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}.
If you include the combinations of 0 out of 4 and 4 out of 4 as well, you get a 16 note scale known as a tesseract which is always solidly rooted with all the other notes otonal to the 1.
Bihexany
Another way of expanding out a hexany is by taking two copies of the same hexany and offsetting them by some other interval. This is particularly useful because the resulting 12 note scale can be mapped onto a standard keyboard with no missed or repeated notes, giving you as many harmonic options as possible without having to buy a custom instrument to play the scale properly. For example, two 1-3-5-9 hexanies separated by a 7/5 would produce a scale of:
{1 21/20 9/8 7/6 5/4 21/16 7/5 3/2 63/40 5/3 7/4 15/8 2/1}
Duohexany
Two different hexanies together is a duohexany. If the hexanies have no intervals in common, this will result in a 12 note scale mappable to a standard keyboard. If they share intervals, the notes per octave will be less than 12.
Pages for individual hexanies
See Category:Hexanies and Gallery of combination product sets
External links
- Ervin Wilson's Hexany by Kraig Grady
- The Tonality Cube Demonstration video by 12tone music.
- Paul Erlich. The Forms of Tonality.