of a section. ...

Princples of chordal space

In 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. In graph theory, a regular graph is a graph where each vertex has the same number of neighbors, i. ...

Cyclic chordal space

Lehrdal asserts that Heinichen's idea, to model chordal space by means of a cyclic graph, is "too impoverished a space" to work as a means of handling the relationships between both the major and the minor triads of common practice music. However, if suitably adapted Heinichen's basic idea forms the basis of one of the most satisfactory models of common practice harmony, or of the bare bones of it at any rate. // Cyclic graph A cycle graph of length 6. ...

We first take note that Heinichen's arrangement violates the principle that chords should be linked if they share a common interval; we can began to fix that by placing the relative minor before, rather than after, the corresponding major chord. Hence, we start out with ... -d-F-a-C-e-G-b-D- ... Any two adjacent chords in this chain are now linked by two intervals, so that the two chords adjacent to a given chord are strongly linked to that chord. Next we note (under the assumption of meantone temperament) that d is also linked by a shared interval (this time the fifth) with D. We therefore draw a line ahead seven steps from the minor triad to the major triad on the same root, or behind seven steps from a major triad to its associated minor triad. We do not, however, draw a line ahead from C major to c# minor, or behind from e minor to Eb major, because these share only one note. Meantone temperament is a system of musical tuning. ...

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. Equal temperament is a scheme of musical tuning in which the octave is divided into a series of equal steps (equal frequency ratios). ...

Image File history File links Chordal12. ...

Cyclic chordal space for 12 equal

We may obtain very similar pictures for any of the other equal temperaments supporting the use of meantone temperament, including in particular 19 equal temperament and 31 equal temperament, 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. Meantone temperament is a system of musical tuning. ...

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 desireable; 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 starting point. The cycle C-a-F-d-D-b-G-e-C is characterisitic 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. Cycle in graph theory and computer science has several meanings: A closed walk, with repeated vertices allowed. ... Just intonation is any musical tuning in which the frequencies of notes are related by whole number ratios. ... In music theory, a comma is a small or very small interval between two enharmonic notes tuned in different ways. ... When you ascend by a cycle of justly tuned perfect fifths (ratio 3:2), leapfrogging 12 times, you eventually reach a note around seven octaves above the note you started on, which, when lowered to the same octave as your starting point, is 23. ...

References

• Lerdahl, Fred (2001). Tonal Pitch Space, pp. 42-43. Oxford: Oxford University Press. ISBN 0195058348.
• Mathieu, W. A. (1997). Harmonic Experience: Tonal Harmony from Its Natural Origins to Its Modern Expression. Inner Traditions Intl Ltd. ISBN 0892815604.