Dekany

A dekany is a 10-note scale built using all the possible combinations of either 2 or 3 intervals (but not a mix of both) from a given set of 5 intervals. It is a particular case of a combination product set (CPS).

In this article, 2-combination dekany refers to a 2)5 CPS and 3-combination dekany refers to a 3)5 CPS; Erv Wilson did not necessarily use these terms.

Examples

2-combination dekany

Here is a step-by-step construction of the canonical 1-3-5-7-11 2-combination dekany (i.e. using 2-combinations of 1/1, 3/1, 5/1, 7/1, and 11/1 with the smallest product as the root):

1. Multiply together each pair of intervals (to find the combinations):
   {1 × 3, 1 × 5, 1 × 7, 1 × 11, 3 × 5, 3 × 7, 3 × 11, 5 × 7, 5 × 11, 7 × 11}
   = {3, 5, 7, 11, 15, 21, 33, 35, 55, 77};
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, 11/3, 15/3, 21/3, 33/3, 35/3, 55/3, 77/3}
   = {1/1, 5/3, 7/3, 11/3, 5/1, 7/1, 11/1, 35/3, 55/3, 77/3};
3. Octave-reduce each element:
   {1/1, 5/3, 7/6, 11/6, 5/4, 7/4, 11/8, 35/24, 55/48, 77/48};
4. Sort the elements in ascending order:
   {1/1, 55/48, 7/6, 5/4, 11/8, 35/24, 77/48, 5/3, 7/4, 11/6};
5. Replace the unison (1/1) by the octave (2/1) for a Scala-compatible octave-repeating scale:
   {55/48, 7/6, 5/4, 11/8, 35/24, 77/48, 5/3, 7/4, 11/6, 2/1}.

3-combination dekany

Here is a step-by-step construction of the canonical 1-3-5-7-11 3-combination dekany (i.e. using 3-combinations of 1/1, 3/1, 5/1, 7/1, and 11/1 with the smallest product as the root):

1. Multiply together each pair of intervals (to find the combinations):
   {1 × 3 × 5, 1 × 3 × 7, 1 × 3 × 11, 1 × 5 × 7, 1 × 5 × 11, 1 × 7 × 11, 3 × 5 × 7, 3 × 5 × 11, 3 × 7 × 11, 5 × 7 × 11}
   = {15, 21, 33, 35, 55, 77, 105, 165, 231, 385};
2. Divide each product by the smallest element of the previous set (to base the scale on 1/1):
   {15/15, 21/15, 33/15, 35/15, 55/15, 77/15, 105/15, 165/15, 231/15, 385/15}
   = {1/1, 7/5, 11/5, 7/3, 11/3, 77/15, 7/1, 11/1, 77/5, 77/3};
3. Octave-reduce each element:
   {1/1, 7/5, 11/10, 7/6, 11/6, 77/60, 7/4, 11/8, 77/40, 77/48};
4. Sort the elements in ascending order:
   {1/1, 11/10, 7/6, 77/60, 11/8, 7/5, 77/48, 7/4, 11/6, 77/40};
5. Replace the unison (1/1) by the octave (2/1) for a Scala-compatible octave-repeating scale:
   {11/10, 7/6, 77/60, 11/8, 7/5, 77/48, 7/4, 11/6, 77/40, 2/1}.