Aureusmic chords: Difference between revisions

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The most typical aureusmic triad is a palindrome in the 2.5.17.19 [[subgroup]] [[19-odd-limit]] since it identifies 5/4 by a stack of two 19/17's:

* 1-19/17-5/4 with steps 19/17-19/17-8/5.

* 1-19/17-5/4 chord with steps 19/17-19/17-8/5.

There is an inversely related pair which is even more squeezed:

* 1-17/16-19/17 with steps 17/16-20/19-8/5, and its inverse

* 1-17/16-19/17 with steps 17/16-20/19-34/19 and its inverse 1-20/19-19/17 with steps 20/19-17/16-34/19.

* 1-20/19-19/17 with steps 20/19-17/16-8/5.

They can be extended to the following ~~inversely related~~ tetrads, ~~all~~ of ~~which seem to be based largely on a sort~~ of ~~secundal harmony:~~

They can be extended to the following tetrads, with two palindromic chords and two pairs of chords in inverse relationship. The palindromic chords are

* 1-19/17-19/16-5/4 with steps 19/17-17/16-20/19-8/5, and its inverse

* 1-17/16-19/17-19/16 chord with steps 17/16-20/19-17/16-32/19;

* 1-20/19-19/17-5/4 with steps 20/19-17/16-19/17-8/5;

* 1-20/19-19/17-20/17 chord with steps 20/19-17/16-20/19-17/10.

* 1-19/17-20/17-5/4 with steps 19/17-20/19-17/16-8/5, and its inverse

* 1-17/16-19/17-5/4 with steps 17/16-20/19-19/17-8/5.

The inversely related pairs of chords are

* 1-19/17-19/16-5/4 with steps 19/17-17/16-20/19-8/5 and its inverse 1-20/19-19/17-5/4 with steps 20/19-17/16-19/17-8/5;

* 1-19/17-20/17-5/4 with steps 19/17-20/19-17/16-8/5 and its inverse 1-17/16-19/17-5/4 with steps 17/16-20/19-19/17-8/5.

Then there are two inversely related pentads:

* 1-17/16-19/17-19/16-5/4 with steps 17/16-20/19-17/16-20/19-8/5, and its inverse

* 1-17/16-19/17-19/16-5/4 with steps 17/16-20/19-17/16-20/19-8/5 and its inverse 1-20/19-19/17-20/17-5/4 with steps 20/19-17/16-20/19-17/16-8/5.

* 1-20/19-19/17-20/17-5/4 with steps 20/19-17/16-20/19-17/16-8/5.

Equal temperaments with aureusmic chords include {{Optimal ET sequence|12, 13, 25, 31, 37, 43, 56, 68, 81, 93, 124, 137, 149, 205, 230 and 286}}.

[[Category:19-odd-limit]]

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 The most typical aureusmic triad is a palindrome in the 2.5.17.19 [[subgroup]] [[19-odd-limit]] since it identifies 5/4 by a stack of two 19/17's:
-* 1-19/17-5/4 with steps 19/17-19/17-8/5.
+* 1-19/17-5/4 chord with steps 19/17-19/17-8/5.
 There is an inversely related pair which is even more squeezed:
-* 1-17/16-19/17 with steps 17/16-20/19-8/5, and its inverse
+* 1-17/16-19/17 with steps 17/16-20/19-34/19 and its inverse 1-20/19-19/17 with steps 20/19-17/16-34/19.
-* 1-20/19-19/17 with steps 20/19-17/16-8/5.
-They can be extended to the following inversely related tetrads, all of which seem to be based largely on a sort of secundal harmony:
+They can be extended to the following tetrads, with two palindromic chords and two pairs of chords in inverse relationship. The palindromic chords are
-* 1-19/17-19/16-5/4 with steps 19/17-17/16-20/19-8/5, and its inverse
+* 1-17/16-19/17-19/16 chord with steps 17/16-20/19-17/16-32/19;
-* 1-20/19-19/17-5/4 with steps 20/19-17/16-19/17-8/5;
+* 1-20/19-19/17-20/17 chord with steps 20/19-17/16-20/19-17/10.
-* 1-19/17-20/17-5/4 with steps 19/17-20/19-17/16-8/5, and its inverse
-* 1-17/16-19/17-5/4 with steps 17/16-20/19-19/17-8/5.
+The inversely related pairs of chords are
+* 1-19/17-19/16-5/4 with steps 19/17-17/16-20/19-8/5 and its inverse 1-20/19-19/17-5/4 with steps 20/19-17/16-19/17-8/5;
+* 1-19/17-20/17-5/4 with steps 19/17-20/19-17/16-8/5 and its inverse 1-17/16-19/17-5/4 with steps 17/16-20/19-19/17-8/5.
 Then there are two inversely related pentads:
-* 1-17/16-19/17-19/16-5/4 with steps 17/16-20/19-17/16-20/19-8/5, and its inverse
+* 1-17/16-19/17-19/16-5/4 with steps 17/16-20/19-17/16-20/19-8/5 and its inverse 1-20/19-19/17-20/17-5/4 with steps 20/19-17/16-20/19-17/16-8/5.
-* 1-20/19-19/17-20/17-5/4 with steps 20/19-17/16-20/19-17/16-8/5.
+Equal temperaments with aureusmic chords include {{Optimal ET sequence|12, 13, 25, 31, 37, 43, 56, 68, 81, 93, 124, 137, 149, 205, 230 and 286}}.
 [[Category:19-odd-limit]]