Aureusmic chords: Difference between revisions

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'''Aureusmic chords''' are [[essentially tempered dyadic chord]]s tempered by the aureusma, [[1445/1444]].

~~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:

Aureusmic chords are of [[Dyadic chord/Pattern of essentially tempered chords|pattern 1a]] in the 2.5.17.19 [[subgroup]] [[19-odd-limit]], meaning that there are 3 triads, 6 tetrads and 2 pentads, for a total of 11 distinct chord structures.

The most typical aureusmic triad is a palindrome since it identifies 5/4 by a stack of two 19/17's:

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

* 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.

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

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

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

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

The inversely related pairs of ~~chords~~ are

The inversely related pairs of tetrads 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;

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

* 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.

* 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:

Then there is an inversely related pair of 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 1-20/19-19/17-20/17-5/4 with steps 20/19-17/16-20/19-17/16-8/5.

* 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.

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}}.

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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 '''Aureusmic chords''' are [[essentially tempered dyadic chord]]s tempered by the aureusma, [[1445/1444]].
-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:
+Aureusmic chords are of [[Dyadic chord/Pattern of essentially tempered chords|pattern 1a]] in the 2.5.17.19 [[subgroup]] [[19-odd-limit]], meaning that there are 3 triads, 6 tetrads and 2 pentads, for a total of 11 distinct chord structures.
+The most typical aureusmic triad is a palindrome since it identifies 5/4 by a stack of two 19/17's:
 * 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 17/16-20/19-34/19 and its inverse 1-20/19-19/17 with steps 20/19-17/16-34/19.
+* 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.
-They can be extended to the following tetrads, with two palindromic chords and two pairs of chords in inverse relationship. The palindromic chords are
+They can be extended to the following tetrads, with two palindromic chords and two pairs of chords in inverse relationship. The palindromic tetrads are
 * 1-17/16-19/17-19/16 chord with steps 17/16-20/19-17/16-32/19;
 * 1-20/19-19/17-20/17 chord with steps 20/19-17/16-20/19-17/10.
-The inversely related pairs of chords are
+The inversely related pairs of tetrads 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;
+* 1-19/17-19/16-5/4 with steps 19/17-17/16-20/19-8/5, and its inverse
-* 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.
+* 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:
+Then there is an inversely related pair of 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 1-20/19-19/17-20/17-5/4 with steps 20/19-17/16-20/19-17/16-8/5.
+* 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.
-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}}.
+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]]