Eikosany: Difference between revisions

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

== Example ==

~~{{todo|inline=~~1~~|add examples|comment=see~~ [[~~hexany~~]] and [[~~dekany~~]]}}

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):

# 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};

# 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};

# [[Octave reduction|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};

# 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};

# 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}.

# 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.

[[Category:Eikosanies| ]]

[[Category:Combination product sets]]

[[Category:20-tone scales]]

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 An '''eikosany''' is a 20-note [[scale]] built using all the possible combinations of 3 [[interval]]s from a given set of 6 intervals. It is a particular case of a [[combination product set]] (CPS).
+== Example ==
-{{todo|inline=1|add examples|comment=see [[hexany]] and [[dekany]]}}
+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):
-{{stub}}
+# Multiply together each pair of intervals (to find the combinations):<br>{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};
+# Divide each product by the smallest element of the previous set (to base the scale on 1/1):{1x3x5=15}<br>= {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};
+# [[Octave reduction|Octave-reduce]] each element:<br>{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};
+# Sort the elements in ascending order:<br>{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};
+# Replace the unison (1/1) by the octave ([[2/1]]) for a Scala-compatible octave-repeating scale:<br>{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}.
+# 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:<br>{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}.<br>In this case, the points of reflection are 297/280 and 35/33.
 [[Category:Eikosanies| ]] <!-- main article -->
 [[Category:Combination product sets]]
 [[Category:20-tone scales]]