Eikosany

An eikosany is a 20-note scale built using all the possible combinations of 3 intervals from a given set of 6 intervals. It is a particular case of a combination product set (CPS).

Example

Here is a step-by-step construction of the simplest possible eikosany, the 1-3-5-7-9-11 one: (i.e. using 1/1, 3/1, 5/1, 7/1, 9/1 and 11/1 with the smallest product as the root):

1. Multiply together each pair of intervals (to find the combinations):
   {1x3x5=15, 1x3x7=21, 1x3x9=27, 1x3x11=33, 1x5x7=35, 1x5x9=45, 1x5x11=55, 1x7x9=63, 1x7x11=77, 1x9x11=99, 3x5x7=105, 3x5x9=135, 3x5x11=165, 3x7x9=189, 3x7x11=231, 3x9x11=297, 5x7x9=315, 5x7x11=385, 5x9x11=495, 7x9x11=693};
2. Divide each product by the smallest element of the previous set (to base the scale on 1/1):{1x3x5=15}
   = {1/1, 7/5, 9/5, 11/5, 7/3, 3/1, 11/3, 21/5, 77/15, 33/5 7/1, 9/1, 11/1, 63/5, 77/5, 99/5, 21/1, 77/3, 33/1, 231/5};
3. Octave-reduce each element:
   {1/1, 7/5, 9/5, 11/10, 7/6, 3/2, 11/6, 21/20, 77/60, 33/20, 7/4, 9/8, 11/8, 63/40, 77/40, 99/80, 21/16, 77/48, 33/32, 231/160};
4. Sort the elements in ascending order:
   {1, 33/32, 21/20, 11/10, 9/8, 7/6, 99/80, 77/60, 21/16, 11/8, 7/5, 231/160, 3/2, 63/40, 77/48, 33/20, 7/4, 9/5, 11/6, 77/40};
5. Replace the unison (1/1) by the octave (2/1) for a Scala-compatible octave-repeating scale:
   {33/32, 21/20, 11/10, 9/8, 7/6, 99/80, 77/60, 21/16, 11/8, 7/5, 231/160, 3/2, 63/40, 77/48, 33/20, 7/4, 9/5, 11/6, 77/40, 2/1}.
6. Note that like any combination product set which uses half of its factors to derive the notes, these scales are always symmetrical by reflection, which makes it easier to doublecheck that your calculations are correct by looking at the pattern of step sizes:
   {33/32, 56/55, 22/21, 45/44, 28/27, 297/280, 28/27, 45/44, 22/21, 56/55, 33/32, 80/77, 21/20, 55/54, 36/35, 35/33, 36/35, 55/54, 21/20, 80/77}.
   In this case, the points of reflection are 297/280 and 35/33.