Hexany

A hexany is a 6-note scale built using all the possible combinations of 2 intervals from a given set of 4 intervals. It is the simplest case of a combination product set.

Example

Here is a step-by-step construction of the canonical 1-3-5-7 hexany (i.e. using 1/1, 3/1, 5/1, and 7/1 with the smallest product as the root):

1. Multiply together each pair of intervals (to find the combinations):
   {1 × 3, 1 × 5, 1 × 7, 3 × 5, 3 × 7, 5 × 7}
   = {3, 5, 7, 15, 21, 35};
2. Divide each product by the smallest element of the previous set (to base the scale on 1/1):
   {3/3, 5/3, 7/3, 15/3, 21/3, 35/3}
   = {1/1, 5/3, 7/3, 5/1, 7/1, 35/3};
3. Octave-reduce each element:
   {1/1, 5/3, 7/6, 5/4, 7/4, 35/24};
4. Sort the elements in ascending order:
   {1/1, 7/6, 5/4, 35/24, 5/3, 7/4};
5. Replace the unison (1/1) by the octave (2/1) for a Scala-compatible octave-repeating scale:
   {7/6, 5/4, 35/24, 5/3, 7/4, 2/1}.

External links

• Wikipedia: Hexany
• Ervin Wilson's Hexany by Kraig Grady