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 non-trivial 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):
- 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}; - 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}; - Octave-reduce each element:
{1/1, 5/3, 7/6, 5/4, 7/4, 35/24}; - Sort the elements in ascending order:
{1/1, 7/6, 5/4, 35/24, 5/3, 7/4}; - 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}.
Stellated Hexanies
A stellated hexany is a 14-tone scale and is also called a dekatesserany. This is formed by adding the combinations of 1 out of 4 and 3 out of 4 intervals to the set. In the case of the example above, that would expand it to a {1/1, 35/32, 5/4, 21/16 3/2, 105/64, 7/4, 15/8} scale. Note that many of the notes are repeated in this case because 1 is one of the factors and 1x3 is identical to 3, etc. The simplest stellated hexany without any repeated notes is the 3-5-7-9 one, which produces a scale of:
{3, 5, 7, 9} {3x5=15, 3x7=21, 3x9=27, 5x7=35, 5x9=45, 7x9=63} {3x5x7=105, 3x5x9=135, 3x7x9=189, 5x7x9=315}
Divided by the smallest element, octave reduced and sorted by order, this is:
{1/1, 35/32, 9/8, 7/6, 5/4, 21/16, 45/32, 35/24, 3/2, 105/64, 5/3, 7/4, 15/8, 63/32}.
Pages for individual hexanies
See Category:Hexanies.
External links
- Ervin Wilson's Hexany by Kraig Grady
- The Tonality Cube Demonstration video by 12tone music.
- Paul Erlich. The Forms of Tonality.