Editing Modulatory space (section)

== Seven-limit modulatory space ==

In similar fashion, we can define a modulatory space for [[seven-limit]] just intonation, by representing 3<sup>a</sup> 5<sup>b</sup> 7<sup>c</sup> in terms of a corresponding [[cubic honeycomb|cubic lattice]]. Once again, however, a more enlightening picture emerges if we represent it instead in terms of the three-dimensional analog of the hexagonal lattice, a lattice called A<sub>3</sub>, which is equivalent to the [[face-centered cubic|face centered cubic lattice]], or D<sub>3</sub>. Abstractly, it can be defined as the integer triples (a, b, c), associated to 3<sup>a</sup> 5<sup>b</sup> 7<sup>c</sup>, where the distance measure is not the usual Euclidean distance but rather the Euclidean distance deriving from the vector space norm
:<math>||(a, b, c)|| = \sqrt{a^2 + b^2 + c^2 + ab + bc + ca}.</math>

In this picture, the twelve non-unison elements of the seven-limit [[tonality diamond]] are arranged around 1 in the shape of a [[cuboctahedron]].