Eikosany - Revision history

Yourmusic Productions: Show example of working.

2026-07-18T15:22:40Z

Show example of working.

← Older revision		Revision as of 15:22, 18 July 2026
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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).		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:Eikosanies\| ]] <!-- main article -->
	[[Category:Combination product sets]]		[[Category:Combination product sets]]
	[[Category:20-tone scales]]		[[Category:20-tone scales]]

BudjarnLambeth: Todo

2024-12-15T01:25:34Z

Todo

← Older revision		Revision as of 01:25, 15 December 2024
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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).		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).


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

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	[[Category:Combination product sets]]		[[Category:Combination product sets]]
	[[Category:20-tone scales]]		[[Category:20-tone scales]]
	~~[[Category:todo:add examples]] <!-- see Hexany/Dekany -->~~
	~~[[Category:todo:expand]]~~

BudjarnLambeth: +stub

2024-01-30T06:32:00Z

+stub

← Older revision		Revision as of 06:32, 30 January 2024
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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).		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).
			{{stub}}

	[[Category:Eikosanies\| ]] <!-- main article -->		[[Category:Eikosanies\| ]] <!-- main article -->

Inthar at 05:56, 13 September 2021

2021-09-13T05:56:08Z

← Older revision		Revision as of 05:56, 13 September 2021
Line 1:		Line 1:
	A '''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).		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).

	[[Category:Eikosanies\| ]] <!-- main article -->		[[Category:Eikosanies\| ]] <!-- main article -->

Fredg999: Fixed internal link

2021-08-08T00:48:51Z

Fixed internal link

← Older revision		Revision as of 00:48, 8 August 2021
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	A '''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).		A '''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).

	[[Category:Eikosanies\| ]] <!-- main article -->		[[Category:Eikosanies\| ]] <!-- main article -->

Fredg999: Created page with "A '''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 combinatio..."

2021-08-08T00:48:31Z

Created page with "A '''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 combinatio..."

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A '''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).

[[Category:Eikosanies| ]] 
[[Category:Combination product sets]]
[[Category:20-tone scales]]
[[Category:todo:add examples]] 
[[Category:todo:expand]]