Euler–Fokker genus

In music theory and tuning, an Euler–Fokker genus (plural: genera), named after Leonhard Euler and Adriaan Fokker, is a musical scale in just intonation whose pitches can be expressed as products of some of the members of some multiset of generating prime factors. Powers of two are usually ignored, because of the way the human ear perceives octaves as equivalent.

An x-dimensional tone-dimension contains x factors. "An Euler-Fokker genus with two dimensions may be represented in a two-dimensional (rectangular) tone-grid, one with three dimensions in a three-dimensional (block-shaped) tone-lattice. Euler-Fokker genera are characterized by a listing of the number of steps in each dimension. The number of steps is represented by a repeated mention of the dimension, so that there arise descriptions such as [3 3 5 5], [3 5 7], [3 3 5 5 7 7 11 11], etc." For example, the multiset {3, 3, 7} yields the Euler–Fokker genus [3, 3, 7], which contains these pitches:

Normalized to fall within an octave, these become: 1/1, 9/8, 21/16, 3/2, 7/4, 63/32.