# Dekany

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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):

- 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}; - 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}; - Octave-reduce each element:

{1/1, 5/3, 7/6, 11/6, 5/4, 7/4, 11/8, 35/24, 55/48, 77/48}; - 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}; - 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):

- 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}; - 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}; - Octave-reduce each element:

{1/1, 7/5, 11/10, 7/6, 11/6, 77/60, 7/4, 11/8, 77/40, 77/48}; - 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}; - 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}.