Editing Modulatory space (section)

== Five-limit modulatory space ==

[[limit (music)|Five limit]] [[just intonation]] has a modulatory space based on the fact that its pitch classes can be represented by 3<sup>a</sup> 5<sup>b</sup>, where a and b are integers. It is therefore a [[free abelian group]] with the two generators 3 and 5, and can be represented in terms of a [[square lattice]] with fifths along the horizontal axis, and major thirds along the vertical axis.

In many ways a more enlightening picture emerges if we represent it in terms of a [[hexagonal lattice]] instead; this is the [[Tonnetz]] of [[Hugo Riemann]], discovered independently around the same time by [[Shohé Tanaka]]. The fifths are along the horizontal axis, and the major thirds point off to the right at an angle of sixty degrees. Another sixty degrees gives us the axis of major sixths, pointing off to the left. The non-unison elements of the 5-limit [[tonality diamond]], 3/2, 5/4, 5/3, 4/3, 8/5, 6/5 are now arranged in a regular hexagon around 1. The triads are the equilateral triangles of this lattice, with the upwards-pointing triangles being major triads, and downward-pointing triangles being minor triads.

This picture of five-limit modulatory space is generally preferable since it treats the consonances in a uniform way, and does not suggest that, for instance, a major third is more of a consonance than a major sixth. When two lattice points are as close as possible, a unit distance apart, then and only then are they separated by a consonant interval. Hence the hexagonal lattice provides a superior picture of the structure of the five-limit modulatory space.

In more abstract mathematical terms, we can describe this lattice as the integer
pairs (a, b), where instead of the usual Euclidean distance we have a Euclidean distance defined in terms of the vector space norm
:<math>||(a, b)|| = \sqrt{a^2 + ab + b^2}.</math>